PYQs

Consider steady, incompressible, laminar flow between two infinite parallel horizontal plates as shown in the figure.

The flow is in the \(x\) - direction, hence there is no velocity component in either the y- or z- direction (i.e., \(v=0\) and \(w\) \(=0\) ). The steady-state continuity. equation becomes \(\frac{\partial u}{\partial x}=0 ....(1)\)

From Eqn. 1 , it can be concluded that the velocity u is a function of both \(y\) and \(z\) only. Since the plates are infinitely wide, it can be argued that the velocity u should not be a function of \(z\), i.e., it must be a function of y only, \(\mathrm{u}=\mathrm{u}(\mathrm{y})\). Applying the Navier-Stokes equations, along with the assumptions that \(v=0, w=0\) and \(\mathrm{u}=\mathrm{u}(\mathrm{y}),\) yields \(\frac{\partial p}{\partial x}=\mu \frac{d^{2} u}{d y^{2}}\) \(\frac{\partial p}{\partial y}=-p g\) \(\frac{\partial p}{\partial z}=0\) [4] Eqn. 4 indicates that the pressure is a function of \(x\) and \(y\). Integrate Eqn. 3 to yield Eqn. 4 indicates that the pressure is a function of \(x\) and \(y\). Integrate Eqn. 3 to yield

\(p=-p g y+g_{1}(x)\) Hence it can be concluded that \(\frac{\partial \mathrm{p}}{\partial}\) is a function of \(x\) only. Now, integrate Eqn. 2 twice with respect to \(\mathrm{y}\), and treat \(\frac{\partial \mathrm{p}}{\partial \mathrm{x}}\) as a constant (with respect to y) to give: \(u=\frac{1}{2 \mu} \frac{\partial p}{\partial x} y^{2}+c_{1} y+c_{2}\) Applying the no-slip conditions (i.e., the fluid is “stuck” to the plates, or \(\mathrm{u}=0\) at \(\mathrm{y}=\pm \mathrm{h}\) ) to determine the coefficients as follows: \(c_{1}=0\) and \(c_{2}=-\frac{h^{2}}{2 \mu} \frac{\partial p}{\partial x}\) The velocity profile becomes \(u=\frac{1}{2 \mu} \frac{\partial p}{\partial x}\left(y^{2}-h^{2}\right)\) which is a parabola.

The total volumetric flow per linear depth can be obtained by integrating the velocity to give \(q=\int_{-h}^{h} u d y=-\frac{2 h^{3}}{3 \mu}\left(\frac{\partial p}{\partial x}\right)\) Note, \(q\) is per linear depth, which is different than \(Q\) which is the total volumetric flow rate. Also note that the flow is negative, i.e. to the left, for a positive pressure gradient, \(\mathrm{dp} / \mathrm{dx}\). This is due to the gradient definition where decreasing pressure to the right is negative.