The limits of Riemann solutions to Euler equations of compressible fluid flow with a source term
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The limits of Riemann solutions to Euler equations of compressible fluid flow with a source term Shouqiong Sheng · Zhiqiang Shao
Received: 17 July 2019 / Accepted: 26 August 2020 © Springer Nature B.V. 2020
Abstract In this paper, we investigate the limits of Riemann solutions to the Euler equations of compressible fluid flow with a source term as the adiabatic exponent tends to one. The source term can represent friction or gravity or both in Engineering. For instance, a concrete physical model is a model of gas dynamics in a gravitational field with entropy assumed to be a constant. The body force source term is presented if there is some external force acting on the fluid. The force assumed here is the gravity. Different from the homogeneous equations, the Riemann solutions of the inhomogeneous system are non self-similar. We rigorously proved that, as the adiabatic exponent tends to one, any two-shock Riemann solution tends to a delta shock solution of the pressureless Euler system with a Coulomb-like friction term, and the intermediate density between the two shocks tends to a weighted δ-mesaure which forms the delta shock; while any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution of the pressureless Euler system with a Coulomb-like friction term, whose intermediate state between the two contact discontinuities is a vacuum state. Moreover, we also give some numerical simulations to confirm the theoretical analysis. Keywords Pressureless limit · Euler equations · Coulomb-like friction term · Non self-similar solution · Delta shock wave
1 Introduction In this paper, we are concerned with the Euler equations of one-dimensional compressible fluid flow with the source term in the following form: u2 + p(ρ) = β, (1.1) ρt + (ρu)x = 0, u t + x 2 where β is a constant, β represents a Coulomb-like friction term or a gravity term or both, ρ denotes the density, u the velocity, and p(ρ) the pressure of the fluid, in which the nonlinear function p(ρ) = (θ/2)ρ γ −1 , θ = (γ − 1)/2 and γ ∈ (1, 2) is a constant. The adiabatic exponent γ plays the important role in different situations. For instance, when γ = −1, p(ρ) = −1/(2ρ 2 ) is the so-called Chaplygin gas pressure, which was introduced by Chaplygin [1], S. Sheng · Z. Shao (B) College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China e-mail: [email protected]
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S. Sheng, Z. Shao
Tsien [2] and von Karman [3] as a suitable mathematical approximation for calculating the lifting force on a wing of an airplane in aerodynamics. Shen [4] considered the pressureless Euler system with the Coulomb-like friction term and obtained the non self-similar Riemann solutions by introducing a new velocity: v(t, x) = u(t, x) − βt,
(1.2)
which was introduced by Faccanoni and Mangeney [5] to study the Riemann problem of the shallow water equations with the Coulomb-like friction term. In addition, some other hyperbolic systems with a Coulomb-like friction term were also considered such as in [6–8]. The Coulomb-like fri
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