Berman flow

In fluid dynamics, Berman flow is a steady flow created inside a rectangular channel with two equally porous walls. The concept is named after a scientist Abraham S. Berman who formulated the problem in 1953.

Flow description
Consider a rectangular channel of width much longer than the height. Let the distance between the top and bottom wall be $$2h$$ and choose the coordinates such that $$x=0, \ y=0$$ lies in the midway between the two walls, with $$y$$ points perpendicular to the planes. Let both walls be porous with equal velocity $$V$$. Then the continuity equation and Navier–Stokes equations for incompressible fluid become


 * $$\begin{align}

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} &=0 \\ u\frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} & = - \frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right), \\ u\frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} & = - \frac{1}{\rho} \frac{\partial p}{\partial y} + \nu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2}\right) \end{align}$$

with boundary conditions


 * $$u(x,\pm h) = 0, \quad \left(\frac{\partial u}{\partial y}\right)_{y=0}=0, \quad v(x,0)=0, \quad v(x,\pm h) = V$$

The boundary conditions at the center is due to symmetry. Since the solution is symmetric above the plane $$y=0$$, it is enough to describe only half of the flow, say for $$y>0$$. If we look for $$v$$ a solution, that is independent of $$x$$, the continuity equation dictates that the horizontal velocity $$u$$ can at most be a linear function of $$x$$. Therefore, Berman introduced the following form,


 * $$\eta = \frac{y}{h}, \quad \psi(x,\eta) = [h\bar u_o-xV]f(\eta), \quad u = \left(\bar u_o - \frac{Vx}{h}\right)f'(\eta), \quad v=V f(\eta)$$

where $$\bar u_0$$ is the average value (averaged cross-sectionally) of $$u$$ at $$x=0$$, that is to say


 * $$\bar u_0 = \frac{1}{2}\int_{-1}^{1} u(0,\eta) d\eta=\frac{\bar u_0}{2} \int_{-1}^1f'(\eta) d\eta=\frac{\bar u_0}{2} [f(1)-f(-1)] = \bar u_0$$

This constant will be eliminated out of the problem and will have no influence on the solution. Substituting this into the momentum equation leads to


 * $$\begin{align}

- \frac{1}{\rho} \frac{\partial p}{\partial x} &= \left(\bar u_o - \frac{Vx}{h}\right) \left(-\frac{V}{h}[f'^2 - ff]- \frac{\nu}{h^2}f' \right), \\ - \frac{1}{\rho} \frac{\partial p}{\partial \eta} &= \nu \frac{dv}{d\eta} - \frac{\nu}{h} \frac{d^2 v}{d\eta^2} \end{align}$$



Differentiating the second equation with respect to $$x$$ gives $$\partial^2 p/\partial x\partial\eta =0$$ this can substituted into the first equation after taking the derivative with respect to $$\eta$$ which leads to


 * $$ f^{iv} + \operatorname{Re} (f'^2 - ff'')'=0$$

where $$\operatorname{Re} = Vh/\nu$$ is the Reynolds number. Integrating once, we get


 * $$f' + \operatorname{Re} (f'^2 - ff) = C$$

with boundary conditions


 * $$f(0)=f''(0)= f(1)-1=f'(1)=0$$

This third order nonlinear ordinary differential equation requires three boundary condition and the fourth boundary condition is to determine the constant $$C$$. and this equation is found to possess multiple solutions. The figure shows the numerical solution for low Reynolds number, solving the equation for large Reynolds number is not a trivial computation.

Limiting solutions
In the limit $$Re\rightarrow 0$$, the solution can be written as


 * $$f(\eta) = \frac{1}{2}(3\eta-\eta^3) + \frac{Re}{280}(3\eta^3-2\eta-\eta^7) + O(Re^2).$$

In the limit $$Re\rightarrow -\infty$$, the leading-order solution is given by


 * $$f(\eta) = \sin \frac{\pi}{2}\eta.$$

The above solution satisfies all the necessary boundary conditions even though Reynolds number is infinite (see also Taylor–Culick flow)

Axisymmetric case
The corresponding problem in porous pipe flows was addressed by S. W. Yuan and A. Finkelstein in 1955.