non-Newtonian Compressible Fluid Research Articles

Local well-posedness and singularity formation in non-Newtonian compressible fluids

We investigate the initial value problem of a very general class of non-Newtonian compressible fluids in which the viscous stress tensor with shear and bulk viscosity relaxes to its Navier–Stokes values. These fluids correspond to the non-relativistic limit of well-known Israel–Stewart-like theories used in the relativistic fluid dynamic simulations of high-energy nuclear and astrophysical systems. After establishing the local well-posedness of the Cauchy problem, we show for the first time in the literature that there exists a large class of initial data for which the corresponding evolution breaks down in finite time due to the formation of singularities. This implies that a large class of non-Newtonian fluids do not have finite solutions defined at all times.

Relative energy inequality and weak-strong uniqueness for an isothermal non-Newtonian compressible fluid

Our paper deals with three-dimensional nonsteady Navier-Stokes equations for non-Newtonian compressible fluids. It contains a de­ri­va­tion of the relative energy inequality for the weak solutions to these equations. We show that the standard energy inequality implies the relative energy inequality. Consequently, the relative energy inequality allows us to achieve a weak-strong uniqueness result. In other words, we present that the weak solution of the Navier-Stokes system coincides with the strong solution emanated from the same initial conditions as long as the strong solution exists. For this purpose, a new assumption on the coercivity of the viscous stress tensor was introduced along with two natural examples satisfying it.

Global existence in critical spaces for non Newtonian compressible viscoelastic flows

We are interested in the multi-dimensional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the “div-curl” structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. In absence of compatible conditions, the new effective flux is introduced, which enables us to capture the dissipation arising from combination of density and deformation tensor. The partial dissipation in non-Newtonian compressible fluids, is weaker than that of classical Navier-Stokes equations.

On Integrability up to the Boundary of the Weak Solutions to a Class of non-Newtonian Compressible Fluids with Vacuum

In this paper, we study the integrability up to the boundary of the weak solutions of non-Newtonian compressible fluid with a nonlinear constitutive equation in ℝ3 bounded domain. Galerkin approximation will be used for existence of weak solutions and by applying the bounded linear operator B, introduced by Bogovskii, we prove the square integrability of the density up to the boundary.

On the Long-Time Behavior of Dissipative Solutions to Models of Non-Newtonian Compressible Fluids

We identify a class maximal dissipative solutions to models of compressible viscous fluids that maximize the energy dissipation rate. Then we show that any maximal dissipative solution approaches an equilibrium state for large times.

Stability of planar rarefaction waves under general viscosity perturbation of the isentropic Euler system

We consider the vanishing viscosity limit for a model of a general non-Newtonian compressible fluid in Rd, d=2,3. We suppose that the initial data approach a profile determined by the Riemann data generating a planar rarefaction wave for the isentropic Euler system. Under these circumstances the associated sequence of dissipative solutions approaches the corresponding rarefaction wave strongly in the energy norm in the vanishing viscosity limit. The result covers the particular case of a linearly viscous fluid governed by the Navier–Stokes system.

A general peridynamics model for multiphase transport of non-Newtonian compressible fluids in porous media

A general state-based peridynamics model is developed to simulate transport of fluids in an arbitrary heterogeneous porous medium. The generality encompasses modeling of multiphase, multi-component flow of non-Newtonian and compressible fluids, which is often encountered in but not limited to subsurface reservoirs. Peridynamic model is especially useful for solving non-local problems, such as crack propagation, since it does not assume spatial continuity of field variables. Thus, the formulation presented here, combined with peridynamics-based damage model, can be used to simulate hydraulic fracturing with complex fluids. To demonstrate its capability to simulate multi-phase flow in porous media, the derived model is verified against the analytical Buckley-Leverett solution for immiscible Newtonian two-phase flow. Further, the non-Newtonian two-phase fluid flow in porous media is verified by simulating the polymer flood process involving immiscible displacement of a Newtonian fluid by a non-Newtonian fluid against a generalized solution obtained by Wu et al. [22]. The non-local solutions are shown to be consistent with the corresponding local solutions in limiting cases. Moreover, mass conservation of all the phases is satisfied, irrespective of discretization and extent of non-locality.

Large-time behaviour of solutions to a class of non-Newtonian compressible fluids

In this paper, we consider the large-time dynamics of weak solutions to a class of compressible fluids with nonlinear constitutive equations in a bounded domain \(\Omega \subseteq \mathbb {R}^3\), the global existence of such solutions has been showed by Feireisl et al. (Math Methods Appl Sci 38:3482–3494, 2015). We study the large time behavior of such solutions after discussing the uniqueness of solutions to the stationary problem.

Asymptotic stability for one-dimensional motion of non-Newtonian compressible fluids

Rarefaction wave solutions for a one-dimensional model system associated with non-Newtonian compressible fluid are investigated in terms of asymptotic stability. The rarefaction wave solution is proved to be asymptotically stable, provided the initial disturbance is suitably small. The proof is given by the elemental L 2 energy method.

Asymptotic Wave Propagation in a non-Newtonian Compressible Fluid with Small Dissipation

The spherical motion of a non-Newtonian compressible fluid is considered and a reductive perturbation method is used to study the point-explosion problem. The material response functions involved in the model under consideration are assumed to be of polynomial form and the resulting Burgers-like equation which governs the far-field approximation is investigated. A qualitative analysis of this equation is made via a numerical integration.

Nonuniform Temperature in Non-Newtonian Compressible Fluid Film Lubrication of Rollers

A theoretical analysis of heavily loaded rigid cylindrical roller bearings lubricated by a thin compressible fluid film, considering cavitation, is presented. The lubricating fluid is assumed to follow the non-Newtonian power law model and the consistency as well as density varies exponentially with pressure and temperature. The modified Reynolds and energy equations (adiabatic case) for a compressible power law fluid are obtained and are solved simultaneously. It is observed that the compressible fluid film pressure is comparatively higher than the incompressible one whereas this trend is reversed in case of temperature. Moreover, consistency and density variations, especially in the pressure peak region, are quite significant.