text:
\section{Introduction}

Computer simulations have become one of the primary tools for investigating and
visualizing fluid dynamics. Many types of simulation methods have been developed, but
improvements are still continuously conducted.
These methods are categorized into two groups.
The first group directly discretizes the Navier-Stokes equation (NSE) or its variants,
which include the finite difference method (FDM), finite volume method (FVM),
finite element method (FEM), and so on~\cite{ferz02,wend09}.
The second group is based on a microscopic or mesoscopic equation of motion
({\it e.g.}, Newton's equation of motion and the Boltzmann equation);
these equations are different from NSE, although large-scale dynamics follow NSE.
In the second group, the lattice Boltzmann method \cite{succ01} and three particle-based methods
({\it i.e.}, direct simulation Monte Carlo (DSMC)\cite{bird98}, dissipative particle dynamics (DPD)
\cite{groo97,espa17}, and multi-particle collision (MPC) dynamics \cite{kapr08,gomp09a})
 have been widely adopted.
These particle-based methods take thermal fluctuations into account and share the main constitution form,
while different implementations are employed for local viscous interactions~\cite{nogu07}.
These viscous interactions must conserve translational momentum to properly generate hydrodynamic flow,
while angular-momentum conservation (AMC) can be switched on or off~\cite{nogu07}.
AMC is required for the calculation of torques and flows
which boundary conditions are given by stress balances,
such as those found in multi-component fluids with different viscosities~\cite{goet07}.
AMC is also required to reproduce the friction of slip boundary conditions~\cite{yang15}.
However, the absence of AMC does not affect the velocity field of a fluid when the boundary conditions
are given by velocities.
Thus, in the particle-based methods, AMC is explicitly imposed or
neglected depending on the application, and AMC's effects on these methods are well
understood.

Compared to the above particle-based methods,
AMC has not been well discussed in the discretized NSE methods.
Recently, the violation of AMC has been an issue in smoothed particle hydrodynamics (SPH)~\cite{mona05,pric12,elle18}.
SPH is a particle-based method, although the equation of motion is derived from
NSE in the Lagrangian frame of reference.
Thus, SPH is positioned in the crossover area between the first and second groups.
The original version of SPH did not include thermal fluctuations,
but  Espa{\~n}ol and Revenga developed smoothed dissipative particle dynamics (SDPD), a version of SPH including
thermal fluctuations~\cite{espa03}.
SDPD can also be considered as an extension of DPD. In conventional
SPH, the viscous forces do not conserve angular momentum.
To recover AMC in SDPD, Hu
and Adam~\cite{hu06} restricted the viscous forces only along the vectors
between two interacting particles, as in DPD.
In contrast, M{\"u}ller {\it et al.}~\cite{mull15}
added an internal degree of freedom,
spin, from the fluid particle model~\cite{espa98} to the SDPD particle and applied NSE with the
spins to derive the particle motion.
The rotational flow of binary
fluids and the vesicle dynamics in shear flow~\cite{mull15} and torque under the no-slip boundary condition~\cite{bian12} are properly reproduced by these AMC modifications of SDPD.
These results are well understood when the derived equations
of motion are viewed as one of the mesoscopic simulation methods described in Ref.~\cite{goet07}.
However, the results are unclear from the viewpoint of NSE discretization, because NSEs
are derived under the AMC condition. For example, Ellero and Espa{\~n}ol mentioned ``As the
SDPD equations are a discretization of NSEs which conserve angular momentum, angular
momentum is conserved in SDPD in the limit of high resolution'' in Ref.~\cite{elle18}.
Thus, it has been considered that the lack of AMC is caused by a numerical error in the discretization,
although its mechanism is not known.

In this paper, we describe how AMC is lost in NSE discretization and how it can be
recovered. We found that the degeneracy in the derivatives of shear and volume stress terms
is the origin; as a result, the lack of AMC does not disappear in the limit of high resolution.
In Sec.~\ref{sec:theory}, we describe NSEs for fluids with and without AMC. Hereafter, we refer to fluids
with and without AMC as $+a$ and $-a$ fluids, respectively~\cite{nogu07}.
We show analytical solutions for the planar and rotational laminar flows of the $+a$ and $-a$ fluids.
Next, in Sec.~\ref{sec:AMC}, we explain the AMC loss process
using the Lagrangian finite volume method (LFVM)~\cite{serr01,spri10,duff11}.
LFVM is the most suitable method for this purpose: because it employs fewer approximations than
SPH, the mechanism is clearly captured.
This mechanism is general and is shared by SPH and other methods.
The details of LFVM in two-dimensional (2D) space are then described in Sec.~\ref{sec:LFVM2D}.
The method in Ref.~\cite{serr01,serr06} is
extended to simulate the $+a$ fluids. The condition of the rigid-body rotation of fluids is
described in Sec.~\ref{sec:rbr}.
We found that it is determined by whether the discretized viscous forces are invariant
for a global rotation, ${\bf v}={\boldsymbol \Omega}\times {\bf r}$.
The implementation of the boundary conditions and numerical details are described
in Secs.~\ref{sec:bd} and \ref{sec:detail}, respectively.
In Sec.~\ref{sec:results},
the simulation results of the planar and rotational laminar flows of the $+a$ and $-a$
fluids are shown. The theoretical predictions given in Sec.~\ref{sec:ana}  are accurately reproduced
by these simulations. Sec.~\ref{sec:sum} contains our summary and discussion.

\section{Theory for fluids with and without angular-momentum conservation}\label{sec:theory}
\subsection{Navier-Stokes Equations}\label{sec:NSE}

In conventional viscous fluids that do conserve angular momentum,
the stress is expressed by a symmetric tensor:
\begin{equation} \label{eq:strs+a}
\sigma_{\alpha\beta}^{(+)}= -P\delta_{\alpha\beta}
+ \eta \left(\frac{\partial v_{\alpha}}{\partial x_{\beta}}
+\frac{\partial v_{\beta}}{\partial x_{\alpha}} \right)  + \lambda(\nabla \cdot {\bf v})\delta_{\alpha\beta},
\end{equation}
where $\alpha,\beta \in \{x,y,z\}$ and P is the pressure.
The superscripts $(+)$ and $(-)$ are used to distinguish the quantities of the $+a$ and $-a$ fluids.
Here, $\eta$ is the viscosity and $\lambda$ is the second viscosity coefficient.
For the $-a$ fluids, the latter half of the shear stress, $\eta\partial v_{\beta}/\partial x_{\alpha}$,
is removed as
\begin{equation} \label{eq:strs-a}
\sigma_{\alpha\beta}^{(-)}= -P\delta_{\alpha\beta}
+ \eta \frac{\partial v_{\alpha}}{\partial x_{\beta}} + \lambda(\nabla \cdot {\bf v})\delta_{\alpha\beta}.
\end{equation}
Note that the asymmetric stress tensor is generally expressed by
the addition of the asymmetric term
$\check{\eta} (\partial v_{\alpha}/\partial x_{\beta}- \partial v_{\beta}/\partial x_{\alpha} )$
to Eq.~(\ref{eq:strs+a}) as described in Ref.~\cite{goet07}.
However, in the simulation methods in which angular momentum is not conserved,
the asymmetric stress tensor typically has the form of Eq.~(\ref{eq:strs-a}).
This is because the asymmetric stress is introduced during simplification of the numerical procedure.
Introduction of asymmetric stress during the discretization of NSEs is described in the next section.
In the $-a$ versions of MPC and DPD, the collisional stress follows Eq.~(\ref{eq:strs-a}),
whereas the kinetic stress maintains the symmetry~\cite{nogu08}.

The velocity evolution equation ({\it i.e.}, NSE) of the $+a$ fluids is given by
\begin{eqnarray}
\rho\frac{D v_{\alpha}}{D t} &=& \frac{\partial \sigma_{\alpha\beta}^{(+)}}{\partial x_\beta} + f_{\rm ex,\alpha} \nonumber \\
&=& -\frac{\partial P}{\partial x_\alpha}
+ \frac{\partial}{\partial x_\beta} \Big( \frac{\eta \partial v_{\alpha}}{\partial x_{\beta}}\Big)  \label{eq:nse0} \\ \nonumber
&& + \frac{\partial}{\partial x_\beta}  \Big(\frac{\eta \partial v_{\beta}}{\partial x_{\alpha}}\Big)
+ \frac{\partial}{\partial x_\alpha} \Big(\frac{\lambda \partial v_{\beta}}{\partial x_{\beta}}\Big) + f_{\rm ex,\alpha},
\end{eqnarray}
where the Einstein notion is used,
 $\rho$ is the mass density, and $D/Dt$ is the Lagrangian derivative: $D/Dt=\partial/\partial t+ {\bf v}\cdot \nabla$.
The last term $f_{\rm ex,\alpha}$ expresses the $\alpha$ component of an external force ${\bf f}_{\rm ex}$,
but hereafter we omit it in NSE for simplicity.
For the $-a$ fluids, the third term is removed.

When the viscosity coefficients $\eta$ and $\lambda$ are constant in the fluid,
the third and fourth terms of Eq.~(\ref{eq:nse0}) are degenerated as
\begin{equation} \label{eq:nse1}
\rho\frac{D {\bf v}}{D t} = -\nabla P + \eta \nabla^2 {\bf v} + \mu \nabla(\nabla\cdot{\bf v}),
\end{equation}
where $\mu=\eta+\lambda$ and $\mu=\lambda$ for the $+a$ and $-a$ fluids, respectively.
Interestingly, the derivatives of the asymmetric tensor
$\partial v_{\beta}/\partial x_{\alpha}$  and the symmetric tensor ($\nabla\cdot{\bf v})\delta_{\alpha\beta}$
are identical.
Therefore, the $+a$ and $-a$ fluids follow the same equation;
when $\lambda^{(-)}= \eta+\lambda^{(+)}$ is used in the NSE of the $-a$ fluids,
no difference in fluid motion is detected.
In other words, Eq.~(\ref{eq:nse1}) by itself does not guarantee AMC.

The $-a$ fluids have an asymmetric stress that generates an artificial torque~\cite{goet07}.
In cylindrical coordinates ($r,\theta,z$), the azimuthal stress is given by
\begin{eqnarray} \label{eq:strs_rq+a}
\sigma_{r\theta}^{(+)} &=& \eta r
\frac{\partial \omega}{\partial r}, \\ \label{eq:strs_rq-a}
\sigma_{r\theta}^{(-)} &=& \eta r
\frac{\partial \omega}{\partial r}
+ \eta\omega,
\end{eqnarray}
for the $+a$ and $-a$ fluids, respectively,
where the angular velocity $\omega=v_{\theta}/r$.
The stress of the $+a$ fluids
depends only on the derivative of the angular velocity $\omega$.
In contrast, the $-a$ fluids have
an additional term that is proportional to $\omega$.
Therefore, when a whole fluid slowly rotates as a rigid body,
a $-a$ fluid receives an artificial torque whereas a $+a$ fluid receives no torque.
This additional torque modifies the torque balance on the interface between two fluids with different viscosities,
such that multi-component $-a$ fluids show different flow behavior than $+a$ fluids; this is described in the next subsection.

\subsection{Analytical Solutions of Laminar Flows}\label{sec:ana}

Since our interest is the viscous term,
we consider planar and circular laminar flows under a low Reynolds number and low Mach number.
The analytical solutions for planar and circular laminar flows are respectively described in Secs.~{\ref{sec:planar} and {\ref{sec:circular}.
Single-component and binary-component fluids are considered.
The viscosity of each component is constant.

