I have a physical problem which involves a groove of width $w$ and height $h$ inscribed at the bottom of a flow channel. Inside the groove is fluid A with viscosity $\mu_0$, and outside the groove is an external immiscible fluid B which causes a shear stress, and subsequently tears fluid A away from the groove. This causes the liquid-liquid interface to curve and deflect inward towards the groove's bottom, due to a pressure drop.
The streamwise velocity in the channel satisfies the Poisson equation:
$\mu_0 \nabla^2u=\frac{dp}{dx}$
The velocuty field $u$ is a result of two contributions: a shear-driven contribution, $u_s$, and a pressure-driven contribution, $u_p$. Since the Poisson equation is linear, we can write:
$\mu_0 \nabla^2 (u_s + u_p) = \frac{dp}{dx}$
Now I want to determine the shear-driven flow field, $u_s$, integrate over the cross section of the groove and obtain the shear-driven flux, $q_s$. In order to do this, I can set the pressure gradient to zero (which result in the Laplace equation, as $\frac{dp}{dx}=0$), and prescribe a given shear stress $\tau$ at the top interface, $\mu_0 \frac{du}{dy} = \tau$, and the no slip condition at the walls and floors of the groove, $u(0,z) = u(y,0) = u(y,w) = 0$.
Similarly, I want to find the pressure-driven flow field, $u_p$, so I can consider the pressure gradient $\frac{dp}{dx}$ as the source term, with no shear, $\mu_0 \frac{du}{dy} = 0$. Again, the no slip condition is applied at the walls and floor of the groove, $u(0,z) = u(y,0) = u(y,w) = 0$, and the result is integrated over the cross section to obtain the pressure-driven flux. I have the result of these terms, and I need some help in the process of acquiring this result.
The shear-driven flux:
$q_s = c_s \frac{\tau w h^2}{\mu_0}$
With:
$c_s = \frac{1}{2}-\frac{4h}{w}\sum_{n=0}^{\infty} \frac{(-1)^n}{\lambda_n^4}tanh\left(\frac{\lambda_n w}{2h}\right)$
The pressure-driven flux:
$q_p = -c_p \frac{wh^3}{\mu_0} \frac{dp}{dx}$
With:
$c_p = \frac{1}{3}-\frac{4h}{w}\sum_{n=0}^{\infty} \frac{1}{\lambda_n^5}tanh\left(\frac{\lambda_n w}{2h}\right)$
Where $\lambda_n = \left(n+\frac{1}{2}\right) \pi$
The coefficients $c_s$ and $c_p$ represent the effect of the aspect ratio of the groove, $\frac{w}{h}$, on the total flow, and result from the integration of the Fourier series velocity solution over the cross-sectional area.
Is there anyone here who can help me acquire this solution?
Particularly I want to know how to acquire that $tanh$ term.
Any help would be extremely helpful!