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Free convective trickling over a porous medium of fractional nanofluid with MHD and heat source/sink
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Yuanjian Lin1,2,
Sadique Rehman3,
Nevzat Akkurt4,
Tim Shedd5,
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Scientific Reports volume 12, Article number: 20778 (2022) Cite this article
Mathematics and computing
Nanofluids are considered as smart fluids that can improve heat and mass transfer and have numerous applications in industry and engineering fields such as electronics, manufacturing, and biomedicine. For this reason, blood-based nanofluids with carbon nanotubes (CNTs) as nanoparticles in the presence of a magnetic field are discussed. The nanofluid traverses the porous medium. The nanofluids move on a vertical plate that can be moved. The free convection heat transfer mode is considered when the heat source and heat fluxes are constant. Convective flows are often used in engineering processes, especially in heat removal, such as geothermal and petroleum extraction, building construction, and so on. Heat transfer is used in chemical processing, power generation, automobile manufacturing, air conditioning, refrigeration, and computer technology, among others. Heat transfer fluids such as water, methanol, air and glycerine are used as heat exchange media because these fluids have low thermal conductivity compared to other metals. We have studied the effects of MHD on the heat and velocity of nanofluids keeping efficiency in mind. Laplace transform is used to solve the mathematical model. The velocity and temperature profiles of MHD flow with free convection of nanofluids were described using Nusselt number and skin friction coefficient. An accurate solution is obtained for both the velocity and temperature profiles. The graph shows the effects of the different parameters on the velocity and temperature profiles. The temperature profile improved with increasing estimates of the fraction parameter and the volume friction parameter. The velocity of the nanofluid is also a de-escalating function with the increasing values of the magnetic parameter and the porosity parameter. The thickness of the thermal boundary layer decreases with increasing values of the fractional parameter.
Nowadays, most researchers and scientists pay great attention to those methods and techniques that are useful for improving heat transfer in various heat exchanger processes. To meet these requirements, researchers have developed a new type of fluid called a nanofluid. A nanofluid is a fluid that contains nanoparticles, which are nanometer-sized particles. Metals, their oxides, carbides and carbon nanotubes are the most commonly used nanoparticles in nanofluids. Nanofluids are helpful and have a wide range of applications, including microelectronics, fuel cells, pharmaceutical processes, cross-race machines, temperature controls, heating systems, exhaust gasses from smokestacks, heat dissipation, and so on. Due to the importance of nanofluids, numerous experimental and theoretical observations are being carried out by many researchers. In a detailed study, Kakac et al.1 investigated how nanofluids increase the thermal conductivity of a base fluid. Due to the high predictability of nanofluids, the problems identical to decay, clumping of new charges, and sedimentation do not occur2. In recent years, researchers have focused on the thermal perspectives of nanofluid because it is practical and has more applications in heat transfer and cooling. Natural convection is the general mode of heat movement. The phenomenon of natural convection allows heat to flow with external aids such as suction devices, fans, and pumps, etc., and these flows are created by changing the density of fluids. It has been observed that as the temperature changes, the density decreases, but the volume increases, so that the heated layer is loses its thickness and rises. In nature, free convection currents usually occur, caused by differences in concentration and density. The most important works and reviews by researchers can be such as Ghosh and Beg3 studied the effects of local thermal non-equilibrium (LTNE) on free convection in a uniformly curved, non-Darcian permeable annulus traversed by nanofluid. Fetecau et al.4 used an isothermal vertical plate to study a fractional nanofluid combining the effects of thermal radiation and natural convection, and found the solution of the temperature and dimensionless velocity using the Laplace transform and Caputo-Fabrizio time derivative. Toki and Tokis5 studied the free convection flow considering the time-dependent heating over a porous medium and used the Laplace transform to find an exact solution. Hussanan et al.6 studied mass and heat transfer using a vertical plate and a Newtonian heater and presented an accurate temperature and velocity analysis that satisfied the boundary conditions. Turkyilmazoglu and Pop7 studied a nanofluid over a vertical flat (infinite) surface in natural convection flow with radiation effect. Pramanik8 found a result for a Casson fluid flowing through a stretching surface exponentially porous under the influence of thermal radiation. Turkilmazgolu9 studied the effect of heat transfer and unsteady flow of a nanofluid through a moving vertical plate. Ge-JiLe et al.10 studied the radiated MHD flow of iron-containing nanoparticles with Brownian motion and thermophoresis through a cone. Kavya et al.11 revealed a hybrid nanofluid with MHD and heat extraction/injection through a shrinking/stretching cylinder with a suspension of MoS4 and copper nanoparticles. The study of a hybrid nanofluid composed of a Newtonian and a non-Newtonian fluid flowing over a stretching sheet was reported by12,13,14,15,16,17.
The magnetic field affects both man-made and natural currents. The magnetic field plays a major role in the pumping, stirring, and levitation of liquid metals and in the generation of electricity in industry. Molten metals are found in the Earth's core, creating a magnetic field known as the geomagnetic field. Sunspots and solar flares form the solar magnetic field. Due to practical applications, the study of MHD with heat transfer is of particular importance, as evidenced by the buoyancy-induced effect in quasi-solid bodies, water bodies, and the atmosphere, e.g., the Earth. Khan et al.15 studied the unsteady MHD flow with free convection in a porous medium with heat diffusion and a sloped wall. Khan et al.16 also considered a porous medium with Newtonian heating and a Casson type sodium alginate based nanofluid and analyzed the unsteady MHD flow. Yigra et al.17 dealt with mass transfer and convective heat in a nanofluid in applied magnetic field, flow past a permeable medium on a stretching sheet, chemical reaction, viscous dissipation and Soret effect. Gaffar et al.18 studied MHD (free convection) flow with ohmic dissipation of Eyring-Powell fluid and Hall/Islip flows in a porous medium on a vertical surface. Mahmoudi et al.19 obtained a result to improve the heat transfer and entropy generation in a flow with natural convection using a copper–water nanofluid and a two-dimensional trapezoidal enclosure with a continuous magnetic field. Khan et al.20 considered a non-compressible fluid (viscous) and worked on the results of MHD flow with free convection in a permeable medium located near an oscillating plate. Jha et al.21 used a vertical annular microchannel in which a magnetic field is present and discussed free convection flow. Sheikholeslami et al.22 inquired about the flow behavior using a constant heat source and a porous medium and obtained results for a nanofluid by increasing the buoyancy forces to enhance the heat transfer. In the applied magnetic field, Fetecau et al.23 studied the natural convection flow with radiation effects. Zeeshan et al.24 studied the spontaneous convection flow through porous media under the influence of MHD and provided pictorial and mathematical results. Ashorynejad et al.25 studied hybrid nanofluid as natural convection flow in an open cavity under the influence of MHD. Turkilmazgolu26 studied the heat transfer and mass properties of electrically conducting fluids over a apartment plate (infinite and vertical) and represented them numerically. Sheikholeslami et al.27 studied the effects of MHD on natural convection in a 2D horizontal annulus for an Al2O3-water nanofluid. Azhar et al.28 discussed a fractional nanofluid as a free convection system with a constant heat flux and a heat source flowing over an endless vertical plate, focusing on the graphical and analytical results.
Wang et al.29 studied the heat and mass transfer of a general MHD-Oldroyd-B bio-nanofluid in a permeable medium with increasing conditions in comparison. Heat transport by free convection is an important branch of fluid dynamics that has been matured for applications such as geothermal, geo- and astrophysics, paramedical sciences, and oil reservoirs, etc. Ramudu et al.30 have studied the influence of Soret and Dufour on Casson MHD fluid flow on an extended surface. The solution of the model is obtained by the Runge–Kutta method (along shooting). Farooq et al.31 presented the free convective flow of an oscillating Maxwell nanofluid with heat and mass transport. The velocity is a decreasing function of the volume fraction, while the temperature profile grows with varying estimates of the volume fraction parameter. Tang et al.32 reported the comparative approach of naturally convective flow of a fractional Maxwell fluid with radiation and uniform heat flux. The well-known integral transform (Laplace transform) is used to solve the fractional Caputo and Caputo-Fabrizio model. The phenomenon of heat absorption/consumption has numerous applications in engineering such as reinforcement of thrust bearings, cooling of metal sheets, recovery of unpolished oil and in medicine etc. Anantha Kumar et al.33 studied the first and second order slips in micropolar fluid flow over a convective surface with MHD and varying heat absorption/consumption. The velocity of the fluid increases as the second order slip is estimated, while the temperature decreases against the second order slip. Anantha Kumar et al.34 studied the MHD Cattaneo-Christov flow with variable heat source/sink over a cone and wedge. The study of non-Newtonian MHD fluid flow with heat absorption/consumption along different geometries was analyzed by35,36,37,41,42,43,44,45,46. Anantha Kumar et al.38 studied the MHD fluid Williamson with variable heat source/sink and chemical reaction on a curved/apartment surface. Also Anantha Kumar et al.39,40 presented the influence of free convection and nonlinear radiation of a micropolar MHD fluid near stagnation with convective surface.
From the literature review, no work has been done on convective heat transport of nanofluids along a porous medium under the effect of magnetism. Such geometries have many applications in science and technology, such as power generation, conductive plates, automobiles, refrigeration, power generation, etc. Blood is used as the base fluid for the suspension of CNTs. Carbon nanotubes (CNTs) as nanoparticles have great applications in the field of nanotechnology due to their unique electrical shape and mechanical properties. Applications of CNTs also include energy storage, conductive films, advanced electrodes, catalyst supports, coatings, biomedical and sensing applications, wearable electronics, solar and structural materials. CNTs have higher conductivity, which they use to build a network of conducting tubes. To identify the memory effect of nanofluids, the fractional derivative (Caputo-Fabrizio model) is solved exactly using Laplace technique (LT). Finally, various physical parameters are explained physically and graphically. The skin fraction and the Nusslet value are also obtained to determine the rate of heat transport and the drag forces of the nanofluid. Zakian's algorithm is used to simulate graphs and tables41.
The research questions are as follows, which is helpful in understanding the novelty and key research findings;
How do the SWCNTs and MWCNTs nanoparticles affect the flow of a viscous nanofluid with free convection?
How does the Lorentz force affect the velocity of the nanofluid when magnetic parameters are used?
How can the exact solution of the fractional model be determined and the memory effect on the nanofluid be established?
How does the porosity parameter behave on the velocity of the nanofluid?
How does the fractional parameter affect the thickness of the thermal boundary layer?
Mathematical statement of the problem
The equations for the free convection flow of an incompressible MHD fluid and the heat transfer in the presence of a heat source/sink at an infinite vertical plate in a porous medium subject to the Boussinesq approximation are as follows,
$$\nabla .{\mathbf{V}} = 0,$$
$$\rho_{nf} \left[ {\frac{{\partial {\mathbf{V}}}}{\partial t} + \left( {{\mathbf{V}}.\nabla } \right){\mathbf{V}}} \right] = \mu_{nf} \nabla^{2} {\mathbf{V}} + {\mathbf{J}} \times {\mathbf{B}} + {\mathbf{r}} + \rho_{nf} g\beta_{nf} \left( {T - T_{\infty } } \right),$$
$$\left( {\rho C_{p} } \right)_{nf} \left[ {\frac{\partial T}{{\partial t}} + \left( {{\mathbf{V}}.\nabla } \right)T} \right] = k_{nf} \nabla^{2} T + {{\varvec{\uptau}}}{\mathbf{.L}} - Q^{ * } \left( {T - T_{\infty } } \right).$$
where, \({\mathbf{r}}\) denotes the Darcy's resistance, \({\mathbf{J}}\) is the current density, \({\mathbf{B}}\) demonstrates the total magnetic field, \({\mathbf{V}}\) denotes the velocity vector i-e \({\mathbf{V}} = \left[ {W\left( {Y,\tilde{t}} \right),0,0} \right],\)\({{\varvec{\uptau}}}{\mathbf{.L}}\) represents the term viscous dissipation, \({\mathbf{L}} = {\text{grad}}{\mathbf{V}}\), \({{\varvec{\uptau}}}\) denotes the Cauchy stress tensor i-e \({{\varvec{\uptau}}} = - {\rm P}{\rm I} + S,\)\({\rm P}\) is the pressure, \({\rm I}\) represents the unit tensor, \(S\) expressed the extra stress tensor, \(\rho_{nf} ,\;\mu_{nf} ,\;\beta_{nf} ,\;\left( {C_{p} } \right)_{nf} ,\;k_{nf}\) are respectively density, absolute viscosity, thermal expansion coefficient of nanofluid, specific heat and nanofluid's thermal conductivity, \(g\) is the gravitational acceleration and \(Q^{ * }\) denotes the coefficient of heat source/sink.
Consider the natural convection flow of electrically conducting and incompressible nano-fluids. The flow medium is an infinite vertical plate. The magnetic field strength B_o acts uniformly and perpendicularly on the plate. At a given time, both the plate and the fluid are in a stationary position with ambient temperature. When the timecomes, the plate starts to move with velocity \({U}_{o}(1-{e}^{-\gamma t})\), provided that no heat enters or leaves the system. Here shows the amplitude of the motion and denotes the dimensional constant. The non-Darcian modal with porous medium is considered. In the energy equation, viscous dissipation is not included because of its small size. The geometry of the flow problem is shown in Fig. 1. In addition, the assumptions made to idealize the above model are examined as follows:
Geometry of the flow problem.
The nanofluid consists of the base fluid blood and nanoparticles called SWCNTs and MWCNTs.
The thermal equilibrium is balanced between the base fluid and the nanoparticles.
The temperature buoyancy force in the momentum equation is a function of density.
The viscous dissipation is ignored in the energy equation.
The resulting magnetic field due to the nanofluid flow is neglected compared to the imposed magnetic field.
The influence of the polarization of the nanofluid is ignored, so no external electric field is applied.
However, one-dimensional and unidirectional flow is studied, and the vertical plate is assumed to be infinite in length, so temperature and velocity are only a function of and Darcy's law for viscous fluids is represented as follows
$${\mathbf{r}} = - \frac{{\mu_{nf} }}{K}W\left( {\tilde{t},Y} \right).$$
$${\text{div}}{\mathbf{B}} = 0,{\text{ curl}}{\mathbf{B}} = \mu_{m} {\mathbf{J}},{\text{ Curl}}{\mathbf{E}} = - \frac{{\partial {\mathbf{B}}}}{\partial t},$$
and utilization of Ohm's law that lead to,
$${\mathbf{J \times B}} = - \left( {\sigma_{nf} B_{0}^{2} W,0,0} \right).$$
The suspension of nanoparticles in a fluid cannot be left uncontrolled; it must be controlled or tightened. Fluid movement and temperature are components of and because they are interdependent. Blood (as the base fluid) plus nanoparticles SWCNTs and MWCNTs form the nanofluid. Table 1 lists the physical and thermal properties of the particles.
Table 1 Shows the thermo-physical characteristics of human blood and nanoparticles.
In response to Eqs. (4)–(6) and all the assumptions mentioned, Eqs. (2) and (3) for nanofluids can be considered as follows28;
$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho }_{nf} \frac{{\partial W(Y,\tilde{t})}}{{\partial \tilde{t}}} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mu }_{nf} \frac{{\partial^{2} W(Y,\tilde{t})}}{{\partial Y^{2} }} - W(Y,\tilde{t})\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\sigma }_{nf} \mathop B\nolimits_{o}^{2} - W(Y,\tilde{t})\frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mu }_{nf} }}{K} + g\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\beta } } \right)_{nf} \left( {T(Y,\tilde{t}) - T_{\infty } } \right),$$