\subsubsection{Planar Laminar Flows}\label{sec:planar}

A periodic force is imposed on fluids  in the $y$ direction
as ${\bf f}_{\rm ex}(x)= f_0 \sin(kx) {\bf e}_y$ with the wavelength $L_x=2\pi/k$
where ${\bf e}_y$ is the unit vector along the $y$ axis.
The $+a$ and $-a$ fluids exhibit identical laminar flows.
The steady flow of the single-component fluid is given by ${\bf v}(x)=(f_0/\eta k^2)\sin(kx){\bf e}_y$~\cite{serr06}.
When the initial velocity is given by ${\bf v}(x)=v_{\rm a}(0)\sin(kx){\bf e}_y$,
the velocity field develops as
\begin{eqnarray}
 v_y(x,t) &=& v_{\rm a}(t)\sin(kx), \qquad v_x=v_z=0,\\
v_{\rm a}(t) &=& \frac{f_0}{\eta k^2}+ \Big( v_{\rm a}(0)- \frac{f_0}{\eta k^2} \Big)\exp\Big(-\frac{\eta k^2}{\rho}t\Big).
 \label{eq:sin1}
\end{eqnarray}

For binary fluids, the velocity $v_y$ in the steady flow is given by
\begin{eqnarray}
 \label{eq:sin2}
v_y(x) &=& v_{\rm a}\sin(kx) + v_{\rm s}, \\ \nonumber
v_{\rm a}&=& \left\{
\begin{array}{ll} f_0/\eta_0 k^2 & {\rm for} \quad n-1/2<x/L_x\le n, \\
                 f_0/\eta_1 k^2 & {\rm for} \qquad \qquad n <x/L_x\le n+1/2,
\end{array}
\right.
\end{eqnarray}
where $v_{\rm s}= (1/\eta_0-1/\eta_1)f_0/\pi k^2$ for a flow in which
 total translational momentum is zero.
Two fluids have the viscosities $\eta_0$ and $\eta_1$ and
fill in the regions of $n-1/2<x/L_x\le n$ and $n <x/L_x\le n+1/2$, respectively, where $n$ is an arbitrary integer.

\subsubsection{Circular Laminar Flows}\label{sec:circular}

The fluids filling a single cylinder and the space between two concentric cylinders are considered
as shown in Fig.~\ref{fig:cyl}(a) and (b), respectively.
The azimuthal velocity in a steady rotational flow is given by \cite{trit88,goet07}
\begin{equation} \label{eq:v_cf}
v_\theta(r) = A_ir+ \frac{B_i}{r},
\end{equation}
where $i=0$ and $i=1$ for the outer and inner fluids, respectively.
The $+a$ fluids in the single cylinder
rotate with a constant angular velocity (similar to that of a rigid body), {\it i.e.},
\begin{equation}
A_0=A_1=\Omega_{\rm out}, \qquad B_0=B_1=0,
\end{equation}
which is independent of the viscosity ratio $\eta_1/\eta_0$,
because the rigid-body rotation yields no gradient of the angular velocity $\partial \omega/\partial r$.
However, the $-a$ fluids have a stress term proportional to the absolute angular velocity $\omega$ [see Eq.~(\ref{eq:strs_rq-a})];
thus, the stress balance at the fluid boundary is expressed as
$(\eta_1-\eta_0)\omega(R_{\rm m})= -2\eta_0B_0/{R_{\rm m}}^2$.
Therefore, the flow is given by Eq.~(\ref{eq:v_cf}) with
\begin{eqnarray}
A_0 &=& \frac{\Omega_{\rm out}}{1-\frac{\eta_1-\eta_0}{\eta_1+\eta_0}\frac{{R_{\rm m}}^2}{{R_{\rm out}}^2}} , \quad B_0= - \frac{{R_{\rm m}}^2\Omega_{\rm out}}{\frac{\eta_1+\eta_0}{\eta_1-\eta_0}-\frac{{R_{\rm m}}^2}{{R_{\rm out}}^2}} ,\nonumber \\
A_1 &=&  \frac{\frac{2\eta_0}{\eta_1+\eta_0}\Omega_{\rm out}}{1-\frac{\eta_1-\eta_0}{\eta_1+\eta_0}\frac{{R_{\rm m}}^2}{{R_{\rm out}}^2}} ,    \quad B_1=0.
\label{eq:sng-a}
\end{eqnarray}

In the  circular Couette flow between two concentric cylinders,
the coefficients $A_i$ and $B_i$ are obtained from the boundary conditions at $R_{\rm m}$,  $R_{\rm in}$, and $R_{\rm out}$.
For the $+a$ fluids, they are given by
\begin{equation}
B_0 = \frac{\Omega_{\rm in} - \Omega_{\rm out}}{ \frac{\eta_0}{\eta_1}\frac{1}{{R_{\rm in}}^2} - \frac{1}{{R_{\rm out}}^2} + \big(1-\frac{\eta_0}{\eta_1}\big)\frac{1}{{R_{\rm m}}^2} }, \quad B_1= \frac{\eta_0}{\eta_1}B_0,
 \label{eq:dw+a}
\end{equation}
and
\begin{equation}
A_0 = \Omega_{\rm out} -  \frac{B_0}{{R_{\rm out}}^2}, \qquad  A_1 = \Omega_{\rm in} -  \frac{B_1}{{R_{\rm in}}^2}.
 \label{eq:dw_cfa}
\end{equation}
Thus, the constant torque $4\pi\eta_0B_0$ propagates from the inner to outer cylinders via the fluids
because of AMC.
However, in the $-a$ fluids, the lack of AMC generates an additional torque, and the flow is modified as
\begin{eqnarray} \label{eq:dw-a}
B_0 &=& \frac{ \frac{2\eta_1}{\eta_1+\eta_0} {R_{\rm in}}^2\Omega_{\rm in} - \big( {R_{\rm in}}^2 + \frac{\eta_1-\eta_0}{\eta_1+\eta_0}{R_{\rm m}}^2 \big)\Omega_{\rm out} } {1-\frac{{R_{\rm in}}^2}{{R_{\rm out}}^2} +  \frac{\eta_1-\eta_0}{\eta_1+\eta_0}\big(\frac{{R_{\rm in}}^2}{{R_{\rm m}}^2}-\frac{{R_{\rm m}}^2}{{R_{\rm out}}^2}\big)  }, \\ \nonumber
B_1 &=& \frac{  \big( {R_{\rm out}}^2 - \frac{\eta_1-\eta_0}{\eta_1+\eta_0}{R_{\rm m}}^2 \big)\Omega_{\rm in} -\frac{2\eta_0}{\eta_1+\eta_0} {R_{\rm out}}^2\Omega_{\rm out}} {\frac{{R_{\rm out}}^2}{{R_{\rm in}}^2}-1 +  \frac{\eta_1-\eta_0}{\eta_1+\eta_0}\big(\frac{{R_{\rm out}}^2}{{R_{\rm m}}^2}-\frac{{R_{\rm m}}^2}{{R_{\rm in}}^2}\big)  }.
\end{eqnarray}
The coefficients $A_0$ and $A_1$ are obtained by substituting Eq.~(\ref{eq:dw-a}) into Eq.~(\ref{eq:dw_cfa}).

\begin{figure}
\includegraphics{fig1.eps}
\caption{
Geometry of (a) a single cylinder and (b) double cylinders for circular laminar flow.
Two fluids with viscosities $\eta_0$ and $\eta_1$ respectively fill
 yellow (white) and blue (gray) regions of the cylinders.
The inner and outer cylinders are rotated with angular velocities $\Omega_{\rm in}$ and $\Omega_{\rm out}$, respectively.
}
\label{fig:cyl}
\end{figure}

\section{Lagrangian finite volume method}\label{sec:LFVM}

\subsection{Angular-Momentum Conservation}\label{sec:AMC}

The finite volume method is widely used to simulate fluids~\cite{ferz02,wend09}.
A fluid is divided into small cells, and the time evolutions of quantities averaged in each cell are considered.
The average gradient is calculated using the surface integral according to the divergence theorem:
\begin{equation}\label{eq:div}
\int_{V_i} \nabla f({\bf r})\ dV = \int_{S_i} f({\bf r})\hat{\bf n}\ dS,
\end{equation}
for a scalar function $f({\bf r})$, where $\hat{\bf n}$ is the unit normal vector of the surface $S_i$ of the $i$-th cell
and points outwards.
A Lagrangian version (LFVM) has been developed for both inviscid fluids~\cite{spri10,duff11} and viscous fluids~\cite{serr01,serr06}.
A Voronoi tessellation is employed in LFVM [see Fig.~{\ref{fig:vor}}(a)].
The mass of the $i$-th cell, $M_i$, is given by
$M_i= \int_{V_i} \rho({\bf r}) dV $.
Since the method is Lagrangian, $M_i$ is constant during the time evolution: $DM_i/Dt=0$.
The mesh-generating point of the $i$-th cell, {\it i.e.}, the vertex of the Delaunay triangulation,
moves with the velocity ${\bf v}_i$.
The velocity evolution is given by integrating Eq.~(\ref{eq:nse0}) as
\begin{equation}\label{eq:FVM0}
M_i \frac{D{\bf v}_i}{Dt} =  \int_{V_i} \nabla {\boldsymbol \sigma}\ dV
= \int_{S_i} {\boldsymbol \sigma}\hat{\bf n}\ dS.
\end{equation}
The momenta of the neighboring Voronoi cells are transported via the stress on the interface surface between the cells.
The right-hand side gives the forces
to the $i$-th Voronoi cell, such that
Eq.~(\ref{eq:FVM0}) is interpreted as Newton's equation of motion for the center of mass of the fluid in the $i$-th cell.

In substituting the symmetric stress of Eq.~(\ref{eq:strs+a}) into  Eq.~(\ref{eq:FVM0}),
the velocity evolution conserves the angular momentum ({\it i.e.}, $+a$ fluids):
\begin{equation}\label{eq:FVM+a}
M_i \frac{D{\bf v}_i}{Dt} = \int_{S_i}  -P\hat{\bf n} + \eta (\hat{\bf n}\cdot \nabla){\bf v}_i + \eta \nabla v_{\rm n} + \lambda(\nabla \cdot {\bf v})\hat{\bf n} \ dS,
\end{equation}
where  $v_{\rm n}= {\bf v}\cdot \hat{\bf n}$  is the velocity normal to the interface.
In contrast,
the integration of NSE (Eq.~(\ref{eq:nse1})) using  Eq.~(\ref{eq:div})
gives a velocity evolution that DOES NOT conserve the angular momentum:
\begin{equation}\label{eq:FVM-a}
M_i \frac{D{\bf v}_i}{Dt} = \int_{S_i}  -P\hat{\bf n} + \eta (\hat{\bf n}\cdot \nabla){\bf v}_i + \mu(\nabla \cdot {\bf v})\hat{\bf n}  \ dS.
\end{equation}
Because the third term in Eq.~(\ref{eq:FVM+a}) is missing, the stress is asymmetric.
Even when the NSE of the $+a$ fluids is employed, the resultant equation of motion is for the $-a$ fluids  with $\lambda^{(-)}= \eta+\lambda^{(+)}$.
This is because both of the degenerated stress terms are integrated as the symmetric volume-compression stress, {\it i.e.},
the term originated from the latter half of the shear stress ($\eta\partial v_{\beta}/\partial x_{\alpha}$)
is also treated as the volume-compression stress ($\lambda(\nabla \cdot {\bf v})\delta_{\alpha\beta}$).
Since the former half of the shear stress is only left, the stress becomes asymmetric.
Thus, the loss of AMC is caused by the treatment of the degenerated stress terms.

This mechanism of AMC loss is commonly utilized in computational fluid dynamics (CFD).
The last viscous term, $\mu \nabla(\nabla\cdot{\bf v})$, of NSE is typically implemented as the gradient of the volume stress.
From the viewpoint of the numerical implementation, it is reasonable
to employ  the viscous term that can be discretized more accurately than the other.
Since it is more accurate than separately discretizing both terms,
numerical accuracy is improved and numerical costs are reduced.
In square meshes, the calculation
of the derivatives along the tangential direction of the cell surface
is as easy as along the normal direction.
However, in non-regular meshes, those along the tangential direction are less accurate.
The above argument for LFVM is straightforwardly applied to FDM, FEM, and SPH.
Effective volumes can be considered for quantities at vertices in FDM,
local variables in shape functions in FEM, quantities for SPH particles.
For SPH, the average in the Voronoi cell is replaced by
the weighting average as $f_i =\sum_j f(r_{ij}) w(r_{ij})$,
where  $r_{ij}$ is the distance between the $i$-th and $j$-th particles and $w(r_{ij})$ is a bell-shaped weight function.
The SPH forces between neighboring particles $f_{ij}w'(r_{ij})$ are interpreted
as the integral of an immersed surface expressed by $w'(r_{ij})$.
A similar surface expression can be found in phase field models~\cite{bibe05}.
Since the equation of motion of the SPH particles is derived from the above discretization of Eq.~(\ref{eq:nse1}),
the resultant equation follows Eq.~(\ref{eq:FVM-a}).
This is the reason for the lack of AMC in conventional SPH,
and AMC is not recovered even if higher-resolution versions are developed in this scheme.

