Past exam of the options II course of the University of Oxford / 2007 / K (Fluid Dynamics and Waves)

Orient the body boundary counterclockwise. The outward normal from the body satisfies $nds=(dy,-dx)$, so pressure givesFor steady irrotational flow, Bernoulli's equation makes $p+\rho |u{|}^2/2$ constant in the connected fluid region. Its constant contribution integrates to zero. Write $q=w^{\prime }=u-iv$. Impermeability says the velocity is tangent to the boundary, so $qdz=\bar{q}d\bar{z}$ there. Consequentlywhich proves Blasius' theorem with this contour convention.

The background stream function is the imaginary part of $w_0(z)=\alpha (z-c{)}^3$. Since its coefficients are real, the circle theorem givesIts derivative isIn $q^2$ the coefficient of $z^{-1}$ comes from the exponent pairs $(2,-3)$ and $(1,-2)$, including both orders. It is $36{\alpha }^{2}c{a}^4+36{\alpha }^2{c}^3{a}^2$. The residue theorem then givesThe vanishing transverse force also agrees with reflection symmetry about the real axis.

A material interface obeys ${\eta }_t+{\varphi }_{j,x}{\eta }_x={\varphi }_{j,y}$ at its actual height. Keeping only first-order terms and evaluating at the undisturbed interface gives ${\varphi }_{1,y}={\varphi }_{2,y}={\eta }_t$ at $y=0$. Linearized unsteady Bernoulli gives the pressure perturbation $-{\rho }_j({\varphi }_{j,t}+g\eta )$, after absorbing time-only constants into the potentials. Pressure continuity therefore givesFor $\vartheta =kx-\omega t$, choose potentials decaying away from the interface:They are harmonic and their normal derivatives equal $A\omega \sin \vartheta ={\eta }_t$ at zero. The dynamic condition reduces toIf the denser fluid is above, the squared frequency is negative. The corresponding perturbations grow or decay exponentially instead of oscillating: this is Rayleigh-Taylor instability, with growing rate $\sqrt{gk({\rho }_2-{\rho }_1)/({\rho }_1+{\rho }_2)}$ in the ideal model without surface tension.