Question

Consider steady, incompressible, laminar flow of a Newtonian fluid in the narrow gap between two infinite parallel plates (Figure \(3(a))\). The top plate is moving at speed \(V\), and the bottom plate is stationary. The distance between these two plates is \(h .\) The gravity acts in the \(z\) direction so the flow can be considered essentially two dimensional in \(x-y\) plane. Assume there acts an applied pressure gradient in the \(x\) -direction with its gradient given by, \(\partial P / \partial x=\) \(\left(P_{2}-P_{1}\right) /\left(x_{2}-x_{1}\right)=\) constant, where \(x_{1}\) and \(x_{2}\) are two arbitrary locations along the \(x\) -axis, and \(P_{1}\) and \(P_{2}\) are the pressures at those two locations.

(i) Prove that the streamwise velocity field \(u\) is given by the following expression.

$$ u=\frac{V y}{h}+\frac{1}{2 \mu} \frac{\partial P}{\partial x}\left(y^{2}-h y\right) $$

(ii) An engineer claims that the, as the gap between walls continues to widen, the applied pressure gradient will eventually determines the drag force the two walls experience. Is his/her claim true? Justify your answer with reason(s).

Answer 1