As described in Sec.~\ref{sec:theory},
the equation of motion, Eq.~(\ref{eq:FVM-a}), reproduces the same behavior as Eq.~(\ref{eq:FVM+a}) in bulk fluids;
Artificial flows are generated depending on the boundary conditions on rigid or deformable objects including contact with other fluids.
When the boundary conditions are imposed by the stress balance in Eq.~(\ref{eq:FVM-a}),
the artificial torque on the boundary induces artificial flows as demonstrated for the  circular laminar flows in Sec.~\ref{sec:circular}.
However, in  deterministic simulations without thermal fluctuations,
the boundary condition is more frequently imposed as a separated condition from the bulk velocity evolution
and the symmetric stress of Eq.~(\ref{eq:strs+a}) is calculated from the flow field.
Thus, the boundary is  implemented in accordance with AMC.
In this case, flows of the $+a$ fluids are obtained even when a simulation method lacks AMC.
Thus, the implementation of the boundaries is a significant step in avoiding the artifacts induced by the lack of AMC.
On the other hand, the angular-momentum-conserving ($+a$) methods have an advantage in that
the bulk and boundary fluids are  consistently implemented, such that the boundary conditions are simplified.

In contrast, in particle-based simulation methods involving thermal fluctuations (DPD, MPC, DSMC, and SDPD),
it is difficult to separately impose the stress balance on the boundaries.
The boundary stress can disturb the thermodynamic balance.
Since the instantaneous flow field is noisy,
a  time average is needed in order to obtain a sufficiently accurate flow field for the stress estimation.
Therefore, the instantaneous stress must be directly calculated from the equation of motion
in order to impose the stress balance on the boundary.
Thus, for these methods, the $+a$ methods are required to simulate the flows with deformable boundaries.

When the viscosity is varied in space and the gradient $\nabla\eta({\bf r})$ or  $\nabla\lambda({\bf r})$ is not negligibly small,
the viscous terms are not  degenerated; thus,
Eqs.~(\ref{eq:nse0}) and (\ref{eq:FVM+a}) are chosen for NSE and LFVM, respectively.
In this case,  the $+a$ methods are required for both deterministic and stochastic simulations.

\begin{figure}
\includegraphics[width=8.5cm]{fig2.eps}
\caption{
Voronoi tessellation of fluids between two concentric cylinders.
(a) The thick gray (magenta and green) lines of the inner and outer polygons represent the Voronoi cell of the vertex $i$ and
the nearest neighbor vertices, respectively.
(b) The boundary of the outer cylinder. The closed circles ($\bullet$) represent the boundary vertices.
(c) The interface between inner and outer fluids.
The thick gray (magenta) lines represent the interface.
}
\label{fig:vor}
\end{figure}

\subsection{Discretization of Two-Dimensional LFVM}\label{sec:LFVM2D}

To see more detailed relations,
we investigated the LFVM algorithm for $+a$ fluids in  2D space.
The first (pressure) term in Eq.~(\ref{eq:FVM+a}) is discretized as
\begin{equation}\label{eq:press}
\int_{S_i}  -P\hat{\bf n}\ dS = \sum_j P_j \frac{\partial A_j}{\partial {\bf r}_{i}},
\end{equation}
where the summation is taken over all Voronoi cells including the $i$-th cell.
The area $A_i$ of the $i$-th Voronoi cell is calculated by
$A_i=\sum_j b_{ij}r_{ij}/4$ with the edge length $b_{ij}= (\cot(\theta_{ikj})+\cot(\theta_{ik'j}) )r_{ij}/2$,
where the summation is taken over the neighbor cells of the $i$-th cell and
the angle $\theta_{ikj}=\arccos(\hat{\bf{r}}_{ik}\cdot \hat{\bf{r}}_{jk} )$
[see Fig.~\ref{fig:vor}(a)].
We use ${\bf r}_{ji}={\bf r}_{j}-{\bf r}_{i}$, $r_{ij}=|{\bf r}_{ij}|$, and $\hat{\bf r}_{ji}= {\bf r}_{ji}/r_{ij}$
in this paper.
The equation for the inviscid fluids (no viscous interactions)
exactly conserve the translational and angular momenta;
thus, the inviscid term does not influence the present AMC issue.
The derivation of this term is given in Appendix~\ref{app:gra}.
The pressure $P_i$ is the function of the mass density $\rho_i=M_i/A_i$ of the $i$-th Voronoi cell.
We employ $P_i= (\rho_0c^2/2)[(\rho_i/\rho_0)^2-1]$ following Ref.~\cite{elle07},
where $c$ is the speed of sound and $\rho_0$ is the equilibrium mass density.

The second  term in Eq.~(\ref{eq:FVM+a}) is discretized as described in~\cite{serr06}
\begin{equation}\label{eq:stgreen}
 \int_{S_i} \eta (\hat{\bf n}\cdot \nabla){\bf v}_i\ dS = \eta  \sum_j \frac{b_{ij}{\bf v}_{ji}}{r_{ij}},
\end{equation}
where ${\bf v}_{ji}= {\bf v}_j-{\bf v}_i$
and the summation is taken over the neighbor cells.
The third and fourth terms in Eq.~(\ref{eq:FVM+a}) are divided into the differentials along the interface and along the normal direction, $\hat{\bf n}$, as
\begin{eqnarray}\label{eq:stt0}
\int_{S_i} \eta  \nabla v_{\rm n}\ dS &=& \eta  \int_{S_i} \frac{\partial v_{\rm n}}{\partial x_{\rm t}}\hat{\bf t}+  \frac{\partial v_{\rm n}}{\partial x_{\rm n}}\hat{\bf n}\ dS, \\ \label{eq:stv0}
\int_{S_i}  \lambda(\nabla \cdot {\bf v})\hat{\bf n}\ dS &=&  \lambda \int_{S_i} \frac{\partial v_{\rm t}}{\partial x_{\rm t}}\hat{\bf n}+  \frac{\partial v_{\rm n}}{\partial x_{\rm n}}\hat{\bf n}\ dS,
\end{eqnarray}
where   $\hat{\bf t}={\bf e}_z\times \hat{\bf n}$ is the unit vector along the interface and $v_{\rm t}= {\bf v}\cdot \hat{\bf t}$.
Equation~(\ref{eq:stv0})  conserves translational and angular momenta by itself since it consists of the central forces between neighboring cells.
On the other hand, the first term  of Eq.~(\ref{eq:stt0}) does not conserve the angular momentum by itself,
but it is supposed to be canceled along with the non-conserving part of Eq.~(\ref{eq:stgreen}).

Since the differential along $\hat{\bf t}$ has not been taken into account in the previous studies,
we examined several discretization methods.
Among them, the following two give the best and second-best results;
 we refer to them as type I and type II, respectively.
The first term in Eq.~(\ref{eq:stt0}) is discretized as
\begin{eqnarray}\label{eq:stt1}
\big(\int_{S_i} \frac{\partial v_{\rm n}}{\partial x_{\rm t}}\hat{\bf t}\ dS\big)^{\rm type\ I}  &=& \frac{1}{3} \sum_{\rm dela} ({\bf v}_{jk}\cdot \hat{\bf n}_{jk})\hat{\bf r}_{jk}, \\ \label{eq:stt2}
\big(\int_{S_i} \frac{\partial v_{\rm n}}{\partial x_{\rm t}}\hat{\bf t}\ dS\big)^{\rm type\ II}  &=& \sum_{\rm dela} \frac{b_{jk}{\bf v}_{jk}\cdot \hat{\bf n}_{jk}}{r_{im} \hat{\bf n}_{jk}\cdot\hat{\bf r}_{mi} }\hat{\bf r}_{jk}.
\end{eqnarray}
The first term in Eq.~(\ref{eq:stv0}) is discretized as
\begin{eqnarray}\label{eq:stv1}
\big(\int_{S_i} \frac{\partial v_{\rm t}}{\partial x_{\rm t}}\hat{\bf n}\ dS\big)^{\rm type\ I}  &=& \frac{1}{3} \sum_{\rm dela} ({\bf v}_{jk}\cdot \hat{\bf r}_{jk})\hat{\bf r}_{mi}, \\ \label{eq:stv2}
\big(\int_{S_i} \frac{\partial v_{\rm t}}{\partial x_{\rm t}}\hat{\bf n}\ dS\big)^{\rm type\ II}  &=& \sum_{\rm dela} \frac{b_{jk}{\bf v}_{jk}\cdot \hat{\bf r}_{jk}}{r_{im}}\hat{\bf r}_{mi}.
\end{eqnarray}
In Eqs.~(\ref{eq:stt1})--(\ref{eq:stv2}),
 the summation is taken over the edges of the Delaunay triangles [the outer thick lines in Fig.~\ref{fig:vor}(a)],
and $\hat{\bf n}_{jk}$ is the unit normal vector of the edge connecting the vertices $j$ and $k$: $\hat{\bf n}_{jk}\cdot {\bf r}_{jk}=0$; $\hat{\bf n}_{jk}$ points outwards: $\hat{\bf n}_{jk}\cdot {\bf r}_{mi}>0$.
When the divergence theorem is directly applied,
the discretizations are slightly different from the type I forms:
the factor $A_i/A_{\rm dela,i}$ appears instead of  $1/3$ in Eqs.~(\ref{eq:stt1}) and (\ref{eq:stv1}),
and $\hat{\bf n}_{jk}$ appears instead of $\hat{\bf r}_{mi}$ in Eq.~(\ref{eq:stv1}),
where   $A_{\rm dela,i}$ is
 the total area of the Delaunay triangles contacting the vertex $i$ [the area surrounded by the outer thick line in Fig.~\ref{fig:vor}(a)].
These modifications are employed to  maintain the translational-momentum conservation and AMC, respectively.
However, the angular momentum is not exactly conserved in the type I form for Eqs.~(\ref{eq:stgreen}) and (\ref{eq:stt1}).
The type II form is constructed to recover AMC; the torque between the vertices $j$ and $k$ in Eq.~(\ref{eq:stgreen})
is canceled by the torque between the vertices $i$ and $m$  in Eq.~(\ref{eq:stt2}).
In a regular lattice in which all Delaunay triangles are equilateral,
the types I and II discretizations coincide.

The second term in Eqs.~(\ref{eq:stt0}) and (\ref{eq:stv0}) is discretized as
\begin{equation}\label{eq:stcen}
 \int_{S_i}  \frac{\partial v_{\rm n}}{\partial x_{\rm n}}\hat{\bf n}\ dS =
\sum_j \frac{b_{ij}}{r_{ij}}({\bf v}_{ji}\cdot \hat{\bf r}_{ji}) \hat{\bf r}_{ji},
\end{equation}
where the summation is taken over the neighbor cells as in Eq.~(\ref{eq:stgreen}).
The translational momentum is conserved by itself in all of the above discretization terms in
Eqs.~(\ref{eq:press}), (\ref{eq:stgreen}), and (\ref{eq:stt1})--(\ref{eq:stcen}).
On the other hand, the angular momentum is conserved by itself in Eqs.~(\ref{eq:press}) and (\ref{eq:stv1})--(\ref{eq:stcen}),
while the angular momentum is approximately or exactly conserved by the combination of  Eq.~(\ref{eq:stgreen}) with  Eq.~(\ref{eq:stt1})
or Eq.~(\ref{eq:stt2}), respectively.

\subsection{Rigid-Body Rotation}\label{sec:rbr}

When a cylinder or box is filled with a fluid and is slowly rotated as shown in Fig.~\ref{fig:cyl},
the whole fluid rotates with a constant angular velocity like a rigid body: ${\bf v}_{\rm rot}= {\boldsymbol \Omega}\times {\bf r}$.
The pressure gradient is balanced by centrifugation force,
and the viscous terms vanish in NSE: $\nabla^2 {\bf v}_{\rm rot}=\nabla(\nabla\cdot {\bf v}_{\rm rot})= 0$.
However, these vanishments are not trivial in the discretized equations.

