Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 37E (Fluid Dynamics II)

Lubrication theory exploits a film thickness $b$ much smaller than its horizontal variation scale $\ell$. Incompressibility then gives vertical velocity of order $(b/\ell )U$, vertical viscous derivatives dominate horizontal ones, and inertia is negligible when $\rho U{b}^2/(\mu \ell )\ll 1$. The leading equations reduce to $\mu u_{yy}=p_x$ and vertical hydrostatic balance $p_y=-\rho g$. Boundary conditions determine the local horizontal velocity profile; integrating it across the film and enforcing mass conservation produces an equation for flux or surface shape. This reduction turns the thin-film problem into a one-dimensional pressure problem while retaining the dominant viscous stresses.

Here $\ell \sim \sqrt{ab}$, so the geometric condition is $b/a\ll 1$. Let $U$ be the signed roller velocity near its top, with magnitude $a|\Omega |$; choose the rotation sense making $U=a\Omega$ positive in the $x$ direction. The small-inertia condition iswith $|U|$ understood for a reversed rotation. The roller's slope and change of tangential speed are small in the gap region. The liquid is Newtonian, the roller imposes no slip and the free surface is shear-free; surface tension is neglected as in the gravity-only model.

For the leading flat film geometry, use $y=-d(x)$ as the roller boundary and $y=0$ as the free surface. The lubrication approximation gives $\mu u_{yy}=p_x$, $u(-d)=U$, $u_y(0)=0$, henceThe corresponding vertical velocity is determined by incompressibility, for example $v(x,y)={\int }_y^0{u}_x(x,s)ds$, so that $v(0)=0$ and $v(-d)=-U{d}^{\prime }$ by differentiating the constant-flux equation. This is the leading tangency condition at the roller. Steadiness makes $Q$ independent of $x$.

For the rotating roller under a nearly flat free surface, the excess pressure tends to zero in the outer fluid on both sides of the gap. ConsequentlySet $x=\sqrt{2ab}s$. The supplied recurrence gives $I_2=\pi /2$ and $I_3=3\pi /8$, soThe extension of the parabolic gap integrals to infinity is the usual matched thin-gap approximation; their dominant contribution comes from $x=O(\sqrt{ab})\ll a$. Taking the surface flat refers to this leading geometry, not setting its small hydrostatic displacement identically to zero in the pressure balance. An exactly flat stress-free surface at exactly constant pressure could not sustain this variable pressure gradient; the displacement needed to do so is checked in part (ii).

Vertical hydrostatic balance with atmospheric pressure at $y=h(x)$ gives $p(x,y)=p_0+\rho g(h(x)-y)$ to this order. Therefore $p_x=\rho g{h}^{\prime }$ and integrating the gradient from part (i), with $h\to 0$ on both sides, givesFor positive $U$ the upstream surface rises and the downstream surface falls. Thus the displacement scale is $\mu |U|\sqrt{a}/(\rho g{b}^{3/2})$, and the flat-geometry consistency condition isIndeed ${\max }_s|s/(1+s^2{)}^2|=9/(16\sqrt{3})$, so this scale controls the maximum, not only a typical value. Together with $b/a\ll 1$ and $\rho |\Omega |\sqrt{a}{b}^{3/2}/\mu \ll 1$, these conditions ensure a thin, weakly displaced, viscosity-dominated gap. The slope is smaller still, of order $(|h|/b)\sqrt{b/a}$.