text:
\section{Introduction}
Drug injection is a popular and efficient way to deliver a drug into biological tissues in order to get more appropriate results. Among the several injection techniques, subcutaneous (SC) injection is a useful as well as highly effective corresponding to the medication of insulin, morphine, diacetylmorphine, goserelin, etc. inside fatty subcutaneous tissue immediately below the dermis layer \citep{kim2017effective}. SC injection becomes advantageous for possible self-administration to the patients who need certain medicine on a regular basis \citep{ogston2014subcutaneous}. Also, this technique becomes an alternative way of drug intake that results in better drug mobility for patients with poor venous access \citep{dychter2012subcutaneous,shapiro2013use}. In the context of safety and efficacy, the SC injection is better than any other techniques such as intravenous or intramuscular injection \citep{allen2013ansel,prettyman2005subcutaneous}. A survey done by \citet{stoner2015intravenous} suggests that the SC route is preferable to the patients as compared to the intravenous route. For those patients who need multiple daily doses of one or more drug(s), SC injection provides a wider range of alternative sites of injection than intramuscular injection \citep{haller2007converting}.\\

\noindent
In order to understand the detailed mechanism of SC injection and mechanical response of the tissue where such injection technique is utilized, it is our primary goal to understand the composition of the tissue at the injection site. A SC tissue is in general a composition of adipose tissue along with extracellular fluid \citep{geerligs2008linear,comley2011deep,derler2012tribology}. According to \citet{shrestha2018fluid,shrestha2020imaging} skin tissue behaves as a deformable porous medium that absorbs fluid as a result of the formation of a cavity under the local expansion of tissue rather than rupturing.\\

\noindent
The fluid flow through the tissue matrix during an intradermal injection is affected by its porosity and permeability. Hence, fluid flow and deformation of solid phases get coupled \citep{barry1992flow}. The porosity and permeability variation during an injection plays an important role to control the accuracy of the amount of fluid injected into the skin at different stages of the injection \citep{shrestha2018fluid}. Consequently, the dosage of a drug to be delivered can be controlled. It would be much important to identify the field equations which govern the above phenomena. In this context, one can go through the classical study of \citet{oomens1987mixture} where skin tissue has been considered as a biphasic mixture of solid ($s$) and fluid ($f$) constituents. The biphasic nature of skin tissue can be described by a set of non-linear field equations. If $\rho_i$ and $\mathbf{V}_{i}$ represent apparent mass density and velocity of $i^{\textrm{th}}$ constituent ($i\in\{s,~f\}$) then the mass balance equation for $i^{\textrm{th}}$ constituent becomes
\begin{equation}\label{IEq1}
\frac{\partial \rho_i}{\partial t} + \nabla. (\rho_i \mathbf{V_i})=m_i,\hspace{0.5cm} i\in\{s,~f\}
\end{equation}
where $m_i$ is the net rate of production of $i^{\textrm{th}}$ constituent per unit volume inside the skin tissue satisfies the following constraints
\begin{equation}\label{IEq2}
\sum_{i=1}^2 m_{i} = 0.
\end{equation}
On the other hand, the balance of momentum for the $i^{\textrm{th}}$ constituent is given by
\begin{equation}\label{IEq3}
\rho_i \frac{D\mathbf{V}_i}{Dt} = \nabla . \mathbf{T}_i + \rho_i \mathbf{F}_i + \mathbf{\Pi}_i,    \hspace{0.5cm} i\in\{s,~f\},
\end{equation}
where ${D\mathbf{V}_i}/{Dt}= {\partial}/{\partial t} + (\nabla. \mathbf{V}_i)$ denotes
 the material derivative, $\mathbf{T}_i$ represents cauchy's stress tensor, $\mathbf{F}_i$ is the external body force corresponding to the $i^{\textrm{th}}$ constituent and $\mathbf{\Pi}_i$ is the interactive force on $i$-th constituent due to the other. In addition, the balance of momentum of the whole tissue matrix leads to
\begin{equation}\label{IEq4}
\sum_{i=1}^{2}(\mathbf{\Pi}_i + m_i \mathbf{V}_i) = 0.
\end{equation}
Similar sets of equations (\ref{IEq1})-(\ref{IEq4}) have been reported in several studies based on mixture theory \citep{rajagopal1995mechanics,ambrosi2002closure,byrne2003modelling,rajagopal2007hierarchy,astanin2008multiphase,dey2021mathematical}. While considering any growth inside a tissue, one can assume the cellular phase to behave as a fluid continuum as the growth process is of large time scale \citep{byrne2003two}. Therefore, a similar fluid momentum equation can govern the dynamics of each of the phases with different viscosity \citep{chen2001influence,byrne2003modelling,ambrosi2002closure}. However, the situation would be different when flow-induced deformation of biological tissues is studied as the deformation is of a small time scale. \citet{barry1990comparison} compared the flow-induced deformation between soft biological tissues and polyurethane sponge through a mathematical model assuming the solid phase to behave as a poroelastic material. In this regard, the dynamics of the whole tissues are governed by the set of equations (\ref{IEq1})-(\ref{IEq4}) stated above. But the Cauchy stress tensor ($\mathbf{T}_s$) corresponding to the solid phase has to follow the stress relation for an elastic material. On the other hand, the fluid stress ($\mathbf{T}_f$) can depend either solely on the pore pressure \citep{barry1990comparison} or both the pore pressure and fluid viscosity \citep{barry1991fluid}. Note that any volumetric change in the tissue due to fluid-induced deformation is infinitesimal in nature. \citet{barry1995injection} studied fluid injection as a point source into a deformable porous layer with both the boundaries are impermeable to fluid flow using biphasic mixture theory. Later, \citet{barry1997deformation} extended this by considering a set of boundary conditions where the upper surface is permeable to fluid flow. In this regard, \citet{li2009mathematical} reviewed several models relevant to the subcutaneous injection of insulin. \citet{shrestha2020imaging} studied fluid injections into soft biological tissues to extract two macroscopic permeability parameters.\\

\noindent
Besides the composition of tissue structure, tissue hydraulic conductivity or tissue permeability plays an important role in order to deliver drug through the injection. If we consider a soft biological tissue as a deformable porous media \citep{barry1992flow}, it may consist either of an isotropic matrix whose permeability is the same along all directions \citep{mow1984fluid,holmes1985theoretical,wei2003flow,dey2016hydrodynamics,shrestha2020imaging}, or an anisotropic matrix whose permeability varies with direction \citep{rees1995effect,rees2002convective,payne2001effect,karmakar2016lifting,karmakar2018effect,rajani2020anisotropic}. In particular, the anisotropic permeability may varies in the principal directions only \citep{kohr2008green,karmakar2017note}. The effects of anisotropic permeability have been observed in the various study of articular cartilage \citep{reynaud2006anisotropic,federico2008anisotropy,iatridis1998degeneration}. \citet{federico2008permeability} studied the effects of anisotropic permeability in a biological tissue filled with interstitial fluid and reinforced by impermeable collagen fibers. It is observed that most of the previous studies are mainly based on the isotropic nature and there are few involving anisotropic. But due to the variations in the distribution of collagen fibers, soft connective tissue can show anisotropic behavior \citep{holzapfel2001biomechanics}. As subcutaneous tissue is a soft connective tissue, it should possess anisotropic permeability. As reported by \citet{kim2017effective} for a fixed flow rate vertical permeability of skin tissue is greater than horizontal permeability and there is no strong evidence such that the converse may not hold. This motivates us to think about the situation when the horizontal permeability is greater than the vertical permeability. Therefore in this study, we consider the anisotropic nature of the subcutaneous tissue region with both the above-mentioned cases which may be an interesting topic.\\

\noindent
Detailed literature review indicates the lacuna about mathematical modeling of fluid flow problems inside soft connective tissues, which include both the anisotropic and deformable nature. Consequently in this article, we introduce a mathematical model to discuss drug delivery through a subcutaneous injection. The subcutaneous tissue region has a biphasic description in terms of two constituents fat adipose cells and interstitial fluid. In addition, the interstitial hydraulic conductivity has anisotropic which varies in the principal directions only.  In this current study, our primary aim is to discuss the mechanical response of the subcutaneous tissue region of skin in terms of the variation of anisotropic hydraulic conductivity, the viscosity of the injected drug, the mean depth of the subcutaneous layer, etc. In addition, we would like to study the pain realized by a patient near the injection site with the help of pressure gradient and shear stress.
\begin{figure}[H]
\centering
\includegraphics[scale=0.7]{Carton.pdf}
\caption{Cartoon diagram of subcutaneous injection: (a) Skin is bunching during injection, (b) Needle injecting after skin bunched \citep{shepherd2018injection}.}
\label{Carton}
\end{figure}
\section{Mathematical formulation}
Subcutaneous injections are used to administer drugs and vaccines into the fatty tissue layer (subcutaneous tissue) sandwiched between the dermis and muscle layer. While injecting a fluid containing drugs or vaccines to a patient, the lifted skin fold must be used to avoid the risk of injecting a drug into the muscle. The best method is to lift the skin of the injection site to pull the fat tissue within the subcutaneous region away from the underlying muscle layer and hold the entire duration of the procedure (Figure \ref{Carton}). Consequently, one would expect the subcutaneous and dermis interface to assume a cosinusoidal wavy curve. Moreover, the above interface may continue to hold such configuration up to a certain duration even after the release of the skin.\\

\begin{figure}[h!]
\centering
\includegraphics[scale=0.7]{Geometry.pdf}
\caption{Schematic of the mathematical model approximating the fluid injection process.}
\label{Geometry}
\end{figure}
\noindent
In this paper, we discuss drug delivery through fluid injection in the subcutaneous (SC) layer with $(0,y_0)$ being the injecting point i.e., the tip of the needle. Consequently, we assume that the SC layer is bounded by permeable upper dermis layer located at $y=R(x)=b\left(1+a \text{cos}(2\pi x/L)\right)$ and permeable lower muscle layer at $y=0$ (Figure \ref{Geometry}). Two major components of the SC layer are interstitial fluid and adipose cells. The cells are oriented such that the overall SC region permeability becomes anisotropic. We consider both interstitial fluid and adipose cell presence in the subcutaneous layer as fluid phases and also the injected fluid has the same viscosity as interstitial fluid. A two-dimensional steady flow of these two incompressible fluids in the subcutaneous layer is considered. We consider the layer with mean width $b$, an amplitude $ba$, and wavelength $L$. The curve $y=R(x)$ represents not only the irregular shape of the subcutaneous layer but within $x\in[-L/2, L/2]$, $y=R(x)$ stands for the uplifted subcutaneous layer. We assumed that the flow within the porous subcutaneous region is governed by the Brinkman equation together with the equation of mass given by \citep{rajagopal2007hierarchy,dey2016hydrodynamics,karmakar2017note,dey2021mathematical}:
\begin{equation}\label{MEq1}
-\varphi_f \nabla P + \lambda_f \nabla (\nabla. \mathbf{u_f}) + \mu_f (\nabla^2 \mathbf{u_f}) - \mu_f \mathbf{K}^{-1}(\mathbf{u}_f-\mathbf{u}_c)=0,
\end{equation}
\begin{equation}\label{MEq2}
-\varphi_c \nabla P + \mu_c (\nabla^2 \mathbf{u_c}) + \mu_f \mathbf{K}^{-1}(\mathbf{u_f}-\mathbf{u_c})=0,
\end{equation}
\begin{equation}\label{MEq3}
\nabla.(\varphi_f \mathbf{u_f})=F(x,y)
\end{equation}
and
\begin{equation}\label{MEq4}
\nabla.(\varphi_c \mathbf{u_c})=0,
\end{equation}
where $\mathbf{u}_{f}=(u_{f}, v_{f})$ and $\mathbf{u_c}=(u_{c}, v_{c})$ are the velocity vector for interstitial fluid and adipose cell respectively; $\varphi_f$ and $\varphi_c$ are the volume fraction for interstitial fluid and adipose cell respectively with $\varphi_f + \varphi_c = 1$; $F(x,y)$ represents the source corresponding to the injected fluid; $P$ is the hydrodynamic pressure; $\mu_f$ and $\mu_c$ are the viscosities of the interstitial fluid and adipose cell respectively. Note that the injected drug becomes the interstitial fluid after injection. Hence, one can think that $\mu_{f}$ is modified to become $\mu^{e}_{f}$ which is the effective interstitial fluid viscosity. \\

