Publication Date:
1969-11-10
Description:
We consider the propagation of waves of small finite amplitude ε in a gas whose internal energy is characterized by two temperatures T (translational) and Ti (internal) in the form e = CvfT + CvfTi, and Ti is governed by a rate equation dTi/dt = (T − Ti)/τ. By means of approximations appropriate for a wave advancing into an undisturbed region x 〉 0, we show that to order εδ, the equation satisfied by velocity takes the non-linear form [igg(aufrac{partial}{partial t}+1 igg) igg{frac{partial u}{partial t}+ igg(a_1+frac{gamma + 1}{2}u igg)frac{partial u}{partial x}-{extstylefrac{1}{2}}lambdafrac{partial^2u}{partial x^2} igg}=(a_1-a_0)frac{partial u}{partial x}, ] where a1, a0 are 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 velocity U in the range a0 〈 U 〈 a1. If U 〉 a1 the 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.
Print ISSN:
0022-1120
Electronic ISSN:
1469-7645
Topics:
Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
,
Physics
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