In our LFVM scheme,
the viscous forces of Eqs.~(\ref{eq:stv1})--(\ref{eq:stcen}) vanish in the rigid-body rotation,
since ${\bf v}_{ji}\cdot \hat{\bf r}_{ji}=0$.
The forces of Eqs.~(\ref{eq:stgreen}) and (\ref{eq:stt1}) also vanish,
owing to the divergence theorem and the fact that the divergence of a constant function is null:
$\sum_j b_{ij} \hat{\bf n}=\sum_j b_{ij} \hat{\bf t}=0$.
However, in the type II scheme, the forces of Eq.~(\ref{eq:stt2}) do not vanish in the rigid-body rotation.
This remaining stress destabilizes the constant rotation and induces an artificial flow
in which the angular velocity depends on the position in the cylinder as demonstrated in Sec.~\ref{sec:results}.
Thus, the  $+a$ fluids  with the type II discretization do not reproduce the rigid-body rotation,
while the angular momentum is strictly conserved.
In contrast, the $+a$ fluids  with the type I discretization reproduce the rigid-body rotation correctly,
while the angular momentum is not strictly conserved.
Although it is better if one method satisfies both conditions,
 we have not found such a discretization scheme.
However, in the type I method, AMC is sufficiently well maintained to reproduce binary flows
(it will be shown in Sec.~\ref{sec:results}) so that the type I method can be employed in applications.

It is surprising that
the conditions of AMC in the equation of motion  and the rigid-body rotation are different in CFD, unlike in molecular dynamics (MD) simulations.
MD has no velocity-dependent forces, and the rotational invariance of potentials guarantees AMC and the rigid-body rotation.
Since the viscous terms depend on the velocity in CFD,
 invariance is needed to account for global rotational displacement
and the global rotation of a constant speed.
Therefore, the viscous terms must be independent of the total velocity
and total angular velocity of the interacting elements (vertices or particles) in CFD.
The $+a$ versions of DPD and MPC satisfy this condition.
The viscous force in DPD is a function of the relative velocity of two DPD particles along the line connecting them: $({\bf v}_{ij}\cdot \hat{\bf r}_{ij})\hat{\bf r}_{ij}$.
The $+a$ types of the MPC collisions are independent of the total velocity and total angular velocity of the MPC particles in each collision cell.
Thus, the viscous interactions of DPD and MPC are independent of the total angular velocity of the interacting particles.
To see this invariance condition more clearly,
let us consider an artificial pairwise force, ${\bf f}_{{\rm rot},ij}=|{\bf v}_{ij}\times \hat{\bf r}_{ij}|\hat{\bf r}_{ij}$, between the $i$-th and $j$-th particles.
This force conserves the translational and angular momenta, since
${\bf f}_{{\rm rot},ji}=-{\bf f}_{{\rm rot},ij}$ and
${\bf r}_{ij}\times {\bf f}_{{\rm rot},ij}=0$. On the other hand, the force amplitude is determined by the angular velocity, $|{\bf v}_{ij}\times \hat{\bf r}_{ij}|$, and is varied by the global rigid-body rotation.
Thus, these two conditions must be distinguished.

In conventional SPH, the viscous force proportional to ${\bf v}_{ji}$ between the neighboring $i$-th and $j$-th particles is employed,
and this force is independent of the positions of the other particles.
Therefore, the SPH viscous force does not vanish in the rigid-body rotation; as a result, the rigid-body rotation cannot be reproduced
except in cases in which SPH particles are initially arranged in a regular lattice.
In SDPD, thermal fluctuations reduce the anisotropic distribution of the particles,
and the deviation from the rigid-body rotation is likely reduced.

\subsection{Implementation at Boundary Wall and Fluid Interface}\label{sec:bd}

We have  explicitly discretized the boundary on a wall and an interface between two fluids,
and the vertices are placed on them as shown in Fig.~\ref{fig:vor}(b) and (c).
Although a mirror image is widely employed to impose the no-slip boundary condition,
the degeneracy of the Delaunay triangles always occurs at a flat boundary,
since the neighboring vertices and their images form an isosceles trapezoid.
To avoid the reduction in the number of the neighbors due to this degeneracy,
we employ the boundary vertices instead.
Since we investigate flows with steady interfaces,
the interface vertices are not added or removed in the simulations.
If one considers flows accompanied by large interface deformations,
the rearrangements of the interface vertices are required.

The vertices on the inner and outer boundaries of the cylinders are moved along the boundaries with constant velocities
$\Omega_{\rm in} R_{\rm in}$ and $\Omega_{\rm out} R_{\rm out}$, respectively.
The area of a Voronoi cell on the boundary is approximately half that of the other cells  [see Fig.~\ref{fig:vor}(b)].

To generate a sharp fluid interface, the viscous forces around the interface cells
are carefully implemented.
The interface Voronoi cell is divided into two regions that each contains one types of fluid,
and the viscous force between the interface vertices depends on the area ratio.
The vertices in the different fluids [{\it e.g.}, the vertices $k$ and $k'$ in Fig.~\ref{fig:vor}(c)]
do not directly interact.
In our scheme, the boundaries are consistently treated with bulk fluids.
When $\eta_1/\eta_0=1$ is inputted, the equation becomes identical to the bulk equation.
The details of the procedure are described in Appendix~\ref{app:interface}.

\begin{figure}
\includegraphics{fig3.eps}
\caption{
Periodic planar laminar flows.
(a) Velocity profile $v_y(x)$ of the steady states at $\eta_1/\eta_0=1$, $2$, and $10$.
(b) Time evolution of the velocity amplitude $v_{\rm a}$ at $\eta_1/\eta_0=1$.
Two fluids with viscosity $\eta_0$ and $\eta_1$ fill
in the regions $-L_x/2<x\le 0$ and $0<x\le L_x/2$, respectively.
The solid lines are obtained by LFVM for the $+a$ type I method.
The dashed lines represent the analytical solutions of (a) Eq.~(\ref{eq:sin2}) and (b) Eq.~(\ref{eq:sin1}),
and they completely overlap with the solid lines.
The velocity and time are normalized by $v_0=f_0/\eta_0 k^2$
and $t_0=\rho/\eta k^2$, respectively.
}
\label{fig:sin2}
\end{figure}

\subsection{Numerical Details}\label{sec:detail}

In LFVM, it is known that the discretization errors become unacceptably large in disordered meshes~\cite{serr06};
this problem is addressed through rearrangement of the Voronoi vertices~\cite{spri10,duff11}.
Here, we do not employ this rearrangement but
maintain a flow condition in which Voronoi vertices remain in semi-regular arrangements, as shown in Fig.~\ref{fig:vor}.
Since our aim is to clarify the AMC effects, the influence of the  rearrangement is avoided.

Equations (\ref{eq:FVM+a}) and (\ref{eq:FVM-a}) are numerically integrated
using the fourth-order Runge--Kutta method with $\Delta t= 0.001$.
The parameters $\rho_{\rm int}=\rho_0=1$ and $\eta_0=1$ are used for all flows,
where $\rho_{\rm int}$ is the density in the initial state.
If not specified, $\mu=0$ is used.

For the periodic planar flow,
a simulation box with side lengths of $L_x=52.1$ and $L_y=53.7$ is employed under the periodic boundary condition.
As the initial state, a regular triangular lattice with $A_i=1$ is created, where
the total number of Voronoi cells $N=2800$.
To simulate laminar flows, $f_0=0.01$ and $c=10v_0$ are used, where $v_0=f_0/\eta_0 k^2$.

In the circular laminar flows,
vertices are aligned on concentric circles with mean area $\langle A_i\rangle=1$ as the initial state
and $c=1$ is used.
For the  single cylinder,
$R_{\rm out}=30.1$, $R_{\rm m}=15.1$, and $\Omega_{\rm out}=0.0001$ are used.
For the double cylinders,
$R_{\rm out}=60.1$, $R_{\rm m}=40.1$, $R_{\rm in}=20.1$, and $\Omega_{\rm out}-\Omega_{\rm in}=0.0001$
are used.
We conducted simulations with other parameter values
to confirm that the results are not sensitive to the above-mentioned parameter choices.

\section{Simulation results}\label{sec:results}

\subsection{Planar Laminar Flows}\label{sec:replanar}

Before showing the AMC effects in circular flows,
we describe the results of the planar laminar flows induced by the periodic force,
and we discuss the accuracy of our simulation methods.

In the single-component fluids,
the steady flow profile and the velocity relaxation from ${\bf v}(0)=0$ or ${\bf v}(0)=2v_0\sin(kx){\bf e}_y$ agree
well with the analytical predictions
as shown in Fig.~\ref{fig:sin2}(a) and (b), respectively.
The simulation data overlap with the analytical curves.
The values of the velocity amplitude $v_a$ are calculated by the Fourier transformation of the velocity $v_y$.
Although Fig.~\ref{fig:sin2} shows only the results of the $+a$ fluids with type I discretization at $\mu=0$,
the other combinations of the methods give very similar results
and the differences  only appear in the amplitudes $v_a$.
The viscosity of the simulation is estimated from this amplitude as $\eta_{\rm sin}=f_0/v_ak^2$
as shown in Fig.~\ref{fig:vis}.
At $\mu=0$, all methods generate a viscosity close to the input viscosity $\eta_0$.
As $\mu$ increases, the viscosity linearly decreases; moreover, the type II discretization
gives a lower slope than that of type I.
From this aspect, the type II discretization is better.
Compared to it, the effects of AMC ($+a$ or $-a$ fluids) are marginal.
To estimate the numerical errors, we also calculate the viscosity at $f_0=0.005$, where the other parameters are same; The difference of $\eta_{\rm sin}$ is negligibly small: $\Delta \eta_{\rm sin}/\eta_0 <0.002$ for $+a$ type II and  $\Delta \eta_{\rm sin}/\eta_0 <0.0002$ for the other three methods.

Hereafter, we only show the results for $\mu=0$.
The simulation results at $\mu>0$ are very similar to those at $\mu=0$,
and the differences can be understood by rescaling the velocity using $\eta_0/\eta_{\rm sin}$.
For the $-a$ fluids,
 the choice between the type I or II viscous term does not need to be considered at $\mu=0$,
since they vanish at $\mu=0$.

For two fluids with different viscosities,
the velocity profile has a kink at the fluid interface as shown in Fig.~\ref{fig:sin2}(a).
Again, all of the methods ($+a$ or $-a$ fluids with type I or type II)
show excellent agreement with the analytical solutions.
Thus,  the planar laminar flows are not influenced by the lack of AMC.

\begin{figure}
\includegraphics{fig4.eps}
\caption{
Viscosity $\eta_{\rm sin}$ calculated from the planar laminar flow
for LFVM with
symmetric ($+a$) and asymmetric ($-a$) viscous terms, using two types of
discretization methods [type I Eqs.~(\ref{eq:stt1}) and (\ref{eq:stv1}) and type II Eqs.~(\ref{eq:stt2}) and (\ref{eq:stv2})].
The dependence on the coefficient, $\mu$, of the last term in Eq.~(\ref{eq:nse1}) is shown.
}
\label{fig:vis}
\end{figure}

\begin{figure}
\includegraphics{fig5.eps}
\caption{
Angular velocity $v_{\theta}/r$ of the single-component $+a$ fluids in the single cylinder.
The fluid using the type I viscous term exhibits the rigid-body rotation,
while the fluid using  the type II term exhibits a non-uniform rotation.
}
\label{fig:sng_dcos}
\end{figure}

\begin{figure}
\includegraphics{fig6.eps}
\caption{
Angular velocity $v_{\theta}/r$ of the binary $-a$ fluids in the single cylinder.
The fluids do not exhibit the rigid-body rotation at $\eta_1/\eta_0 \ne 1$.
The symbols $\circ$, $\triangle$, $\Box$, and $\diamond$ represent
the simulation data for $\eta_1/\eta_0 = 1$, $2$, $4$, and $10$, respectively.
The solid lines represent the analytical solution of Eq.~(\ref{eq:v_cf}) with Eq.~(\ref{eq:sng-a}).
}
\label{fig:sng-a}
\end{figure}

\begin{figure}
\includegraphics{fig7.eps}
\caption{
Angular velocity $v_{\theta}/r$ of the binary $+a$ fluids in the double cylinders.
The solid lines represent the analytical solution of Eq.~(\ref{eq:v_cf}) with Eqs.~(\ref{eq:dw+a}) and (\ref{eq:dw_cfa}).
The data ($\times$) for $\Omega_{\rm out}=0$ and $\Omega_{\rm in}=-0.0001$ overlap
with those for $\Omega_{\rm out}=0.0001$ and $\Omega_{\rm in}=0$
($\circ, \triangle, \Box$, and $\diamond$ for $\eta_1/\eta_0 =1, 2, 4$, and $10$, respectively).
}
\label{fig:dw+a}
\end{figure}

\begin{figure}
\includegraphics{fig8.eps}
\caption{
Angular velocity $v_{\theta}/r$ of the binary $-a$ fluids in the double cylinders
for (a) $\eta_1/\eta_0 = 2$ and (b) $\eta_1/\eta_0 = 10$.
The symbol $\circ$ ($\times$) represents
the simulation data at $\Omega_{\rm out}=0.0001$ and $\Omega_{\rm in}=0$ ($\Omega_{\rm out}=0$ and $\Omega_{\rm in}=-0.0001$).
The solid lines represent the analytical solution of Eq.~(\ref{eq:v_cf}) combined with Eqs.~(\ref{eq:dw_cfa}) and (\ref{eq:dw-a}).
}
\label{fig:dw-a}
\end{figure}

\subsection{Circular Laminar Flows}\label{sec:recircular}

First, a single-component fluid filling a single cylinder is considered.
The type I $+a$ fluid and the $-a$ fluid exhibit a rigid-body rotation;
the angular velocities are constant (see the upper lines in Figs.~\ref{fig:sng_dcos} and \ref{fig:sng-a}).
However, the type II $+a$ fluid exhibits an artificial flow as described in Sec.~\ref{sec:rbr}
(see the lower line in Fig.~\ref{fig:sng_dcos}).
Therefore, the invariance of the viscous forces under the global rotation
is significant in the simulation of rotational flows.
For this reason, the type I method is better than the type II method,
although the type II method provides better viscosity estimation for $\mu>0$.
The invariance under the global rotation is more important than
the exact AMC in the equation of motion.
The type I method maintains AMC not exactly but sufficiently well
to reproduce binary flows as shown below.
Hereafter, we use only the type I method for the $+a$ fluid.