\noindent
Considering the anisotropic orientation of fat and connective tissues within the subcutaneous tissue region, the permeability tensor $\mathbf{K}$ should possess both non-zero off-diagonal entries along with dissimilar diagonal elements. In other words, $\mathbf{K}$ may depend on the anisotropic angle $\phi$ between the horizontal direction and the principal axis \citep{degan2002forced,karmakar2016effect}. For simplicity, one can take the permeability tensor $\mathbf{K}$ in the principal directions only. Therefore as given in \citep{kohr2008green,karmakar2016effect,karmakar2017note,karmakar2018effect}, $\mathbf{K}$ follows
\begin{equation}
\mathbf{K} =
\begin{bmatrix}
K_1 & 0\\
0 & K_2
\end{bmatrix},
\end{equation}
with $K_1$ and $K_2$ are the permeabilities along the $x$ and $y$ directions (i.e. in principal directions) respectively.
\subsection{Boundary conditions:}
In order to proceed the solution of the problem we consider the following boundary conditions:\\\\
(i) on $y=R(x),$
\begin{equation}\label{BC1}
\mathbf{u}_{f}.\hat{\mathbf{t}}= 0 \text{    ,    } \mathbf{u}_{f}.\hat{\mathbf{n}}= V_D(x),
\end{equation}
where $\hat{\mathbf{t}}$ and $\hat{\mathbf{n}}$ are the unit tangent and normal vector respectively on $y=R(x)$. On the other hand, on $y=R(x),$
\begin{equation}\label{BC2}
u_{c}=0 \text{     and    } v_{c}=0.
\end{equation}
(ii) on $y=0,$
\begin{equation}\label{BC3}
u_{f} = \lambda_{s_{f}} \frac{\partial u_f}{\partial y} \text{    ,    } v_{f} = V_M(x),
\end{equation}
\begin{equation}\label{BC4}
u_{c}=\lambda_{s_{c}} \frac{\partial u_c}{\partial y} \text{     and    } v_{c}=0 .
\end{equation}
where $\lambda_{s_{f}}$ and $\lambda_{s_{c}}$ are the slip coefficients. The structural difference between the subcutaneous and muscle layer suggests the permeability variation between them. According to \citet{kim2017effective} corresponding to the same flow rate, the horizontal permeability of the subcutaneous layer is found to be higher than that of the muscle layer. On the other hand, during the injection procedure, a horizontal movement in the subcutaneous tissue may be noted at the subcutaneous and muscle interface (SM interface) along with the muscle layer due to the downward fluid pressure generated from the injection site. Such motion can be characterized by the boundary conditions Eq.(\ref{BC3})-(\ref{BC4}) as proposed by \citet{beavers1967boundary,jones1973low,karmakar2018squeeze,hill2008poiseuille} at the SM interface are regarded as slip conditions where the parameter $\lambda_s$ (called slip coefficient) is directly proportional to the length scale same as $\sqrt K_1$. In particular, we consider $\lambda_{s_{f}}=\varphi_{f}\lambda_{s}$ and $\lambda_{s_{c}}=\varphi_{c}\lambda_{s}$ (where $\lambda_{s}$ is the common slip coefficient at the interface of subcutaneous and dermis layers). Lastly, we define $V_D(x)$ and $V_M(x)$ which represent the vertical permeation through the subcutaneous-dermis and subcutaneous-muscle interface respectively.\\

\noindent
(iii) Flux condition:
Let $Q$ be the volumetric flow rate across the region which is given by
\begin{equation}\label{BC5}
Q=\int_{0}^{R(x)} (\varphi_{f}u_{f} + \varphi_{c} u_{c}) \hspace{0.01 cm}dy .
\end{equation}
\subsection{Non-dimensionalisation}
We introduce the following non-dimensional variables:\\
$x'=x/L$, $y'=y/b$, $\delta = b/L$, $u_{i}'=u_i/(Q/b)$, $v_{i}'=v_i/(Q/L)$, $p'=p/(\mu_f QL/K_{1}b),$ for $i=f, c$ in Eqs. (\ref{MEq1})-(\ref{MEq4}) to make them dimensionless. Accordingly, we obtain following dimensionless parameters within the resulting dimensionless governing equations: $\mu_r=\mu_f/\mu_c$, $\text{Da}=K_1/b^2$, $\lambda^2=K_1/K_2$, $V'_D=V_D/(Q/L)$, $V'_M=V_M/(Q/L)$, $\lambda'_s=\lambda_s/L$ and $F'=F/(Q/bL)$. \\

\noindent
Here, $\delta$ denotes the aspect ratio of the SC layer, $\text{Da}$ is the Darcy number which is the ease of fluid percolation in the horizontal direction, $\mu_r$ is the ratio of the interstitial fluids to the adipose cells and $\lambda^2$ is the ratio of horizontal permeability to the vertical permeability, which can be referred to as the anisotropic ratio.  \\

