One-dimensional model and numerical solution to the viscous and heat-conducting micropolar real gas flow with homogeneous boundary conditions

Local existence theorem for micropolar viscous real gas flow with homogeneous boundary conditions

“…The Lagrangian description was used for simplicity reasons. As mentioned in the introduction of this paper, we are dealing with a one-dimensional model of a viscous and thermally conductive micropolar real gas flow derived in Baši ć-Šiško et al 20 The governing initial and boundary value problem in the Lagrangian description is given by It should be noted that the equations are local forms of the conservation laws for mass, momentum, torque, and energy, respectively. Here, 𝜌 = 𝜌(x, t), v = v(x, t), 𝜔 = 𝜔(x, t), and 𝜃 = 𝜃(x, t) are mass density, velocity, microrotation, and absolute temperature, respectively.…”

“…The Faedo-Galerkin approximations ( 12)-( 14) are described in detail in Baši ć-Šiško et al, 20 where they were used to find the numerical solution to our problem.…”

“…As mentioned in the introduction of this paper, we are dealing with a one‐dimensional model of a viscous and thermally conductive micropolar real gas flow derived in Bašić‐Šiško et al 20 The governing initial‐boundary value problem in the Lagrangian description is given by $$$$ {\partial}_t\rho =-\frac{1}{L}{\rho}^2{\partial}_xv, $$$$ $$$$ {\partial}_tv=-\frac{R}{L}{\partial}_x\left({\rho}^p\theta \right)+\frac{\lambda +2\mu }{L^2}{\partial}_x\left(\rho {\partial}_xv\right), $$$$ $$$$ {j}_I{\partial}_t\omega =\frac{c_0+2{c}_d}{L^2}{\partial}_x\left(\rho {\partial}_x\omega \right)-4{\mu}_r\frac{\omega }{\rho }, $$$$ …”

“…The Lagrangian description was used for simplicity reasons. As mentioned in the introduction of this paper, we are dealing with a one‐dimensional model of a viscous and thermally conductive micropolar real gas flow derived in Bašić‐Šiško et al 20 The governing initial and boundary value problem in the Lagrangian description is given by…”

“…The aim of this work is to prove the local existence of the generalized solution for the viscous and thermally conductive micropolar real gas flow with homogeneous boundary conditions, where we assume the generalized form of the pressure function. This model and the corresponding numerical scheme based on the Faedo-Galerkin method were derived in Baši ć-Šiško et al 20 In our proof, we follow the ideas from Antontsev et al 21 and Mujakovi ć, 12 where classical and micropolar fluid flow were considered, respectively. The proof is divided into a sequence of lemmas and is based on the a priori estimates of the approximate solutions.…”