The type I $+a$ binary fluids maintain constant angular velocity
 in a single cylinder (see Appendix~\ref{app:interface}).
In contrast, in the $-a$ binary flow, the outer fluid exhibits
the angular-velocity gradients that are well predicted by the analytical solutions in Sec.~\ref{sec:circular},
as shown in Fig.~\ref{fig:sng-a}.

The results from the simulations with double cylinders
also have excellent agreement with the analytical solutions for both $+a$ and $-a$ fluids, as shown in Figs.~\ref{fig:dw+a} and \ref{fig:dw-a}.
In the $+a$ fluids, the velocity gradient is independent of the absolute angular velocity;
the simulation results of the outer-wall rotation ($\Omega_{\rm out}=0.0001$ and $\Omega_{\rm in}=0$)
 completely overlap with those of the inner-wall rotation ($\Omega_{\rm out}=0$ and $\Omega_{\rm in}=-0.0001$).
On the other hand, the velocity gradient of the $-a$ fluids depends on the absolute angular velocity.
Thus, the AMC of the viscous interactions is significant in simulating multi-component fluids with different viscosities.

\section{Summary and discussion}\label{sec:sum}

We have clarified the conditions of angular-momentum conservation and the invariance
under global rotation in fluid dynamics simulations. In NSE, the viscous terms for volume
change and half of the shear deformation are degenerated for a fluid with a constant
viscosity, although they are the gradients of symmetric and asymmetric stresses,
respectively. To maintain the AMC of the discretized equation of motion, these degenerated
terms must be separately implemented in accordance with their stress origins. If the
degenerated terms are implemented together as the gradient of the symmetric stress, the
resultant forces for the shear deformation lack AMC. On the other hand, the invariance
of the viscous forces in a global rotation with a constant speed is a necessary condition for
reproducing the rigid-body rotation of a fluid in a slowly rotating container.

When applying CFD, the torque balance on wall boundaries and fluid interfaces
is essential in reproducing flows involving a rotation. When the boundary implementation
lacks AMC, multi-component fluids with different viscosities generate
artificial flows. To avoid them,
one has the following two choices.
(i) A $+a$ CFD method is employed for all of the fluids.
(ii) The bulk fluids are solved by a $-a$ method but
the boundary fluids are solved by the stress balance where the symmetric viscous stress
is calculated from the velocity field.

To calculate low Mach number flows of liquids, the incompressible assumption is widely
employed. In incompressible fluids, the last term of NSE is removed ($\mu=0$) and the
incompressible constraint is imposed. Thus, the stress becomes asymmetric. However, the
incompressible constraint implicitly transports momentum
 and a fluid element can interact with itself via the periodic boundary.
Although it originates from a symmetric stress (pressure),
 it is not completely obvious whether it loses AMC. This is an important
issue that should receive further study.

We developed LFVM to demonstrate the effects of AMC in binary fluids. The
discretizations of the viscous stresses and the fluid interfaces were newly implemented. The
simulation results show excellent agreement with the analytical solutions. In developing
other fluid dynamics methods, the steady rotational flows presented in this study are
suitable for checking whether a method reproduces AMC flows.

The LFVM implementation in this study is difficult to directly apply to SPH,
since the SPH interaction is constructed in a  pairwise manner.
In the $+a$ version of SDPD~\cite{mull15}, the spin variable is introduced.
Instead of imposing the asymmetric stress in the equation of motion,
the asymmetric viscous force is stored in the spin velocity
and released with a time delay.
It can be interpreted that the lacked asymmetric stress is implicitly
imposed via the spin rotation.

\begin{acknowledgments}
This work was supported by JSPS KAKENHI Grant Number JP17K05607
and MEXT as ``Exploratory Challenge on Post-K computer'' (Frontiers of Basic Science: Challenging the Limits).
\end{acknowledgments}

\begin{appendix}

\section{Gradient Discretization}\label{app:gra}

The volume integral of the gradient in a Voronoi cell
is expressed by the surface integral of the cell, as described in Eq.~(\ref{eq:div}).
When the surface value is approximated as the mean value of two contacting cells, $f_s= (f_i+f_j)/2$,
the gradient is given by
\begin{equation}\label{eq:gra1}
(\nabla f)_i =  \frac{1}{V_{i}}\int_{V_i} \nabla f({\bf r})\ dV = \frac{1}{V_{i}}\sum_j \frac{S_{ij}}{2}(f_i+f_j) \hat{\bf r}_{ji},
\end{equation}
where $V_i$ is the volume (area in 2D) of the $i$-th Voronoi cell
and $S_{ij}$ is the area (length in 2D) of the interface between the $i$-th and $j$-th Voronoi cells.
The summation is taken over the cells neighboring the $i$-th cell.
However, this discretization is inaccurate.
To improve the accuracy,
Serrano {\it et al.}~\cite{serr05} and Springel~\cite{spri10} independently derived
the following form:
\begin{equation} \label{eq:gra2}
(\nabla f)_i = \frac{1}{V_{i}} \sum_j S_{ij}\Big[(f_i+f_j) \frac{\hat{\bf r}_{ji}}{2} + (f_j-f_i)\frac{{\bf c}_{ij}}{r_{ij}}\Big].
\end{equation}
The vector ${\bf c}_{ij}$ of the center of the interface
from the middle position $({\bf r}_i+{\bf r}_j)/2$ between the $i$-th and $j$-th Voronoi cells is given by
\begin{equation}\label{eq:cij}
{\bf c}_{ij} = \frac{1}{S_{ij}}\int_{S_{ij}} \Big( {\bf r} - \frac{{\bf r}_i+{\bf r}_j}{2} \Big)\ dS.
\end{equation}
Although Eq.~(\ref{eq:gra2}) works quite well,
the second term does not conserve the angular momentum,
whereas it conserves the translational momentum.

\begin{figure}
\includegraphics{fig9.eps}
\caption{
Angular velocity $v_{\theta}/r$ of the binary $+a$ fluids in the single cylinder at $\eta_1/\eta_0 =10$.
Three implementations representing the viscosity stress on the fluid interface are compared for the type I discretization.
}
\label{fig:sng_trav}
\end{figure}

We introduce an AMC discretization form:
\begin{equation} \label{eq:gra3}
(\nabla f)_i = - \frac{1}{V_{i}} \sum_j  f_j\frac{\partial V_j}{\partial {\bf r}_i},
\end{equation}
where the summation is taken over all cells.
Equation~(\ref{eq:gra3}) is derived from the Lagrangian equation for the pressure in Ref.~\cite{serr05} and from
$\int \partial f(r)/\partial {\bf r}_i dV=0$ and $\int f(r) dV= \sum_i f_{i}V_i$ in Ref.~\cite{spri10}.
Nevertheless,  Eq.~(\ref{eq:gra2}) was employed instead, since
Eq.~(\ref{eq:gra3}) was considered to be identical to Eq.~(\ref{eq:gra2}).
However, they are not exactly equal.
Equation~(\ref{eq:gra3}) conserves the translational and  angular momenta,
since $V_i$ is invariant for the translation and rotation of the coordinates, unlike  Eq.~(\ref{eq:gra2}).
The first and second terms in Eq.~(\ref{eq:gra2}) represent the translation and rotation of the interface $S_{ij}$, respectively;
thus, the change in the interface area (length in 2D) is not taken into account.

\section{Viscous Interactions on Fluid Interface}\label{app:interface}

The type I viscous forces are modified on the fluid interface as follows.
The viscous forces along the interface [between the vertices $i$ and $j$ in Fig.~\ref{fig:vor}(c)] are calculated
using Eqs.~(\ref{eq:stgreen}) and (\ref{eq:stcen}) with
\begin{equation} \label{eq:etam}
\eta= \frac{b_{ij0}\eta_0+b_{ij1}\eta_1}{b_{ij}},
\end{equation}
and $\lambda$ is similarly averaged, where the Voronoi edge is divided into two regions, each with one type of fluid:
$b_{ij0}= \cot(\theta_{ikj})r_{ij}/2$ and $b_{ij1}= \cot(\theta_{ik'j})r_{ij}/2$ [see Fig.~\ref{fig:vor}(c)].
For the viscous forces of Eqs.~(\ref{eq:stt1}) and (\ref{eq:stv1}),
the viscosity of the Delaunay triangle $ijk$ is employed.
Since the stresses $\sigma_{kij}$ and $\sigma_{k'ij}$ on the vertices $k$ and $k'$ for the triangles $ijk$ and $ijk'$  in Fig.~\ref{fig:vor}(c)
are not balanced at $\eta_1\ne \eta_0$,
the remaining stresses are imposed on the interface vertices $i$ and $j$: $ \sigma_{i}^{\rm fi}=\sigma_{j}^{\rm fi}= -(\sigma_{kij}+\sigma_{k'ij})/2$.
This means that the edge $ij$ exerts stress on the vertex $k$ and receives the opposite stress by Newton's third law.
Using these implementations,
the velocity fields around the interface are properly reproduced.
The binary fluids with $\eta_1/\eta_0=10$ exhibit constant-speed rotation,
as shown in line (i) in Fig.~\ref{fig:sng_trav}.

To clarify the improvements provided by the above implementation, we examined the following two modifications [(ii) and (iii)].
(ii) $\eta= (\eta_0+\eta_1)/2$ is employed instead of Eq.~(\ref{eq:etam}):
The velocity of the inner fluid slightly deviates as shown in line (ii) in Fig.~\ref{fig:sng_trav},
since the area of the outer fluid is larger than $1/2$ on convexly curved interfaces.
(iii) The average viscosity of Eq.~(\ref{eq:etam}) is employed for the stresses on the vertices $k$ and $k'$
to balance $\sigma_{kij}$ and $\sigma_{k'ij}$:
A gap in the angular velocity appears at the interface as shown in line (iii) in Fig.~\ref{fig:sng_trav}.
Since the type II stress term involves the interaction between $k$ and $k'$,
the type II fluids exhibit a similar gap at the interface.

Thus, the correct implementation of the interface is important in reproducing the flow behavior
involving a zero or negligibly small angular-velocity gradient.
For flows involving the angular-velocity gradient such as the  circular Couette flows shown in Fig.~\ref{fig:dw+a},
this gap is significantly reduced.

