Recall Internal Laminar Flows


My memory about the classic flow problems faded in the past several years. The notes here are my redemption.

Consider the steady-state, fully-developed, incompressible, viscous laminar flow between two infinite long parallel plates. \(x, y, z\) are the axes of the Cartesian coordinate system and all flow properties remain the same in the \(z\) direction. The flow direction is toward \(+x\). Let \(u, v\) be the velocity in the \(x, y\) directions, respectively, \(p\) the pressure, and \(\tau_{\xi\eta}\) the stress. Let the lower plate be at \(y = 0\) and the upper plate be at \(y = a\), where \(a\) is a given constant.

Consider the \(x\) component of the momentum equation on a infinitesimal square control volume that’s \(\mathrm{d} x\) wide and \(\mathrm{d} y\) high:

\[\begin{split}& \left[p - \left(p + \frac{\partial p}{\partial x}\right)\right]\mathrm{d} y - \left[\tau_{yx} - \left(\tau_{yx} + \frac{\partial \tau_{yx}}{\partial y} \mathrm{d} y \right) \right] \mathrm{d} x = 0 \\ \Rightarrow & \frac{\partial\tau_{yx}}{\partial y} = \frac{\partial p}{\partial x}\end{split}\]

Assume Newtonian fluid:

\[\tau_{yx} = \mu\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right)\]

where \(\mu\) is the dynamic viscosity coefficient. Because the flow is fully developed, the vertical velocity \(v\) is void and thus \(\partial v / \partial x = 0\):

\[\tau_{yx} = \mu\frac{\partial u}{\partial y}\]

Then we obtain the following equation for \(u\):

\[\mu\frac{\partial^2 u}{\partial y^2} = \frac{\partial p}{\partial x}\]

Let \(\partial p / \partial x\) be constant (this can be justified) and apply the conditions of \(u(0) = u(a) = 0\):

\[u(y) = \frac{1}{2\mu}\frac{\partial p}{\partial x} y \left(y-a\right) = - \frac{a^2}{8\mu}\frac{\partial p}{\partial x} + \frac{1}{2\mu}\frac{\partial p}{\partial x} \left(y - \frac{a}{2}\right)^2\]

For \(\partial p / \partial x < 0\), \(u \ge 0\) and the peak velocity is \(u(y = a/2) = -(a^2/8\mu)\partial p / \partial x\).