Nonhomogeneous Fluids Research Articles

Well-posedness theory for non-homogeneous incompressible fluids with odd viscosity

Several fluid systems are characterised by time reversal and parity breaking. Examples of such phenomena arise both in quantum and classical hydrodynamics. In these situations, the viscosity tensor, often dubbed “odd viscosity”, becomes non-dissipative. At the mathematical level, this fact translates into a loss of derivatives at the level of a priori estimates: while the odd viscosity term depends on derivatives of the velocity field, no parabolic smoothing effect can be expected.In the present paper, we establish a well-posedness theory in Sobolev spaces for a system of incompressible non-homogeneous fluids with odd viscosity. The crucial point of the analysis is the introduction of a set of good unknowns, which allow for the emerging of a hidden hyperbolic structure underlying the system of equations. It is exactly this hyperbolic structure which makes it possible to circumvent the derivative loss and propagate high enough Sobolev norms of the solution. The well-posedness result is local in time; two continuation criteria are also established.

Global well-posedness for 2D nonhomogeneous asymmetric fluids with magnetic field and density-dependent viscosity

Global well-posedness for 2D nonhomogeneous asymmetric fluids with magnetic field and density-dependent viscosity

On the Cauchy problem of 3D nonhomogeneous micropolar fluids with density-dependent viscosity

<p>In this paper, we considered the global well-posedness of strong solutions to the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible micropolar fluids with density-dependent viscosity and vacuum. Based on the energy method, some key a priori exponential decay-in-time rates of strong solutions are obtained. As a result, the existence and large-time asymptotic behavior of strong solutions in the whole space $ \mathbb{R}^3 $ are established, provided that the initial mass is sufficiently small. Note that this result is proven without any compatibility conditions.</p>

Existence of large-data global weak solutions to kinetic models of nonhomogeneous dilute polymeric fluids

We prove the existence of large-data global-in-time weak solutions to a general class of coupled bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials for nonhomogeneous incompressible dilute polymeric fluids in a bounded domain in $ \mathbb{R}^d $, $ d = 2 $ or $ 3 $. The class of models under consideration involves the Navier–Stokes system with variable density, where the viscosity coefficient depends on both the density and the polymer number density, coupled to a Fokker–Planck equation with a density-dependent drag coefficient. The proof is based on combining a truncation of the probability density function with a two-stage Galerkin approximation and weak compactness and compensated compactness techniques to pass to the limits in the sequence of Galerkin approximations and in the truncation level.

Global well-posedness to the 3D Cauchy problem of nonhomogeneous micropolar fluids involving density-dependent viscosity with large initial velocity and micro-rotational velocity

Global well-posedness to the 3D Cauchy problem of nonhomogeneous micropolar fluids involving density-dependent viscosity with large initial velocity and micro-rotational velocity

Symmetrization and asymptotic stability in non-homogeneous fluids around stratified shear flows

Symmetrization and asymptotic stability in non-homogeneous fluids around stratified shear flows

Fast Rotating Non-homogeneous Fluids in Thin Domains and the Ekman Pumping Effect

In this paper, we perform the fast rotation limit varepsilon rightarrow 0^+ of the density-dependent incompressible Navier–Stokes–Coriolis system in a thin strip Omega _varepsilon :=,{mathbb {R}}^2times , left. right] -ell _varepsilon ,ell _varepsilon left[ right. ,, where varepsilon in ,left. right] 0,1left. right] is the size of the Rossby number and ell _varepsilon >0 for any varepsilon >0. By letting ell _varepsilon longrightarrow 0^+ for varepsilon rightarrow 0^+ and considering Navier-slip boundary conditions at the boundary of Omega _varepsilon , we give a rigorous justification of the phenomenon of the Ekman pumping in the context of non-homogeneous fluids. With respect to previous studies (performed for flows of contant density and for compressible fluids), our approach has the advantage of circumventing the complicated analysis of boundary layers. To the best of our knowledge, this is the first study dealing with the asymptotic analysis of fast rotating incompressible fluids with variable density in a 3-D setting. In this respect, we remark that the case ell _varepsilon geqslant ell >0 for all varepsilon >0 remains largely open at present.

