Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 18C (Fluid Dynamics)

Use the standard ideal-fluid assumptions of incompressible flow, inviscid flow, gravity $-g{\mathbf{e}}_y$, and no surface tension. Irrotational flow gives ${\mathbf{u}}_j=\nabla {\phi }_j$ in each layer, and incompressibility gives Laplace equationThe rigid-wall boundary conditions are ${\phi }_{1y}=0$ at $y=h_1$ and ${\phi }_{2y}=0$ at $y=-h_2$. At the moving interface, each fluid has the same normal velocity as the interface:The dynamic condition is continuity of pressure, $p_1=p_2$ at $y=\eta$. Bernoulli equation in each fluid reads ${\phi }_{jt}+\frac{1}{2}|\nabla {\phi }_j{|}^2+p_j/{\rho }_j+gy=C_j(t)$. Choosing the potential gauges to subtract a common reference pressure, the interface condition becomesTangential velocities need not agree across an inviscid interface.

Linearize about the flat interface at rest, $\eta =0$, ${\phi }_j=0$. Both potentials still satisfy Laplace equation, now on the fixed strips $0<y<h_1$ and $-h_2<y<0$, with the same rigid-wall conditions. Discarding products of perturbations and evaluating the interface conditions at $y=0$ givesFor a normal mode proportional to $e^{i(kx-\omega t)}$, the vertical profiles satisfy ${\Phi }_j^{\prime \prime }-k^2{\Phi }_j=0$, the interface velocity is $-i\omega \eta$, and the wall-normal derivatives vanish. Smallness of the amplitude relative to wavelength and layer depths justifies this linearization.

Put $\kappa =|k|>0$. Profiles satisfying the rigid-wall conditions areThe kinematic conditions at zero giveConsequently ${\phi }_{1t}(0)$ has amplitude ${\omega }^2{\eta }_0\coth (\kappa h_1)/\kappa$, whereas ${\phi }_{2t}(0)$ has amplitude $-{\omega }^2{\eta }_0\coth (\kappa h_2)/\kappa$. Substituting into the dynamic condition yields the dispersion relation for interfacial gravity waves between rigid boundaries:It is positive because the heavier fluid is below. As $k\to 0$, ${\omega }^2\sim g({\rho }_2-{\rho }_1)k^2/({\rho }_1/h_1+{\rho }_2/h_2)$, while in the deep-layer limit it becomes $g({\rho }_2-{\rho }_1)|k|/({\rho }_1+{\rho }_2)$. The spatially uniform mode has zero restoring frequency.