$$\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } C_{p} } \right)_{nf} \frac{{\partial T(Y,\tilde{t})}}{{\partial \tilde{t}}} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{nf} \frac{{\partial^{2} T(Y,\tilde{t})}}{{\partial Y^{2} }} - Q^{ * } (T(Y,\tilde{t}) - T_{\infty } ),Y,\tilde{t} > 0.$$
Here \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\sigma }_{nf}\), \(K\) are respectively the electrical conductivity of nanofluid, permeability of porous medium, \(T\left( {Y,\tilde{t}} \right)\) is the temperature of the nano-fluid and \(W(Y,\tilde{t})\) denotes the velocity of nanofluid.
The expressions of \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho }_{nf}\), \(\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\beta } } \right)_{nf}\),\(\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } C_{p} } \right)_{nf} ,\frac{{\kappa_{nf} }}{{\kappa_{f} }},{\text{ and }}\mu_{nf}\), \(\frac{{\sigma_{nf} }}{{\sigma_{f} }}\) are;
$$\left\{ \begin{gathered} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho }_{nf} = \ddot{\varphi }\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho }_{s} + (1 - \ddot{\varphi })\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho }_{f} ,\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } C_{p} } \right)_{nf} = (1 - \ddot{\varphi })\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } C_{p} } \right)_{f} + \ddot{\varphi }\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } C_{p} } \right)_{s} , \hfill \\ \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\beta } } \right)_{nf} = (1 - \ddot{\varphi })\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\beta } } \right)_{f} + \ddot{\varphi }\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\beta } } \right)_{s} ,\mu_{nf} = \mu_{f} \left( {1 - \ddot{\varphi }} \right)^{ - 2.5} , \hfill \\ \frac{{\kappa_{nf} }}{{\kappa_{f} }} = \frac{{(\kappa_{s} + 2\kappa_{f} ) - 2\ddot{\varphi }\left( {\kappa_{f} - \kappa_{s} } \right)}}{{(\kappa_{s} + 2\kappa_{f} ) + \ddot{\varphi }\left( {\kappa_{f} - \kappa_{s} } \right)}} = g\left( {\ddot{\varphi }} \right), \hfill \\ \frac{{\sigma_{nf} }}{{\sigma_{f} }} = 1 + \frac{{\left( {\frac{{\sigma_{s} }}{{\sigma_{s} }} - 1} \right)\ddot{\varphi }}}{{\left( {\frac{{\sigma_{s} }}{{\sigma_{s} }} + 2} \right) - \left( {\frac{{\sigma_{s} }}{{\sigma_{s} }} - 1} \right)\ddot{\varphi }}}. \hfill \\ \end{gathered} \right\}$$
Here \(\ddot{\varphi }\), \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho }_{f}\), \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho }_{s}\), \(C_{p}\) \(\kappa_{f} ,\kappa_{s} ,\mu_{f}\) represent the volume fraction of the nanoparticles, the density of the base fluid, the density of the solid particles or the specific heat at constant pressure, the thermal conductivity of the base fluid, the thermal conductivity of the base fluid and the viscosity of the base fluid.
For the prescribed Pde's (Eq. 7 and Eq. 8), the corresponding boundary conditions and initial conditions are as follows;
$$W(Y,0) = 0,\;\;T(Y,0) = T_{\infty } \;\;Y \ge 0,$$
$$W\left( {0,\tilde{t}} \right) = U_{o} \left( {1 - e^{{ - \gamma \tilde{t}}} } \right),k_{{nf}} \left. {\frac{{\partial T\left( {Y,\tilde{t}} \right)}}{{\partial Y}}} \right|_{{Y = 0}} = - q_{w} ,\tilde{t} > 0,$$
$$W(Y,\tilde{t}) = 0,T(Y,\tilde{t}) = T_{\infty } ,\;\;{\text{as}}\;\;Y \to \infty .$$
\(q_{w}\) represents heat passing from surface of wall.
Now incorporate the unit less parameters
$$\left[ \begin{gathered} Y^{*} = \frac{Y}{N},\tilde{t}^{*} = \frac{{\upsilon_{f} }}{{N^{2} }}\tilde{t},W^{*} = \frac{N}{{\upsilon_{f} }}W,\Theta = \frac{{k_{f} }}{{q_{w} }}(T - T_{\infty } ) \hfill \\ N = \left( {\frac{{k_{f} \upsilon_{f}^{2} }}{{g\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\beta }_{f} q_{w} }}} \right)^{{^{\frac{1}{4}} }} ,U_{o} = \frac{{\upsilon_{f} }}{N},\gamma^{*} = \frac{{\gamma N^{2} }}{{\upsilon_{f} }}, \hfill \\ \end{gathered} \right]$$
and by neglecting \(*\) from Eqs. (7), (8) and from Eqs. (10–12), we get the unit less form given as;
$$\vartheta_{1} \frac{{\partial W(Y,\tilde{t})}}{{\partial \tilde{t}}} = \frac{{\partial^{2} W(Y,\tilde{t})}}{{\partial Y^{2} }} - \vartheta_{3}^{ * } W(Y,\tilde{t}) - \vartheta_{4}^{ * } W(Y,\tilde{t}) + \vartheta_{2} \Theta (Y,\tilde{t}),Y,\tilde{t} \ge 0$$
$$\vartheta_{3} \frac{{\partial \Theta (Y,\tilde{t})}}{{\partial \tilde{t}}} = \frac{{\partial^{2} \Theta (Y,\tilde{t})}}{{\partial Y^{2} }} - \vartheta_{4} \Theta (Y,\tilde{t}),Y,\tilde{t} \ge 0,$$
$$W(Y,0) = 0,\Theta (Y,0) = 0,Y > 0,$$
$$W(0,\tilde{t}) = U_{o} \left( {1 - e^{{ - \gamma \tilde{t}}} } \right),g(h,\ddot{\varphi })\left. {\frac{{\partial \Theta (Y,\tilde{t})}}{{\partial Y}}} \right|_{{Y = 0}} = - 1,\tilde{t} > 0,$$
$$W(Y,\tilde{t}) = 0,\Theta (Y,\tilde{t}) \to 0,\;\;{\text{as}}\;\;Y \to \infty .$$
\(\vartheta_{1}\), \(\vartheta_{2}\), \(\vartheta_{3}^{ * }\), \(\vartheta_{4}^{ * }\), \(\vartheta_{3}\), and \(\vartheta_{4}\) are values in the equations before which can be expressed as;
$$\left\{ \begin{gathered} \vartheta_{1} = \frac{1}{{p(\ddot{\varphi })}}\left( {(1 - \ddot{\varphi }) + \ddot{\varphi }\frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho }_{s} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho }_{f} }}} \right),\vartheta_{2} \frac{1}{{p(\ddot{\varphi })}}\left( {(1 - \ddot{\varphi }) + \ddot{\varphi }\frac{{(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\beta } )_{s} }}{{(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\beta } )_{f} }}} \right), \hfill \\ \vartheta *_{3} = \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\sigma }_{f} B_{o}^{2} N^{2} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho }_{f} \upsilon_{f} p(\ddot{\varphi })}} = \frac{{\sigma_{nf} }}{{\sigma_{f} }}\frac{{M^{2} }}{{p(\ddot{\varphi })}}, \, \vartheta *_{4} = \frac{{N^{2} }}{K} = K_{p} \hfill \\ \vartheta_{3} = \frac{\Pr }{{g(\ddot{\varphi })}}\left( {(1 - \ddot{\varphi }) + \ddot{\varphi }\frac{{(\rho C_{p} )_{s} }}{{(\rho C_{p} )_{f} }}} \right),Pr = \frac{{\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mu } C_{p} } \right)_{f} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{f} }},\vartheta_{4} = \frac{{\upsilon_{f} Q^{ * } }}{{g(\ddot{\varphi })\sqrt {g\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\beta }_{f} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{f} } q_{w} }} \hfill \\ \end{gathered} \right\},$$
where \(\Pr ,M,K_{p}\) are respectively represents the Prandtl number, the Magnetic factor and the inverse permeability.
To obtain a fractional model, we include the Caputo-Fabrizio time derivative in Eqs. (7) and (8):
$$\vartheta_{1} {}^{CF}\delta_{t}^{{\hat{\alpha }}} W(Y,\tilde{t}) = \frac{{\partial^{2} W(Y,\tilde{t})}}{{\partial Y^{2} }} - \vartheta_{3}^{ * } W(Y,\tilde{t}) - \vartheta_{4}^{ * } W(Y,\tilde{t}) + \vartheta_{2} \Theta (Y,\tilde{t}),0 < \hat{\alpha } \le 1,Y,\tilde{t} \ge 0,$$
$$\vartheta_{3} {}^{CF}\delta_{t}^{{\hat{\beta }}} \Theta (Y,\tilde{t}) = \frac{{\partial^{2} \Theta (Y,\tilde{t})}}{{\partial Y^{2} }} - \vartheta_{4} \Theta (Y,\tilde{t}),0 < \hat{\beta } \le 1,Y,\tilde{t} \ge 0.$$
The Caputo-Fabrizio time fractional derivative and their Laplace transform are given by;
$${}^{CF}\delta_{\tau }^{\chi } Z(Y,\tau ) = \frac{1}{1 - \tau }\int\limits_{0}^{\tau } {\frac{\partial Z(Y,r)}{{\partial r}}} \, e^{{\left( { - \frac{\chi (\tau - \chi )}{{1 - \chi }}} \right)}} dr,0 < \chi < 1,$$
$$L\left\{ {{}^{CF}\delta_{\tau }^{\chi } Z(Y,\tau )} \right\} = \frac{{r\overline{Z}(Y,r) - \overline{Z}(Y,0)}}{(1 - \chi )r + \chi }.$$
'L' denotes the LT.
Solution of the problem
Temperature field
Taking LT on (21) and make use of respective transformed ICs and BCs along with Eq. (23), we obtained
$$\frac{{\partial^{2} \overline{\Theta } (Y,r)}}{{\partial Y^{2} }} = \left[ {\frac{\eta q + \chi }{{r + \psi }}} \right]\overline{\Theta } (Y,r)$$
$$\eta = \frac{{\vartheta_{3} }}{{(1 - \hat{\beta })}} + \vartheta_{4} ,\chi = \frac{{\vartheta_{4} \hat{\beta }}}{{1 - \hat{\beta }}},\psi = \frac{{\hat{\beta }}}{1 - v},\hat{\beta } \in (0,1),$$
\(r\) represents the Laplace frequency and \(\beta\) is the fractional parameter.
$$\left. {\frac{{\partial \bar{\Theta }(Y,r)}}{{\partial Y}}} \right|_{{Y = 0}} = \frac{{ - 1}}{{g(\ddot{\varphi })r}},\bar{\Theta }(Y,r) \to 0,\;\;Y \to \infty .$$
For the solution of Eq. (24) and utilizing Eq. (25), we get
$$\overline{\Theta }\left( {Y,r} \right) = \frac{1}{{g(\ddot{\varphi })}}\frac{1}{r}\overline{\Omega } (Y,r;\eta ,\chi ,\psi ),$$
$$\overline{\Omega } (Y,r;\eta ,\chi ,\psi ) = \frac{1}{r}\frac{1}{{\sqrt {\frac{\eta r + \chi }{{r + \psi }}} }}e^{{ - Y\sqrt {\frac{\eta r + \chi }{{r + \psi }}} }} = \frac{1}{r}\frac{1}{{\sqrt {d_{{\hat{\beta }}} (r)} }}e^{{ - Y\sqrt {d_{{\hat{\beta }}} (r} )}} ,$$
$$\, d_{{\hat{\beta }}} (r) = \frac{\eta r + \chi }{{r + \psi }} = \frac{{\left[ {\vartheta_{3} + (1 - \hat{\beta })\vartheta_{4} } \right] + \vartheta_{4} \hat{\beta }}}{{(1 - \hat{\beta })r + \vartheta_{4} }},$$
Now we have to find \(\overline{\Omega }\) which is solved by using Laplace inverse over \(\overline{\Omega }\) but the given function is not a simple function, it is a compound function and can be defined as;
If \(\overline{F}(r)\) be a function then the Laplace inverse \(\overline{F}(r)\) of is \(F(\tilde{t})\). Then the LIT of \(F(d(r))\) is represented by;
$$\, L^{ - 1} \{ F(d(q))\} = \int\limits_{0}^{\infty } {F(m)s(m,\tilde{t})} dm, \;\; \text{and}$$
$$s(m,\tilde{t}) = L^{ - 1} \left\{ {e^{( - md(r))} } \right\}$$
Taking LIT to Eq. (28) and utilizing Faltung product present in Eq. (23), here \(\overline{F}(r) = \left( {\frac{1}{\sqrt r }} \right)e^{ - Y\sqrt r }\) and \(d(r) = d_{\beta } (r)\), we acquired the Laplace inverse of \(\overline{\Omega }(Y,r;\eta ,\chi ,\psi )\), we have
$$\Omega (Y,\tilde{t};\eta ,\chi ,\psi ) = \left\{ \begin{gathered} \frac{1}{\sqrt \pi }H(\tilde{t})\int\limits_{0}^{\infty } {\frac{1}{\sqrt \pi }} e^{{\left( {\frac{{ - Y^{2} }}{4m} - \eta m} \right)}} dm + \hfill \\ \sqrt {\frac{\eta \psi - \chi }{\pi }} \int\limits_{0}^{\infty } {e^{{\left( {\frac{{ - Y^{2} }}{4m} - \eta m} \right)}} } \times \hfill \\ \int\limits_{0}^{\infty } {\frac{{e^{ - \psi n} }}{\sqrt n }J_{1} \left( {2\sqrt {(\eta \psi - \chi )mn} } \right)dndm} \hfill \\ \end{gathered} \right\},$$
The unit step Heaviside function H(t) and the Bessel modified function of order one and first types are expressed in the preceding equation. The more reliable form of Eq. (30) is given below;
$$\Omega (Y,\tilde{t};\eta ,\chi ,\psi ) = \frac{1}{\sqrt \eta }H(\tilde{t})e^{ - Y\sqrt \eta } + \sqrt {\frac{\eta c - b}{\pi }} \int\limits_{0}^{\infty } {e^{{\left( {\frac{{ - Y^{2} }}{4m} - \eta m} \right)}} } \times \int\limits_{0}^{{\tilde{t}}} {\frac{{e^{ - \psi n} }}{\sqrt n }J_{1} \left( {2\sqrt {(\eta \psi - \chi )mn} } \right)dndm} ,$$
Applying LIT upon Eq. (26) we acquired
$$\Theta (Y,\tilde{t}) = \frac{1}{{g(h,\ddot{\varphi })}}\frac{1}{r}\Omega (Y,\tilde{t};\eta ,\chi ,\psi ).$$
Nusslet number
The Nusselt number Nu, is taken from23,
$$Nu\left( r \right) = \frac{{Nq_{w} }}{{k_{f} (T_{w} - T_{\infty } )}} = \frac{1}{{\Theta (Y,\tilde{t})}}|_{Y = 0} = \frac{1}{{L^{ - 1} \{ \overline{\Theta } (0,r)\} }},$$
$$Nu\left( {\tilde{t}} \right) = \frac{{g(r,\ddot{\varphi })}}{{\Omega (0,\tilde{t};\eta ,\chi ,\psi )}},$$
The expression \(\Omega (0,\tilde{t};\eta ,\chi ,\psi )\) has been acquired by using
$$\Omega (0,\tilde{t};\eta ,\chi ,\psi ) = L^{ - 1} \left\{ \begin{gathered} \frac{1}{r}\sqrt {\frac{r + \psi }{{\eta r + \chi }}} = \frac{1}{\sqrt \eta }e^{{\left( {\frac{{ - (\eta \psi + \chi )\tilde{t}}}{2\eta }} \right)}} \times J_{o} \left( {\frac{{ - (\eta \psi + \chi )\tilde{t}}}{2\eta }} \right)H(\tilde{t}) + \hfill \\ \frac{\psi }{\sqrt \eta } \times \int\limits_{0}^{{\tilde{t}}} {e^{{\left( {\frac{ - (\eta \psi + \chi )m}{{2\eta }}} \right)}} J_{o} \left( {\frac{ - (\eta \psi + \chi )m}{{2\eta }}} \right)} dm. \hfill \\ \end{gathered} \right\}$$
To find the thermal boundary layer thickness in term of fractional derivative. We will integrate thermal layer Eq. (24) from \(Y \to 0\) to \(Y \to \infty\)
$$\Phi_{T} (\tilde{t}) = \int\limits_{0}^{{\Phi_{1T} }} {\Theta (Y,\tilde{t})dY,}$$
By utilizing the ICs and BCs in Eqs. (17) and (18), we acquired
$$\vartheta_{3} {}^{CF}\delta_{{\tilde{t}}}^{{\hat{\beta }}} \Phi_{T} (\tilde{t}) + \vartheta_{4} \delta_{{\tilde{t}}}^{{\hat{\beta }}} \Phi_{T} (\tilde{t}) = \frac{1}{{g(h,\ddot{\varphi })}},$$
After solving Eq. (37) and using respective ICs and BCs, we acquired
$$\Phi_{T} \left( {\tilde{t}} \right) = \frac{1}{{\vartheta_{4} g(h,\ddot{\varphi })}}\left\{ {1 - \frac{{\vartheta_{3} }}{{\vartheta_{3} + (1 - \hat{\beta })\vartheta_{4} }}e^{{\left( {\frac{{ - \vartheta_{4} \hat{\beta }\hat{t}}}{{\vartheta_{3} + (1 - \hat{\beta })\vartheta_{4} }}} \right)}} } \right\},$$
For \(\hat{\beta } \to 1\) (integer order derivative), Eq. (38) becomes.
$$\Phi_{T} \left( {\tilde{t}} \right) = \frac{1}{{\vartheta_{4} g(h,\ddot{\varphi })}}\left\{ {1 - e^{{\left( {\frac{{ - \vartheta_{4} \tilde{t}}}{{\vartheta_{3} }}} \right)}} } \right\}.$$
Dimensional less velocity and skin Friction coefficient
Taking LT of Caputo-Fabrizio derivative of Eq. (23) upon Eq. (20) and their respective ICs and BCs and incorporate Eq. (27), we acquired
$$\frac{{\partial^{2} \overline{W} (Y,r)}}{{\partial Y^{2} }} = \frac{{sr\overline{W} (Y,r)}}{r + j} + \vartheta_{3}^{ * } \overline{W} (Y,r) + \vartheta_{4}^{ * } \overline{W} (Y,r) - \vartheta_{2} \frac{1}{{g(\ddot{\varphi })r}}\frac{1}{{\sqrt {d_{{\hat{\beta }}} (r)} }}e^{{ - Y\sqrt {d_{{\hat{\beta }}} (r} )}}$$
$$s = \frac{{\vartheta_{1} }}{{1 - \hat{\alpha }}}{, }j{ = }\frac{{\hat{\alpha }}}{{1 - \hat{\alpha }}},$$
After solving Eq. (40) and using ICs and BCs, we acquired
$$\overline{W} (Y,r) = \left\{ \begin{gathered} \frac{\gamma }{r(r + \gamma )}\overline{\zeta } (Y,r;s,j,\vartheta_{3}^{ * } ,\vartheta_{4}^{ * } ) + a_{2} \overline{D} \left( r \right))\overline{\Theta } (Y,r) \hfill \\ \, - a_{2} \overline{\Theta } (0,r)\overline{D} (r)\overline{\zeta } (Y,r;s,j,\vartheta_{3}^{ * } ,\vartheta_{4}^{ * } ) \hfill \\ \end{gathered} \right\},$$
$$\overline{\zeta } (Y,r;s,j,\vartheta_{3}^{ * } ,\vartheta_{4}^{ * } ) = e^{{ - Y\sqrt {U_{{\hat{\alpha }}} (r)} }} ,{\text{ Where }}U_{{\hat{\alpha }}} (r) = \frac{sr}{{r + j}} + \vartheta_{3}^{ * } + \vartheta_{3}^{ * } ,$$
$$\begin{gathered} \overline{D} (r) = \left\{ \begin{gathered} \frac{(r + \psi )(r + j)}{{(s + \vartheta_{3}^{ * } + \vartheta_{4}^{ * } - \eta )r^{2} - r(\eta j + \chi - s\psi - \vartheta_{3}^{ * } \psi + \vartheta_{4}^{ * } - j\psi \vartheta_{3}^{ * } - \vartheta_{4}^{ * } j) - (j\chi - \vartheta_{3}^{ * } j\psi - \vartheta_{4}^{ * } j\psi )}} \hfill \\ = \frac{1}{s - \eta }\left( {1 + \frac{{l_{11} }}{{r - q_{1} }} - \frac{{l_{12} }}{{r - q_{2} }}} \right){ , } \quad s \ne \eta . \, \hfill \\ \end{gathered} \right\} \hfill \\ \, \hfill \\ \end{gathered}$$
where \(l_{11} = \frac{{(q_{1} + \psi )(q_{1} + j)}}{{q_{1} - q_{2} }},l_{12} = \frac{{(q_{2} + \psi )(q_{2} + j)}}{{q_{1} - q_{2} }}.\)
$$q_{1,2} = \frac{{\left\{ \begin{gathered} \eta j + \chi - s\psi - \vartheta_{3}^{ * } \psi - \vartheta_{4}^{ * } \psi - \vartheta_{3}^{ * } j - \vartheta_{4}^{ * } j \pm \hfill \\ \sqrt {(\eta j + \chi - s\psi - \vartheta_{3}^{ * } \psi - \vartheta_{4}^{ * } \psi - \vartheta_{3}^{ * } j - \vartheta_{4}^{ * } j)^{2} + 4(\chi j - \vartheta_{3}^{ * } j\psi - \vartheta_{4}^{ * } j\psi )(s + \vartheta_{3}^{ * } + \vartheta_{4}^{ * } - \eta )} \hfill \\ \end{gathered} \right\}}}{{2(s + \vartheta_{3}^{ * } + \vartheta_{4}^{ * } - \eta )}}$$
are the polynomial roots
$$p(r) = (s + \vartheta_{3}^{ * } + \vartheta_{3}^{ * } - \eta )r^{2} - r(\eta j + \chi - \psi s - \vartheta_{3}^{ * } \psi + \vartheta_{4}^{ * } - \vartheta_{3}^{ * } \psi j - \vartheta_{4}^{ * } \psi ) - (\eta j - \vartheta_{3}^{ * } \psi j - \vartheta_{4}^{ * } \psi j)$$
Applying Laplace inverse transform to Eq. (41) using Eq. (29) i.e. Eq. of compound function. with \(E(r) = e^{ - Y\sqrt r } {\text{ and }}U_{{\hat{\alpha }}} (r) = \frac{sr}{{r + j}} + \vartheta_{3}^{ * } + \vartheta_{3}^{ * } ,\) we acquired
$$\zeta (Y,\tilde{t};s,j,\vartheta_{3}^{ * } ,\vartheta_{4}^{ * } ) = \delta (\tilde{t})e^{{ - Y\sqrt {s + \vartheta_{3}^{ * } + \vartheta_{4}^{ * } } }} + \zeta_{o} (Y,\tilde{t};s,j,\vartheta_{3}^{ * } ,\vartheta_{4}^{ * } ),$$
$$\zeta_{o} \left( {Y,\tilde{t};s,j,\vartheta_{3}^{ * } ,\vartheta_{4}^{ * } } \right) = \frac{{Y\sqrt {sj} e^{{ - j\tilde{t}}} }}{{2\sqrt {\pi \tilde{t}} }}\int\limits_{0}^{\infty } {\frac{{e^{{^{{\left( {\frac{{ - Y^{2} }}{4x}} \right. - x\left( {s + \vartheta_{3}^{ * } + \vartheta_{3}^{ * } } \right)}} }} }}{x} \times J_{1} \left( {2\sqrt {sjx\tilde{t}} } \right)dx.}$$
The LIT of \(\overline{D}(r)\), present in Eq. (42) is,
$$D(\tilde{t}) = \frac{1}{s - \eta }\left( {\delta (\tilde{t}) + l_{11} e^{{q_{1} \tilde{t}}} - l_{12} e^{{q_{2} \tilde{t}}} } \right) \,$$
Applying the Laplace inverse transformation on Eq. (41) and Faltung theorem, we acquired
$$\begin{aligned} W(Y,\tilde{t}) & = \left( {1 - e^{{ - \gamma \tilde{t}}} } \right)e^{{ - Y\sqrt {s + \vartheta_{3}^{ * } + \vartheta_{4}^{ * } } }} + \vartheta_{2} \int\limits_{0}^{{\tilde{t}}} {\left( {1 - e^{{ - \gamma (\tilde{t} - k)}} } \right)\zeta_{o} (Y,k;s,\vartheta_{3}^{ * } ,\vartheta_{4}^{ * } )dk + \vartheta_{2} \int\limits_{0}^{{\tilde{t}}} {D(\tilde{t} - k)\Theta (Y,k)dk} } \hfill \\ & \quad - \vartheta_{2} \int\limits_{0}^{{\tilde{t}}} {\int_{0}^{k} {\Theta (0,\tilde{t} - k)D(k - \varsigma )\zeta_{o} (Y,\varsigma ;s,\vartheta_{3}^{ * } ,\vartheta_{4}^{ * } )d\varsigma dk.} } \, \hfill \\ \end{aligned}$$
Skin friction coefficient is a basic physical quantity of relevance that is defined as
