Generalized Navier-Stokes Equations Research Articles

Weak solutions for steady, fully inhomogeneous generalized Navier-Stokes equations

We consider the question of existence of weak solutions for the fully inhomogeneous, stationary generalized Navier–Stokes equations for homogeneous, shear-thinning fluids. For a shear rate exponent p∈(2dd+1,2), previous results require either smallness of the norm or vanishing of the normal component of the boundary data. In this work, combining previous methods, we propose a new, more general smallness condition.

Finite Element Discretization of the Steady, Generalized Navier–Stokes Equations with Inhomogeneous Dirichlet Boundary Conditions

Finite Element Discretization of the Steady, Generalized Navier–Stokes Equations with Inhomogeneous Dirichlet Boundary Conditions

REV-Scale study of miscible density-driven convection in porous media

Miscible density-driven convection in porous media has important implications for the long-term security of geological CO2 sequestration. In this study, a REV-scale lattice Boltzmann equation method based on the generalized Navier–Stokes equations was used to simulate density-driven convection in porous media, Thus, the effects of the Rayleigh number, the Darcy number, the Schmidt number, and the porosity of porous media can be discussed separately. The results show that density-driven convection only occurs when the Rayleigh–Darcy–Schmidt number RaD-S exceeds 102. The larger the Ra, the more disordered the concentration field, the earlier the convective phenomenon begins, and the more significant the convective mixing; The larger the Da, the finer the generated fingers. These findings provide important insights for the development of geological sequestration technologies.

Averaging principle for stochastic 3D generalized Navier–Stokes equations

In this paper, the multiscale stochastic 3D generalized Navier–Stokes equations are studied. By using Khasminkii’s time discretization approach and the technique of stopping time, the strong averaging principle for stochastic 3D generalized Navier–Stokes equations is proved in the space [Formula: see text].

Regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term

<abstract><p>This paper was devoted to establishing some regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term. We focused on considering the role of the damping term in regularity criteria and the global existence brought by these criteria.</p></abstract>

Global regularity of the 3D generalized Navier-Stokes equations with damping term

Global regularity of the 3D generalized Navier-Stokes equations with damping term

Solutions for the Navier-Stokes equations with critical and subcritical fractional dissipation in Lei-Lin and Lei-Lin-Gevrey spaces

In this article, we prove the existence of a unique global solution for the critical case of the generalized Navier-Stokes equations in Lei-Lin and Lei-Lin-Gevrey spaces, by assuming that the initial data is small enough. Moreover, we obtain a unique local solution for the subcritical case of this system, for any initial data, in these same spaces. It is important to point out that our main result is obtained by discussing some properties of the solutions for the heat equation with fractional dissipation.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/78/abstr.html

The initial value problem for the equations of motion of fractional compressible viscous fluids

In this paper we consider the initial value problem to the fractional compressible isentropic generalized Navier-Stokes equations for viscous fluids with one Levy diffusion process in which the viscosity term appeared in the fluid equations is described by the nonlocal fractional Laplace operator. We give one detailed spectrum analysis on a linearized operator and the decay law in time of the solution semigroup for the linearized fractional compressible isentropic generalized Navier-Stokes equations around a constant state by the Fourier analysis technique, which is shown that the order of the fractional derivatives plays a key role in the analysis so that the spectrum structure involved here is more complex than that of the classical compressible Navier-Stokes system. Based on this and the elaborate energy method, the global-in-time existence and one optimal decay rate in time of the smooth solution are obtained under the assumption that the initial data are given in a small neighborhood of a constant state.

A linear mass and energy conserving numerical scheme for two-phase flows with thermocapillary effects

In this paper, a thermodynamically consistent phase-field model is employed to simulate the thermocapillary migration of a droplet. The model equations consist of a general Navier–Stokes equation for the two-phase flows, a Cahn–Hilliard equation for the diffuse interface, and a heat equation, and meanwhile satisfy the balance laws of mass, energy and entropy. In particular, the total energy of the system includes kinetic energy, potential energy and internal energy, which leads to a highly coupled and nonlinear equation system. We therefore develop a linear mass and energy conserving, semi-decoupled numerical method for the numerical simulations. As the model contains a heat (energy) equation, a simple error term introduced by the temporal discretization of the momentum equation can be absorbed into the heat equation, such that the numerical solutions satisfy the conservation laws of mass and energy exactly at the temporal discrete level. Several numerical tests are carried out to validate our numerical method.