On Rayleigh–Taylor instability in Navier–Stokes–Korteweg equations

This paper focuses on the Rayleigh–Taylor instability in the two-dimensional system of equations of nonhomogeneous incompressible viscous fluids with capillarity effects in a horizontal periodic domain with infinite height. First, we use the modified variational method to construct (linear) unstable solutions for the linearized capillary Rayleigh–Taylor problem. Then, motivated by the Grenier’s idea in (Grenier in Commun. Pure Appl. Math. 53(9):1067–1091, 2000), we further construct approximate solutions with higher-order growing modes to the capillary Rayleigh–Taylor problem and derive the error estimates between both the approximate solutions and nonlinear solutions of the capillary Rayleigh–Taylor problem. Finally, we prove the existence of escape points based on the bootstrap instability method of Hwang–Guo in (Hwang and Guo in Arch. Ration. Mech. Anal. 167(3):235–253, 2003), and thus obtain the nonlinear Rayleigh–Taylor instability result. Our instability result presents that the Rayleigh–Taylor instability can occur in the fluids with capillarity effects for any capillary coefficient kappa >0 if the critical capillary coefficient is infinite. In particular, it improves the previous Zhang’s result in (Zhang in J. Math. Fluid Mech. 24(3):70–23, 2022) with the assumption of smallness of the capillary coefficient.

A sediment transport theory based on distortion-free-boundary nonhomogeneous fluid flows

Over the last 50 years several sediment transport models in coastal environments based on Shallow Water(SW) type models have been developed in the literature. The water flow over an abrupt moving topography quickly spatially variable becomes accelerated and strongly varied arising the turbulence (distortion). The acceleration and strong variation of the flow facilitate the transport of a large quantity of sediments present at the bottom while modifying it. The mathematical models based on SW type models widely used to describe the sediment transport phenomena do not account the distortion effects. Indeed, it is well-known that the SW models are derived from first order approximation of long wave theory. The acceleration and strong variation of the water flow near the bottom is due to the distortion of the horizontal velocity profile along the vertical direction. One can regard distortion as a combination of strain and rotation. The effect of the rotational component is to weaken the effect of the strain somewhat. In this work, we put in place a king theory of sediment transport derived from the second order approximation of long wave theory that can describe sediment transport processes in distortion-free-boundary nonhomogeneous fluid flows. The derived model accounts the distortion (fluctuation with great correlation lengths) that creates the turbulence. Moreover, the model differentiates the fluid velocity from sediment velocity (phase-lag) near the sediment bed. The proposed theory significantly reduces the modeling errors observed in several sediment transport models based on nonhomogeneous shallow water equations and has a great potential to increase the predictive power of sediment transport models in rivers, lakes, coastal flows, ocean basins and so on. The proposed theory improves several existing sediment transport theories recently developed in the literature and can be apply with some degree confidence.

Regularity criteria for nonhomogeneous magneto-micropolar fluid equations with vacuum

Applying delicate energy estimates, we show regularity criteria of strong solutions to the three-dimensional (3D) nonhomogeneous magneto-micropolar fluid equations with vacuum.

Strong solutions for the Cauchy problem to three-dimensional nonhomogeneous incompressible micropolar fluid equations with vacuum

This paper is concerned with a Cauchy problem for the three-dimensional (3D) nonhomogeneous incompressible micropolar fluid equations in the whole space. We first establish a weak Serrin-type blowup criterion for the strong solutions. It is shown that for the Cauchy problem of the 3D nonhomogeneous micropolar equations, the strong solution exists globally if the velocity satisfies the weak Serrin's condition. In particular, this criterion is independent of the micro-rotational velocity. Then as an immediate application, we prove that the Cauchy problem of micropolar fluid equations has a unique global strong solution, provided that the kinematic viscosity is sufficiently large, or the upper bound of initial density or initial kinetic energy is small enough.

Local well-posedness and regularity criterion for nonhomogeneous magneto-micropolar fluid equations without angular viscosity

Local well-posedness and regularity criterion for nonhomogeneous magneto-micropolar fluid equations without angular viscosity

Global Well-Posedness for 3D Nonhomogeneous Micropolar Fluids with Density-Dependent Viscosity

Global Well-Posedness for 3D Nonhomogeneous Micropolar Fluids with Density-Dependent Viscosity

A Priori Estimates for a System Modelling Nonhomogeneous Asymmetric Fluids

In this article, we prove some a priori estimates for a system of partial differential equations arising in the nonstationary flow of a nonhomogeneous incompressible asymmetric fluid in a bounded domain with smooth boundary. The unknowns of the system are the velocity field of the fluid particles, the angular velocity of rotation of the fluid particles, the mass density of the fluid and the pressure distribution. For the density functions we consider the application of the Helmholtz decomposition.