$$C_{f} = \left\{ \begin{gathered} \frac{{\hat{\tau }_{w} }}{{\hat{\rho }_{f} ({{\hat{\upsilon }_{f} } \mathord{\left/ {\vphantom {{\hat{\upsilon }_{f} } N}} \right. \kern-\nulldelimiterspace} N})^{2} }} = p(\ddot{\varphi })\frac{{\partial^{2} W(Y,\tilde{t})}}{{\partial Y^{2} }}|_{Y = 0} = p(\ddot{\varphi })L^{ - 1} \left( {\frac{{\partial^{2} W(Y,\tilde{t})}}{{\partial Y^{2} }}|_{Y = 0} } \right) \hfill \\ = p(\ddot{\varphi })L^{ - 1} \left( {\vartheta_{2} \overline{\Theta } {(0,}r{)}\overline{D} {(}r{)}\overline{Q} {(}r{) - }\frac{{\vartheta_{2} }}{{g(h,\ddot{\varphi })}}\frac{1}{r}\overline{D} {(}r{)} - \frac{\gamma }{r(r + \gamma )}\overline{Q} {(}r{)}} \right) \hfill \\ \end{gathered} \right\},$$
$$Q(r) = \sqrt {\frac{sr}{{r + j}} + \vartheta_{3}^{ * } + \vartheta_{4}^{ * } } \,$$
The LIT of skin friction coefficient is;
$$C_{f} = p(\ddot{\varphi })\left\{ {\vartheta_{2} \int\limits_{0}^{{\tilde{t}}} {\int\limits_{0}^{k} {D{(}k - \varsigma {)}\Theta {(0,}\tilde{t} - k{)}Q{(}\varsigma } } {)}d\varsigma dk - \frac{{\vartheta_{2} }}{{g(h,\ddot{\varphi })}} \times \int\limits_{0}^{{\tilde{t}}} {D{(}k{)}dk} - \int\limits_{0}^{{\tilde{t}}} {Q(k)(1 - e^{{ - \gamma (\tilde{t}{ - }k)}} )dk} } \right\},$$
$$Q(\tilde{t}) = \sqrt \vartheta \left( {\frac{\vartheta j - \chi }{{2\vartheta }}J_{o} \left( {\frac{\vartheta j - \chi }{{2\vartheta }}\tilde{t}} \right) + \delta (\tilde{t})} \right)e^{{\left( {\frac{ - \vartheta j + \chi }{{2\vartheta }}\tilde{t}} \right)}} + \frac{\chi - \vartheta j}{{2\sqrt \vartheta }}J_{o} \left( {\frac{\vartheta j - \chi }{{2\vartheta }}\tilde{t}} \right)e^{{\left( {\frac{ - \vartheta j + \chi }{{2\vartheta }}\tilde{t}} \right)}} .$$
Numerical results and discussions
In the following section, a detailed graphical description of the results obtained in the previous section is given. Figures 2, 3, 4 and 5 show the behavior of various parameters with respect to the temperature curve. Figure 2 shows the physical observation of the fractional parameter on the temperature field. It shows that the temperature of nanofluids increases with the increase of the estimated fractional parameter. Physically, this behavior is due to the kernel of the fractional operator. The kernel studied the memory of the function and is capable of impectvely capturing the memory effect through the process. Thus, temperature of nanofluid elevates. From Fig. 3 it can be seen that the temperature of the nanofluid increases with increasing value of the volume fraction physically, this result is due to the high thermal conductivity of CNTs, which causes the thermal conductivity of the base fluid to increase when CNTs are added to it. Consequently, the temperature profile grows. This result highlights the importance of nanoparticles in the heating and cooling process. Figure 4 shows the temperature of the outline when the heat injector or the heat sink is is associated to the system. The temperature field falls with the intensifying estimations of. In associated graph, represents heat consumption, represents heat injection and represents that no heat is consumed or supplied. Physically, the addition of heat means an increase in the temperature of the nanofluid, while the consumption of heat means a decrease in the temperature of the nanofluid. In this process, heat is consumed because the temperature is lowered. Figure 5 shows the transient effect on the temperature curve. The nanofluid temperature curve increases as the time period increases. The temperature of the nanofluid is high near the plate and finally reaches zero asymptotically away from the plate. Figures 6, 7, 8, 9, 10, 11 and 12 show the characteristics of various relevant parameters on the velocity contour. Figure 6 describes the effect of the fractional parameter on velocity. It is worth noted that the nanofluid's velocity boost with accelerating the fractional parameter. Physically, it is due to the higher value of momentum boundary layer, the velocity is boosted.
Temperature profile for different values of \(\beta\).
Temperature profile for different values of \(\ddot{\varphi }\).
Temperature profile for different values of \(Q\).
Temperature profile for different values of \(t\).
Velocity profile for different values of \(\beta\).
Velocity profile for different values of \(\alpha\).
Velocity profile for different values of \(\ddot{\varphi }\).
Velocity profile for different values of \(M\).
Velocity profile for different values of \(K\).
Velocity profile for different values of \(\gamma\).
Velocity profile for different values of \(t\).
Figure 7 shows the effects of the fractional parameter on the velocity contour. The higher the fractional parameter is estimated, the higher the velocity of the nanofluids. Figure 8 shows the behavior of the volume fractional parameter on the velocity contour. From Fig. 8, it can be seen that the velocity and momentum of the boundary layer of the nanofluids increases. Physically, the resistance between the particles of the nanofluid is low due to the higher temperature, so the velocity increases. This is also due to the fact that the suspension of CNTs in the base fluid reduces the viscous forces and leads to an increase in the momentum boundary layer. Figure 9 shows the characteristics of the magnetic factor on the velocity sketch. The velocity of the nanofluid decreases at a higher value of the magnetic factor. This is because the magnetic field acts on electrically isolated nanofluids, which behave as a source for generating Lorentz drag forces. Because of these drag forces, the velocity of the nanofluids decreases. As the fluids move away from the plate, the Lorentz force weakens and the fluid comes to rest. Figure 10 shows the influence of the inverse permeability parameter on the velocity of the nanofluid. The momentum boundary layer thickness and velocity decreases with a larger estimate of permeability parameters . Physically, due to the high porosity of the medium, the resistance in the nanofluid particles increases, causing the velocity to decrease. In Fig. 11, the influence of on the velocity of the nanofluid is shown. It can be seen that the velocity increases as the estimate of increases. The velocity is initially higher, later it asymptotically approaches zero. Physically, this happens because there is inverse relation between and viscous forces. As we elevate the estimation of, the viscous forces reduce. As a result, velocity of nanofluid rises. Figure 12 shows that the velocity of nanofluids increases with increasing time value. The momentum boundary layer is raised for a higher estimate of the transient effects. Figure 13 shows the effect of SWCNTs and MWCNTs on the temperature distribution. The temperature of SWCNTs is higher than that of MWCNTs due to the high thermal conductivity of SWCNTs. Figure 14 shows the comparison between the velocity of SWCNTs and MWCNTs. It shows that the velocity of MWCNTs is larger than that of SWCNTs. Figure 15 is the contour plot for the thickness of the thermal layer. The thickness of the thermal boundary layer decreases as we increase the estimates of the fractional parameters Table 2 shows the properties of various relevant parameters on the Nusselt number of SWCNTs and MWCNTs. It can be seen that the heat transport rate increases with the increase of heat source/sink and the time while decrease occurs against fractional parameter and volume fraction From Table 3, it can be seen that the skin fraction (drag forces) increases with the increase in the fractional parameter while the function against the other fractional parameter for both SWCNTs and MWCNTs. Similarly, skin friction is dominant with the increasing value of the magnetic factor, permeability parameter and heat source or sink. Moreover, the drag forces are de-escalates with the escalation of volume fraction time and furthermore, the skin fraction of MWCNTs is lower than the SWCNTs.
Analysis of SWCNTs and MWCNTs on \(\Theta \left( {Y,t} \right).\)
Analysis of SWCNTs and MWCNTs on \(W\left( {Y,t} \right).\)
Analysis of \(\beta\) on thermal boundary layer thickness.
Table 2 Demonstrated the influence of various parameters on Nusslet number of both SWCNTs and MWCNTs.
Table 3 Depicts the impact of various pertinent parameters on skin fraction of both SWCNTs and MWCNTs.
The main topic of this research is to investigate the MHD and permeability effects on CNT-based nanofluids. SWCNTs and MWSNTs are suspended in blood (base fluid). Laplace transform is a very powerful mathematical tool used in various fields of physics and electrical power engineering. The Laplace transform is very important in circuit analysis, system modeling, analog signal processing, digital signal processing, process control, and radioactive decay, etc. Laplace transform technique is used to solve the non-dimensional fractional model. The exact solution for the velocity, temperature and thermal layer thickness is obtained by the above method. Zakian's algorithm is used for the simulations and inverse Laplace transform. The physical parameters such as skin fraction (drag force) and Nusselt number (heat transfer rate) are also studied. The conclusions of this study are presented below:
The temperature of the nanofluid is higher due to the increasing estimate of the volume fraction parameter \(\ddot{\varphi }\), fractional parameter \(\beta\), and time \(t\).
The greater the value of the heat source or heat sink, the lower the temperature curve.
The velocity of nanofluids in escalating function as we estimate the volume escalated fractional parameter \(\ddot{\varphi }\), fractional parameters \(\alpha {\text{ and }}\beta\), time \(t\), and \(\gamma .\)
The velocity of the nanofluid is deescalated to increase the estimate of the magnetic factor \(M\) and permeability parameter \(K\) due to high drag forces.
Nanofluid temperature is higher for SWCNTs, while the reverse effects on velocity are seen.
The thermal boundary layer increases against the fractional parameter \(\beta .\)
The heat transport rate is lower for both SWCNTs and MWCNTs as a function of fraction parameters \(\beta\) and \(\ddot{\varphi }\) higher as a function of heat source/sink and time.
Reinforcement occurs in the skin fraction with the growing estimate of \(M,K,\alpha {\text{ and }}\gamma\) while boosting the value of \(\beta ,\ddot{\varphi },Q{\text{ and }}t,\) reduces the skin fraction.
In future, we will study, what will be the effects of various fractional operators on free convective trickling over a porous medium of nanofluids with MHD and heat source/sink.
All data generated or analyzed during this study are included in this article.
\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\tilde{\rho }}_{nf}\) :
Density of nanofluid
\({\mathbf{r}}\) :
Darcy's resistance
\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\tilde{\mu }}_{nf}\) :
Absolute viscosity of nanofluid
\(B_{0}^{2}\) :
\({\mathbf{J}}\) :
\({\mathbf{B}}\) :
Total magnetic field
\(\alpha ,\beta\) :
Fractional parameters
\(K\) :
Permeability of medium
\(M\) :
Magnetic parameter
\(S\) :
Extra stress tensor
\({\rm I}\) :
Unit tensor
\(T_{\infty }\) :
\(\nu_{f}\) :
Kinematic viscosity
\(Y\) :
Space coordinates
\(T\left( {Y,\tilde{t}} \right)\) :
LT:
Laplace transforms
\(f\) :
Base fluid
\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\tilde{\sigma }}_{nf}\) :
Electrical conductivity of nanofluid
\(\left( \beta \right)_{nf}\) :
Thermal expansion of nanofluid
\(g\) :
Acceleration due to gravity
\(\left( {C_{p} } \right)_{nf}\) :
Specific heat of nanofluid
\(Q^{ * }\) :
Coefficient of heat source/sink
\(k_{nf}\) :
Thermal conductivity of nanofluid
\(\gamma\) :
Dimensional constant
\({{\varvec{\uptau}}}\) :
\(\Pr\) :
Prandtl number
\({\rm P}\) :
\(U_{0}\) :
Amplitude of the motion
\(\varphi\) :
Volume fraction parameter
\(q_{w}\) :
Heat passing from surface of wall
\(\tilde{t}\) :
\(W\left( {Y,\tilde{t}} \right)\) :
Velocity field
\(nf\) :
Solid nano-particles
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Fetecau, C., Akhtar, S., Pop, I. & Fetecau, C. Unsteady general solution for MHD natural convection flow with radiative effects, heat source and shear stress on the boundary. Int. J. Numer. Meth. Heat Fluid Flow 27(6), 1266–1281 (2017).
Zeeshan, A., Ellahi, R. & Hassan, M. Magnetohydrodynamic flow of waterethylene glycol based nanofluids with natural convection through a porous medium. Eur. Phys. J. Plus 129(12), 261 (2014).
Ashorynejad, H. R. & Shahriari, A. MHD natural convection of hybrid nanofluid in an open wavy cavity. Results Phys. 9, 440–455 (2018).
Turkyilmazoglu, M. Exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids. Chem. Eng. Sci. 84, 182–187 (2012).
Sheikholeslami, M., Gorji-Bandpy, M. & Vajravelu, K. Lattice Boltzmann simulation of magnetohydrodynamic natural convection heat transfer of Al2O3–water nanofluid in a horizontal cylindrical enclosure with an inner triangular cylinder. Int. J. Heat Mass Transf. 80, 16–25 (2015).
Azhar, W. A., Vieru, D. & Fetecau, C. Free convection flow of some fractional nanofluids over a moving vertical plate with uniform heat flux and heat source. Phys. Fluids 29(8), 082001 (2017).
Wang, F. et al. Comparative study of heat and mass transfer of generalized MHD Oldroyd-B bio-nano fluid in a permeable medium with ramped conditions. Sci. Rep. 11(1), 1–32 (2021).
Venkata Ramudu, A. C., Anantha Kumar, K., Sugunamma, V. & Sandeep, N. Impact of Soret and Dufour on MHD Casson fluid flow past a stretching surface with convective–diffusive conditions. J. Therm. Anal. Calorim. 147(3), 2653–2663 (2022).
Farooq, A. et al. Closed-form solution of oscillating Maxwell nano-fluid with heat and mass transfer. Sci. Rep. 12(1), 1–13 (2022).
Tang, R. et al. A comparative study of natural convection flow of fractional maxwell fluid with uniform heat flux and radiation. Complexity 2021, 526 (2021).
Kumar, K. A., Sugunamma, V., Sandeep, N. & Mustafa, M. Simultaneous solutions for first order and second order slips on micropolar fluid flow across a convective surface in the presence of Lorentz force and variable heat source/sink. Sci. Rep. 9(1), 1–14 (2019).
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Kumar, K. A., Sugunamma, V., Sandeep, N. & Reddy, J. R, Numerical examination of MHD nonlinear radiative slip motion of non-newtonian fluid across a stretching sheet in the presence of a porous medium. Heat Transfer Res. 50, 12 (2019).
Anantha Kumar, K., Sugunamma, V. & Sandeep, N. Influence of viscous dissipation on MHD flow of micropolar fluid over a slendering stretching surface with modified heat flux model. J. Therm. Anal. Calorim. 139(6), 3661–3674 (2020).
Anantha Kumar, K., Sugunamma, V. & Sandeep, N. Effect of thermal radiation on MHD Casson fluid flow over an exponentially stretching curved sheet. J. Therm. Anal. Calorim. 140(5), 2377–2385 (2020).
Kumar, K. A., Reddy, J. R., Sugunamma, V. & Sandeep, N. MHD flow of chemically reacting Williamson fluid over a curved/flat surface with variable heat source/sink. Int. J. Fluid Mech. Res. 46, 5 (2019).
Kumar, A., Sugunamma, V. & Sandeep, N. Impact of non-linear radiation on MHD non-aligned stagnation point flow of micropolar fluid over a convective surface. J. Non-Equilib. Thermodyn. 43(4), 327–345 (2018).
Anantha-Kumar, K., Sugunamma, V. & Sandeep, N. Physical aspects on unsteady MHD-free convective stagnation point flow of micropolar fluid over a stretching surface. Heat Transfer Asian Res. 48(8), 3968–3985 (2019).
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Article CAS MATH Google Scholar
This work is partially supported by the Doctoral Special fund for Science and Technology Program of Nanachang Institute of Science & Technology (No. NGKJ-21-06). Abdulaziz N. Alharbi would like to acknowledge the financial support of Taif University Researchers Supporting Project number (TURSP-2020/319), Taif University, Taif, Saudi Arabia.
Nanchang Normal College of Applied Technology, Nanchang, 330108, China
Yuanjian Lin
Nanchang Institute of Science and Technology, Nanchang, 330108, China
Division of Mathematical and Physical Sciences, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan
Sadique Rehman
Rare Earth Elements Applications and Research Center, Munzur University, 62000, Tunceli, Turkey
Nevzat Akkurt
Engineering Technologist, Office of the CTO, Dell Technologies, Austin, TX, USA
Tim Shedd
Department of Mathematics, COMSATS University Islamabad, Wah Campus, Wah, 47040, Pakistan
Muhammad Kamran
Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari, 61100, Pakistan
Muhammad Imran Qureshi
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand
Thongchai Botmart
Department of Physics, College of Science, Taif University, P. O. Pox 11099, Taif, 21944, Saudi Arabia
Abdulaziz N. Alharbi
Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad, Pakistan
Aamir Farooq
Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah, 11952, Saudi Arabia
Ilyas Khan
Formal analysis, N.A. and Y.L.; investigation, N.A., T.B. and I.K.; methodology, A.F. and I.K.; project administration, N.A., A.N.A. and A.F.; resources, M.I.Q., T.S. and T.B.; software, Y.L., S.R. and T.S.; supervision, M.K., A.N.A. and T.S.; visualization, T.S. and T.B.; writing—original draft, S.R. and A.F.; writing—review & editing, S.R., N.A., T.S. and A.F.
Correspondence to Thongchai Botmart.
Lin, Y., Rehman, S., Akkurt, N. et al. Free convective trickling over a porous medium of fractional nanofluid with MHD and heat source/sink. Sci Rep 12, 20778 (2022). https://doi.org/10.1038/s41598-022-25063-y
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Scientific Reports (Sci Rep) ISSN 2045-2322 (online)


