Problem 2

(1) A thin layer of viscous fluid of thickness \(h(\theta)\) (shown shaded, and with exaggerated thickness, in the diagram) covers the outer surface of an infinitely long cylinder whose circumference rotates at a speed $U$ about a horizontal axis through the centre of the cylinder, in the presence of a gravitational acceleration \(g\). Show that under the lubrication approximation, the fluid velocity satisfies the equation \[ \nu \frac{\partial^2 u}{\partial z^2} = g \cos(\theta) \]

Problem2

where \(z\) is the distance from the surface of the cylinder in the normal direction, \(u(\theta, z)\) is the velocity of the fluid in the \(\theta\) direction, and \(\nu\) is the kinematic viscosity. [hint: think in terms of velocity components parallel and normal to the film. You do not need N-S equations in cylindrical coordinates for this problem.]

[7]

(2) Write down the boundary conditions at \(z=0\) and \(z=h(\theta)\) and show that \[ u(\theta, z) = U + \left( \frac{z^2}{2} - zh \right) \frac{g}{\nu} \cos(\theta) \] [3]


(3) Obtain an expression for the volume flux \[ Q(\theta) = \int_0^{h(\theta)} u \, dz \] across any section (with \(\theta = \text{constant}\)). Hence show that \[ \frac{1}{H^2} - \frac{1}{H^3} = \alpha \cos(\theta), \] where \[ H(\theta) = Uh(\theta)/Q, \qquad \alpha = \frac{gQ^2}{3\nu U^3}. \] Explain why \(Q\) is constant. [7]


(4) Find a perturbative solution for \(h(\theta)\) assuming \(\alpha \ll 1\), to first order in \(\alpha\). [hint: try a solution of the form \(H = A_0 + A_1[\alpha]\). [3]


(5) Show that a solution \(H(\theta)\) can exist for all values of \(\theta\) only if \(\alpha \leq 4/27\). [hint: it is easier to find the answer graphically]}. Show that the maximum film thickness the system can stably accommodate is \( h_{\text{max}} = \frac{3Q}{2H} \) Explain the physical mechanism behind this phenomenon. [5]