arXiv subjects

Geng Lai

Publications and source records attributed to Geng Lai.

5 recordsLinked to original sources

arXiv2026

Oblique wave interactions in 2D steady supersonic flows of Bethe-Zel'dovich-Thompson fluids

This paper studies steady supersonic flow in a 2D semi-infinite divergent duct. We assume that the flow satisfies the slip boundary condition on the walls of the duct, and the state of the flow is given at the inlet of the divergent duct. When the fluid is a polytropic ideal gas, the problem can be reduced to some interactions of rarefaction simple waves, and the existence of a global classical solution inside the divergent duct can be established using the method of characteristics. In this paper we assume that the fluid is a nonconvex Bethe-Zel'dovich-Thompson (BZT) fluid. This type of fluid may significantly differ from polytropic ideal gases. For instance, physically admissible rarefaction shocks can occur. Depending on the oncoming flow state and the flare angles of the divergent duct, thirteen distinct types of oblique wave interactions may occur, including oblique composite waves consisting of shocks and centered simple waves. This paper systematically studies these oblique wave interactions and constructs global, piecewise smooth, supersonic solutions within the divergent duct using characteristic decomposition and hodograph transformation methods. We also obtain the detailed structures of these solutions in addition to their existence. The results and methods of this paper are also applicable to some 2D Riemann problems for gases with nonconvex equations of state.

math.AP↗

arXiv2025

Two-dimensional steady supersonic ramp flows of Bethe-Zel'dovich-Thompson fluids

Two-dimensional (2D) steady supersonic ramp flows are important and well-studied flow patterns in aerodynamics. In the paper [41], Vimercati, Kluwick, and Guardone constructed various self-similar composite wave solutions consisting of centered simple waves and oblique shocks for the 2D steady supersonic flow of BZT fluids past compressible and rarefactive ramps. In the present paper, we study the stabilities of the self-similar fan-shock-fan and shock-fan-shock composite waves in 2D steady supersonic flows of BZT fluids. In contrast to ideal gases, the flow downstream (or upstream) of a shock of a BZT fluid may be sonic in the sense of the flow velocity relative to the shock front, and the formulation of boundary conditions for the shocks is usually unknown in advance. In order to study the stabilities of the composite waves, we establish some a priori estimates of different shock types by comparing the shock speed with the acoustic speed along the shocks and applying Liu's extended entropy condition, and to solve some post-sonic and pre-sonic shock free boundary problems. The sonic shocks are envelopes of one of the acoustic families of characteristics, and not characteristics. This results in a fact that the flow downstream (or upstream) of a sonic shock is not C1smooth up to the shock boundary. We use a weighted characteristic decomposition method and a hodograph transformation method to overcome the difficulty caused by the singularity on sonic shocks of 2D steady full Euler equations and potential flow equations, respectively. Some new iteration schemes are also proposed to solve the post-sonic and pre-sonic shock free boundary problems.

math.AP↗

arXiv2025

On the expansion of a wedge of van der Waals gas into vacuum III: interaction of fan-shock-fan composite waves

This paper studies the expansion into vacuum of a wedge of gas at rest. This problem catches several important classes of wave interactions in the context of 2D Riemann problems. When the gas at rest is a nonideal gas, the gas away from the sharp corner of the wedge may expand into the vacuum as two symmetrical planar rarefaction fan waves, shock-fan composite waves, or fan-shock-fan composite waves. Then the expansion in vacuum problem can be reduced to the interactions of these elementary waves. Global existences of classical solutions to the interaction of the fan waves and the interaction of the shock-fan composite waves were obtained by the author in [21,22]. In the present paper we study the third case: interaction of fan-shock-fan composite waves. In contrast to the first two cases, the third case involves shock waves in the interaction region and is actually a shock free boundary problem. Differing from the transonic shock free boundary problems arising in 2D Riemann problems for ideal gases, the type of the shocks for this shock free boundary problem is also a priori unknown. This results in the fact that the formulation of the boundary conditions on the shocks is also a priori unknown. By calculating the curvatures of the shocks and using the Liu's extended entropy condition, we prove that the shocks in the interaction region must be post-sonic (in the sense of the flow velocity relative to the shock front). We also prove that the shocks are envelopes of one out of the two families of wave characteristics of the flow behind them, and not characteristics. By virtue of the hodograph transformation method and the characteristic decomposition method, we construct a global-in-time piecewise smooth solution to the expansion in vacuum problem for the third case.

math.AP↗

arXiv2020

Global non-isentropic rotational supersonic flows in a semi-infinite divergent duct

Supersonic flows for the two-dimensional (2D) steady full Euler system are studied. We construct a global non-isentropic rotational supersonic flow in a semi-infinite divergent duct. The flow satisfies the slip condition on the walls of the duct, and the state of the flow is given at the inlet of the duct. The solution is constructed by the method of characteristics. The main difficulty for the global existence is that uniform a priori $C^1$ norm estimate of the solution is hard to obtain, especially when the solution tends to vacuum state. We derive a group of characteristic decompositions for the 2D steady full Euler system. Using these decompositions, we obtain the uniform a priori estimates of the derivatives of the solution. A sufficient condition for the appearance of vacuum is given. We also show that if there is a vacuum then the vacuum is always adjacent to one of the walls, and the interface between gas and vacuum must be straight.

math.AP↗

arXiv2020

Self-similar solutions of the spherically symmetric Euler equations for general equations of state

The study of spherically symmetric motion is important for the theory of explosion waves. In this paper, we construct rigorously self-similar solutions to the Riemann problem of the spherically symmetric Euler equations for general equations of state. We used the assumption of self-similarity to reduce the spherically symmetric Euler equations to a system of nonlinear ordinary differential equations, from which we obtain detailed structures of solutions besides their existence.

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