"Snowmobile Racing's Early Years 1967-1973" has 212 pages with over 450 of snowmobiling's most spectacular photos from snowmobile racing's vintage "Early Years".
Featuring: Rare early machines, ads of the era, oval racing, exotic speed machines, and cross country. Photos of faces, places, and machines with special informational insights of a bygone era available nowhere else. All from the exclusive, extensive photo archives of Race & Rally / SnowTech Magazine.
This brand new book, "Snowmobile Racing'sEarly Years", is a photographic presentation by SnowTech Magazine. These are truly spectacular images of a time now barely existing in the memories of those who participated in this bygone era.
This 212 page book of over 450 photos is an opportunity to revisit the sights, sensations and emotions through the expert photography of the Race & Rally staff from 1967-1973. A period of exploding industry growth and disappointing manufacturer failures all wrapped up in a fascinating package. A once-in-a-lifetime opportunity to collect a very special volume of photography dangerously close to remaining stored away forever. A snowmobiling photo book without comparison – or any possibility of duplication.
Something to immediately enjoy, to treasure, to revisit periodically, to share and discuss endlessly with snowmobiling friends. A "coffee table" book needing a chain (just kidding) to keep it from being stolen.
"My gosh, what a wonderful surprise and such a great edition. The early years were really full of excitment and fun – and once in a while real sorrow. The pictures …. of races all over North America really brought back memories. Thank you so much for sending my copy so quickly. I will be reading it over and over. Here is another check. Please send a copy to my son.
This is the ideal Christmas gift for a snowmobiling family member or friend.
Visa, Mastercard, or Discover accepted!
*Sorry We do not accept American Express.