\end{appendix}

redpajama_set_name: RedPajamaArXiv


balance,enquiry,prepaid card balance. X. Forex Cards / Travel Currency : Options to check the available balance on your prepaid card. a.
retirement and investment balances. Citibank purchases using your Citibank Banking card to earn the benefits you enjoy with Citibank checking!
Forex cards are a comfortable & safe way of carrying currency from country to country. Click here to know How to check balance and change PIN on ForexPlus Cards.
2011-02-17 · How to Check Your Credit Card Balance. Checking the balance of prepaid credit cards should be very similar to checking the balance of those that are not.
Learn about our high yield and fee-free checking accounts and choose the right account for you. Skip to main content. Credit Cards Checking Minimum balance. $0.
I received a letter by Citibank stating I needed to pay a remaining balance and provided a own a Citibank card checking account with Citibank.
Check Citibank account balance by sending SMS. You must have Citibank Credit/Debit card number to get started. Indian Bank missed call balance checking.
Reviews, rates, fees and customer service info for the Citibank Simple Checking. Compare to other cards and apply online in seconds.
In checking as in life, needs vary. That is why Citigold® International at Citibank offers you checking account options, each with its own unique benefits. Meet us.


text:
R. Douglas Caney scrapbook on the atomic bomb
Richard Douglas Caney (1918-1994) was a chemical engineer with E.I. du Pont de Nemours & Company, working in the Explosives Department on the Manhattan Project. To remember his participation in the Manhattan Project, Caney prepared a scapbook of newspaper clippings of articles that flooded the media after the atomic bombs were dropped and the secrecy around the project ended.
Caney, R. Douglas, 1918-1994 (Person)
1 volume(s)
Richard Douglas Caney (1918-1994) was a chemical engineer with E.I. du Pont de Nemours & Company, working in the Explosives Department on the Manhattan Project.
Caney was born on May 6, 1918, and graduated from Princeton University in 1939. He spent his entire career as a chemical engineer with E.I. du Pont de Nemours & Company, first in the Photo Products Department and then in the Atomic Energy Division of the Explosives Department, where he was assigned to work on the Manhattan Project at Oak Ridge, Tennessee, and Hanford, Washington.
At the close of the war, he returned to Wilmington, Delaware, where he worked in the Textile Fibers Department at DuPont until retiring in 1980. He died in a hospital near Wilmington on April 11, 1994.
To remember his participation in the Manhattan Project, R. Douglas Caney prepared a scapbook of newspaper clippings of articles that flooded the media after the atomic bombs were dropped and the secrecy around the project ended. As can be imagined, the articles all have a triumphalist tone, celebrating the end of the war and the success of the multinational scientific and engineering endeavor, while at the same time downplaying issues such as nuclear safety and radiation disease. At the same time, the press also revealed the existence of the Hanford plant and the new town of Richland, Washington, and some of the clippings are from local sources. There is also a souvenir booklet on the Army-Navy "E" Award given to project participants.
E.I. du Pont de Nemours & Company. Atomic Energy Division
Manhattan Project (U.S.)
Nuclear weapons industry
R. Douglas Caney scrapbook on the atomic bomb, 2604. Manuscripts and Archives.
R. Douglas Caney scrapbook on the atomic bomb, 2604. Manuscripts and Archives. https://findingaids.hagley.org/repositories/3/resources/1432 Accessed January 30, 2023.

redpajama_set_name: RedPajamaCommonCrawl


Many artifacts are woven for use in tents, huts and dwellings. Among the most numerous are rectangular weavings which are folded in half and stitched and then become cushions and pillows. A small opening is left on a long side sufficient for various fillings to be inserted by hand. Bags of all sizes range from the tiny which contain precious items of jewellery, spices or dried fruit to the huge which are filled with grain.
Saddle bags for transport of goods by horse or donkey are common.
Finely woven bags are still made as dowry pieces.
Many items of clothing are woven for both every day, ceremonies and special occasions.
ANATOLIAN SADDLE BAG A rare semi-antique shepherd's bag from the region of Fetiyhe in south west Turkey. The long opening in the centre allows for the shepherd's head to go through and the bags hang on his front and back. The black and white photo shows a similar piece worn by a rural shepherd. This region in Ottoman Empire times was known as "Megri" but the name was changed around 1920. The intense shades of indigo, madder and mustard yellow are only found in this region and it is therefore probable they are natural dyes and sourced from nearby plants. It is in impeccable condition complete with the original small loops at each corner which allows it to be fastened to the body when worn. Reference 285. Size 131 x 47cms. 4ft. 4ins. x 1ft. 7ins.. Euros 450.
ANATOLIAN SADDLE BAG A rare semi-antique shepherd's bag from the region of Fetiyhe in south west Turkey. The long opening in the centre allows for the shepherd's head to go through and the bags hang on his front and back. The black and white photo shows a similar piece worn by a rural shepherd. This region in Ottoman Empire times was known as "Megri" but the name was changed around 1920. The intense shades of indigo, madder and mustard yellow are only found in this region and it is therefore probable they are natural dyes and sourced from nearby plants. It is in impeccable condition complete with the original small loops at each corner which allows it to be fastened to the body when worn. Reference 285. Size 131 x 47cms. 4ft. 4ins. x 1ft. 7ins. SHEPHERD WITH BAG.
SPECIAL OFFER............MOROCCAN BERBER HEAD DRESS which is for a young girl and is called "Bonnet d`enfant". These precious pieces form part of the dress for ceremonial occasions and this one originates from Berbers in the High Atlas mountain region of Ait Ouaouzguite, south of Marrakech. Similar pieces can be seen in "Splendeurs du Maroc", plate 278, page 208, and "Taznacht" plate 65, page 77. Reference 2692, size 41 x 21 cms., or 16 ins x 9 ins. Euros 65.
MOROCCAN BERBER HEAD DRESS which is for a young girl and is called "Bonnet d`enfant". These precious pieces form part of the dress for ceremonial occasions and this one originates from Berbers in the High Atlas mountain region of Ait Ouaouzguite, south of Marrakech. Similar pieces can be seen in "Splendeurs du Maroc", plate 278, page 208, and "Taznacht" plate 65, page 77. Reference 2692, size 41 x 21 cms., or 16 ins. x 9 ins. CLOSE UP.
MOROCCAN BERBER HIGH ATLAS TENT BAG. Known as "aalau" which is a squarish, partly piled bag usually woven by Berber women of the Ait Bou Ichaouen in the eastern High Atlas. The bag is given to a daughter prior to marriage and is subsequently used to contain either jewellery or clothing or sometimes ready-spun wool. It is fastened at the top by hand plaited rope which passes through a line of slits. This piece has been opened out and when in use would be folded in half and both sides joined together. The tassels at the midway stage would then hang down. The top row consists of an interesting design of 5 Berber tents. Weaving of this type is well described in "Ait Bou Ichaouen-Weaving of a Nomadic Berber Tribe" by Alfred and Suzanne Saulniers. It is a fascinating and collectable piece. Reference 2620. Size opened out 84 x 56cms. 2ft.9in. x 1ft.10in. Euros 170.
MOROCCAN BERBER HIGH ATLAS TENT BAG. Known as "aalau" which is a squarish, partly piled bag usually woven by Berber women of the Ait Bou Ichaouen in the eastern High Atlas. The bag is given to a daughter prior to marriage and is subsequently used to contain either jewellery or clothing or sometimes ready-spun wool. It is fastened at the top by hand plaited rope which passes through a line of slits. This piece has been opened out and when in use would be folded in half and both sides joined together. The tassels at the midway stage would then hang down. The top row consists of an interesting design of 5 Berber tents. Weaving of this type is well described in "Ait Bou Ichaouen-Weaving of a Nomadic Berber Tribe" by Alfred and Suzanne Saulniers. It is a fascinating and collectable piece. Reference 2620. Size opened out 84 x 56cms. 2ft.9in. x 1ft.10in. REVERSE.
SPECIAL OFFER.............MOROCCAN BERBER MAN'S CAPE. Hand woven from undyed, hand spun wool. This is the work of Berbers in the region of Marmoucher in the Middle Atlas mountains and would have been used by a Berber man, probably a shepherd, for protection against the biting winter cold. These capes are known as "jilaba" and there is good information with colour photos in "Splendeurs du Maroc" pages 155 and 197 and also "North African Textiles" on page 86. This example has very fine and detailed design incorporated in the weave. Reference 2605. Size 160cms. high x 95cms. opened at bottom. 5ft.8in. x 3ft.2in. Euros 190.
MOROCCAN BERBER MAN'S CAPE. Hand woven from undyed, hand spun wool. This is the work of Berbers in the region of Marmoucher in the Middle Atlas mountains and would have been used by a Berber man, probably a shepherd, for protection against the biting winter cold. These capes are known as "jilaba" and there is good information with colour photos in "Splendeurs du Maroc" pages 155 and 197 and also "North African Textiles" on page 86. This example has very fine and detailed design incorporated in the weave. Reference 2605. Size 160cms. high x 95cms. opened at bottom. 5ft.8in. x 3ft.2in. DETAIL OF HOOD.
MOROCCAN BERBER HIGH ATLAS BAG. Sometimes these bags, known as "aalau" are woven in pairs and used as a gift from the bride's mother to her daughter and son-in-law. It is the work of the Ait Bou Ichaouen Berbers of the far east High Atlas. It is old and in great condition with original tassels, overcasting and handle. It was probably subsequently used to store jewellery and precious items. The design is of repeated Berber fertility symbols. Reference 2646. Size 37 x 36cms. 1ft.2in.square. Euros 67.
MOROCCAN BERBER HIGH ATLAS BAG. Sometimes these bags, known as "aalau" are woven in pairs and used as a gift from the bride's mother to her daughter and son-in-law. It is the work of the Ait Bou Ichaouen Berbers of the far east High Atlas. It is old and in great condition with original tassels, overcasting and handle. It was probably subsequently used to store jewellery and precious items. The design is of repeated Berber fertility symbols. Reference 2646. Size 37 x 36cms. 1ft.2in.square. CLOSE UP.
BALOUCH / BELOUCH BAG FACES. A rare pair from the region of Firdous which lies in the desert region of Khorrasan in north east Persia. Saddle bags from this Belouch group are particularly rare. The kilim borders with embroidered designs of rows of goats are most unusual and I know of only one other piece with this feature which was shown in HALI Edition 94 on page 122. The wools are soft and lustrous. Reference 292A and 292B. Size 71 x 73cms. each. 2ft.4in. x 2ft.5in. each. Euros 595 the pair.
BALOUCH / BELOUCH BAG FACES. A rare pair from the region of Firdous which lies in the desert region of Khorrasan in north east Persia. Saddle bags from this Belouch group are particularly rare. The kilim borders with embroidered designs of rows of goats are most unusual and I know of only one other piece with this feature which was shown in HALI Edition 94 on page 122. The wools are soft and lustrous. Reference 292A and 292B. Size 71 x 73cms. each. 2ft.4in. x 2ft.5in. each. Euros 595 the pair. CLOSE UP showing row of goats which each bag has in the top kilim border.
RARE MOROCCAN BERBER SHOES dating from the first half of the 20th century and which are hand made from camel leather. The embroidery is hand stitched silk. They would have been used for ceremonial or important occasions. They were not for daily wear. They are from a Berber group in the Djebel Siroua region of the High Atlas pre-Sahara in southern Morocco. Two similar pairs can be seen in "Spendeurs of Maroc" published by Musee Royal de l'Afrique Centrale as plates 247 and 248 which are part of a private collection of rare Berber artifacts. Reference 2578. Size 25cms.10ins. long. Euros 265.
RARE MOROCCAN BERBER SHOES dating from the first half of the 20th century and which are hand made from camel leather. The embroidery is hand stitched silk. They would have been used for ceremonial or important occasions. They were not for daily wear. They are from a Berber group in the Djebel Siroua region of the High Atlas pre-Sahara in southern Morocco. Two similar pairs can be seen in "Spendeurs of Maroc" published by Musee Royal de l'Afrique Centrale as plates 247 and 248 which are part of a private collection of rare Berber artifacts.
MOROCCAN BERBER HOOD. The small size of the portion to cover the head, together with the delicate embroidered designs, suggest that the is a "Bonnet d�Enfant" for a young girl. The material is cream colored felt with designs and original tassels for fastening in fine cotton. It is quite old but in unblemished condition and may have been kept for ceremonial or special occasion use. Reference 2813. Size 62 x 60 cms., or 24 x 22 ins. Euros 58.
MOROCCAN BERBER HOOD. The small size of the portion to cover the head, together with the delicate embroidered designs, suggest that the is a "Bonnet d�Enfant" for a young girl. The material is cream colored felt with designs and original tassels for fastening in fine cotton. It is quite old but in unblemished condition and may have been kept for ceremonial or special occasion use. Reference 2813. Size 62 x 60 cms., or 24 x 22 ins. SIDE VIEW.
MOROCCAN BERBER HIGH ATLAS BAG / PILLOW from the Berber group Ait Sergrouchan in the eastern High Atlas mountains close to the Algerian border. This is a very old piece with fine embroidered design embellished with many metal sequins. Very fine white cotton has been used for emphasis. This work is very little known but there is a piled rug from this tribe which has some elements of these designs and which is shown as Plate 34 in "Maroc - Tapis de Tribus". Reference 2641. Size 66 x 30cms. 2ft.2in. x 1ft. Euros 105.
MOROCCAN BERBER HIGH ATLAS BAG / PILLOW from the Berber group Ait Sergrouchan in the eastern High Atlas mountains close to the Algerian border. This is a very old piece with fine embroidered design embellished with many metal sequins. Very fine white cotton has been used for emphasis. This work is very little known but there is a piled rug from this tribe which has some elements of these designs and which is shown as Plate 34 in "Maroc - Tapis de Tribus". Reference 2641. Size 66 x 30cms. 2ft.2in. x 1ft. REVERSE.
MOROCCAN BERBER MIDDLE ATLAS SADDLE BAG which is a very rare example from the Berber tribe Beni M'guild. The deep purple used for the foundation kilim and which is plied together with deep indigo for the original fastening ropes is often used in the tribe's piled carpets. The bag faces are embellished with multi-coloured woollen decorations and the same wools have been used for the extensive tassel work. A spontaneous and unique tribal artifact. Reference 2642. Size 100 x 37cms. 3ft.3in. x 1ft.3in. without tassels. Euros 365.
MOROCCAN BERBER MIDDLE ATLAS SADDLE BAG which is a very rare example from the Berber tribe Beni M'guild. The deep purple used for the foundation kilim and which is plied together with deep indigo for the original fastening ropes is often used in the tribe's piled carpets. The bag faces are embellished with multi-coloured woollen decorations and the same wools have been used for the extensive tassel work. A spontaneous and unique tribal artifact. Reference 2642. Size 100 x 37cms. 3ft.3in. x 1ft.3in. without tassels. CLOSE UP.
TUNISIAN BERBER EMBROIDERY which is small, very finely stitched and would have been used as a tray cover. The inner rows of flowers are depictions of long petalled tulips which have been borrowed from Ottoman designs. The background is very fine white linen. Reference 2815. Size 33 x 50 cms., or 13 x 20 ins. Euros 35.
TUNISIAN BERBER EMBROIDERY which is small, very finely stitched and would have been used as a tray cover. The inner rows of flowers are depictions of long petalled tulips which have been borrowed from Ottoman designs. The background is very fine white linen. Reference 2815. CLOSE UP.
BERBER HIGH ATLAS TENT BAG which is the very fine work of the little known Ait bou Ichaouen Berbers in the eastern High Atlas Mountains which lie close to the border wth Algeria. There are at least ten major variations of flat weave techniques used by this Berber tribe which is extraordinary considering the primiive nature of both geography and lifestyle which prevail. Fortunately the weaving of both flatweaves and piled rugs was researched and documented in the book "Ait bou Ichaouen" by the husband and wife team of Alfred and Susanne Saulniers. This fine bag was probably one of a pair and given as a dowry gift. The small narrow opening along the top edge is just big enough to allow a hand to put items inside for storage. The design on the front is a lattice of hand tied knots. Reference 2820. Size 13 x 29 inches. Euros 190.
BERBER HIGH ATLAS TENT BAG which is the very fine work of the little known Ait bou Ichaouen Berbers in the eastern High Atlas Mountains which lie close to the border wth Algeria. There are at least ten major variations of flat weave techniques used by this Berber tribe which is extraordinary considering the primiive nature of both geography and lifestyle which prevail. Fortunately the weaving of both flatweaves and piled rugs was researched and documented in the book "Ait bou Ichaouen" by the husband and wife team of Alfred and Susanne Saulniers. This fine bag was probably one of a pair and given as a dowry gift. The small narrow opening along the top edge is just big enough to allow a hand to put items inside for storage. The design on the front is a lattice of hand tied knots. Reference 2820. Size 13 x 29 inches. REVERSE.
MOROCCAN MIDDLE ATLAS DOUBLE SADDLE BAG from the region of Zemmour. PUBLISHED IN "ORIENTAL RUGS: AN ILLUSTRATED LEXICON OF MOTIFS, MATERIALS AND ORIGINS" by Peter Stone, page 316. The bag itself has a very fine embroidered design on both panels and has been further embellished by extensive use of metal sequins. See `Splendeurs du Maroc` - Musee Royal de l`Afrique Centrale, 1998, page 148. Reference 2012. Size 104 x 37cms, 3ft 5ins x 1ft 3ins, Euros 385.
THIS DOUBLE BAG IS PUBLISHED on page 316 of this very important book.
Vats of dye in the city of Fez.
This is where I buy wool for my restoration studio.
MOROCCAN BERBER MIDDLE ATLAS SADDLE BAG which is of exceptional fineness and was probably woven as a dowry piece. It has a plain, fine kilim background upon which design has been embroidered and many sequins added. There is a similar piece attributed to the early to mid 20th century from the region of Zaine in the Middle Atlas which is colour plate 58 in "Arts and Crafts of Morocco". Reference 2550. Size 84 x 34cms. 2ft.9in. x 1ft.2in. CLOSE UP.
MOROCCAN BERBER MIDDLE ATLAS SADDLE BAG which is of exceptional fineness and was probably woven as a dowry piece. It has a plain, fine kilim background upon which design has been embroidered and many sequins added. There is a similar piece attributed to the early to mid 20th century from the region of Zaine in the Middle Atlas which is colour plate 58 in "Arts and Crafts of Morocco". Reference 2550. Size 84 x 34cms. 2ft.9in. x 1ft.2in. Euros 190.
MOROCCAN BERBER HIGH ATLAS BAG FACE, which is an extremely rare example of work from the Berber of Ait bou Ichaouen, probably the Ait Hammou group. The weaving has significant religious designs and the owner may have been one of the men who appear in the two diamonds, the lower one of which contains the name "Abdullah". There are also the words for "Allah", "The King" and "Our Land". The wording used is in the Amazigh (Berber) language and one very old inhabitant of the village I found it in said it was in the Tifinagh dialect. The letter "H" and number "2" are also shown and these are both used in written Amazigh. Abdullah is well protected by the inclusion of a design representing a dagger to his right. A most fascinating, rare and important piece. Reference 2663. Size 84 x 45 cms., 2ft. 9ins. x 1ft. 6ins. CLOSE UP OF INSCRIPTIONS.
MOROCCAN BERBER HIGH ATLAS BAG FACE, which is an extremely rare example of work from the Berber of Ait bou Ichaouen, probably the Ait Hammou group. The weaving has significant religious designs and the owner may have been one of the men who appear in the two diamonds, the lower one of which contains the name "Abdullah". There are also the words for "Allah", "The King" and "Our Land". The wording used is in the Amazigh (Berber) language and one very old inhabitant of the village I found it in said it was in the Tifinagh dialect. The letter "H" and number "2" are also shown and these are both used in written Amazigh. Abdullah is well protected by the inclusion of a design representing a dagger to his right. A most fascinating, rare and important piece. Reference 2663. Size 84 x 45 cms., 2ft. 9ins. x 1ft. 6ins. Euros 675.
I checked every single one of the bags and trappings brought by this man to the Monday morning market, but could not find anything really great.
Monday morning rug market in the Middle Atlas. Rugs have been gathered in the villages for the past week and now the negotiations begin. I work here frequently.
MOROCCAN BERBER TENT BAG. Known as "Anoual", this type of piled bag acts as a cushion or pillow and is usually part of a dowry. It is an old example in remarkably good condition with all original fastening loops intact. The bag is from the Berbers of Ait bou Ichaouen in the eastern High Atlas close to the border with Algeria. There are entirely different designs on each side. One is extremely rare and depicts a flight of Potez 25 bi-planes used by the French colonial government to bomb the rebellious Ait Hammou Berbers in the early 1930s. The Berbers have long memories. The incident and designs which commemorate it are described in the excellent research book "Ait bou Ichaouen". The second side depicts a large "evil eye" motif which in Berber superstition protects the owner from bad luck. This is an important and rare example of its kind. Reference 2664. Size 42 x 88 cms., or 17 x 35 inches. Euros 188.


