# Layered flow with viscosity contrast *This section was contributed by Cedric Thieulot.* The idea behind this benchmark is to construct an analytical solution to the incompressible Stokes equation in the case where the viscosity field showcases a viscosity contrast at location $y=y_0$ whose amplitude and width can be controlled. The viscosity is defined as {math} \eta(y)=\frac{1}{\frac{1}{\pi} \tan^{-1} (\frac{y-y_0}{\beta} ) + 1/2 + \epsilon}  where $\beta$ and $\epsilon$ are chosen by the user. Viscosity profiles for different values of $\beta$ and $\epsilon$ are shown in {numref}fig:layeredflow1 and {numref}fig:layeredflow2. The set up of this benchmark allows testing how discretizations deal with abrupt changes in the viscosity (if $\beta$ is small) as well as large changes in the viscosity (if $\epsilon$ is small). {figure-md} fig:layeredflow1 Layered flow benchmark: Viscosity profiles for various $\beta$ values, using $y_0 = 1/3$ and $\epsilon = 0.0$.  {figure-md} fig:layeredflow2 Layered flow benchmark: Viscosity profiles for various $\beta$ values, using $y_0 = 1/3$ and $\epsilon = 0.2$.  The flow is assumed to take place in an infinitely long domain (in the horizontal direction) and bounded by $y=-1$ and $y+1$. At the bottom we impose $v_x(y=-1)=0$, while we impose $v_x(y=+1)=1$ at the top. The density is set to 1 while the gravity is set to zero. Under these assumptions, the flow velocity and pressure fields are given by: {math} \begin{aligned} v_x(x,y)&=&\frac{1}{2\pi} \left( -\beta C_1 \log [\beta^2 + (z-z_0)^2] + 2 (z-z_0) C_1 \tan^{-1} \frac{z-z_0}{\beta} + \pi (1+2\epsilon) z C_1 + C_2 \right), \nonumber\\ v_y(x,y) &=& 0, \nonumber\\ p(x,y) &=& 0, \end{aligned}  where $C_1$ and $C_2$ are integration constants: {math} \begin{aligned} C_1 &=& 2\pi \Bigl[ \beta \log [\beta^2 + (1+z_0)^2] - 2(1+z_0) \tan^{-1} \frac{1+z_0}{\beta} \\ &&\qquad\qquad -\beta \log [\beta^2 + (1-z_0)^2] + 2(1-z_0) \tan^{-1} \frac{1-z_0}{\beta} + 2\pi (1+2\epsilon) \Bigr]^{-1},\\ C_2 &=& \left[ \beta \log [\beta^2 + (1+z_0)^2] - 2(1+z_0) \tan^{-1} \frac{1+z_0}{\beta} + \pi(1+2\epsilon) \right]C_1. \end{aligned}  The viscosity and velocity fields are shown in {numref}fig:layeredflow3 and {numref}fig:layeredflow4 for $\beta=0.01$ and $\epsilon=0.05$. {figure-md} fig:layeredflow3 Velocity fields for $\beta=0.01$ and $\epsilon=0.05$ at uniform level 8 resolution, using $y_0=1/3$.  {figure-md} fig:layeredflow4 Viscosity fields for $\beta=0.01$ and $\epsilon=0.05$ at uniform level 8 resolution, using $y_0=1/3$.  [2]: #fig:layeredflow1 [4]: #fig:layeredflow2