Non-linear wave propagation in a relaxing gas

Abstract

We consider the propagation of waves of small finite amplitude ε in a gas whose internal energy is characterized by two temperaturesT(translational) andT_i(internal) in the forme=C_vfT+C_vfT_i, andT_iis governed by a rate equationdT_i/dt= (T−T_i)/τ. By means of approximations appropriate for a wave advancing into an undisturbed regionx> 0, we show that to order εδ, the equation satisfied by velocity takes the non-linear form\[ \bigg(\tau\frac{\partial}{\partial t}+1\bigg)\bigg\{\frac{\partial u}{\partial t}+\bigg(a_1+\frac{\gamma + 1}{2}u\bigg)\frac{\partial u}{\partial x}-{\textstyle\frac{1}{2}}\lambda\frac{\partial^2u}{\partial x^2}\bigg\}=(a_1-a_0)\frac{\partial u}{\partial x}, \]wherea₁,a₀are the frozen and equilibrium speeds of sound in the undisturbed region, δ = ½(1 − (a²₀/a²₁)), and λ is the diffusivity of sound due to viscosity and heat conduction (λ may be neglected except when discussing the fine structure of a discontinuity). Some numerical solutions of this model equation are given.When ε is small compared with δ, it is also possible to construct a solution for the flow produced by a piston moving with a constant velocity by means of a sequence of matched asymptotic expansions. The limit reached for large times for either compressive or expansive pistons is the expected non-linear solution of the exact equations. For a certain range of advancing piston speeds, this is a fully dispersed wave with velocityUin the rangea₀<U<a₁. IfU>a₁the solution is discontinuous, and indeterminate in the absence of viscosity; a singular perturbation technique based on λ is then used to determine the structure of the wave head.