Also, put one off one's stroke. Interfere with one's progress, distract or disturb one, as in The interruption put her off her stride for a moment, and she took several seconds to resume her train of thought , or The noise of the airplanes overhead put her off her stroke, and she missed the next ball . The first term, first recorded in 1946, alludes to the regular pace of a walker or runner; the variant, first recorded in 1914, alludes to the regular strokes of a rower. Also see throw off the track.


Terrestrial, lithophytic or sometimes epiphytic plants. Rhizome long creeping; rhizome scales entire to subentire, brown. Lamina deltoid to braodly ovate, 2- to 4-pinnatifid, coriaceous, glabrous. Pinnae costae without raised edges on the upper surface; ultimate segments rhomboid and auricled, basal lower pinnae oblique; veins free. Sori round, dorsal or subterminal on veins; indusia peltate or orbicular-reniform, long-stalked, spores monolete.
Derivation of name: rumohra: named after Karl von Rumohr Holstein, German botanist of the 19th century.
Worldwide: c. 5 species, circum-austral distribution.
We have 1 taxa in the database for Rumohra.


See below for what's new. There's a separate section for past favorites including old "best of" lists dating back to 1996.
This page is dedicated to recent and new releases that have caught my ear recently. Since I try to avoid repeating particular releases on my regular program, I thought it best to set aside a separate page to emphasize notable releases and shows. Although the idea is to focus on what's happened over the past year, I have included some releases from the previous due to a number of factors, including the occasional vagueness of independent label distribution, slow word of mouth on non-distributed releases and my own never-ending sloth. All releases are available in the CD format unless otherwise stated (some may be available in analog formats as well). Recent additions are marked with a complicated system involving a variable amount of asterisks.
My section about forthcoming releases isn't really up to date, sorry!
News! (7/25/04)- I've finally had a chance to organize myself and wrap up 2003. See the "past favorites" link above to check out my favorites of 2003. In the meantime, the list below is all nice and tidy now (except for the forthcoming releases, of course).


text:
Q: Evaluate an expression, users can enter as many numbers and operators as they want This is the Programming Project 7.12 of the C Programming - A Modern Approach. The program is expected to evaluate an expression (for example 1+2.5*3) and return its result (10.5 in this case, note that the expression is evaluated from left to right, and no operator takes precedence over any other operator).
This is what I've tried (the expression is 1 + 2.5 *3, for example): store 1 in a, + in ch, and 2.5 in b. Then let result = a, compute the "new" result by either adding, subtracting, dividing or multiplying with b. Then continue reading the expression, storing * in ch, and 3 in b. Run the while loop again until the scanf detects a new-line character. I wonder what is wrong with my method, or my code.
#include <stdio.h>
int main(void)
{
    float a, b, result;
    char ch;
    printf("Enter an expression: ");
    scanf("%f%c%f", &a, &ch, &b);
    result = a;
    while (ch != '\n') {
        if (ch == '+')
            result += b;
        else if (ch == '-')
            result -= b;
        else if (ch == '*')
            result *= b;
        else if (ch == '/') {
            result /= b; }
        scanf("%c", &ch);
        scanf("%f", &b);
    }
    printf("Value of expression: %.2f", result);
    return 0;
}

It doesn't return anything :(

A: Even after you press enter after putting complete expression.
scanf("%c", &ch);
scanf("%f", &b);

Will still wait to input float because of order of scanf.
Just break immediately once you press enter as below.
while (ch != '\n') {
    if (ch == '+')
        result += b;
    else if (ch == '-')
        result -= b;
    else if (ch == '*')
        result *= b;
    else if (ch == '/') {
        result /= b; }
    scanf("%c", &ch);
    if (ch == '\n') break;
    scanf("%f", &b);
}

I would suggest you to use fgets to read the complete expression at
  once and process the expression afterwards also your current code is very naive wouldn't even work for complex expression which includes (, ) etc, you may want to use LIFO data structure by processing the read expression in reverse order.

