Nonlinear acoustic phenomena in viscous, thermally relaxing fluids:
shock bifurcation and the emergence of diffusive solitons

Dr. P. M. Jordan
Naval Research Laboratory

Abstract. In this talk we will consider the propagation of
finite-amplitude acoustic waves in fluids that exhibit both viscosity
and thermal relaxation.  Under the assumption that the thermal flux
vector is given by the Maxwell-Cattaneo law, which is a well known
generalization of Fourier's law that includes the effects of thermal
inertia, we derive the weakly nonlinear equation of motion in terms of
the acoustic potential.  We then use singular surface theory to
determine how an input signal in the form of a shock wave evolves over
time, and for different values of the Mach number.  Then, numerical
methods are used to illustrate our analytical findings.  In
particular, it is shown that the shock amplitude exhibits a
transcritical bifurcation; that a stable, nonzero equilibrium solution
is possible; and that a Taylor shock (i.e., a diffusive soliton), in
the form of a “tanh” profile, can emerge from the input shock wave.
Finally, an application related to the kinematic-wave theory of
traffic flow is noted.  (Work supported by ONR/NRL funding.)