Central Colored High School, Shreveport, Louisiana, Caddo Parish was a victim of planned obsolescence, as evidenced by its name. Its doors were opened in 1917, however, the true impact of its place in Louisiana African American high school history was not realized, perhaps, because 12th grade was not offered until 1949, one year before its closure. Rising from the Central Colored High School's ashes like a phoenix, Booker T. Washington High School was the capable successor. Central Colored High School, a shadow of itself during its days of glory, stands in the United States Registry of Historic Places.


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Did Interior Abandon NEPA for Antofagasta?
New documents show top officials at the Department of the Interior planned to review Antofagasta's mineral leases near the Boundary Waters under the National Environmental Policy Act, or NEPA, before renewing them. That plan appears to have been abandoned after meetings with Chilean mining company executives in spring of 2018.
The latest Boundary Waters documents in response to my FOIA lawsuit come from Daniel Jorjani, who was Deputy Solicitor at the Department of the Interior when these records were created. The release consists of 122 heavily redacted pages, mostly emails and briefings that circulated as the Department of Interior was preparing to announce that it had reinstated Antofagasta's mineral leases on May 2, 2018.
These records show that the Bureau of Land Management decided against any "proactive" statement (like a press release) on the reinstatement, and opted instead to create an "if-asked" statement for the press. Russell Newell drafted the if-asked statement and Associate Solicitor Karen Hawbecker reviewed and edited it on Monday, April 30. Deputy Solicitor Jorjani approved Hawbecker's edits at 5:30PM the same day.
Newell's draft and Hawbecker's edits of the if-asked statement are both fully redacted, but we know what the if-asked statement said because Dylan Brown, a journalist writing for E & E News, asked.
Lori Mashburn, White House Liaison at the Department of the Interior, included the official response to Brown's query in her May 4 Daily Update for Cabinet Affairs. The Update went to Jorjani, David Bernhardt, Doug Domenech and other political appointees as well as Russell Newell.
At the end of April, 2018, the Department understood that the lease renewals would require "review under the National Environmental Policy Act." That is also the understanding of the law set forward by the plaintiffs in a lawsuit over the lease renewals currently before the US District Court for the District of Columbia:
The National Environmental Policy Act ("NEPA") requires that agencies take a "hard look" at the environmental impacts of their actions before the actions occur, and that they prepare an Environmental Impact Statement ("EIS") for "major Federal actions significantly affecting the quality of the human environment." 42 U.S.C. § 4332(2)(C); Kleppe v. Sierra Club, 427 U.S. 390, 410 n.21 (1976). Courts have clarified that in the mineral leasing context, an agency must prepare an EIS analyzing the ultimate effect of mineral development when it issues a lease without reserving absolute authority to prevent development on the lease.
But when it came to renewing Antofagasta's mineral leases, one year later, the Department of the Interior set NEPA aside. Instead of taking a hard look, as required by NEPA, they issued an EA or Environmental Assessment — which is really only a first step in determining whether a project will have significant environmental impact.
Why the change of plan? As I've written here and elsewhere, the Department of the Interior seems to have abandoned plans for an EIS after meetings with executives from Antofagasta in spring of 2018.
In a March 6 meeting summary included with a previous release of documents, Antofagasta officials explicitly stated that an EIS would interfere with their plans. They wanted a Categorical Exclusion; they would settle for an EA. That is exactly what they got.
So it is difficult to avoid the conclusion that top Interior officials knew renewing the leases would require review under NEPA, but they deliberately set aside US law in order to do the bidding of Chilean mining executives.
The August documents are now online here, and all the Boundary Waters documents I've obtained to date are here.
Read more about the Boundary Waters reversal here.
This entry was posted in Business and Society, Lake Superior, Louis V. Galdieri's Blog, Prosperity, The Boundary Waters Reversal and tagged Boundary Waters, corruption, good government, kleptocracy, NEPA, regulatory capture on August 11, 2020 by lvgaldieri.
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@FollowAlisonT I'm not ignoring this question. I've been thinking about it on and off all day. Still no clear answer. 9 hours ago
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redpajama_set_name: RedPajamaCommonCrawl


