Past exam of the theoretical mechanics course of the University of Oxford / 2010 / B / 4

The material derivative $\boxed{D/Dt={\partial }_t+\mathbf{u}\cdot \nabla }$ measures change while following a moving gas parcel. The four equations respectively express mass conservation, momentum conservation, internal-energy balance, and the ideal gas law. The first accounts for density change under expansion or compression. The second is Newton's law with pressure as the only force. In the third the specific internal energy is $c_{v}T$ and pressure work $-p\nabla \cdot \mathbf{u}$ heats compression and cools expansion. The fourth relates thermodynamic pressure, density and temperature; $R$ is the gas constant per unit mass.

Continuity gives $D\log \rho /Dt=-\nabla \cdot \mathbf{u}$. Using $p=\rho RT$, the temperature equation gives $D\log T/Dt=-(R/c_v)\nabla \cdot \mathbf{u}$. ThereforeConsequentlyEach smooth parcel evolves isentropically under these adiabatic inviscid equations; the argument does not cover entropy-producing shocks.

Conservation of the parcel's entropy makes $p/{\rho }^{\gamma }$ constant along its trajectory. Its initial value is $p_0/{\rho }_0^{\gamma }$, soThis is a relation following the specified parcel; parcels with different initial entropy can have different constants.

Linearization about the uniform resting state givesFor irrotational motion write ${\mathbf{u}}^{\prime }=\nabla \varphi$. The momentum equation integrates to $p^{\prime }=-{\rho }_0{\varphi }_t$ after choosing the time-dependent potential gauge. Substituting into the pressure equation yieldsThe constant is the adiabatic sound speed, also given by $(\partial p/\partial \rho {)}_S^{1/2}$ at the base state.

Rigid impermeable walls require zero normal fluid velocity, so the potential satisfies a homogeneous Neumann condition. In the cylinder,It is also single-valued and periodic in $\theta$, and regular on the axis. No tangential no-slip boundary condition is imposed on this inviscid acoustic potential.

Use $\varphi =F(r)\Theta (\theta )Z(z)e^{-i\omega t}$. Periodicity gives $\Theta =\cos m\theta$ or $\sin m\theta$, $m\ge 0$, with only the cosine branch for $m=0$. The end-wall conditions give $Z=\cos (\ell \pi z/h)$, $\ell \ge 0$. With radial separation constant ${\kappa }^2$, the remaining equation isFor $\kappa >0$, boundedness on the axis selects $F=J_m(\kappa r)$. The cylindrical wall then requires $J_m^{\prime }(\kappa a)=0$. Thus the modes are real and imaginary parts ofHere take the positive derivative zeros and include the separate constant radial mode $m=0$, $k=0$. For zero radial separation constant, regularity gives $r^m$ when $m>0$, which fails the wall condition, and a constant for $m=0$. This avoids counting a formal derivative zero at the origin that would make $J_m(0r)$ identically zero. The completely constant mode has zero frequency and no velocity; the remaining modes are acoustic oscillations. The self-adjoint Neumann Laplacian has these separated modes as a complete system, so no additional nonzero natural frequencies occur.