Determination of pressure from nonlinear equations of nonhomogeneous non-newtonian fluids

In this paper, the initial-boundary problem for a system of nonlinear equations modified by p-laplacian diffusion and nonlinear damping term, in which describe the motion of a non-homogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids is considered. Usually, a pressure does not include in the definition of weak generalized solution to equations of hydrodynamics for incompressible fluids, because weak solutions consider in the solenoidal space due to the incompressibility (continuity) equation. In the classical hydrodynamic equations, after determining a velocity and a density, a pressure can be determined by using the theorem of a representation of the space L2(Ω) into the direct sum of two orthogonal subspaces. The recovering of a pressure from the nonlinear equations of hydrodynamics modified by p-laplacian and nonlinear damping term requires a new approches. Here, under a suitable conditions on the data of the problem, a pressure has been uniquelly determined from the initial-boundary value problem for the system of nonlinear modified equations describing the motion of nonhomogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids. The main functional spaces and necessary axuillary conclusions are introduced. A space of generalized weak solutions is identified. The pressure uniquelly recovered by using the known de-Ram lemma.

On the mass-conserving Allen-Cahn approximation for incompressible binary fluids

This paper is devoted to the global well-posedness of two Diffuse Interface systems modeling the motion of an incompressible two-phase fluid mixture in presence of capillarity effects in a bounded smooth domain Ω⊂Rd, d=2,3. We focus on dissipative mixing effects originating from the mass-conserving Allen-Cahn dynamics with the physically relevant Flory-Huggins potential. More precisely, we study the mass-conserving Navier-Stokes-Allen-Cahn system for nonhomogeneous fluids and the mass-conserving Euler-Allen-Cahn system for homogeneous fluids. We prove existence and uniqueness of global weak and strong solutions as well as their property of separation from the pure states. In our analysis, we combine the energy and entropy estimates, a novel end-point estimate of the product of two functions, a new estimate for the Stokes problem with non-constant viscosity, and logarithmic type Gronwall arguments.

Kelvin-Voigt equations for incompressible and nonhomogeneous fluids with anisotropic viscosity, relaxation and damping

Kelvin-Voigt equations for incompressible and nonhomogeneous fluids with anisotropic viscosity, relaxation and damping

On Rayleigh–Taylor instability in nonhomogeneous incompressible elasticity fluids

This paper focuses on the Rayleigh–Taylor instability in the system of equations of the two-dimensional nonhomogeneous incompressible elasticity fluid in a horizontal periodic domain with infinite height. First, we use variational method to construct (linear) unstable solutions for the linearized elastic Rayleigh–Taylor problem. Then, motivated by the Grenier's idea in [10], we further construct approximate solutions with higher-order growing modes to the elastic Rayleigh–Taylor problem due to the absence of viscosity in the system, and derive the error estimates between both the approximate solutions and nonlinear solutions of the elastic Rayleigh–Taylor problem. Finally, we prove the existence of escape points based on the bootstrap instability method of Hwang–Guo in [25], and thus obtain the nonlinear Rayleigh–Taylor instability result, which presents that the Rayleigh–Taylor instability can occur in elasticity fluids with small elasticity coefficient.

Transiently linked nonlinear FENE two species dynamics

Transiently breaking/reforming networks occur in a variety of mixtures such as wormlike micelles, telechelic polymers, and mucins. Experiments on these preparations show nonlinear and nonhomogeneous fluid behaviors including shear thinning, shear thickening, shear banding and elastic recoil. In this paper elastically active FENE bead–spring breaking/reforming single/double dumbbells are modeled and simulated in a homogeneous imposed shear at the mesoscale. The topology of the network is tracked allowing for dependence of macroscale flow properties on the local material behavior. The work builds on and is related to that of earlier authors such as van den Brule et al. [JNNFM, 1993], Cifre et al. [JNNFM, 2003], and Adams et al. [JOR, 2018]. The model simulation results exhibit viscous shear thinning and nonmonotonicity in the steady state flow curve. The latter indicates that the model will generate shear banding under full flow conditions. The study shows that the negative slope in the flow curve is associated with the enhanced detachment and subsequent attachment of FENE dumbbells, which results in a continuous reconfiguration of the dumbbells of the chain. • Transiently connected FENE dumbbells. • A stochastic mesoscale model of a physically networked FENE dumbbell fluid. • Network topology is tracked in simulations of a transiently networked FENE fluid. • Predictions of a FENE dumbbell network model in homogeneous shear. • A mesoscale model of breaking/reforming kinetics in a networked fluid.

Convective Flow of Non-homogeneous Fluid Conveying Nano-Sized Particles with Non-Fourier Thermal Relaxation: Application in Polymer Coating

Convective Flow of Non-homogeneous Fluid Conveying Nano-Sized Particles with Non-Fourier Thermal Relaxation: Application in Polymer Coating