\noindent
Correspondingly, the non-dimensional governing equations can be written as (omitting dash)\\
\begin{equation}\label{NEq1}
 -\varphi_{f} \alpha^{2}\frac{\partial p}{\partial x}+\delta^2\left(\frac{\lambda_{f}}{\mu_{f}}\right)\frac{\partial}{\partial x} \left(\frac{\partial u_f}{\partial x} + \frac{\partial v_f}{\partial y}\right) + \left( \delta^2 \frac{\partial^2 u_f}{\partial x^2} + \frac{\partial^2 u_f}{\partial y^2}  \right) - \alpha^2(u_f - u_c) = 0
\end{equation}
\begin{equation}\label{NEq2}
 -\varphi_{f}\alpha^{2} \frac{\partial p}{\partial y}+ \delta^2\left(\frac{\lambda_{f}}{\mu_{f}}\right)\frac{\partial}{\partial y} \left(\frac{\partial u_f}{\partial x} + \frac{\partial v_f}{\partial y}\right) + \delta^2 \left( \delta^2 \frac{\partial^2 v_f}{\partial x^2} + \frac{\partial^2 v_f}{\partial y^2}  \right) - \delta^2 \lambda^2 \alpha^2(v_f - v_c) = 0
\end{equation}
\begin{equation}\label{NEq3}
-\varphi_c \alpha^{2}\mu_r\frac{\partial p}{\partial x} + \left(\delta^2\frac{\partial^2 u_c}{\partial x^2}+\frac{\partial^2 u_c}{\partial y^2}\right)+ \alpha^2\mu_r(u_f-u_c)=0
\end{equation}
\begin{equation}\label{NEq4}
-\varphi_c\alpha^{2}\mu_r\frac{\partial p}{\partial y} + \delta^2 \left (\delta^2 \frac{\partial^2 v_c}{\partial x^2}+\frac{\partial^2 v_c}{\partial y^2}\right)+ \delta^2 \lambda^2 \alpha^2\mu_r(v_f-v_c)=0
\end{equation}
\begin{equation}\label{NEq5}
\varphi_f\left(\frac{\partial u_f}{\partial x} + \frac{\partial v_f}{\partial y}\right)= F(x,y)
\end{equation}
\begin{equation}\label{NEq6}
\varphi_c\left(\frac{\partial u_c}{\partial x} + \frac{\partial v_c}{\partial y}\right)= 0
\end{equation}\\
where $\alpha^2=1/\text{Da}$. Here we consider stokes hypothesis by taking $2\lambda_f + 3 \mu_f = 0$ i.e. $\frac{\lambda_{f}}{\mu_{f}}=-3/2$ \citep{dey2016hydrodynamics}.
Also the corresponding boundary conditions are (dropping dash) \\\\
(i) on $y=R(x)=1+a\text{cos}(2\pi x),$\\
\begin{equation}\label{BC6}
 u_{f}=2\pi a \delta^{2}\frac{\sin(2\pi x)}{\sqrt{1+4\pi^{2}a^{2}\delta^{2}\sin(2\pi x)}}V_D(x), \hspace{0.1cm} v_{f}= \frac{1}{\sqrt{1+4\pi^{2}a^{2}\delta^{2}\sin(2\pi x)}}V_D(x),
 \end{equation}
 \begin{equation}\label{BC7}
 u_{c}=0 \text{     and    } v_{c}=0  .
 \end{equation}
(ii) on $y=0,$
\begin{equation}\label{BC8}
u_{f} = \varphi_{f} \lambda_s \frac{\partial u_f}{\partial y}, \hspace{0.01cm} v_{f}=V_{M}(x) ,
\end{equation}
\begin{equation}\label{BC9}
u_{c} = \varphi_{c} \lambda_s \frac{\partial u_c}{\partial y} \text{     and    } v_{c}=0  .
\end{equation}
(iii) Flux condition: The non-dimensional volumetric flow rate is given by \\
\begin{equation}\label{BC10}
1=\int_{0}^{R(x)} (\varphi_{f}u_{f}+\varphi_{c}u_{c}) \hspace{0.01 cm}dy .
\end{equation}
\section{Perturbation approximation}
In order to solve the above boundary value problem, we can use the perturbation method to find the approximate solution \citep{tsangaris1984laminar,wei2003flow,karmakar2017note}. We assume that the aspect ratio ($\delta$) of the region is small , i.e. $\delta^2 \ll 1$, and this allows us to apply perturbation theory. Accordingly, with respect to the small parameter $\delta^2$, we can expand the velocity and pressure in a perturbation series as:
\begin{equation}\label{PEq1}
(u_i, v_i, p)= (u_{i_0}, v_{i_0}, p_{0}) + \delta^2 (u_{i_1}, v_{i_1}, p_1) + \text{O}(\delta^4), \hspace{0.9cm}   \text{for } i=f, c.
\end{equation}
The first order correction is $\delta^2$, since no terms of order $\delta$ appear in the governing equations and boundary conditions. The flow field is solved by collecting the similar powers of $\delta^2$.
\subsection{The leading-order problem}
The governing equations reduces to
\begin{equation}\label{Eq1}
-\varphi_f \alpha^2\frac{\partial p_0}{\partial x} + \frac{\partial^2 u_{f_0}}{\partial y^2} - \alpha^2(u_{f_0} - u_{c_0}) = 0,
\end{equation}
\begin{equation}\label{Eq2}
-\varphi_f \alpha^2\frac{\partial p_{0}}{\partial y} = 0,
\end{equation}
\begin{equation}\label{Eq3}
-\varphi_c \mu_{r} \alpha^2\frac{\partial p_0}{\partial x} + \frac{\partial^2 u_{c_0}}{\partial y^2} + \alpha^2\mu_r(u_{f_0} - u_{c_0}) = 0,
\end{equation}
\begin{equation}\label{Eq4}
-\varphi_{c} \mu_{r} \alpha^2\frac{\partial p_{0}}{\partial y} = 0,
\end{equation}
\begin{equation}\label{Eq5}
\varphi_f\left(\frac{\partial u_{f_0}}{\partial x} + \frac{\partial v_{f_0}}{\partial y}\right)= F(x,y),
\end{equation}
\begin{equation}\label{Eq6}
\varphi_c\left(\frac{\partial u_{c_0}}{\partial x} + \frac{\partial v_{c_0}}{\partial y}\right)= 0.
\end{equation}\\
The corresponding boundary conditions are\\\\
(i)  on $y=R(x),$\\
\begin{equation}\label{BC11}
 u_{f_0}=0, v_{f_0}=V_D(x),
 \end{equation}
\begin{equation}\label{BC12}
 u_{c_0}=0 \text{     and    } v_{c_0}=0  .
\end{equation}
(ii) on $y=0$\\
\begin{equation}\label{BC13}
u_{f_0}= \lambda_s \frac{\partial u_{f_0}}{\partial y}, v_{f_0}=V_M(x),
\end{equation}
\begin{equation}\label{BC14}
u_{c_0}= \lambda_s \frac{\partial u_{c_0}}{\partial y}\text{     and    } v_{c_0}=0  .
\end{equation}
(iii) Flux condition
\begin{equation}\label{BC15}
1=\int_{0}^{R(x)} (\varphi_{f}u_{f_0} + \varphi_{c} u_{c_0}) \hspace{0.01 cm}dy .
\end{equation}
\begin{theorem}
The vertical permeations $V_D(x)$ and $V_M(x)$ along subcutaneous-dermis(SD) interface and subcutaneous-muscle(SM) interface respectively are equal when
$$1=\int_{0}^{R(x)} (\varphi_{f}u_{f}+\varphi_{c}u_{c}) \hspace{0.01 cm}dy$$.
\end{theorem}
\begin{proof}
If we differentiate
$$\int_{0}^{R(x)} (\varphi_{f}u_{f_0} + \varphi_{c} u_{c_0}) \hspace{0.01 cm}dy=1,$$
under the integration sign using Leibnitz rule, we obtain
$$(\varphi_{f}u_{f_0} + \varphi_{c} u_{c_0})(x,R(x)) \frac{dR(x)}{dx} + \int_{0}^{R(x)} (\varphi_{f} \frac{\partial u_{f_0}}{\partial x} + \varphi_{c} \frac{\partial u_{c_0}}{\partial x}) \hspace{0.01 cm}dy = 0,$$
which implies
$$\left(\varphi_{f}u_{f_0}(x,R(x)) + \varphi_{c} u_{c_0}(x,R(x))\right) \frac{dR(x)}{dx} + \int_{0}^{R(x)} \left(\varphi_{f} (-\frac{\partial v_{f_0}}{\partial y}) + \varphi_{c} (-\frac{\partial v_{c0}}{\partial y})\right) \hspace{0.01 cm}dy = 0.$$
After some simplification,
$$(\varphi_{f}v_{f_0} + \varphi_{c} v_{c_0})(y=R(x)) - (\varphi_{f}v_{f_0} + \varphi_{c} v_{c_0})(y=0) = 0.$$
Hence, $$V_D(x)=V_M(x).$$
\end{proof}
\subsection{The \text{O}($\delta^2$) problem}
The governing equations corresponding to the first order are
\begin{equation}\label{Eq7}
-\varphi_{f} \alpha^{2}\frac{\partial p_1}{\partial x} +\left(\frac{\lambda_{f}}{\mu_{f}}\right) \frac{\partial}{\partial x}\left(\frac{\partial u_{f_0}}{\partial x}+\frac{\partial v_{f_0}}{\partial y}\right) + \left(\frac{\partial^2 u_{f_0}}{\partial x^2} + \frac{\partial^2 u_{f_1}}{\partial y^2}\right) - \alpha^2(u_{f_1}-u_{c_1})=0,
\end{equation}
\begin{equation}\label{Eq8}
-\varphi_{f} \alpha^{2}\frac{\partial p_1}{\partial y}+\left(\frac{\lambda_{f}}{\mu_{f}}\right) \frac{\partial}{\partial y}\left(\frac{\partial u_{f_0}}{\partial x}+\frac{\partial v_{f_0}}{\partial y}\right) + \frac{\partial^2 v_{f_0}}{\partial y^2} -  \alpha^2 \lambda^2(v_{f_0}-v_{c_0}) = 0,
\end{equation}
\begin{equation}\label{Eq9}
-\varphi_{c} \mu_{r} \alpha^{2}\frac{\partial p_1}{\partial x} + \left(\frac{\partial^2 u_{c_0}}{\partial x^2} + \frac{\partial^2 u_{c_1}}{\partial y^2}\right) + \mu_{r}\alpha^2(u_{f_1}-u_{c_1})=0,
\end{equation}
\begin{equation}\label{Eq10}
-\varphi_{c} \mu_{r} \alpha^{2}\frac{\partial p_1}{\partial y} + \frac{\partial^2 v_{c_0}}{\partial y^2}  + \mu_{r}\alpha^2\lambda^2(v_{f_0}-v_{c_0})=0,
\end{equation}
\begin{equation}\label{Eq11}
\varphi_{f}\left(\frac{\partial u_{f_1}}{\partial x} + \frac{\partial v_{f_1}}{\partial y} \right) = 0,
\end{equation}
\begin{equation}\label{Eq12}
\varphi_{c}\left(\frac{\partial u_{c_1}}{\partial x} + \frac{\partial v_{c_1}}{\partial y} \right) = 0.
\end{equation}\\
Corresponding boundary conditions reduces to\\\\
(i)  on $y=R(x),$\\
\begin{equation}\label{BC16}
 u_{f_1}=2\pi a^{2}\sin(2\pi x)V_{D}(x),\hspace{0.2cm} v_{f_1}=-2\pi^{2} a^{2}\sin(2\pi x)V_{D}(x),
\end{equation}
\begin{equation}\label{BC17}
u_{c_1}=0 \text{     and    } v_{c_1}=0  .
\end{equation}
(ii) on $y=0,$\\
\begin{equation}\label{BC18}
u_{f_1}=  \lambda_s \frac{\partial u_{f_1}}{\partial y},\hspace{0.2cm} v_{f_1}=0,
\end{equation}
\begin{equation}\label{BC19}
u_{c_1}= \lambda_s \frac{\partial u_{c_1}}{\partial y} \text{     and    } v_{c_1}=0  .
\end{equation}
(iii) Flux condition \\
\begin{equation}\label{BC20}
0=\int_{0}^{R(x)} (\varphi_{f}u_{f_1} + \varphi_{c} u_{c_1}) \hspace{0.01 cm}dy.
\end{equation}
\subsection{Details of the source term $F(x,y)$}
In Eq (\ref{Eq5}), $F(x,y)$ represents the tip of the needle of the syringe at some point inside the subcutaneous tissue region. One can think of a point source at the point $(0,y_{0})$ (see Figure \ref{Geometry}) which can be expressed as
\begin{equation}\label{Eq13}
F(x,y)=m_{0}\delta(x)\delta(y-y_{0}),
\end{equation}
where $m_{0}$ represents the strength of the point source. One can solve the leading order and first order equations using finite difference scheme by discretizing the domain with keeping the point $(0,y_{0})$ outside the meshgrid. However, one can attempt the solution in the regions $y<y_0$ and $y>y_0$ for all values of $x$ through analytically. Since $y=y_{0}$ is a line on which injection point (tip of needle) must lie, thus the following conditions at $y=y_0$ can be used to match the solution :
$$u_{f_0}(x,y^+_{0}) = u_{f_0}(x,y^-_{0}),$$
$$u_{c_0}(x,y^+_{0}) = u_{c_0}(x,y^-_{0}),$$
and\\
$$\frac{\partial u_{f_0}(x,y)}{\partial y}\Big|_{y=y^+_0} = \frac{\partial u_{f_0}(x,y)}{\partial y}\Big|_{y=y^-_0}  ,$$
$$\frac{\partial u_{c_0}(x,y)}{\partial y}\Big|_{y=y^+_0} = \frac{\partial u_{c_0}(x,y)}{\partial y}\Big|_{y=y^-_0}  .$$
\subsection{Subcutaneous tissue velocity and stream function}
We define subcutaneous tissue velocity or composite velocity $\mathbf{u}=(u,v)$ of the mixture of interstitial fluid and adipose cells presents in the subcutaneous tissue as
\begin{equation}\label{Eq14}
  \mathbf{u} = \varphi_f\mathbf{u}_{f} + \varphi_c\mathbf{u}_{c}
\end{equation}
If $\Phi_f$ and $\Phi_c$ are the stream function of the interstitial fluid and adipose cells respectively presents in the subcutaneous region, then their relation with the velocity components are
 $$u_{f}=\frac{\partial \Phi_{f}}{\partial y}, u_{c}=\frac{\partial \Phi_{c}}{\partial y}  \text{       and      } v_{f} = - \frac{\partial \Phi_{f}}{\partial x}, v_{c} = - \frac{\partial \Phi_{c}}{\partial x} .$$
 Also, we define the composite stream function of the mixture (since we consider both is in fluid phase) as
\begin{equation}\label{Eq15}
  \Phi = \varphi_f\Phi_{f} + \varphi_c\Phi_{c}
\end{equation}
In the subcutaneous region, the quantity of interstitial fluid is much larger than the quantity of adipose cells. Thus the subcutaneous tissue velocity or composite velocity and stream function can be considered in macroscopic level as the velocity and stream function for interstitial fluid, which we can assume from literature \citep{barry1997deformation,barry1991fluid}.
\begin{theorem}
If $\Phi_i (i=f,c)$ are the stream functions defined as
  \[\Phi_i = \begin{cases}
      \Phi_i^{(1)}, & 0\leq y<y_0 \\
      \Phi_i^{(2)}, & y_0 < y \leq R(x)
   \end{cases}
\]
and $\exists$ $\psi_{i}$ satisfying $\psi_{i}=\int u_{i} \hspace{0.03 cm} dy$ such that
\[\psi_i = \begin{cases}
      \psi_i^{(1)}, & 0\leq y<y_0 \\
      \psi_i^{(2)}, & y_0 < y \leq R(x)
   \end{cases}
\]
Then two functions $f_{i}(x)$ and $g_{i}(x)$ can be obtained satisfying the following relation
$$\Phi_i^{(1)}(x,y) = \psi_{i}^{(1)}(x,y) + f_{i}(x) ,$$
$$\Phi_i^{(2)}(x,y) = \psi_{i}^{(2)}(x,y) + g_{i}(x) .$$
\end{theorem}
\begin{proof}
The stream function is related to the velocity components by the relations
$$u_{i}=\frac{\partial \Phi_{i}}{\partial y} \text{       and      } v_{i} = - \frac{\partial \Phi_{i}}{\partial x} .$$
Considering the first relation $$u_{i}=\frac{\partial \Phi_{i}}{\partial y} ,$$
or $$\Phi_{i} = \int u_{i}\hspace{0.03 cm} dy + h(x) ,$$
where $h(x)$ is the integrating constant. Let $\psi_{i}(x,y)=\int u_{i}\hspace{0.03 cm} dy$.

\noindent
Since $u_{i}$ and $v_{i}$ are defined in two regions (say $u_{i}^{(1)}$, $u_{i}^{(2)}$  and $v_{i}^{(1)}$, $v_{i}^{(2)}$), thus we have stream function in the regions as
$$\Phi_i^{(1)}(x,y) = \psi_{i}^{(1)}(x,y) + f_{i}(x), \hspace{1 cm} \text{if} \hspace{0.3 cm} 0\leq y<y_0 $$
$$\Phi_i^{(2)}(x,y) = \psi_{i}^{(2)}(x,y) + g_{i}(x). \hspace{1 cm}~~~~~~ \text{if} \hspace{0.3 cm} y_0\leq y<R(x) $$
Next we have to find the functions $f_{i}(x)$ and $g_{i}(x)$.\\

Since stream function is continuous, thus we have
$$\Phi_i^{(1)}(x,y_0^{-})=\Phi_i^{(2)}(x,y_0^{+}) ,$$
which gives $$f_{i}(x) = \psi_{i}^{(2)}(x,y) - \psi_{i}^{(1)}(x,y) + g_{i}(x) .$$
Also since $$\Phi_i^{(2)}(x,y) = \psi_{i}^{(2)}(x,y) + g_{i}(x) ,$$
Upon taking derivative in both sides with respect to $x$
$$ \frac{\partial \Phi_i^{(2)}}{\partial x} = \frac{\partial \psi_{i}^{(2)}}{\partial x} + g_{i}'(x) ,$$
which gives
$$ g_{i}'(x) = - \frac{\partial \psi_{i}^{(2)}}{\partial x} - v_{i}^{(2)}(x,y) .$$
Now integrating both sides, we get
$$g_{i}(x)=-\psi_{2}(x,y) - \int v_{i}^{(2)}(x,y) \hspace{0.03 cm} dx ,$$
which satisfies throughout the considered region.\\
Thus at $y=R(x)$, \\
$$g_{i}(x) = -\psi_{2}(x,R(x)) - \int v_{i}^{(2)}(x,R(x)) \hspace{0.03 cm} dx .$$
Since $v_{i}^{(2)}(x,R(x))$ is a known function (which we obtain using boundary conditions), thus we get $g_{i}(x)$ and using this we can oblation $f_{i}(x)$ easily from the continuity condition of stream function.
\end{proof}

\noindent
The details solution of the leading order and \text{O}($\delta^2$) problem corresponding to interstitial fluid and adipose cells are shown in Appendix 1 and Appendix 2 respectively.
\section{Results and Discussion}
In this study, a two dimensional flow induced by fluid injection has been considered within the anisotropic SC tissue region which is bounded by permeable dermis layer from topside and permeable muscle layer from bottom. Here we assume that the principal components of subcutaneous layer are adipose cells (fat tissue) and interstitial fluids with large proportional of fluid part. Consequently, we can choose $\varphi_f$ within the range $0.8\leq \varphi_f \leq 0.9$ throughout the study. The flow in the considered region is governed by the equations (\ref{NEq1})-(\ref{NEq6}) (in non-dimensional form). Corresponding flow parameters such as $\lambda$, $\mu_r$, $\delta$, $a$, $\text{Da}$, $\lambda_s$ are reported in Table \ref{table_1} with their reference ranges. For example, \citet{shrestha2020imaging} have reported the optimum value of permeability of skin tissue lies within the range $0.59\times 10^{-14}$ $\textrm{m}^{2}$ to $2.10\times 10^{-12}$ $\textrm{m}^{2}$. Therefore, one can consider $K_{1}$ and $K_{2}$ lie between the above range. Consequently, $\lambda$ lies within the range $0.53\leq \lambda \leq 2$. Except for $\lambda=1$ (isotropic), anisotropic behaviour is exhibited for all values of $\lambda$ within the above range.
\begin{table}[h!]
\centering
\caption{Various parameters involved in this study with their range.}
\label{table_1}
\begin{tabular}{c c c}
 \hline
 Parameter & Range & Remark \\
 \hline
 Anisotropic ratio ($\lambda$) & $0.5\leq\lambda\leq2$ & \citet{shrestha2020imaging}\\