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Review: "Shang-Chi" raises the bar for Marvel movies to come
Courtesy of Marvel Studios
"Shang-Chi" is a sturdy stand-alone film that is delightful no matter an audience member's knowledge of the Marvel franchise.
By Sydney Brumfield — Staff Writer
Prepare to be blown away by "Shang-Chi and the Legend of the Ten Rings," director Destin Daniel Cretton's Marvel movie with top-tier action scenes and brilliant characters. Filled with dialogue that perfectly walks the line between witty, impactful and humorous, "Shang-Chi" is a sturdy stand-alone film that is delightful no matter an audience member's knowledge of the Marvel franchise.
Shang-Chi, (Simu Liu) an ex-assassin hiding away in San Francisco under the name Shaun, is working as a valet with his best friend Katy (Awkwafina). Their life is thrown into mayhem when Shang-Chi's father Xu Wenwu (Tony Leung Chie-Wai) — the founder of the international crime syndicate the "Ten Rings" — sends his soldiers after Shang-Chi. This opens the door to Shang-Chi's horrendous past of being trained as an assassin since the age of seven. Now, he must attempt to protect his family as well as uncover and foil the true motives of his father. In order to save the world, Shang-Chi is forced to confront his past or become defined by it.
"Shang-Chi" is composed of the most gripping action sequences created by Marvel in years. The choreography, stunts, visual effects and cinematography of the fight scenes are newfangled, launching the audience into physical conflicts unlike anything ever seen. The smaller level fights between Shang-Chi and other characters have a large emphasis on the environment. They explore the impact of the location on a fight scene and exceed expectations by having captivating sequences. These elaborate sequences occur in multiple high intensity settings, like on an out of control bus and on the scaffolding attached to the side of a skyscraper in Macau. The large ensemble fight sequences feel like a beautiful dance as the camera delicately jumps between the main cast of characters. This allows for the viewer to admire the action but not become overwhelmed by it.
Cretton creates a full cast of characters that manage to be both nuanced and complex, but all of them compelling. Shang-Chi is a unique hero character who the audience gets to watch grow and fulfill his destiny. Being able to watch his struggle both in gaining physical endurance and tackling the ghosts of his past make him an enthralling character. Just as impactful, Shang-Chi is not cocky or arrogant like a majority of other hero characters in this franchise. He is allowed to be timid and emotionally vulnerable which is refreshing to see in a strong male-lead. Even the supporting and antagonistic characters all have goals, obstacles and personal development. Most exciting is that even the antagonist Xu Wenwu is complex with a gray history and a heart wrenching desire to be reunited with the love of his life.
The story of "Shang-Chi " is fresh, tight and nothing like the typical Marvel film. The viewer is not expected to know the extended backstory of the characters much like they are in most other Marvel Cinematic Universe (MCU) films. Instead, the story is structured through juxtaposition of the characters' pasts as they become relevant to the present-day plot line. This allows for the viewers to learn a lot about the characters in a meaningful and direct way. The only downside to this structuring is that when having to periodically switch between the past and present, it makes moments in the second act feel slower. These scenes feel more expositional because they provide the audience with necessary background information that is typically delivered in the beginning of the film.
Cretton utilizes excellent full-circle plotting that whisks viewers off on an epic journey and then places them right back into the real world by the end of the film. "Shang-Chi" is an excellent addition to the MCU Universe and an outstanding visual experience.
Sydney Brumfield can be reached at sbrumfield@ithaca.edu
Review: "Candyman" crafts terror from real-world fears
By Jackson Noel | Sep 7, 2021
Review: Addison Rae stars in insipid Netflix remake
By Elijah de Castro | Sep 7, 2021
Review: Highly anticipated sci-fi movie lacks substance
By Sydney Brumfield | Aug 31, 2021
Destin Daniel CrettonMarvelMCUShang-ChiShang-Chi and the Legend of the Ten RingsSimu Liu


