V. A. Shargatov

Stability of discontinuity structures described by a generalized KdV–Burgers equation

The stability of discontinuities representing solutions of a model generalized KdV–Burgers equation with a nonmonotone potential of the form φ(u) = u4–u2 is analyzed. Among these solutions, there are ones corresponding to special discontinuities. A discontinuity is called special if its structure represents a heteroclinic phase curve joining two saddle-type special points (of which one is the state ahead of the discontinuity and the other is the state behind the discontinuity).The spectral (linear) stability of the structure of special discontinuities was previously studied. It was shown that only a special discontinuity with a monotone structure is stable, whereas special discontinuities with a nonmonotone structure are unstable. In this paper, the spectral stability of nonspecial discontinuities is investigated. The structure of a nonspecial discontinuity represents a phase curve joining two special points: a saddle (the state ahead of the discontinuity) and a focus or node (the state behind the discontinuity). The set of nonspecial discontinuities is examined depending on the dispersion and dissipation parameters. A set of stable nonspecial discontinuities is found.

Stability of nonstationary solutions of the generalized KdV-Burgers equation

The stability of nonstationary solutions to the Cauchy problem for a model equation with a complex nonlinearity, dispersion, and dissipation is analyzed. The equation describes the propagation of nonlinear longitudinal waves in rods. Previously, complex behavior of traveling waves was found, which can be treated as discontinuity structures in solutions of the same equation without dissipation and dispersion. As a result, the solutions of standard self-similar problems constructed as a sequence of Riemann waves and shocks with a stationary structure become multivalued. The multivaluedness of the solutions is attributed to special discontinuities caused by the large effect of dispersion in conjunction with viscosity. The stability of special discontinuities in the case of varying dispersion and dissipation parameters is analyzed numerically. The computations performed concern the stability analysis of a special discontinuity propagating through a layer with varying dispersion and dissipation parameters.

Uniqueness of self-similar solutions to the Riemann problem for the Hopf equation with complex nonlinearity

Solutions of the Riemann problem for a generalized Hopf equation are studied. The solutions are constructed using a sequence of non-overturning Riemann waves and shock waves with stable stationary and nonstationary structures.

Dynamics of water evaporation fronts

The evolution and shapes of water evaporation fronts caused by long-wave instability of vertical flows with a phase transition in extended two-dimensional horizontal porous domains are analyzed numerically. The plane surface of the phase transition loses stability when the wave number becomes infinite or zero. In the latter case, the transition to instability is accompanied with reversible bifurcations in a subcritical neighborhood of the instability threshold and by the formation of secondary (not necessarily horizontal homogeneous) flows. An example of motion in a porous medium is considered concerning the instability of a water layer lying above a mixture of air and vapor filling a porous layer under isothermal conditions in the presence of capillary forces acting on the phase transition interface.

Flow Structure behind a Shock Wave in a Channel with Periodically Arranged Obstacles

We study the propagation of a pressure wave in a rectangular channel with periodically arranged obstacles and show that a flow corresponding to a discontinuity structure may exist in such a channel. The discontinuity structure is a complex consisting of a leading shock wave and a zone in which pressure relaxation occurs. The pressure at the end of the relaxation zone can be much higher than the pressure immediately behind the gas-dynamic shock. We derive an approximate formula that relates the gas parameters behind the discontinuity structure to the average velocity of the structure. The calculations of the pressure, velocity, and density of the gas behind the structure that are based on the average velocity of the structure agree well with the results of gas-dynamic calculations. The approximate dependences obtained allow us to estimate the minimum pressure at which there exists a flow with a discontinuity structure. This estimate is confirmed by gas-dynamic calculations.

Unsteady Flows in Deformable Pipes: The Energy Conservation Law

We derive a quasi-one-dimensional energy equation that corresponds to the flow of a compressible viscous fluid in a deformable pipeline. To describe the flow of such a fluid in a pipeline, we couple this equation with the previously derived continuity and momentum equations as well as with the equations of state for the internal energies of the fluid, the pipe deformations, pressure, and the cross-sectional area of the pipe. The derivation of the equations is based on averaging over the pipeline cross section. The equations obtained are designed for numerical simulations of long-distance transportation of a compressible fluid.