 Viscosity ratio ($\mu_r$) & $0<\mu_r<1$ & Considered \\

 Darcy number ($Da$) & $10^{-3}\leq Da\leq 5\times10^{-3}$ & \citet{dey2016hydrodynamics}\\

 Amplitude of the wavy layer ($a$) & $0.1\leq a\leq 0.5$ & Considered  \\

 Aspect ratio of the region ($\delta$) & $0.1\leq\delta\leq 0.3$ & \citet{karmakar2017note}\\

 Slip coefficient ($\lambda_s$) & $0.001\leq\lambda_s\leq0.05$ & \citet{dey2016hydrodynamics}\\
 \hline
\end{tabular}
\end{table}
On the other hand, the viscosity ratio $\mu_r=\mu_f/\mu_c$ can be chosen to lie within the range $0<\mu_{r}<1$. This is because the adipose cellular phase is highly viscous as compared to the interstitial fluid within the similar continuum description. Since the SC tissue region behaves as a deformable Brinkman medium, the associated Darcy number $(Da)$ which is the ease of fluid percolation in the horizontal direction can be considered in the range $10^{-3}\leq Da\leq 5\times10^{-3}$ \citep{karmakar2017note}. Due to the pinching and holding of the skin during the injection procedure, a co-sinusoidal wavy bunch with amplitude $ba$ is created within the SC region. As $b$ is fixed, the height of the lifted skin can vary with the magnitude of $a$. Considering all possible heights of the lifted skin, $a$ is considered within the range $0.1\leq a\leq0.45$. Various skin pinching heights for SC injection are shown in Table \ref{table_2}.\\

\begin{table}[h!]
\centering
\caption{Various types of skin lifting and the corresponding height of the skin produced for SC injection}
\label{table_2}
\begin{tabular}{c c}
 \hline
 Nature of skin lifting & Height of the skin produced for SC injection \\
 \hline
 Deep lifting ($a=0.4$) & $R(0)=1.4$  \\

 Medium lifting ($a=0.3$) & $R(0)=1.3$ \\

 Low lifting ($a=0.1$) & $R(0)=1.1$ \\
 \hline
\end{tabular}
\end{table}
\noindent
On the other hand, $\delta$ is the ratio between the depth of the SC region and the length of the SC tissue trapped inside the thumb and pointer of the medical staff during the injection. Essentially, $\delta$ can be chosen as a perturbation parameter since $\delta^2\ll1$ corresponds to this situation. Similar parameter can be observed in the study of \citet{karmakar2017note} and correspondingly, we opt the magnitude of $\delta$ within the range $0.1\leq\delta\leq 0.3$ (see Table \ref{table_3}). Finally, we specify the slip coefficient $\lambda_s$ within the range $0.001\leq \lambda_{s} \leq 0.05$ as the value of it is up to $\text{O}(\sqrt{\text{Da}})$. Note that $\lambda_{s}\rightarrow 0$ makes the adipose cellular phase rigid towards the squeezing effect at the subcutaneous-muscle (SM) interface due to the fluid pressure at the line of injection. Each of these parameters mentioned above is supposed to have a significant impact on the hydrodynamic behavior of the SC region during fluid injection. In the upcoming sections, we are going to discuss in detail.
\begin{table}[h!]
\centering
\caption{Various tissue anisotropic ratio ($\lambda$) magnitude, aspect ratio ($\delta$) value and corresponding $\delta^2\lambda^2$.}
\label{table_3}
\begin{tabular}{c c c}
 \hline
 $\lambda$ & $\delta^2(\ll1)$ & $\delta^2\lambda^2(\ll1)$ \\
 \hline
 0.5 & 0.09 & 0.0255 \\
 1 & 0.09 & 0.09 \\
 1.25 & 0.09 & 0.141\\
 1.5 & 0.09 & 0.2025\\
 1.75 & 0.09 & 0.2756\\
 2 & 0.09 & 0.36\\
 \hline
\end{tabular}
\end{table}
\subsection{Effect of the skin pinching height $a$}
During the SC injection procedure, the portion of the skin pinched up takes a bell-shaped cosine curve. If $a$ is its amplitude, this represents the height of the skin pinched up. Figures (\ref{contour_a1})-(\ref{contour_a4}) show composite streamlines ($\Phi$) for four different values $a = 0.1$, $a=0.2$, $a=0.3$ and $a=0.4$ respectively where other parameters are $\text{Da}=3\times10^{-3}$, $\delta=0.3$, $\lambda = 2$, $\mu_r = 0.01$, $\lambda_{s} = 0.05$. It can be observed that the closed contours formed near the line of injection (i.e., primary eddy structure) due to the development of high pressure near it (Figure \ref{pressure_various_a}). In addition, one can notice that streamlines patterns are quite simple corresponding to $a=0.1$ (i.e. low height of skin lift) (Figures (\ref{contour_a1})). But for a higher magnitude of $a$ (i.e., higher depth of skin lifting), there is a tendency of formation of eddy structure within the lifted portion (i.e., secondary eddy structure) of the SC region (Figure (\ref{contour_a4})). If we locate a particular contour say $c=0.6$ corresponding to the primary eddy, its size is found to increase with $a$ within the primary eddy. On the other hand, several contours (e.g., $c=0.03,0.035,0.039$, etc.) are developed within the lifted portion to generate secondary eddy. This is due to the constant transfer of energy from large to small eddies until it is dissipated. The formation of eddies helps proper mixing of injected fluid within the interstitial fluid better than pure molecular diffusion. Hence, SC tissue anisotropy causes better mixing of injected fluid with interstitial fluid. Tissue anisotropy reduces the chance of trapping of injected fluid within the folds present at the subcutaneous-dermis (SD) interface unless the said interface is irregular and causes adequate mixing (for better convective transport of injected drug) with the interstitial fluid by creating secondary eddies.\\

\begin{figure}
\centering
\begin{subfigure}{.4\textwidth}
\includegraphics[width=\textwidth]{contour_a1.pdf}
\caption{$a=0.1$}
\label{contour_a1}
\end{subfigure}
\begin{subfigure}{.4\textwidth}
\includegraphics[width=\textwidth]{contour_a2.pdf}
\caption{$a=0.2$}
\label{contour_a2}
\end{subfigure}
\begin{subfigure}{.4\textwidth}
\includegraphics[width=\textwidth]{contour_a3.pdf}
\caption{$a=0.3$}
\label{contour_a3}
\end{subfigure}
\begin{subfigure}{.4\textwidth}
\includegraphics[width=\textwidth]{contour_a4.pdf}
\caption{$a=0.4$}
\label{contour_a4}
\end{subfigure}
\caption{Composite streamlines within the subcutaneous layer for various parameters: $\lambda=2$, $Da=3\times10^{-3}$, $\delta=0.3$, $\mu_r = 0.01$ and $\lambda_s=0.05$, when $y_0=0.45$ is the line of injection.}
\end{figure}
\noindent
To justify the behaviour of the composite streamlines, one can study the axial composite velocity ($u$) profiles as shown in Figure (\ref{velocity_a}) corresponding to four different skin pinching heights $a$ as depicted through Figures (\ref{contour_a1})-(\ref{contour_a4}). The axial composite velocity shows a varied behavior between region of the lifted skin $\left(\left\{(x,y)|-\infty <x < \infty, 1\leq y \leq 1.4\right\}\right)$ and the region just above the line of injection $\left(\left\{(x,y)|-\infty <x < \infty, 0.45 < y \leq 1\right\}\right)$. This incident can justify the dissipation of a contour within primary eddy after getting larger in size with increased magnitude of $a$ and simultaneous creation of new small contours within the secondary eddy. Increased magnitude of axial composite velocity with respect to $a$ justifies significant scope of axial convective transport to take place corresponding to higher skin pinching. In addition, the formation of secondary eddy corresponding to $a=0.4$ is due to change in sign of $u$ in comparison with the cases $a=0.1, 0.2, 0.3$.\\

\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{velocity_a_new.pdf}
\caption{Axial composite velocity profile at $x=0$ for $a=0.1, 0.2, 0.3, 0.4$.}
\label{velocity_a}
\end{figure}
\noindent
Beside the study of composite streamlines and axial composite velocity, it is also necessary to explore the pressure gradient and shear stress for various $a$ at three different positions $y=y_{0}$ (line of injection), $y=0$ (SM interface) and $y=R(x)$ (SD interface). One can calculate shear stress as
\begin{equation}\label{SEq11}
\tau_{xy} = \frac{\partial u}{\partial y} + \delta^2 \frac{\partial v}{\partial x}.
\end{equation}
Figures (\ref{pressure_various_a})-(\ref{shear_various_a}) illustrate that for all four magnitudes of $a$, the pressure gradient is found to be maximum at the line of injection but, average shear stress is the maximum in the case of SM interface. Next lower magnitude of pressure gradient and average shear stress are found to attain at the SM interface and the line of injection respectively. Both the shear stress and pressure gradient are minimum at SD interface for all $a$ as compared to the other two positions. Moreover, the impact of increased $a$ is behind the diminishing of both the pressure gradient and average shear stress at the SD interface. However, this matter is rather difficult to predict in the case of SM interface and line of injection except at a close neighborhood of $x=0$. The higher pressure gradient at $y=y_{0}$ and higher average shear stress at $y=0$ helps injected fluid to move away from the line of injection and lateral spreading of injected fluid at SM interface due to the consideration of slip property. If one can associate the experience of the intensity of pain realization with the variation of pressure gradient and shear stress developed at various locations within the SC tissue, a patient may realize less pain in case of higher skin lifting at a close neighborhood of $x=0$ for most of the cases considered here. In addition, decisions can be made towards the experience of lowest intensity pain at the SM interface for various heights of skin lifting.
\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\textwidth]{pressure_various_a.pdf}
\caption{}\label{pressure_various_a}
\end{subfigure}
\begin{subfigure}{.48\textwidth}
\includegraphics[width=\textwidth]{shear_various_a.pdf}
\caption{}\label{shear_various_a}
\end{subfigure}
\caption{(a) Pressure gradient (b) Shear stress profiles for $a=0.1,0.2,0.3,0.4$ at three positions $y=0$ (SM interface), $y=y_{0}$ (line of injection), and $y=R(x)$ (SD interface).}
\end{figure}
\subsection{Effect of anisotropic permeability}
Figures (\ref{contour_lambda_0.5})-(\ref{contour_lambda_2}) display the streamline patterns corresponding to $\Phi$ inside SC region for a wide variety of tissue anisotropy ($\lambda$). Among all considered values of $\lambda$, Figure (\ref{contour_lambda_1}) shows the isotropic nature of the SC region. In case of Figure (\ref{contour_lambda_0.5}), the horizontal permeability $K_1$ is smaller than the vertical permeability $K_2$ and rest of the Figures are plotted for $K_1>K_2$. Keeping $\text{Da}$ is fixed, an increase in $\lambda$ results in a reduction of the tissue anisotropy towards the vertical direction. In other words, an increase in $\lambda$ causes a reduction in permeability towards the vertical direction. Consequently, the streamlines in the upstream follow the shape of the SD interface until $\lambda\simeq1.5$ (see Figures (\ref{contour_lambda_0.5})-(\ref{contour_lambda_1.25})) where the skin is pinched up. Thereafter, the flow in the upstream starts to digress following the shape of the interface (Figure \ref{contour_lambda_1.5}). Corresponding to $\lambda=1.75, 2$, a development of secondary eddy structure causing flow circulation can be seen at the position where the skin is lifted (see Figures (\ref{contour_lambda_1.75}) and (\ref{contour_lambda_2})). As discussed in the previous section, these secondary eddy structures aftermaths good mixing of injected fluid with the interstitial fluid. Hence, the movement of the injected drug within the SC tissue region becomes vigorous in case of larger tissue anisotropy and high lifting of the skin. \\

