## Q. 3.3

Flow Between Parallel Plates Consider viscous fluid flow between horizontal or tilted plates a small constant distance h apart. For the test section 0 ≤ x ≤ l of interest, the flow is fully-developed and could be driven by friction when the upper plate is moving at $\rm u_0 = ¢$, or by a constant pressure gradient, dp/dx, and/or by gravity (see Couette flow, Example 2.8)

 Sketch Assumptions Concepts • Steady laminar unidirectional flow • Reduced N-S equation where $\vec {\bf v}$ = (u, 0, 0) and $-\frac{\partial \rm p}{\partial\rm x} \approx \frac{\Delta \rm p}{\rm l} =¢$ • Constant h, ∂p / ∂x, and fluid properties • u = u(y) only

## Verified Solution

Although this problem looks like a simple case of a moving plate on top of a flowing film, the situation is more interesting due to the additional pressure gradient, $\frac{\partial p}{\partial x} \gtreqqless 0$ and the boundary condition u(y = h) = $\rm u_0=¢$. With the postulates: v = w = 0, u = u(y) only, $-\frac{\partial p}{\partial x} \approx \frac{\Delta \rm p}{\rm l} =¢$, and $\rm f _{body} = ρg \sin \phi$, continuity is fulfilled and the x-momentum equation (see App. A, Equation Sheet) reduces to:

$0=\frac{1}{\rho }\left\lgroup\frac{\Delta{\rm p}}{l}\right\rgroup +\nu\,\frac{\mathrm{d}^{2}{\rm u}}{\mathrm{d}{\rm y}^{2}}+\rm g\sin\phi$                  (E.3.3.1a)

or

${\frac{\mathrm{d}^{2}\mathrm{u}}{\mathrm{dy}^{2}}}={\frac{-1}{\mathrm{\mu}}}\left[\left\lgroup{\frac{\Delta\mathrm{p}}{l}}\right\rgroup +\mathrm{\rho g}\sin\ \phi\right]\ =¢$                      (E.3.3.1b)

subject to u(y = 0) = 0 and u(y = h) = $\rm u_0$ . Again, we have to solve an ODE of the form u′′ = K . Introducing a dimensionless pressure gradient

${\mathrm{P}}\equiv-{\frac{{\mathrm{h}}^{2}}{2\mu{\mathrm{\mu}}_{0}}}\left[\left\lgroup{\frac{\Delta{\mathrm{P}}}{l}}\right\rgroup +{\mathrm{pg}}\mathrm{}\,\,\sin\phi\right]$                    (E.3.3.2)

we can write the solution u(y), known as Couette flow, in a more compact form, i.e.,

$\frac{\mathrm{u(y)}}{\mathrm{u}_{0}}=\frac{\mathrm{y}}{\mathrm{h}}-\mathrm{P}\left\lgroup\frac{\mathrm{y}}{\mathrm{h}}\right\rgroup \left\lgroup1-\frac{\mathrm{y}}{\mathrm{h}}\right\rgroup$                          (E.3.3.3)

Graph ($\phi$ = 0):

• The pressure gradient, P or $\frac{\partial \rm p}{\partial\rm x}$, greatly influences u(y). Clearly, $\frac{\partial \rm p}{\partial\rm x}$ < 0 implies a “favorable” and $\frac{\partial \rm p}{\partial\rm x}$ > 0 an “adverse” pressure gradient
• For $\rm u_0=0$ , we have flow between parallel plates, i.e.,
$\mathrm{u}({\rm y})=+\frac{{\rm h}^{2}}{2\mathrm{\mu}}\Bigg[\left\lgroup\frac{\Delta{\rm p}}{ I}\right\rgroup +\rho{\rm g}\sin{\phi}\Bigg]\Bigg[\frac{{\rm y}}{\rm h}-\left\lgroup\frac{{\rm y}}{{\rm h}}\right\rgroup^{2}\Bigg]$                      (E.3.3.4)