Global bounded variation solutions describing Fanno–Rayleigh fluid flows in nozzles

Abstract

In this paper, we investigate the initial-boundary value problem of compressible Euler equations including friction and heating that model the transonic Fanno–Rayleigh flows through symmetric variable area nozzles. In particular, the case of contracting nozzles is considered. A new version of a generalized Glimm scheme (GGS) is presented for establishing the global existence of entropy solutions with bounded variation. Modified Riemann and boundary Riemann solutions are applied to design this GGS, which is constructed using the contraction matrices acting on the homogeneous Riemann (or boundary-Riemann) solutions. The extended Glimm–Goodman’s type of wave interaction estimates are investigated to determine the stability of the scheme and the positivity of gas velocity that results in the existence of the weak solution. The limit of approximation solutions serves as an entropy solution. Moreover, a quantitative relation between the shape of the nozzle, friction, and heat is proposed for the global existence result in the contracting nozzle. Numerical simulations of the contraction-expansion and expansion-contraction nozzles are presented to validate the scheme.

Global transonic solutions of hot-Jupiter model for exoplanetary atmosphere

“…On the other hand, from the 3rd equation of (1.6) the evolution of the internal energy ω of the gas will be affected by the tidal force. We find the model is not reasonable, since 3GMsr d 3 acts as an outer force of the gas in 2nd equation of (1.5), so it should only affect the kinetic energy of the gas, not the internal energy of the gas. For this reason, we modify the model as (1.2).…”

“…To initiate the generalised Glimm scheme, we adopt the same approach as in [3,12]. We choose ∆r and ∆t are positive constants small enough so that the CFL condition (2.1) is satisfied and…”

The global existence of L∞$L^{\infty }$ solutions to isentropic Euler equations in general nozzle

“…was studied for the polytropic gas 𝑃(𝜌) = was studied in [3] , under suitable conditions among the initial data, 𝑎(𝑥) and 𝛼(𝑥), by using a new version of a generalized Glimm scheme, where 𝜌, 𝑢, 𝐸 are, respectively, the density, velocity, total energy and pressure of the gas, 𝛼 is the coefficient of friction, and 𝑞(𝑥) is a given function representing the heating effect from the force outside the nozzle.…”

“…\end{equation}$$was studied for the polytropic gas $$ P(\rho ) = \frac{1}{\gamma } \rho ^{\gamma }$$ and γ is limited in (3, ∞) for a technical difficulty; and the initial‐boundary value problem of the compressible Euler equations with friction and heating $$$$\begin{equation} {\left\lbrace \def\eqcellsep{&}\begin{array}{l}(a(x)\rho )_{t}+(a(x)\rho u)_{x}=0, \\[3pt] (a(x) \rho u)_t+(a(x) \rho u^2+ a(x) P)_x=a^{\prime }(x)P- \alpha \sqrt {a(x)} \rho u |u|, \\[3pt] (a(x)E)_{t}+(a(x)u(E+P))_{x}= \beta a(x) q(x) - \alpha \sqrt {a(x)} \rho u^{2} |u|, \end{array} \right.} \end{equation}$$$$was studied in [3] , under suitable conditions among the initial data, $$a(x)$$ and $$\alpha (x)$$…”

Global existence and strong trace property of entropy solutions by the source-concentration Glimm scheme for nonlinear hyperbolic balance laws

“…In recent papers [1,10], the IBVP of hyperbolic balance laws for Fanno-Rayleigh flows and hydrodynamic escape problem were studied. As in this paper, the authors in [1,10] also applied Glimm's method to obtain the global existence of the entropy solutions. However, they didn't deal with the boundary regularity of solutions.…”