\begin{figure}
\centering
\begin{subfigure}{.39\textwidth}
\includegraphics[width=\textwidth]{contour_lambda1.pdf}
\caption{$\lambda=0.5$}
\label{contour_lambda_0.5}
\end{subfigure}
\begin{subfigure}{.39\textwidth}
\includegraphics[width=\textwidth]{contour_lambda2.pdf}
\caption{$\lambda=1$}
\label{contour_lambda_1}
\end{subfigure}
\begin{subfigure}{.39\textwidth}
\includegraphics[width=\textwidth]{contour_lambda3.pdf}
\caption{$\lambda=1.25$}
\label{contour_lambda_1.25}
\end{subfigure}
\begin{subfigure}{.39\textwidth}
\includegraphics[width=\textwidth]{contour_lambda4.pdf}
\caption{$\lambda=1.5$}
\label{contour_lambda_1.5}
\end{subfigure}
\begin{subfigure}{.39\textwidth}
\includegraphics[width=\textwidth]{contour_lambda5.pdf}
\caption{$\lambda=1.75$}
\label{contour_lambda_1.75}
\end{subfigure}
\begin{subfigure}{.39\textwidth}
\includegraphics[width=\textwidth]{contour_lambda6.pdf}
\caption{$\lambda=2$}
\label{contour_lambda_2}
\end{subfigure}
\caption{Composite streamlines within subcutaneous layer for various parameters: $a=0.4$, $Da=3\times10^{-3}$, $\delta=0.3$, $\mu_r = 0.01$ and $\lambda_s=0.05$, when $y_0=0.45$ is the line of injection.}
\end{figure}
\noindent
The responsibility of tissue anisotropy in order to generate secondary eddy can be discussed in terms of axial composite velocity $u$. Figure (\ref{velocity_lambda}) represents profiles of $u$ versus $y$ at $x=0$ for various $\lambda$ when $a=0.4$, $\mu_r = 0.01$, $\lambda_{s} = 0.05$, $\text{Da}=3\times10^{-3}$, and $\delta=0.3$. At the SD interface, $u$ becomes zero due to the no slip condition while the interstitial fluid and adipose cells exhibit horizontal motion at the SM interface as a result of the slip velocity associated with the squeezing effect under high pressure developed due to SC injection. Except the region $\left\{(x,y)|-\infty <x < \infty, 0.4\leq y \leq 1\right\}$, $u$ increases with $\lambda$ while opposite behavior is noted within the stated region. Such contrasting behavior of $u$ is due to the fixed volumetric flow rate across the SC region. Moreover, $u$ does not change its sign for $\lambda=0.5, 1, 1.25, 1.5$ but it change from positive to negative for $\lambda=1.75, 2$ indicating the development of secondary eddy structure near the SD interface where the skin is lifted.\\

\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{velocity_lambda.pdf}
\caption{Axial composite velocity versus $y$ for $\lambda=0.5,1,1.25,1.5,1.75,2$ when $x=0$.}
\label{velocity_lambda}
\end{figure}
\noindent
Figure (\ref{pressure_various_lambda_best}) shows the behavior of pressure gradient for various $\lambda$ at three positions $y=0$, $y=y_0$ and $y=R(x)$ with the other parameters are fixed as above. Clearly, a drop in the pressure gradient near $x=0$ can be observed for $\lambda\geq 1.75$ corresponding to $y=0$, $y=y_0$. However, better prominency in the pressure drop is noticed for $y=y_0$. But corresponding to $y=R(x)$, a rising of pressure gradient is observed near $x=0$ for $\lambda\geq 1.75$. Such rise is more faster in case of $\lambda< 1.75$. This opposite nature of pressure gradient between $y=y_{0}$ and $y=R(x)$ explains the development of primary and secondary eddies corresponding to $\lambda=1.75, 2$. On the other hand, the pressure gradient maintains parabolic profile for $\lambda\leq 1$ in a neighbourhood of $x=0$ corresponding to all three positions mentioned above. That is in case of $\lambda\leq 1$, the pressure gradient maximizes on the line $x=0$ for the three cases $y=0$, $y=y_0$ and $y=R(x)$. More precisely, among the three points $(x=0,y=0)$, $(x=0,y=y_{0})$, $(x=0, y=R(0))$, the highest pressure gradient is attained at $(x=0,y=y_{0})$. Hence this phenomena results the development of primary eddy in the neighbourhood of $(x=0,y=y_{0})$. The energy associated with the contour $c=0.7$ (eddy formed) corresponding to Figures (\ref{contour_lambda_0.5})-(\ref{contour_lambda_1}) are the maxima among the all six cases associated with $\lambda$. With increase in $\lambda$, the contour $c=0.7$ increases in size and hence dissipates the energy. This happens for all other eddies. This dissipated energy is responsible for development of secondary eddies. \\

\noindent
Besides the pressure gradient, the shear stress distributions for various $\lambda$ are also helpful to discuss the generation of eddies through their fluctuating nature. Figure (\ref{shear_various_lambda_best}) depicts that among the three positions $y=0$ (SM interface), $y=y_0$ (line of injection) and $y=R(x)$ (SD interface), the maximum shear stress may be realized at SM interface for various $\lambda$. With increasing $\lambda$, the shear stress profiles corresponding to the three above positions exhibit oscillatory nature. The sensitivity is measured by the oscillatory nature of the shear stress field with $x$. In other words, increased $\lambda$ (more precisely $\lambda>1$) imparts sensitivity in the shear stress profiles. The maximum sensitivity in the shear stress profile can be observed at the line of injection $y=y_{0}$. In addition, between SD and SM interfaces the shear stress behaves exactly opposite to each other for all values of $\lambda$ considered here. However, such behavior of the shear stress field between the line of injection and SM interface is similar with higher sensitivity in the former case. Therefore, a rapid behavioral change in the shear stress field with the SD interface and the line of injection results creation of secondary eddies near the SD interface from primary eddies. Higher shear stress close to the SM interface does not allow eddies to move towards it. Hence, the formation of secondary eddies with $\lambda>1$ takes place near the SD interface or at the lifted portion of the skin only. On the other hand, the larger shear stress developed at the SM interface with higher tissue anisotropy may consequence the patient to realize pain that is generated from the SM interface. Such pain generated from the other portion of the injection site would be expected to be less.
\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\textwidth]{pressure_various_lambda_best.pdf}
\caption{}\label{pressure_various_lambda_best}
\end{subfigure}
\begin{subfigure}{.485\textwidth}
\includegraphics[width=\textwidth]{shear_various_lambda_best.pdf}
\caption{}\label{shear_various_lambda_best}
\end{subfigure}
\caption{(a) Pressure gradient (b) Shear stress variations with $x\in[-0.5,0.5]$ for $\lambda=0.5,1,1.75,2$ at three different locations (i) SM interface ($y=0$) (ii) line of injection ($y=y_{0}$) (iii) SD interface ($y=R(x)$).}
\end{figure}
\subsection{Effect of viscosity ratio $\mu_r$}
In general, the interstitial fluid is less viscous as compared to that of the adipose cellular phase. Consequently, $\mu_r$ is less than 1 (i.e., $\mu_f<\mu_c$). In this situation, the cellular phase experiences less drag from the interstitial fluid side. In other words, the cellular phase can exert higher interstitial resistance towards interstitial fluid movement. Figures (\ref{contour_mur_0.01})-(\ref{contour_mur_0.1}) illustrate composite streamlines for $\mu_r=0.1$, $\mu_r=0.05$ and $\mu_r=0.01$. Only primary eddies are developed near the SD interface for $\mu_r=0.1$. But with the decreasing $\mu_r$, a significant viscosity difference between adipose cells and interstitial fluids is developed. The development of secondary eddy can be observed at the lifted portion for $\mu_r=0.05$ which becomes prominent with a further decrement of $\mu_r$. We can locate the smallest contour $c=0.7$ for $\mu_r=0.1$ near the line of injection, which subsequently increases in size for decrease of $\mu_r$ (Figure (\ref{contour_mur_0.05})). Hence $\lambda \geq 1.75$, $a\geq0.4$ and $\mu_r\leq0.01$ are the conditions to be satisfied simultaneously in order to developed eddy structure at the lifted portion of the skin. \\

\begin{figure}
\centering
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\textwidth]{contour_mur3.pdf}
\caption{$\mu_r=0.1$}
\label{contour_mur_0.1}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\textwidth]{contour_mur2.pdf}
\caption{$\mu_r=0.05$}
\label{contour_mur_0.05}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\textwidth]{contour_mur1.pdf}
\caption{$\mu_r=0.01$}
\label{contour_mur_0.01}
\end{subfigure}
\caption{Patterns of composite streamlines within subcutaneous layer for various parameters: $\lambda=2$, $Da=3\times10^{-3}$, $\delta=0.3$, $a = 0.4$ and $\lambda_s=0.05$, when $y=y_0(=0.45)$ is the line of injection.}
\end{figure}
\noindent
Figure (\ref{velocity_mur}) shows axial composite velocity at $x=0$ for various values of the parameter $\mu_r$. We can observe that the composite velocity near the SD and SM interface is decreased with the increasing magnitude of $\mu_r$ but the opposite phenomenon is noticed near the line of injection. Also, axial composite velocity change its sign for $\mu_r=0.01$ and does not change for $\mu_r=0.05$ and $\mu_r=0.1$ which becomes the root cause of prominent secondary eddy structure development corresponding to $\mu_r=0.01$. The development of two eddies may be discussed with the help of pressure gradient and shear stress at the three positions of $y$ i.e., $y=0$, $y=y_0$, and $y=R(x)$.\\

\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{velocity_mur.pdf}
\caption{Axial composite velocity versus $y$ for $\mu_r=0.01,0.05,0.1$ when $x=0$.}
\label{velocity_mur}
\end{figure}
\noindent
Figure (\ref{pressure_various_mur}) exhibits pressure gradient for various $\mu_r$ corresponding to $y=0$, $y=y_0$ and $y=R(x)$ when the other parameters are $a=0.4$, $\lambda=2$, $\lambda_s=0.05$ and $y_0=0.45$. We can observe the rise of the pressure gradient with the increasing magnitude of $\mu_r$ in all the three positions of $y$. Moreover, a rapid drop in the pressure gradient may be detected for decrease in $\mu_{r}$ below $0.05$. In particular for $y=0$ and $y=R(x)$, the maximum pressure gradient is observed near $x=0$. However, a pressure drop may be observed for $y=y_{0}$ close to $x=0$ for three above values of $\mu_{r}$ and such pressure drop becomes significant for lower magnitude of $\mu_{r}$. The existence of this pressure drop can be explained using the fact of certain reduction of axial composite velocity with $\mu_{r}$ (see Figure (\ref{velocity_mur})). Therefore, one can summarize that upon injection of low viscous fluid (much lower as compared to the adipose cells), a pressure drop is developed near $x=0$. This pressure drop is on the other hand equivalent to the rise in pressure gradient near $x=0$ corresponding to $y=R(x)$. This opposite nature of pressure gradient between $y=y_{0}$ and $y=R(x)$ explains the development of primary and secondary eddies corresponding to $\mu_{r}=0.01$. \\

\noindent
Further we focus on the Figure (\ref{shear_various_mur}) which discusses the shear stress distribution for various $\mu_r$ and $y=0$, $y=y_0$ and $y=R(x)$ when the other parameters are fixed as above. Like the pressure gradient, low $\mu_{r}$ is sensitive towards the shear stress field. The response of the shear stress field is not much significant near the SM interface (i.e, near $y=0$). However, near the line injection (i.e, $y=y_{0}$) the sensitivity is highest which is measured by the oscillatory nature of the shear stress field with $x$. Lastly, a moderate oscillation in the shear stress field for $y=R(x)$ depicts moderate sensitivity. In all the cases, the shear stress field becomes most sensitive for $\mu_{r}=0.01$. It is expected that this sensitivity becomes even higher in case of further lower $\mu_{r}$. Therefore the creation of primary and secondary eddies are due to the rapid change in the shear stress field during the injection of low viscous fluid. Consequently, from the observations of pressure gradient and shear stress field we may conclude that besides the higher tissue anisotropy ($\lambda\geq1.75$), a larger difference in viscosity between the injected fluid and adipose cell (i.e., $\mu_r<0.05$) causes adequate circulation within the interstitial space.
\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\textwidth]{pressure_various_mur.pdf}
\caption{}\label{pressure_various_mur}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\textwidth]{shear_various_mur.pdf}
\caption{}\label{shear_various_mur}
\end{subfigure}
\caption{(a) Pressure gradient (b) Shear stress variation with $x\in[-0.5,0.5]$ for $\mu_r=0.01,0.05,0.1$ and at three different locations (i) SM interface ($y=0$) (ii) line of injection ($y=y_{0}$) (iii) SD interface ($y=R(x)$).}
\end{figure}
\subsection{Effect of slip coefficient $\lambda_s$}
During the fluid injection process, the interstitial fluid and adipose cells residing close to the SM interface are expected to realize an axial momentum along $x$ axis away from the point $(x,y)=(0,0)$. To realize such phenomena, we have assumed slip length $\lambda_{s}>0$. Figures (\ref{contour_slip1})-(\ref{contour_slip3}) depict the variation of composite streamlines corresponding to $\lambda_{s}=0.001$ (nearly no slip interface), $\lambda_{s}=0.01$ and $\lambda_{s}=0.05$ respectively when the others are $a=0.4$, $\lambda=2$, $\delta=0.3$, $\mu_{r}=0.01$ and $y_{0}=0.5$ is the line of injection. If we consider two particular contours say $c=0.9$ and $c=0.03$ within the primary and secondary eddy respectively then it is observed that with increases in $\lambda_{s}$, size of the contour $c=0.9$ reduces but opposite behaviour is observed for the contour $c=0.03$ with development of new contours. Consequently, the increased $\lambda_s$ becomes favourable situation for the convective transport of injected fluid due to less chance of trapping within the lifted portion of the skin and any fold at the SD interface. \\