redpajama_set_name: RedPajamaStackExchange


text:
Formula 1 , Opinion
Racing Drivers Can't Agree On Whether Verstappen Should Have Been Penalised
Posted on October 23, 2017 | by Dan Thorn
Max Verstappen's pass on Kimi Raikkonen was a definite highlight of the US GP, but when stewards realised he'd exceeded track limits to pull it off, he was swiftly given a penalty.
Was it the right thing to do? Should stewards have been more lenient in the name of entertainment? Several drivers (current and former) took to social media to say which side of the fence they're sitting on.
Not fair ! This was not a track limit abuse from @Max33Verstappen , this was just a mega move – nothing else ! #grrr
— alex wurz (@alex_wurz) October 22, 2017
.@AussieGrit live on @C4F1: "It's a shit decision" – no bleeper here! #USGP
— F1 Broadcasting (@f1broadcasting) October 22, 2017
Brilliant drive from @Max33Verstappen – such a shame to be penalised for that. So many people off track all the time. Makes no sense!
— Karun Chandhok (@karunchandhok) October 22, 2017
After watching videos could equally argue Kimi forced him there. Best overtake of #USGP. Feel sorry for @Max33Verstappen you got robbed @F1
— Mario Andretti (@MarioAndretti) October 22, 2017
Verstappen passing RAI was awesome! He cut the corner because RAI was turning into him and it looked like RAI would cause a crash #nopenalty
— Nico Rosberg (@nico_rosberg) October 23, 2017
There's an outside shot… he's off.
Stunning shot, from the edge of the 18 yard box hits the bar.
Crowd applaud.. doesn't make it a goal.
— Alex Brundle (@AlexBrundle) October 22, 2017
Not correct Alex, the problem is that all the drivers drive outside track limits which is not a skill. That's called track limit abuse.
— Johnny Herbert (@johnnyherbertf1) October 22, 2017
You know I hate it when track limits aren't enforced but they have always penalised going off to overtake. Mega move but not on track. Shame
— Tiff Needell (@tiff_tv) October 22, 2017
So, who's right? It's a really difficult situation. By the letter of the law, the rules say that drivers can't gain a lasting advantage by leaving the track. Verstappen definitely did that, but there are so many other occasions across the weekend where drivers left the circuit and weren't penalised.
Lewis Hamilton was off the track on his pole lap. Sebastian Vettel was off when he took the lead at turn one. Carlos Sainz ran wide on the exit of Turn 18 after getting past Sergio Perez. Valtteri Bottas used the run off several times when battling other drivers.
The FIA and the stewards seem to be stuck in this rut where they only see it as gaining an advantage where one driver is overtaking another, and doing so on the inside of the corner. To them, running wide on the exit of a corner or extending the track to maintain speed doesn't constitute 'gaining a lasting advantage'. But racing drivers are a clever bunch – if they didn't gain time from going off the track, they wouldn't be doing it.
Given that we'd spent all weekend seeing drivers go off the track, to see Verstappen suddenly get penalised for doing it is what created the shock. It's why fans are angry and it's why so many drivers are coming out in support of Max.
F1 has been making a huge effort this year to attract new fans and to penalise Verstappen for his awesome, last lap, crowd-pleasing pass in this scenario – regardless of how 'technically correct' the penalty is – goes against that.
Of course, in an ideal world, situations like this shouldn't even be allowed to happen, but modern circuit design is making it very, very difficult. Just because it's really hard to police it at a circuit like COTA, it doesn't mean they should be able to pick and choose when and where they dish out penalties.
I f*****g hate track limits, either enforce everywhere, or don't enforce anywhere. That's why I like old school American tracks..
— Marino Franchitti (@MFranchitti) October 22, 2017
So, was it right to penalise Max? As an individual incident, it's hard to argue against it. Allowing it to stick could have set a very bad precedent. But given the way things had gone across the weekend, and the negative reaction from drivers, team bosses and fans alike, could they not have allowed this one to stick, and then clarified things in the future? Because it's pretty clear that F1 needs to have a good hard think about how it deals with track limits – and it's been that way for a long, long time.
Now we hate to be overly biased here at WTF1, so we'll finish by sharing a tweet on the subject from a level-headed, neutral source who has no vested interests whatsoever in what the outcome was.
This is bulshitt. Sorry. This sucks
— Jos Verstappen (@MaVic009) October 22, 2017


text:
"""
Test log proxy and kblogging
"""
import logging
import os
import signal
import time
import unittest
from biokbase.narrative.common import log_proxy as proxy

__author__ = "Dan Gunter <dkgunter@lbl.gov>"

class MainTestCase(unittest.TestCase):
    # name of var lines up with cmdline arg 'dest', don't change
    conf = "/tmp/kbase_logforward.conf"
    vb = 0
    smpcfg = "sample"
    meta = None

    def setUp(self):
        self._config(["db: test", "collection: kblog"])
        if proxy.g_log is None:
            proxy.g_log = logging.getLogger(proxy.LOGGER_NAME)

    def _config(self, lines):
        text = "\n".join(lines)
        with open(self.conf, "w") as f:
            f.write(text)

    def test_run_proxy(self):
        pid = os.fork()
        if pid == 0:
            print("Run child")
            proxy.run(self)
        else:
            time.sleep(1)
            print("Wait for child to start")
            # let it start
            time.sleep(4)
            # send it a HUP to stop it
            print("Send child ({:d}) a HUP".format(pid))
            os.kill(pid, signal.SIGHUP)
            # wait for it to stop
            print("Wait for child ({:d}) to stop".format(pid))
            cpid, r = os.waitpid(pid, 0)
            print("cpid, r: {}, {}".format(cpid, r))
            self.assertTrue(r < 2, "Bad exit status ({:d}) from proxy".format(r))

    def test_configuration(self):
        # empty
        self._config([])
        self.assertRaises(ValueError, proxy.DBConfiguration, self.conf)
        # missing collection
        self._config(["db: test"])
        self.assertRaises(KeyError, proxy.DBConfiguration, self.conf)
        # bad db name
        self._config(["db: 1test", "collection: kblog"])
        self.assertRaises(ValueError, proxy.DBConfiguration, self.conf)
        # bad db name
        self._config(["db: test.er", "collection: kblog"])
        self.assertRaises(ValueError, proxy.DBConfiguration, self.conf)
        # too long
        self._config(
            [
                "db: aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa",
                "collection: bbbbbbbbbbbbbbbbbbcccccccccccccccccccccccc"
                "ddddddddddddddddddddddddddddddd",
            ]
        )
        self.assertRaises(ValueError, proxy.DBConfiguration, self.conf)
        # bad collection
        self._config(["db: test", "collection: kb$log"])
        self.assertRaises(ValueError, proxy.DBConfiguration, self.conf)
        # user, no pass
        self._config(["db: test", "collection: kblog", "user: joe"])
        self.assertRaises(KeyError, proxy.DBConfiguration, self.conf)

class LogRecordTest(unittest.TestCase):
    def setUp(self):
        if proxy.g_log is None:
            proxy.g_log = logging.getLogger(proxy.LOGGER_NAME)

    def test_basic(self):
        for input in {}, {"message": "hello"}:
            kbrec = proxy.DBRecord(input)
        kbrec = proxy.DBRecord({"message": "greeting;Hello=World"})
        self.assertEqual(kbrec.record["event"], "greeting")
        self.assertEqual(kbrec.record["Hello"], "World")

    def test_strict(self):
        for inp in (
            {"xanthium": 12},
            {12: "xanthium"},
            {"message": "Hello=World;greeting"},
        ):
            proxy.DBRecord(inp)
            self.assertRaises(ValueError, proxy.DBRecord, inp, strict=True)

if __name__ == "__main__":
    unittest.main()

redpajama_set_name: RedPajamaGithub


As you may know, I really enjoy birthday cake.
It doesn't have to be my birthday, or a holiday, or anything like that, I just like cake. My favorite kind is just a plain, simple, vanilla cake with chocolate frosting. And that is what I made. YUM YUM YUM YUM YUM.
I know I have posted a pretty similar recipe before, but this recipe is SO much better than that one. It's 10,000x moister, fluffier, and just altogether more delicious. Believe me, it took me a while to finally get this recipe together. I'm SUPER forgetful, so I used the wrong kind of oil and messed up the proportions a couple of times before I finally got it right.
Also thank you to Elana's Pantry for the delicious vegan frosting recipe... I had to eat a couple spoonfuls before it went on the cake.
Combine all wet ingredients in a small bowl.
Pour the wet mixture into the dry mixture and stir until combined.
Yummy yummy! I hope you enjoy your cake!


text:
Q: Make PNG Logo bigger Size has anyone an Idea, how I can resize our Logo?
It has a height of 52px but it only shows a height of 39px when I inspect it.
http://www.loccoz.com/

A: I inspected img tag of yours, there is a setting of max-height:65% for .navbar-brand-image > img. Increase the percentage to max-height:100% property. it will work.


Here is a fun robot for Ghost in the Shell fans that can be controlled from an iOS/Android device. The Tachikoma Robot can move around thanks to its feet and wheels. It has various postures and speech recognition/synthesis tech. Its battery lasts up to 6 hours.
The above video shows this robot in action. You can get yours here.


Sunland Bark and Topsoil has been building soils and selling quality landscape materials to homeowners, farmers and contractors in Skagit, Island, Whatcom and Snohomish Counties since 1974. Farmers depend on us for sawdust and shavings for their animals and fields. Landscape contractors and homeowners rely on us for timely delivery of soil, mulches, bark and rock.
We specialize in producing soil blends using local materials, such as sandy loam from Fidalgo Island, organic egg farm chicken compost, and aged bark mulch and sawdust from Whatcom County. We are able to produce weed free soils by using excavated sandy loam, instead of native topsoil. We use no yard waste compost in our soils. The mulches we sell are blended onsite using a variety of reclaimed materials for optimum nutrition and weed control.
We also have a constant supply of gravels, decorative rock and various bark sizes. Call or come by to see what we have to offer for your project. Our knowledgeable sales staff will be glad to help you.


Are you confused about the many types of Dusts available in the land of cake decorating? Petal Dust, Pearl Dust, Luster Dust, Sparkle Dust, Highlighter Dust and more..
This Thursdays 'Tip Thursday' on Facebook is about The Different Types of cake decorating Dusts. I do hope you find this useful.
Visit my Facebook page for more such 'Tip Thursday's.


text:
SportJunior Sport
sport, junior-sport, youth girls, wagga and district, competition, afl, wagga
The Wagga and District Youth Girls competition will get underway for a second season on Wednesday. A new competition structure will be used this year with under 12s, under 14s and under 16s competitions replacing last year's format of under 13s and 16s. There is also new clubs involved with teams from Temora-Cootamundra, Ganmain-Grong Grong-Matong and Collingullie-Glenfield Park-The Rock Yerong Creek being entered. The competitions get underway on Wednesday night and will run for six weeks, with the grand finals being held on September 26. Games will be held at Robertson Oval and Bolton Park from 4.30pm until 8pm. While you're with us, did you know that The Daily Advertiser is now offering sports and local events as part of its growing email newsletter service? Our sports email will keep the sports-mad up-to-date with a round-up of all the weekend's sports action - sign up here.
https://nnimgt-a.akamaihd.net/transform/v1/crop/frm/8tYDWUpBiaA8SfdG6xkddz/a9ce55d3-3419-4368-b949-510e098092eb.jpg/r0_122_3241_1953_w1200_h678_fmax.jpg
New teams and more age groups for this year's Wagga and District Youth Girls competition
Wagga Tigers' Harriet Priest and Turvey Park's Maggie Hallcroft compete in last year's Wagga Youth Girls competition.
The Wagga and District Youth Girls competition will get underway for a second season on Wednesday.
A new competition structure will be used this year with under 12s, under 14s and under 16s competitions replacing last year's format of under 13s and 16s.
There is also new clubs involved with teams from Temora-Cootamundra, Ganmain-Grong Grong-Matong and Collingullie-Glenfield Park-The Rock Yerong Creek being entered.
The competitions get underway on Wednesday night and will run for six weeks, with the grand finals being held on September 26.
Games will be held at Robertson Oval and Bolton Park from 4.30pm until 8pm.
While you're with us, did you know that The Daily Advertiser is now offering sports and local events as part of its growing email newsletter service? Our sports email will keep the sports-mad up-to-date with a round-up of all the weekend's sports action - sign up here.
Discuss "Wagga Youth Girls competition to go again"