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Bambūyān (persiska: بمبویان, Bambeyūn, Bambīūn) är en ort i Iran.   Den ligger i provinsen Kerman, i den sydöstra delen av landet,  km sydost om huvudstaden Teheran. Bambūyān ligger  meter över havet och antalet invånare är .

Terrängen runt Bambūyān är platt. Den högsta punkten i närheten är  meter över havet,  km väster om Bambūyān. Runt Bambūyān är det glesbefolkat, med  invånare per kvadratkilometer. Bambūyān är det största samhället i trakten. Trakten runt Bambūyān är ofruktbar med lite eller ingen växtlighet.

I trakten råder ett hett ökenklimat. Årsmedeltemperaturen i trakten är  °C. Den varmaste månaden är juli, då medeltemperaturen är  °C, och den kallaste är januari, med  °C. Genomsnittlig årsnederbörd är  millimeter. Den regnigaste månaden är mars, med i genomsnitt  mm nederbörd, och den torraste är juni, med  mm nederbörd.

Kommentarer

Källor

Orter i Kerman (provins)

redpajama_set_name: RedPajamaWikipedia


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Edmilson Alves (nacido el 17 de febrero de 1976) es un futbolista brasileño que se desempeña como centrocampista.

Jugó para clubes como el Londrina, Fortaleza, Oita Trinita, Ceará, Vissel Kobe y Roasso Kumamoto.

Trayectoria

Clubes

Referencias

Futbolistas de Brasil
Futbolistas del Clube Atlético Juventus
Futbolistas del Londrina Esporte Clube
Futbolistas del Fortaleza Esporte Clube
Futbolistas del Oita Trinita
Futbolistas del Ceará Sporting Club
Futbolistas del Vissel Kobe
Futbolistas del Roasso Kumamoto


text:
package com.twu.biblioteca.model;

public interface User {
    boolean validate(String libraryNumber, String password);
}

redpajama_set_name: RedPajamaGithub


$15.75 $19.95 Save 21% Current price is $15.75, Original price is $19.95. You Save 21%.
Multiple Sclerosis is a disease a mystery writer would love: it seemingly appears out of nowhere, its signs can be so subtle they are not recognized, and its symptoms can become more severe and more frequent in some people, while others suffer only one or two mild episodes throughout their lives. Today, the mystery of MS is unraveling, with new disease-modifying medications, more sophisticated diagnostic tests, and better, easier, more comfortable adaptations for an improved quality of life both at work and at home. Multiple Sclerosis: A New Journey takes you through the newest research and treatments to help you find a straight course for your disease. Inside this one easy-to-understand, cross-referenced, and compassionate volume, you will find everything you and your loved ones need to know about diagnosis, symptoms, progression, disease-modifying therapy, rehabilitation, disability awareness, and more. You will find, in short, everything you need to lead a better and more independent life.
Richard C. Senelick, MD, is the medical director at the Health-South Rehabilitation Institute of San Antonio (RIOSA). He also serves as program director of its Brain Injury Program. A native of Illinois, he completed his undergraduate and medical school training at the University of Illinois in Chicago. A neurologist who specializes in neurorehabilitation, he subsequently completed his neurology training at the University of Utah in Salt Lake City. He has authored numerous publications, including co-authoring Living with Brain Injury: A Guide for Families, The Spinal Cord Injury Handbook for Patients and Their Families, and The Disability Awareness Handbook for Families, Co-Workers, and Friends.


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Q: UI-Router nested state view not rendering (rendering parent view) Using ui-router in angular, I have setup a 'users' state where all the users in my application are returned in an ng-repeat. I am trying to create the show state for the users profile page however the view of the child state is inheriting the parent view 'templates/users/index.html' (Here is my app.js:
  .state('users', {
    abstract: true,
    name: 'users',
    url: '/users',
    views: {
      nav: {
        templateUrl: 'templates/navbar.html',
        controller: 'MainCtrl'
      },
      content: {
        templateUrl: 'templates/users/index.html',
        controller: 'UsersCtrl'
      }
    }
  })
    .state('users.show', {
      url: '/show/:id',
      views: {
        nav: {
          templateUrl: 'templates/navbar.html',
          controller: 'MainCtrl'
        },
        content: {
          templateUrl: 'templates/users/show.html',
          controller: 'UsersCtrl'
        }
      }
    });

Here is my index.html:

  Search: <input ng-model="searchText">

 {{ user.name }}

 {{ user.email }}

    <button ng-click="createConversation({sender_id: current_user.id, recipient_id: user.id})" >Message</button>

The url in the browser is http://localhost:8080/#/users/show/2 however the html file being inherited in the users.show state is the parents users (index.html) file.
Any help appreciated!