\noindent
In the above context, it is necessary to analyze the stress field concerning $\lambda_{s}$. It is found from Figure (\ref{shear_various_slip}) that as expected, the SM interface is more prone to respond to the shear stress developed there as compared to the SD interface and the line of injection where the impact of $\lambda_{s}$ is quite marginal. However, at the SM interface, an increase in the shear stress has resulted in a decrease in $\lambda_s$. The developed shear stress at the SM interface during fluid injection gets dissipated along with the interface when $\lambda_s$ attains the higher magnitude. More precisely, corresponding to $\lambda_s=0.001$, such dissipation of stress is lesser as compared to other cases. Hence with the increasing $\lambda_s$, a patient may realize less pain as a result of shear stress dissipation. Consequently, larger $\lambda_s$ (e.g., $\lambda_{s}=0.05$) at the SM interface induces a favorable situation by reducing the shear stress during fluid injection and increasing the scope of convective transport to develop near SD interface.
\begin{figure}
\centering
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\textwidth]{contour_slip1.pdf}
\caption{$\lambda_s=0.001$}
\label{contour_slip1}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\textwidth]{contour_slip2.pdf}
\caption{$\lambda_s=0.01$}
\label{contour_slip2}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\textwidth]{contour_slip3.pdf}
\caption{$\lambda_s=0.05$}
\label{contour_slip3}
\end{subfigure}
\caption{Patterns of composite streamlines within subcutaneous layer for various parameters: $\lambda=2$, $Da=3\times10^{-3}$, $\delta=0.3$, $a = 0.4$ and $\mu_r=0.01$, when $y=y_0(=0.45)$ is the line of injection.}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{shear_various_slip.pdf}
\caption{Shear stress variation with $x\in[-0.5,0.5]$ for $\lambda_s=0.001,0.01,0.05$ and at three different locations (i) SM interface ($y=0$) (ii) line of injection ($y=y_{0}$) (iii) SD interface ($y=R(x)$).}
\label{shear_various_slip}
\end{figure}
\subsection{Hydrodynamic behaviour near the line of injection}
We expect that the viscous force becomes negligible near the line of injection within the SC layer due to the weakening impact of the hydrodynamic boundary layer. One can compare this situation with the significant domination of viscous forces close to the wall for a Poiseuille type flow within a channel or tube. Moreover, such domination is found to be relevant in the case of Poiseuille type flow in fluid overlying a porous medium \citep{hill2008poiseuille} and 2D flow through a wavy anisotropic porous channel \citep{karmakar2017note} where flows near the boundary obeys the Brinkman equation but are far from the boundary viscous forces become less effective and hence the flow satisfies Darcy equation. Consequently in this study, the second-order derivative terms in the momentum equation for the interstitial fluid motion and the terms containing $\mu_f/\mu_c$ can be dropped. Subsequently, we obtain the following equations for the leading order:\\
\begin{equation}\label{HEq2}
(u_{f_0} - u_{c_0}) ~\thicksim - \varphi_f \frac{\partial p_0}{\partial x},
\end{equation}
and
\begin{equation}\label{HEq3}
(v_{f_0} - v_{c_0}) ~\thicksim  \varphi_f \frac{{\partial^2 p}_{0}}{{\partial x}^2} y.
\end{equation}
From the volumetric flow rate balance, one can obtain the pressure gradient for leading order as
\begin{equation}\label{HEq4}
p_{0x}~\thicksim \frac{1}{\varphi^2_f}\left(\frac{R(x)}{2}-\frac{1}{R(x)}\right).
\end{equation}
Similarly, we can obtain the \text{O}($\delta^2$) equations as
\begin{equation}\label{HEq5}
(u_{f_1}-u_{c_1})~\thicksim \varphi_f \lambda^2 \frac{p_{0xxx}}{2}y^2 + d_1(x),
\end{equation}
and consequently, using the volumetric flow rate across the cross section, $d_1(x)$ is determined as
\begin{equation}\label{HEq6}
d_1(x) ~\thicksim -\varphi_f \lambda^2 \frac{p_{0xxx}}{6}R(x)^2 - \frac{1}{\varphi_f}\frac{R(x)}{2}.
\end{equation}
Finally, we have the pressure gradient for \text{O}($\delta^2$) as
\begin{equation}\label{HEq7}
p_{1_x} ~\thicksim -\lambda^2 \frac{p_{0xxx}}{2}\left(y^2 - \frac{R(x)^2}{3}\right) + \frac{1}{\varphi^2_f}\frac{R(x)}{2}.
\end{equation}
Thus, the pressure gradient up to \text{O}($\delta^2$) is
\begin{equation}\label{HEq8}
\frac{\partial p}{\partial x} ~\thicksim \frac{\partial p_0}{\partial x} - \delta^2 \lambda^2 \frac{p_{0xxx}}{2}\left(y^2 - \frac{R(x)^2}{3}\right) + \frac{\delta^2}{\varphi^2_f}\frac{R(x)}{2}.
\end{equation}
Although, a similar hydrodynamic analysis as above has been reported in a large number of studies dealing with hydrodynamic boundary layer behavior, but the validity of this analysis is not beyond any argument. The utility of the above analysis aims to deduce the normal and tangential stresses in a convenient way (without involving tedious calculations). Hence to check the validity, we can compare the pressure gradient as in Eq.(\ref{HEq8}) with that of obtained from the main calculations near the line of injection (or far away from the interfaces). Therefore through Figure \ref{compare}, we observe that both the pressure gradients show similar behavior both qualitative and quantitative points of view (black and red lines). Moreover, in the same graph we try to establish the validity of the present model by plotting the pressure gradient obtained from the study of \citet{karmakar2017note} corresponding to the fixed anisotropy ratio $\lambda=2$ and $\varphi_{f}=1$. The blue line represents the pressure gradient variation versus $x$ showing a nice qualitative agreement with the present study. Note that the anisotropic geometry is the key feature of both the studies. \\

\noindent
Using the calculated hydrodynamic variables in this section, we can express normal fluid stresses along two principle directions as follows
\begin{equation}\label{HEq10}
\tau_{xx}=-p+2\frac{\delta^2}{\alpha^2}\frac{\partial u_f}{\partial x} \hspace{0.3cm}  \text{       and        } \hspace{0.3cm}  \tau_{yy}=-p+2\frac{\delta^2}{\alpha^2}\frac{\partial v_f}{\partial y}.
\end{equation}
\noindent
Figures (\ref{normal1_near_lambda}) and (\ref{normal2_near_lambda}) represent normal stresses $\tau_{xx}$ and $\tau_{yy}$ respectively for various $\lambda$ within the range $0.3\leq y \leq 0.6$ enclosing the line of injection. We observe that both the normal stresses increase with $\lambda$. Therefore both along the $x$ and $y$ directions, the intensity of the pain at the injection site increases with tissue anisotropy. On the other hand, Figure (\ref{shear_near_lambda}) shows opposite phenomena corresponding to the behavior of shear stress variation concerning $\lambda$ within the above range of $y$. The pain generated due to the shear stress decreases with an increase in tissue anisotropy. The inverse behavior of normal and shear stress fields maintains the mechanical equilibrium within the SC layer which becomes imbalanced due to the tissue anisotropy variation. The behavior of the shear stress field can be justified from the longitudinal pressure gradient ($\partial p/\partial x$) variations concerning $y$ for various $\lambda$ near the line of injection. The rate of shear stress variation along the direction normal to the line of injection is proportional to $\partial p/\partial x$.
\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\textwidth]{normal1_near_lambda.pdf}
\caption{}\label{normal1_near_lambda}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\textwidth]{normal2_near_lambda.pdf}
\caption{}\label{normal2_near_lambda}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\textwidth]{shear_near_lambda.pdf}
\caption{}\label{shear_near_lambda}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\textwidth]{pressure_near_lambda.pdf}
\caption{}\label{pressure_near_lambda}
\end{subfigure}
\caption{Normal stress components (a) $\tau_{xx}$ (b) $\tau_{yy}$; (c) shear stress $\tau_{xy}$ and (d) Pressure gradient $\left(\frac{\partial p}{\partial x}\right)$ for $\lambda=0.5,1,1.75,2$.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{compare_pressure.pdf}
\caption{Comparison of pressure gradients obtained (i) from the present study (black line) near the line $y=y_0$ (line of injection), (ii) in the limiting case $\mu_f\rightarrow0$ (red line) close to the line of injection (see Eq.(\ref{HEq8})) and (iii) from the study of \citet{karmakar2017note} which discusses the hydrodynamics of fluid flow through a wavy anisotropic porous channel (blue line).}
\label{compare}
\end{figure}
\section{Concluding Remarks}
A two-dimensional fluid injection model within the subcutaneous layer (SCL) has been investigated using biphasic mixture theory as SCL is composed of mainly adipose cells and interstitial fluids. It is observed that the distribution of composite streamlines, pressure gradient, and shear stress depends on the four parameters $a$ (skin pinching height), $\lambda$ (anisotropic ratio), $\mu_r$ (viscosity ratio), and $\lambda_S$ (slip coefficient). The overall hydrodynamic analysis reveals the creation of primary eddy structures near the line of injection due to high pressure around. There is also a creation of secondary eddy structures noticed at the lifted portion of the SCL mainly due to tissue anisotropy. However, the viscosity of the injected fluid and the skin pinching height plays an additional role in the creation of secondary eddy. Moreover, the generation of secondary eddy may be taken place if the SD interface is not regular. Both the primary and secondary eddies play a significant role to homogenize the injected fluid with the interstitial fluid when (i) the anisotropy ratio of SCL is greater than unity (ii) low viscosity ratio and (iii) higher skin pinching height.  \\

\noindent
The distribution of pressure gradient and shear stress reveals that corresponding to the low viscosity ratio, both the pressure gradient and shear stress magnitude is found to be lower than that of in case of higher viscosity ratio. As both pressure gradient and shear stress act as a marker of pain generation inside SCL \citep{mueller2005pressure}, the high viscous injected fluid causes more realization of pain to a patient receiving SC injection. On the other hand, our analysis reveals that lower skin pinching height and low anisotropy ratio of SCL can be held responsible for the realization of more pain. Moreover, it is observed that shear stress at the SM interface is high corresponding to a low slip coefficient. That is SCL can affect the SM interface more by imparting more shear stress generated from the fluid injection. However, this slip coefficient does not have a significant impact on pressure gradient throughout the SCL.\\

