Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.08212

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

Geng Lai
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Geng Lai. 2025-09-10. Two-dimensional steady supersonic ramp flows of Bethe-Zel'dovich-Thompson fluids. https://arxiv.org/abs/2509.08212

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Well-posedness of the two-dimensional unsteady Prandtl system in Sobolev space with degenerate critical points

This paper is devoted to the well-posedness of classical Prandtl equations in a finite order Sobolev space. For a initial data with degenerate critical points and general outflow, we obtain the local-in-time existence and uniqueness of the solution to the Prandtl equations in a Sobolev space, by introducing a new iteration scheme and linear cancelation. This result shows that Oleinik's monotonicity condition is not a necessary condition for the Prandtl equations to be well-posed in Sobolev spaces and provides evidence to demonstrate that zero shear stress does not necessarily lead to boundary layer separation in two-dimensional unsteady boundary layers.

math.AP↗
arXiv2026

Global existence and time decay for a bipolar Euler-Poisson system with one pressureless and undamped fluid

We study the Cauchy problem for a three-dimensional bipolar Euler--Poisson system in which one fluid is pressureless and undamped, while the other is subject to momentum relaxation. For sufficiently small smooth perturbations of a constant equilibrium, we prove the global existence and uniqueness of smooth solutions under an irrotationality assumption on the initial velocity of the pressureless fluid, together with algebraic time-decay estimates. The main difficulty is that the velocity of the pressureless fluid is dissipated only indirectly through the Poisson coupling, and this mechanism degenerates strongly at high frequencies, leading to a regularity-loss structure. We overcome this difficulty by combining refined Green-function estimates, a low--middle--high frequency decomposition, and high-order nonlinear energy estimates adapted to the asymmetric regularity hierarchy. The result establishes a global small-data theory for this asymmetric regime, in which pressure and damping are simultaneously absent from the same fluid.

math.AP↗
arXiv2026

Boundary layer of 2D Chemotaxis Navier-Stokes equations with logarithmic Sensitivity. II. viscous vanishing limit

This is the second part of a two-part work concerning boundary layer solutions to the coupled Chemotaxis-Navier-Stokes system in the two-dimensional half-space. In the present work, we address the convergence of boundary layer solutions to singular chemotaxis-fluid equations under slip boundary conditions with respect to the chemical diffusion-viscosity parameter $\varepsilon$ in the two-dimensional half-plane. More precisely, we show that the boundary layer for $\varepsilon>0$ (viscous convection coefficient) converges to the superposition of the outer layer (solution with $\varepsilon=0$) and the inner layer as $\varepsilon\rightarrow0$. The outer and inner profiles are explicitly derived as in the first part\cite{WWZ}. Furthermore, the well-posedness results of the coupled Chemotaxis-Navier-Stokes system in conormal Sobolev spaces will be presented in Appendix. They answer the question mentioned in the first part of the two-part work. This study could help the understanding of the chemotactic movement of aerobic bacteria to the water-air surface observed experimentally in fluids, and enrich the theoretical results of boundary layer in chemotactic fluid models.

math.AP↗