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Work in the frame translating with the bubble. The bubble is the fixed circle $r=a$, while the velocity tends to $-U{\mathbf{e}}_x$ at infinity. The potential flow around a circular cylinder has velocity potentialIt is harmonic for $r>a$ and givesThus $u_r=0$ on $r=a$, satisfying the kinematic boundary condition, and $\mathbf{u}\to -U{\mathbf{e}}_x$ as $r\to \infty$. The laboratory-frame flow is obtained by adding $U{\mathbf{e}}_x$.

Writing the velocity as $(u,v)=(y,ax)$ givesso the flow is incompressible. With the conventiona stream function isThe streamlines are its level sets. For $a>0$, they are the hyperbolaswith separatrices $y=\pm \sqrt{a}x$ and a saddle at the origin. For $a<0$, they are concentric ellipsestraversed clockwise. This is the streamline classification of a planar linear saddle or centre.

The scalar vorticity isHence the flow is irrotational exactly whenThen $(u,v)=(y,x)=\nabla \varphi$, and a velocity potential is

When $\mathbf{B}=0$, part (b) gives $\mathbf{V}=2\mathbf{A}/3$ and $\mathbf{\omega }=0$. The total boundary velocity is thereforeThe decaying velocity potentialis harmonic for $r>a$, andhas exactly this value at $r=a$. It is a force-free potential-dipole field and decays as $r^{-3}$.

When $\mathbf{A}=0$, define the degree-three harmonic polynomialAn appropriate decaying harmonic potential isIndeed, the tangential boundary value is proportional towhich combines the prescribed slip with the rigid rotation found in part (b). Since $H_3/r^7=O(r^{-4})$ and multiplication of its gradient by $\mathbf{x}$ preserves that order, the exterior velocity decays as

Use a velocity potential proportional to $e^{ik(x-Vt)+ik{\beta }_{1}y}$, where
${\beta }_1=\sqrt{(V/c_1{)}^2-1}$. The wall condition gives amplitude $-Vϵ/{\beta }_1$. With $p^{\prime }=-{\rho }_1{\partial }_t\varphi$, acoustic energy flux givesFor $V<c_1$ the vertical wave number is imaginary: the disturbance is evanescent and carries no mean vertical energy flux.

Uniform irrotational flow of speed $U$ past a circular cylinder of radius $a$, with zero circulation, has velocity potentialOn the cylinder, the tangential speed is $-2U\sin \theta$, so the Bernoulli equation gives

Adding circulation $\kappa$ gives the possibly multivalued velocity potentialThe velocity remains single-valued, and on the cylinder its tangential component is $-2U\sin \theta +\kappa /(2\pi a)$.

For constant $P=w\rho$ and velocity potential $v_i=i{k}_i\theta$, stress-energy conservation gives the displayed density equation in synchronous gauge in cosmology. Here $h$ is the trace of the spatial metric perturbation, and primes denote conformal time. In particular cold baryons obey ${\delta }_b^{\prime }=k^2{\theta }_b-h^{\prime }/2$. A positive sign in front of $k^2{\theta }_b$ on the left contradicts this potential convention.