Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 81 / 4

Measure $z$ from the rigid tube wall towards the air core, so $0<z<h$. To leading order in $h_0/a$ and small interface slope, the curvature entering the Young–Laplace equation isThe air pressure is spatially uniform at the order retained. Thus the linear liquid pressure perturbation associated with thickness amplitude $H$ is $p_1=-{\sigma }_0(a^{-2}-k^2)H$. The circumferential curvature favors growth at long wavelength; the curvature opposes it at short wavelength.

The lubrication theory momentum equation is $\mu u_{zz}=p_x$. With the wall no-slip boundary condition $u(0)=0$ and zero surface shear stress, $u_z(h)=0$, its solution and flux areLeading thin-film mass conservation is $h_t+q_x=0$. For a mode $e^{\beta t+ikx}$, this gives the capillary instability of an annular liquid liningTherefore the unstable band is $\boxed{0<k^2<a^{-2}}$. Maximization with respect to $k^2$ gives $k_{\ast }^2=1/(2a^2)$ andThese curvature coefficients use $a$ instead of $a-h_0$ consistently at leading order in the thin-lining approximation.

Let the positive undisturbed surface tension be $\bar{\sigma }={\sigma }_0-A{\Gamma }_0>0$. An insoluble surfactant perturbation $g$ produces ${\sigma }_1=-Ag$. The tangential Marangoni stress changes the surface condition to $\mu u_z(h)={\sigma }_x$. Solving the lubrication theory momentum equation now givesAt leading order, conservation of insoluble surfactant on a moving interface reduces to ${\Gamma }_t+(\Gamma u_s{)}_x=0$. Corrections from the changing circumference are smaller by $h_0/a$. Likewise the normal-pressure contribution $-{\sigma }_1/a$ gives a coupling smaller by $h_0/a$ than the retained tangential Marangoni stress, so it is omitted at the same order.

For thickness and concentration amplitudes $H,g$ proportional to $e^{\beta t+ikx}$, put $D=a^{-2}-k^2$ and use $p_1=-\bar{\sigma }DH$. The two conservation equations becomeSet $s=k^2{a}^2$, $\lambda =h_0^3\bar{\sigma }/(3\mu a^4)$ and $\alpha =h_{0}A{\Gamma }_0/(\mu a^2)$. The uncoupled capillary growth and elastic relaxation rates are $L=\lambda s(1-s)$ and $S=\alpha s$. The off-diagonal product is $3LS/4$, so the determinant is $(\beta -L)(\beta +S)+3LS/4$. Thus the surfactant stabilization of an annular liquid lining relation isThis proves the dispersion relation before its lettered deductions. The physical assumption $\bar{\sigma }>0$ is needed for the claimed capillary instability band.

For positive $\lambda ,\alpha$ and $0<s<1$, the constant term of the quadratic equation is negative. Its roots are therefore real with opposite signs, so one has positive growth. At $s>1$, both the coefficient of $\beta$ and the constant term are positive. If the roots are real they are both negative; if complex they have negative real parts. Thus the growing band remains $\boxed{0<k^2{a}^2<1}$. At $s=0$ or $1$ there is a neutral root. If $\alpha =0$, the result reduces continuously to the clean-film capillary branch with a passive zero root.

For $\alpha /\lambda \ll 1$, follow the unstable root near the clean-film value $L=\lambda s(1-s)$. Write $\beta =L+\Delta$ in $(\beta -L)(\beta +S)+3LS/4=0$. At first order in $S$, $L\Delta +3LS/4=0$, givingThis expansion is uniform near the maximizing wavenumber. Differentiation gives $s_{\ast }=1/2-3\alpha /(8\lambda )+O((\alpha /\lambda {)}^2)$. Evaluating the growth there yieldsThe change in maximizing wavenumber only affects the maximum at second order. The first-order decrease therefore has exactly the requested coefficient, relative to the clean-film rate with its surface tension replaced by $\bar{\sigma }$.

The quarter-rate limit is the strong-elasticity limit $\lambda /\alpha \ll 1$, not the weak-elasticity limit. The original PDF repeats $\alpha /\lambda \ll 1$ here; that is inconsistent with the preceding small-$\alpha$ result. In fact at $\alpha =0$ the maximum is $\lambda /4$, directly contradicting a quarter of that value in the same limit. The converted TeX repairs this particular ratio.

For $\alpha \gg \lambda$ and wavenumbers near the maximum, $S=\alpha s\gg L$. In the quadratic equation, the growing root is $O(\lambda )$, so its square and the $-L\beta$ term are small compared with $S\beta$. The leading balance is $S\beta -LS/4=0$. ConsequentlyThe second root is a rapidly damped surfactant-relaxation mode. Thus the stated quarter-rate result is correct after reversing the printed small-parameter ratio.

Without surfactant, capillary pressure drives a surface flow towards a growing bulge. Advection compresses the insoluble surfactant in converging parts of that motion, increasing concentration and lowering surface tension. The resulting gradient supplies a tangential Marangoni stress that opposes the surface motion. This additional resistance slows capillary-driven redistribution of liquid even after the base-tension decrease has already been accounted for.

In the strong-elasticity limit, concentration gradients suppress tangential interface motion. The surface behaves as effectively immobile at leading order. Setting $u_s\approx 0$ in the velocity formula gives ${\sigma }_x\approx h{p}_x/2$, henceone-quarter of the clean shear-free mobility $h^3/(3\mu )$. This mobility reduction explains the quarter-rate limit; it slows the unstable capillary mode without removing the long-wave unstable band.