\noindent
\textbf{Conflict of interest:} Authors have no conflict of interest. \\\\
\noindent
\textbf{Acknowledgements:} First author A. S. Pramanik acknowledges University Grants Commission (UGC), Govt. of India for providing junior research fellowship (NET-JRF). Second author B. Dey acknowledges University research assistance (Ref. 1516/R-2020 dated 01.06.2020) of University of North Bengal for supporting this work. \\\\
\section*{Appendix 1}
\label{appendix1}
\subsection*{Solution to the leading-order problem}
With the boundary conditions the solution of the equations (\ref{Eq1})-(\ref{Eq6}) are
$$u_{f_0}(x,y)=\frac{X_0(x,y) + Y_0(x,y)}{1+\mu_r}\hspace{0.2cm}, \hspace{0.2cm} u_{c_0}(x,y)=\frac{X_0(x,y) - \mu_rY_0(x,y)}{1+\mu_r} ,$$
where $X_0(x,y)$ and $Y_0(x,y)$ are given by\\
\[X_{0}(x,y) = \left\{
  \begin{array}{lr}
L_1 {p_{0}}_{x}y^2 + A_1^{(0)}(x) y + A_{2}^{(0)}(x) ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \text{if  } 0<y<y_{0}\\\\
L_1 {p_{0}}_{x}\left(R(x)-y\right)^2 + A_{3}^{(0)}(x) (R(x)-y) + A_{4}^{(0)}(x) ,~~~~~~~~~~~~~~~~ \text{if  } y_0<y<R(x)
\end{array}
\right.
\]\\
\[Y_{0}(x,y) = \left\{
  \begin{array}{lr}
B_1^{(0)}(x) \text{cosh}\left(\beta y\right) + B_2^{(0)}(x) \text{sinh}\left(\beta y\right) + L_2 p_{0x} ,~~~~~~~~~~~~~~~~~~~~~~~~~~~ \text{if  } 0<y<y_{0}\\\\
B_3^{(0)}(x) \text{cosh}\left(\beta (R(x)-y)\right) + B_4^{(0)}(x) \text{sinh}\left(\beta (R(x)-y)\right) + L_2 p_{0x},~~ \text{if  } y_0<y<R(x)
\end{array}
\right.
\]
 in which $A_i^{(0)}(x)$, $B_i^{(0)}(x)$ ($i=1,2,3,4$) are constants of integration which can calculated using the boundary conditions with the condition of continuity at $y=y_0$. $L_1$, $L_2$ and $\beta$ are given by the followings
$$L_1=\frac{\mu_r \alpha^2}{2} \text{      ,       } L_2 = -\frac{(\phi_f - \phi_c \mu_r)}{1+\mu_r} \text{      and       } \beta^2 = (1+\mu_r)\alpha^2.$$
Also using the equation of continuity, we have\\
$$v_{f0}(x,y)=\frac{V_0(x,y) + W_0(x,y)}{1+\mu_r}\hspace{0.2cm}, \hspace{0.2cm} v_{c0}(x,y)=\frac{V_0(x,y) - \mu_rW_0(x,y)}{1+\mu_r} ,$$
where $V_0(x,y)$ and $W_0(x,y)$ are given by\\
\[V_0(x,y) = \left\{
  \begin{array}{lr}
 - \frac{1}{3}L_1 p_{0xx}y^3 - \frac{1}{2}\left(A^{(0)}_{1}(x)\right)_{x} y^2 - \left(A_{2}^{(0)}(x)\right)_{x} y + c_1^{(0)}(x),~~~~~~~~~~~~~~~~\text{if  } 0<y<y_{0}\\\\
 - \frac{1}{3}L_1 p_{0xx}\left(R(x)-y\right)^3 - \frac{1}{2}\left(A^{(0)}_{3}(x)\right)_{x} (R(x)-y)^2 - \left(A_{4}^{(0)}(x)\right)_{x} (R(x)-y) + c_2^{(0)}(x),\\
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{if  } y_0<y<R(x)
\end{array}
\right.
\]\\
\[W_0(x,y) = \left\{
  \begin{array}{lr}
  - \frac{1}{\beta}\left(B_1^{(0)}(x)\right)_{x} \text{sinh}\left(\beta y\right) - \frac{1}{\beta}\left(B_2^{(0)}(x)\right)_{x} \text{cosh}\left(\beta y\right) - L_2 p_{0xx} y + d_1^{(0)}(x),\\
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{if  } 0<y<y_{0}\\\\
  - \frac{1}{\beta}\left(B_3^{(0)}(x)\right)_{x} \text{sinh}\left(\beta (R(x)-y)\right) - \frac{1}{\beta}\left(B_4^{(0)}(x)\right)_{x} \text{cosh}\left(\beta (R(x)-y)\right)\\ - L_2 p_{0xx} (R(x)-y)  + d_2^{(0)}(x), ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{if  } y_0<y<R(x)
 \end{array}
\right.
\]
in which $c_i^{(0)}(x)$, $d_i^{(0)}(x)$ ($i=1,2$) are constants of integration which can be calculated using the boundary conditions.
\section*{Appendix 2}
\label{appendix2}
\subsection*{Solution to the \text{O}($\delta^2$) problem}
The general solution of the \text{O}($\delta^2$) problem is
$$u_{f1}(x,y)=\frac{X_1(x,y) + Y_1(x,y)}{1+\mu_r}\hspace{0.2cm}, \hspace{0.2cm} u_{c1}(x,y)=\frac{X_1(x,y) - \mu_rY_1(x,y)}{1+\mu_r} ,$$
where $X_1(x,y)$ and $Y_1(x,y)$ are given by
\[X_1(x,y) = \left\{
  \begin{array}{lr}
  E_1(x)y^4 + E_2(x)y^3 + E_3(x) y^2 + A_1^{(1)}(x) y^2 + A_2^{(1)}(x)y + A_3^{(1)}(x),~~~~~\text{if  } 0<y<y_{0} \\\\
  E_1(x)(R(x)-y)^4 + E_4(x)(R(x)-y)^3 + E_5(x) (R(x)-y)^2 +A_1^{(1)}(x) (R(x)-y)^2 \\ + A_4^{(1)}(x)(R(x)-y) + A_5^{(1)}(x), ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \text{if  } y_0<y<R(x)
 \end{array}
\right.
\]\\
\[Y_1(x,y) = \left\{
  \begin{array}{lr}
  B_1^{(1)}(x) + B_2^{(1)}(x)\text{cosh}(\beta y) + B_3^{(1)}(x)\text{sinh}(\beta y) + F_1(x) y \text{sinh}(\beta y) \\ +
  F_2(x) y \text{cosh}(\beta y) + F_3(x) (\beta^2 y^2 + 2) + F_4(x)y + F_{5}(x),~~~~~~~~~~~~~~~~~~\text{if  } 0<y<y_{0}\\\\
  B_1^{(1)}(x) + B_4^{(1)}(x)\text{cosh}(\beta (R(x)-y)) + B_5^{(1)}(x)\text{sinh}(\beta (R(x)-y)) \\
  + F_{6}(x) (R(x)-y) \text{sinh}(\beta (R(x)-y)) + F_{7}(x) (R(x)-y) \text{cosh}(\beta (R(x)-y)) \\
  + F_3(x) (\beta^2 (R(x)-y)^2 + 2) + F_{8}(x)(R(x)-y) + F_{5}(x), ~~~~~~~~~~~~~~~~~\text{if  } y_0<y<R(x)
 \end{array}
\right.
\]
in which $A_i^{(1)}(x)$, $B_i^{(1)}(x)$ ($i=1,2,3,4,5$) are constants of integration which can calculated using the boundary conditions plus condition of continuity at $y=y_0$. $E_{i}(x)$($i=1,2,3,4,5$) and $F_{i}(x)$ ($i=1,2,3,4,5,6,7,8$) are explicitly given in the Appendix 3. \\

\noindent
Hence, the general solution of the considered problem is determined upto \text{O}($\delta^2$) as
$$u_{f}(x,y)=u_{f0}(x,y) + \delta^2 u_{f1}(x,y) + \text{O}(\delta^4), $$
$$u_{c}(x,y)=u_{c0}(x,y) + \delta^2 u_{c1}(x,y) + \text{O}(\delta^4). $$
\section*{Appendix 3}
\label{appendix3}
$E_1(x)=-\frac{1}{6}L_1 p_{0xxx}$ , \hspace{0.5cm}  $E_2(x)=-\frac{1}{3} (A_1^{(0)}(x))_{xx}$, \hspace{0.5cm} $E_3(x)=- (A_2^{(0)}(x))_{xx}$ , \\
$E_4(x)=-\frac{1}{3} (A_3^{(0)}(x))_{xx}$, \hspace{0.5cm} $E_5(x)=- (A_4^{(0)}(x))_{xx}$ , \hspace{0.5cm} $F_1(x)=\frac{(\lambda^2-2)}{2\beta}(B_1^{(0)}(x))_{xx}$, \\
$F_2(x)=\frac{(\lambda^2-2)}{2\beta}(B_2^{(0)}(x))_{xx}$ , \hspace{0.5cm} $F_3(x)=\frac{\lambda^2}{2\beta^2} L_2 p_{0xxx}$ , \hspace{0.5cm} $F_4(x)=\lambda^2 (d_1^{(0)}(x))_{x}$ ,\\
$F_{5}(x)= -\frac{2}{\beta^2}L_{2} p_{0xxx}$ , \hspace{0.5cm} $F_{6}(x)=\frac{(\lambda^2-2)}{2\beta}(B_3^{(0)}(x))_{xx}$ , \hspace{0.5cm} $F_{7}(x)=\frac{(\lambda^2-2)}{2\beta}(B_4^{(0)}(x))_{xx}$, \\
$F_{8}(x)=\lambda^2 (d_2^{(0)}(x))_{x}$ .
\bibliographystyle{unsrtnat}

redpajama_set_name: RedPajamaArXiv


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If anyone in the world of rock can rest on his laurels, it's Paul McCartney.
McCartney was half of the most famous songwriting tandem of all time. McCartney and John Lennon wrote pop-rock standards, which will one day be regarded in the same manner as what William Shakespeare penned.
The 78-year-old has written or co-written a staggering 32 songs that have hit the top of the Billboard Hot 100 – a feat that is without parallel. Beatles compositions such as "And I Love Her," "Blackbird," "Eleanor Rigby" and "Yesterday" are among the most covered songs ever.
But the numbers only tell part of the story. McCartney is a remarkably gifted writer and recording artist who also is a wonderfully melodic bass player who can play a variety of instruments.
Just like on 1970's "McCartney" and 1980's "McCartney 2," Sir James Paul McCartney plays every instrument on "McCartney 3," which drops Dec. 18.
Before even listening to the album, the question is why decide to do all of the heavy lifting for such an album while on the edge of octogenarian status?
But this is McCartney, who before the pandemic was touring regularly, delivering glorious three-hour concerts. Music is McCartney's passion, and he still has plenty to express throughout the moving, melodic and at times masterful "McCartney 3."
The songs are well-structured but also playful. McCartney is performing as if he has nothing to lose, and that is so.
There's some surprising grit and power during "Slidin,' " which includes a solid, memorable riff. "The Kiss of Venus" is catchy and pretty. Thumbs up for the sturdy blues rocker "Lavatory Lil" and the moody "Deep Down."
The gorgeous "Seize the Day" is a welcome optimistic track about enjoying every moment we have on this planet. It's not easy to consider that during a pandemic, but McCartney urges his fans to look at the bright side.
Casual aficionados occasionally posit that McCartney was the uncool Beatle, which couldn't be further from the truth. The reality is that McCartney was the most-talented Beatle who also was the most experimental member of the Fab Four. Which former Beatle recorded a techno album under the pseudonym the Fireman? Yes, it was McCartney.
As a Beatle, he gave more than he received, and he had the greatest range of anyone in the band. McCartney easily traveled from the hard rock of "Helter Skelter" to the dancehall of "When I'm 64." His ability to tackle any genre continued during a solo career that found McCartney recording standards and collaborating with hip-hop innovator Kanye West.
McCartney's greatest sin is arguably longevity. Sure, the living legend has recorded some clunkers, but that's what happens after releasing 38 post-Beatle albums.
"McCartney 3" is far from a clunker. The album hooks listeners from the start. The member of the Order of the British Empire shows off his guitar chops on the leadoff track, "Long Tailed Winter Bird," and concludes with the cute bookend cut "Winter Bird, When Winter Comes."
The first two of McCartney's solo albums in which he played each instrument followed the end of an era. "McCartney" was released months after the Beatles broke up. "McCartney 2" was finished less than a year after his '70s band, Wings, called it a day.
Does "McCartney 3" signify the end of the charismatic philanthropist's lengthy solo career? That's doubtful. After 57 years as a professional musician, the odds are that the restless McCartney will make new albums and one day return to the concert trail.
Perhaps McCartney will visit some new markets, such as Spokane, since he has a penchant for performing in cities that he's yet to visit. McCartney did just that in 2014 when he played Missoula's Washington-Grizzly Stadium.
McCartney and his longtime band deliver stellar sets comprised of Beatles, Wings and solo material. It wouldn't be surprising if McCartney assembled a tour to mark his 80th birthday in 2022. That would certainly be a concert to look forward to post-pandemic!
Knowing McCartney's track record, there will probably be yet another new album for him to tour behind. Enjoy McCartney, who is unlike any other recording artist.
The old man once known as the "Cute Beatle" should be admired for all of the chances he's taken and reveled in for making albums like "McCartney 3," where he simply relaxes and offers off-the-cuff pleasure.
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Here's the fact none of us want to face: We will probably need long-term care at some point in our lives.

