Approximation of the incompressible non‐Newtonian fluid equations by the artificial compressibility method

Abstract

This paper studies the approximation of the non-Newtonian fluid equations by the artificial compressibility method. We first introduce a family of perturbed compressible non-Newtonian fluid equations (depending on a positive parameter ) that approximates the incompressible equations as ! 0 C . Then, we prove the unique existence and convergence of solutions for the compressible equations to the solutions of the incompressible equations.From the Stokes law (see, for example, [1], P 13 ), we see that if the dependence of u on e.u/ is linear, for example,then the Equations (1.1) and (1.2) are the well-known incompressible Navier-Stokes equations. The incompressibility of the fluid is described by div u D 0. An incompressible fluid, the behavior of which is characterized by the Stokes law (1.3), is called Newtonian fluid. Fluids that cannot be adequately described by (1.3) are usually called non-Newtonian fluids.The constitutive relations for an isothermal, nonlinear bipolar viscous two-dimensional (2D) fluid that were introduced in [1, 2] have the form ij .u/ D pı ij C 2 0 Á C je.u/j 2 Compared with the incompressible non-Newtonian fluid Equations (1.7) and (1.8), Equations (1.11) and (1.12) overcome the computational difficulties connected with the constraint 'div u D 0' . It is easier to approximate than the original incompressible equations as the constraint 'div u D 0' has been replaced by the evolution Equation (1.12).Obviously, the perturbed compressible non-Newtonian fluid Equations (1.11)-(1.14) turn to be the incompressible non-Newtonian fluid Equations (1.7)-(1.10) as D 0. Following this clue, the problems are now the following:Now, we can define a continuous functional B.u, v/ from V V to V 0 by hB.u, v/, vi D b.u, v, w/, 8 Especially, we set B.u/ D B.u, u/.

Approximation of a Class of Incompressible Third Grade Fluids Equations

“…Equations (29)-(30) are the constructed slightly compressible third grade fluids equations. This artificial compressibility method has been used by Zhao and You [15] and Zhao [16] to approximate the incompressible convective Brinkman-Forchheimer equations and a class of incompressible non-Newtonian fluids equations.…”

Statistical solutions and degenerate regularity for the micropolar fluid with generalized Newton constitutive law

“…The asymptotic behavior of solutions for the non-Newtonian fluids and micropolar fluids has been extensively studied (see, e.g., previous studies [26][27][28][29][30][31][32][33][34][35]). Concerning statistical solutions, Zhao et al [17] studied trajectory statistical solu-tions for 3D incompressible micropolar fluids, and Zhao et al [16] studied the statistical solutions and partial degenerate regularity for the 2D nonautonomous magneto-micropolar fluids.…”