A: The nested view users.show will actually render within the user view. So you just need to place another <ui-view> within templates/users/index.html where the show view will render.

redpajama_set_name: RedPajamaStackExchange


The human mind is an incredible factor and one that's nonetheless shrouded in thriller, at the same time as we make new developments in medication and the examine of the thoughts. One such examine has discovered that our mind perform might very effectively be depending on the time of yr or the passing of seasons. After all, for a few of us, this might not be a lot of a shock in any respect, particularly in case you are an individual who prefers one season over one other. Now, that desire might very effectively be tied into how our brains are wired and the way they carry out throughout a sure time of the yr.
The outcomes of the examine had been significantly fascinating and appeared to point a particular correlation between the best way our brains perform in relation to the time of yr or the season.
Current research are actually exhibiting a hyperlink between the success price when performing a psychological job and the time of yr through which that job is carried out. In truth, some members studied confirmed a marked distinction when performing the identical job all year long.
The bodily a part of the duty itself remained largely unchanged throughout these research, however the necessary discovering was how the mind reacted to the duty itself. The examine in query was performed over the course of the yr with particular consideration given to the time of yr through which the examine was utilized. When the members had been examined, they had been in a managed setting to forestall any exterior or bodily cues from distorting the outcomes. Mind scans had been carried out whereas the members carried out a cognitive job and the mind's exercise was noticed.
Parts of the mind that appeared to be most affected by the seasons included the flexibility to course of info in addition to the flexibility to retrieve older info. General, the best way an individual accomplished the duty didn't change, so there have been no bodily modifications that might be immediately noticed; nevertheless, on the mind scan a special image was rising. Totally different seasons truly impacted how simply the mind retrieved info and solved the issue exhibiting that some seasons even have the next "value" relating to the power wanted to work out an issue.
One of the apparent observations was the truth that the mind appeared to "perk up" in the summertime months, particularly in late June. This peak appeared to coincide with the summer time solstice when the times had been at their longest. When winter rolled round, mind exercise appeared to lull, once more coinciding with the seasons. This time, it was the equinox that had an influence on mind perform.
For these questioning in regards to the connections with info retrieval and reminiscence, the mind carried out the most effective within the spring and fall. These findings truly present a correlation with earlier colleges of thought regarding how effectively the mind features at numerous instances of the yr. One principle means that the quantity of meals we devour all through these durations is partly accountable, whereas one other means that the modifications are linked to the quantity of daylight or the size of the day through the seasons studied.
Clearly there's a connection between the seasons and the best way we really feel; in reality, almost everybody can relate to feeling a sure approach throughout a specific a part of the yr. For instance, these residing within the North the place winter could be lengthy and drawn out have reported extra melancholy and unhappiness compared with these residing within the Southern areas. These findings additionally appear to align properly with the idea that mind exercise could be impacted by the point of yr.
What stays to be understood is the exact approach through which the mind processes the time of yr and the way these processes then suggestions into the best way the mind performs. The precise motive behind the efficiency of the mind and the time of yr will not be readily obvious and requires extra examine. It's, nevertheless, simply one other fascinating side of the human thoughts and goes to indicate how little we really know relating to the human mind itself.


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* Special Events * (1)
Other Community Events (Not City-Sponsored) (1)
- * Special Events * (1) Other Community Events (Not City-Sponsored) (1)
Mayor's History Lecture Series: Abraham Lincoln with guest lecturer Mike Gross
April 18, 2019, 6:30 PM - 8:00 PM @ Community Center
Mayor's History Lecture Series: Abraham Lincoln with guest lecturer Mike Gross2019-04-18T18:30:00
This History Lecture Series is presented by Mayor Brian Johnson, with support and light refreshments provided by the Kennedale Historical Society.
Michael Gross earned his Baccalaureate degree from the University of the Cumberlands and his Master's from the University of North Texas. In the history of KISD, he is the only three-time winner of "Teacher of the Year". Mr. Gross was named a Horace Mann Fellow and has won exclusive scholarships with the Gilder Lehrman Institute of American History, the Lincoln Forum, The Civil War Institute at Gettysburg College.
Community Center316 W Third StKennedaleTX76060
Kennedale Historical Society Meeting
April 18, 2019, 6:00 PM @ Kennedale Chamber of Commerce or the Kennedale Public Library
Since 2012, we've been working to preserve the historic dwellings, businesses and sites of our town, while sharing the story of Kennedale's rich history. We'd love for you to join us! Meetings are the third Thursday of each month, unless otherwise noted.


Political ambition, betrayal, the physical and psychological deterioration of the powerful, supernatural visions, witches, and rampant paranoia. No this isn't the latest blockbuster from Hollywood, this is Macbeth. As powerful today as it was the day it was first performed, Shakespeare's shortest tragedy is taught around the world as part of English Literature curriculum.
You will likely be assigned a Macbeth essay at some point in your scholarly career and when that time comes (perhaps it is here) you will want to be fully prepared to write a stellar essay that will be sure to impress your teacher.
We will be covering a number of aspects that will awe your instructor, including a number of possible essay topics, but if you find that you still need some more ideas, you may want to get help from this company.
Your thesis statement is the heart of your paper. While a well-thought-out thesis can lead to an A grade, a weak, poorly constructed thesis will pull your grade down no matter how well-rounded the rest of the paper is. A well written thesis should be an argument, some interpretation or evaluation, which is not plainly evident and has the potential to be debated. For example, simply stating a character trait or factual information about the play will not form a very good thesis. However, stating that a particular trait or some factual information helped to shape a theme or give us insight into the character's motivations form the basis for a great thesis.
No matter which direction you take, whether it be to interpret some part of the play or to evaluate different aspects of the literature, you will want to make sure that the statement could just as easily be disproved. If someone so chose, they could attempt to disprove your thesis statement. It is your job in the essay to prove it to be true.
Some of the most memorable characters in the play are the witches. How are the witches characterized and what purpose do the serve the overall theme?
What is elision and why does Shakespeare choose to use this technique?
One of the most widely recognized scenes in Macbeth is the sleepwalking scene in Act 5. Why does Shakespeare choose to have Lady Macbeth speak in prose and how do her words fit into the overall context of the work.
When Banquo is told by the witches that he will be the father of future kings his reaction reveals much about his character. Discuss.
Banquo has an important dramatic purpose both alive and dead. Compare the dramatic importance of this duality.
What is the dramatic purpose of the Ghost Scene and why does Shakespeare use this method for advancing the story?
The destruction of the natural order is a central theme to Macbeth. Analyze the ways in which Shakespeare demonstrates this idea through the use of specific examples.
Macbeth is his by far his worst enemy. Describe how this and other factors culminate in his downfall.
What role does the supernatural play in Macbeth? How does this shape Macbeth's decisions?
These topics will certainly assist you in impressing your teacher, or you can use them as a jumping-off point for your own ideas. Either way, you are well on your way to a stellar Macbeth essay!


The City of St. Catharines is responsible for the maintenance, repair and replacement of traffic signs on City-owned streets.
Missing or damaged traffic signs can be reported to the City by calling 905.688.5600.
Niagara Region is responsible for the maintenance, repair and replacement of traffic signals on City-owned streets.
Damaged traffic signals can be reported to the Niagara Region by calling 905.984.3690.
Hydro poles are the property of Alectra Utilities. Call 905.984.8961 and you will be directed to the Alectra Utilities customer service department.
To report a hydro outage call 905.684.8111.
Telephone poles are the property of Bell Canada. For enquiries dial #611 and you will be directed to Bell Canada's customer service department.
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It is helpful if you can provide the specific pole number, located on the yellow tag attached to the pole.


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text:
NC Attorney General Stein Hears WNC Concerns On Mission Health-HCA Deal
BPR News | By Lilly Knoepp
Published November 15, 2018 at 12:28 PM EST
Lilly Knoepp
Asheville-Buncombe NAACP President Carmen Ramos-Kennedy, Mitchell County physican Arch Woodard and former Governor Jay Nixon were part of the panel at the listening session.
Former Missouri Governor Jay Nixon visited Asheville last month to hear from concerned residents about Mission Health's proposed sale to HCA. Here's more on what has happened since that meeting.
Risa Larsen is a member of the SEARCH committee. Based in Yancey and Mitchell counties, SEARCH stands for Sustaining Essential and Rural Community Healthcare. They have been one of the most outspoken groups outside of Asheville voicing their concerns about nonprofit Mission Health's sale to for-profit HCA.
"We always try to make clear, we're not against the sale. We just want it to have positive outcomes for every who is served by this hospital system," says Larsen.
Last month, the group brought former Missouri governor Jay Nixon to Asheville. Nixon handled a similar deal in Missouri during his time as attorney general and governor there. He attended a listening session to hear from concerned citizens and organizations from the 18 Western North Carolina counties in Mission Health's footprint. After visiting Asheville, Nixon headed to Raleigh where he spoke to state Attorney General Josh Stein, who must approve the sale before it can be finalized.
"I would say that we accomplished even more than we might have hoped for bringing Governor Nixon down from Missouri," says Larsen.
The meeting was closed but Nixon says he echoed the points made at the Listening Session.
He particularly focused on how to improve the setup of the Dogwood Health Trust through greater transparency, a more diverse board and a clear mission. The SEARCH team and Nixon also met with Jennifer Harrod, special deputy attorney general and general counsel Swain Wood.
During that meeting, SEARCH says that Nixon shared the importance of making the Dogwood Health Trust foundation board of directors equitable as well as separate from Mission Health and HCA.
"If you sell a house, you don't get to live in it," says Nixon according to SEARCH meeting minutes.
Larsen says SEARCH hopes that Stein will make the changes recommended by Nixon.
"We're sympathetic to their cause but we need them to know that we need these rural hospitals. And we need them to vibrant to bring industry back into these areas," says Larsen.
SEARCH was founded in 2017 when Spruce Pine's Labor and Delivery Unit closed. No matter the outcome of the sale, Larsen says SEARCH will continue working to improve healthcare in the area.
Highlands Mayor Patrick Taylor and Franklin Mayor Bob Scott also met with the Attorney General's team on November 9th to discuss their concerns about the sale.
Their meeting focused on the importance of rural hospitals and the structure of Dogwood Health Trust. Additionally, Taylor recommended a buy-back option for Highlands-Cashiers Hospital if HCA should ever try to sell it, according to The Highlander.
Attorney General Stein still has not commented on the sale. The deal is expected to be finalized by the end of the year.
News Mission Healthwestern north carolinaBPR NewsBPRMission Health-HCA Deal
Lilly Knoepp serves as BPR's first fulltime reporter covering Western North Carolina. She is a native of Franklin, NC who returns to WNC after serving as the assistant editor of Women@Forbes and digital producer of the Forbes podcast network. She holds a master's degree in international journalism from the City University of New York and earned a double major from UNC-Chapel Hill in religious studies and political science.
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