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Immediately following news that Windows Phone 7.5 "Mango" has been released to manufacturers, Toshiba-Fujitsu became the first vendor in the world to announce a Mango phone. The stylish smartphone will launch with a less-than-memorable IS12T moniker in September, and it will be a KDDI exclusive in Japan at that time. Beyond being the first Mango phone announced, the IS12T will also be the first Windows Phone-powered smartphone to launch in Japan. Spec highlights include a 3.7-inch WVGA display, a 1GHz Snapdragon processor, 32GB of storage and a 13.2-megapixel camera. Toshiba-Fujitsu's announcement is the first in what we hope will be a long line of new smartphone announcements from Microsoft's Windows Phone partners. New devices are expected later this year from vendors including Samsung, HTC, Dell and of course Nokia.
Her left, or your left?
I'll take the yellow one…..
The phone people, the phone!
These Japanese women are far prettier than the Korean ones that are always in Samsung propaganda. With that being said, I am probably the only one that would have a chance in real life as I am really good looking and a charmer at that.
Now give me a WP7 OS on a Samsung Galaxy II platform and its own.
Now THIS should be breaking news.


text:
Brandon Williams, out on bond for alleged drug possession, arrested again
Written by Kristen Kitchens
Submitted photo Brandon Earl Williams
Brandon Earl Williams, who has been out on bond for drug charges, was recently busted by the Coffee County Drug Unit on similar offenses, including multiple counts of possession of a controlled substance and methamphetamine.
According to a copy of a CCDU incident report, on October 27, around 3:27 p.m., a K-9 deputy made a traffic stop on Williams, 35, for a cracked windshield on Old Bell Lake Road in Douglas.
After a brief investigation, Williams was reportedly found to be in possession of methamphetamine, hydrocodone, oxycodone, and tramadol. Williams was booked into the Coffee County Jail that afternoon on one count of possession of methamphetamine, four counts of possession of a controlled substance, and driving with a suspended or revoked license.
In June, Williams was charged with several counts, including two counts of possession of a controlled substance (morphine and Pregabalin), possession of a Schedule IV controlled substance (Alprazolam), possession of a firearm or knife during the commission of or attempt to commit certain felonies, crossing state lines or county guard lines with weapons without consent (pistol), prowling or loitering, driving with a suspended or revoked license, driving under the influence of drugs, and open container. He was released on bond the following month.
According to the Coffee County Jail roster on Wednesday morning, Williams is still being held on his most recent charges.
Coffee County Drug Unit
Coffee County Sheriff's Office
Schedule IV controlled substances
possession of a firearm
Shoplifting incident leads to multi-county chase, crash
Domestic dispute erupts after wife finds husband at mistress's home, husband arrested on aggravated assault charges
Douglas woman back in jail after allegedly failing to meet conditions of bond
Another traffic stop, another meth arrest
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Get $5 worth of Android apps/games for free when you watch your first movie/TV show on Prime Instant Video
- Oct. 2nd 2014 11:31 am ET
AmazonFreeDealsAndroidandroid apps
Last month Amazon began allowing all Android users to watch Prime Instant Video with the updated Amazon app. Through October 16th, anyone who watches their first movie or TV show on the app will receive 500 Amazon Coins ($5 value) for free. These coins can be used to purchase apps/games from the Amazon App Store for Android. Note: it's not clear whether or not you must be a Prime member to take advantage of this promo, but it's worth a shot either way.
Here's how to get your free Amazon Coins:
Download the Amazon app for Android phones
Visit Prime Instant Video player from within the Amazon app by tapping the menu icon, and selecting Prime Instant Video from the menu
Find what you'd like to watch, and tap Play – you'll be prompted to make a one-time install of the Prime Instant Video player to enable playback
Tap Open after installing the Prime Instant Video player and then tap Play
You will receive your 500 Free Amazon Coins within 72 hours of streaming your movie or TV show from your Android phone
T&C of the promo:
Promotion is valid for a limited only. You must watch a television show or movie on the Prime Instant Video player mobile app you downloaded through the Amazon Appstore prior to October 16, 2014 at 11:59 PM PST to receive promotional Amazon Coins
Promotion is available only to U.S. customers
Promotional Amazon Coins expire on the last day of the calendar month one year after we give them to you (for example, promotional Amazon Coins we give you on May 15 will expire on May 31 of the following year)
Amazon Coins will be added to your account within 72 hours of your first viewing of a television show or movie on the Prime Instant Video player mobile app you downloaded through the Amazon Appstore during the promotion period
Amazon Coins can only be used to purchase eligible products through the Amazon Appstore. Coins are not transferable and cannot be redeemed for cash
Your use of Amazon Coins is subject to the Amazon Coins Terms and Conditions and the Amazon Appstore for Android Terms of Use
We reserve the right to modify or cancel the promotion at any time.
Amazon is the world's largest online retailer, offering daily savings across nearly every category imaginable. In recent years, Amazon has grown to encompass more areas…
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text:
When Reading Gets Harder
A new focus on the way words connect — as a pathway to building the reading skills teens need for challenging texts
For years, we've thought that the answer to boosting adolescent reading comprehension lay in building students' vocabulary. Teens often struggle with the jargon and advanced terminology they encounter as they move into middle and high school, so educators have designed curricula and interventions that explicitly teach these complex words.
But these strategies aren't always fully effective, according to literacy researcher Paola Uccelli. As she writes, many of these interventions have yielded "significant growth in vocabulary knowledge yet only modest gains in reading comprehension." Too many teens still struggle to understand assigned texts.
When Connecting Words are Unfamiliar
Uccelli's research explores a new approach. By focusing on how words connect in academic texts — and by recognizing that this connecting language is a possible source of difficulty for adolescent readers — teachers may be better able to equip middle and high school students with the tools to comprehend the texts they're reading for higher-order learning.
Her work identifies a set of language features that are common in academic text but rare in informal spoken language. She's found that many of the most common language features of middle school texts are unknown to large proportions of students, even by eighth grade. This is relevant for pedagogy, she says, because many of these language challenges are invisible to educators, curriculum designers, and researchers.
Understanding How Words and Passages Relate
Too often, says Uccelli, students learn about language in isolation, memorizing vocabulary in one lesson and studying grammar and sentence structure in another. Of course, students need to understand technical terms — "isotope" in a chemistry textbook, "colonize" in a history article — but this knowledge is insufficient if students don't understand how pieces of language relate to each other to create meaning.
In another article, Uccelli showcases the difficulties that academic texts can pose for students:
The scientific agreement today is that the Earth's surface temperatures have increased in recent decades. Moreover, most scientists agree that it is extremely likely that humans are causing this problem through activities that increase concentrations of greenhouse gases.
To understand the text, students have to know the definitions of "greenhouse gases" and "concentrations." But those clarifications won't elucidate the whole passage: Students also need to understand that "this problem" refers to increasing surface temperatures, and that "moreover" signifies that the second sentence builds on the first.
Knowing how to unlock these connecting concepts is particularly important when reading academic texts, which use language far differently than we use it as speakers. When we talk through complicated subjects, we can use simple sentence structures, aided by tools such as hand movements, intonation, and enactment. But when there are only words on a page to explain these ideas precisely and concisely, the language structures often get complex, and students can grow confused when faced with these dense constructions. Paradoxically, the academic language features meant to support precise communication can obstruct comprehension if readers are unfamiliar with them.
Strategies for Practice
How can teachers help students understand the nuances in language — the ways sentences build on each other, convey complex arguments, and relay contrasting information? Uccelli offers strategies to highlight these connections:
When introducing new vocabulary in a passage, think beyond the nouns, adjectives, and verbs to also explain the meanings of the "connectives." Phrases and words like "in contrast," "in other words," "nevertheless," or "consequently" can signal a change in the meaning of a passage and provide clues to the meaning of what follows. Demonstrate to students how to use these connectives in their own writing, too.
Explain "chains of reference." Academic texts often describe a very complex idea in several sentences, and then refer to that idea subsequently with a short phrase such as "this problem" or "this situation." But in conversation, people rarely use this construct. Students need to learn that the short phrases often refer to a previously mentioned complex idea, and that they therefore have to hold that idea in their minds to continue to understand the rest of the text. Tracking these "chains of reference" by linking them with a pencil or underlining them with color markers can help.
More advice on boosting adolescent reading comprehension, from the Harvard Graduate School of Education's Catherine Snow
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K-12 Language and Literacy Learning and Teaching
Learning to Read to Learn
Three ways to enhance adolescent reading comprehension in a digital world
Tagged: K-12, Language and Literacy, Learning and Teaching
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Q: BASE64 plugin not working in ionic 3 I need to convert image to base 64 format which I capture either from camera or from gallery and then send that string to my REST Api.I have tried many things but the most stable solution I came across was using Base 64 package-https://ionicframework.com/docs/native/base64/
But neither it giving me any error nor it is generating the string.
Below is my code:
  this.camera.getPicture(options).then((imagePath) => {
      this.base64.encodeFile(imagePath).then((base64Image: string) => {
        console.log(base64Image);
      }, (err) => {
        console.log(err);
        this.base64Image=err;
      });
      if (this.platform.is('android') && sourceType === this.camera.PictureSourceType.PHOTOLIBRARY) {
        this.filePath.resolveNativePath(imagePath)
          .then(filePath => {
            let correctPath = filePath.substr(0, filePath.lastIndexOf('/') + 1);
            let currentName = imagePath.substring(imagePath.lastIndexOf('/') + 1, imagePath.lastIndexOf('?'));
            this.copyFileToLocalDir(correctPath, currentName, this.createFileName());
          });
      }
      else {
        var currentName = imagePath.substr(imagePath.lastIndexOf('/') + 1);
        var correctPath = imagePath.substr(0, imagePath.lastIndexOf('/') + 1);
        this.copyFileToLocalDir(correctPath, currentName, this.createFileName());
      }
    }, (err) => {
      this.presentToast('Error while selecting image.');
    });

A: You don't need a plugin for encoding, the camera plugin returns a BASE64 string with the correct options, this is a piece of my working code.
let opcoes = {
  maximumImagesCount: 1,
  sourceType: 1,
  encodingType: this.camera.EncodingType.JPEG,
  destinationType: 0, // USE THIS TO RETURN BASE64 STRING
  correctOrientation: true
};

this.camera.getPicture(opcoes).then(res => {
  this.picture= res;
  console.log('check the string', res);
  // IT RETURNS ONLY THE IMAGE STRING WITHOUT THE data:image, IF YOU WANT TO
  // SHOW YOUR PICTURE JUST APPEND THE STRING BELLOW TO YOUR RETURNED STRING AND USE IT TO DISPLAY
  this.pictureForExibition = 'data:image/jpeg;base64,' + res;
}, err => {
  console.log(err);
});

In your HTML
<!-- JUST USE AN *ngIf TO CHECK IF THE PROPERTY ISN'T NULL OR IT'LL THROW ERROR FOR THE INVALID SRC -->
<img [src]="pictureForExibition" *ngIf="pictureForExibition" alt="base64 image" />

Hope this helps

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The Bottle, an upscale eatery that features Southern Cuisine with a French influence, specializes in steaks, chops and a variety of the freshest fish available. The changing seasonal menu includes raw oysters, jumbo lump Maryland style crab cakes, veal chops, lamb racks and Hereford Choice Beef. The Bottle - a name referencing the near mythic, long-shuttered Green Bottle Grill - invites guests an experience in southern hospitality that will not soon be forgotten.


Rubenerd: Cool uses for RequestPolicy!
After talking about scary stuff in my previous post on the RequestPolicy extension for Firefox and other Mozilla browsers, now I get to talk about cool, fun stuff!
Its an effective advertisement blocker!
There are many different solutions to combating advertising on pages such as AdBlock Plus, GlimmerBlocker and the like, but all require the use of regularly updated blacklists to work effectively. You're probably subscribed to several lists yourself.
I didn't even think of it when I installed it, but given the vast majority of advertising is hosted off-site (which makes it a capable attack vector, as well as being irritating) RequestPolicy also blocks this content. As the Ghostery extension does with web bugs, it also lets me see where all the advertising is coming from on a page which isn't useful in and of itself, but its still terribly interesting.
One of the features I liked about TinyURL was the ability to "preview" URLs before you proceeded to visit them. This helps guard against people who use URL shorteners to obfuscate dodgy addresses.
RequestPolicy also blocks automatic redirects, instead rendering retro 301 and 302 redirect errors with a link to the new location. I can preview its address before visiting anything I may not have wanted to, and I get a sneak peak into how websites are constructed which also isn't useful in and of itself, but still terribly interesting. I've used that same sentence somewhere before recently.
It puts a red flag in the status bar!