On Exact Solutions to Problems of Steady Flow of Incompressible Viscous Fluid in a Channel and in a Circular Pipe

Turning to the well-known Couette flow, we again, but now in a new formulation, consider the steady motion of an incompressible viscous fluid in a channel with two parallel flat walls, one of which is stationary while the other moves at a constant speed in its plane. In the classical formulation of this problem on a layered (laminar) fluid flow in a channel, the wall velocity and the constant pressure drop along the channel are independent quantities and are considered predetermined. However, when a solid body moves along the fluid surface, the fluid particles are set in motion. This occurs due to the arising pressure generated in the fluid caused by the movement of the solid body. Therefore, here the pressure or pressure drop is not predetermined but is determined as a result of solving the corresponding two-dimensional boundary value problem of a steady flow of a viscous fluid in a channel. For this, the corresponding boundary value problem of a fluid flow in the channel is formulated based on the linearized Navier-Stokes equations obtained from the general nonlinear Navier-Stokes equations for a steady flow of a viscous fluid neglecting the convective terms, which is true for small Reynolds numbers. Using the Fourier integral transform method, an exact (closed) solution to this boundary value problem is constructed; the velocity and pressure components are determined. Then it is shown that the resulting solution, different from the known Couette flow solution, also satisfies the original nonlinear Navier-Stokes equations, the continuity equation, and boundary conditions. A comparative analysis of the new solution with the Couette flow solution is carried out. The well-known Hagen-Poiseuille problem is also discussed from the point of view of establishing the conditions for the laminar axisymmetric and steady flow of an incompressible viscous fluid in a straight circular pipe According to the study's results, to implement laminar flow under the indicated conditions, the pressure drop along the pipe must be constant. The study demonstrates that this condition is simultaneously sufficient, and therefore, the necessary and sufficient condition is established for a laminar flow in a straight circular pipe.

Weak solutions to the generalized Navier–Stokes equations with mixed boundary conditions and implicit obstacle constraints

This paper is concerned with the investigation of a generalized Navier–Stokes equation for non-Newtonian fluids of Bingham-type (GNSE, for short) involving a multivalued and nonmonotone slip boundary condition formulated by the generalized Clarke subdifferential of a locally Lipschitz superpotential, a no leak boundary condition, and an implicit obstacle inequality. We obtain the weak formulation of (GNSE) which is a generalized quasi-variational–hemivariational inequality. By introducing an Oseen model as an auxiliary (intermediated) problem and employing Kakutani-Ky Fan theorem for multivalued operators as well as the theory of nonsmooth analysis, an existence theorem to (GNSE) is established.

Thermal convection in a higher-gradient Navier–Stokes fluid

We discuss models for flow in a class of generalized Navier–Stokes equations. The work concentrates on producing models for thermal convection, analysing these in detail, and deriving critical Rayleigh and wave numbers for the onset of convective fluid motion. In addition to linear instability theory we present a careful analysis of fully nonlinear stability theory. The theories analysed all possess a bi-Laplacian term in addition to the normal spatial derivative term. The theories discussed are Stokes couple stress theory, dipolar fluid theory, Green–Naghdi theory, Fried–Gurtin–Musesti theory, and a second theory of Fried and Gurtin. We show that the Stokes couple stress theory and the Fried–Gurtin–Musesti theory involve the same partial differential equations while those of Green–Naghdi and dipolar theory are similar. However, we concentrate on boundary conditions which are crucial to understand all five theories and their differences.

FLUID FLOW MODELLING WITH FREE SURFACE

Fluid is a substance that can flow in the form of a liquid or a gas. Based on the movement of the fluid is divided into static and dynamic fluids. This study discusses fluid dynamics, namely modelling fluid flow accompanied by a free surface and an obstacle in the fluid flow. Fluid modelling generally makes some basic assumptions into mathematical equations. The assumptions are incompressible, steady-state and irrotational. The steps to obtain a fluid flow model are using Newton’s second law, the law of conservation of mass, and the law of conservation of momentum to obtain the general Navier-Stokes equation, the designing the Euler free surface equation, the Bernoulli equation, then making a free surface representation and linearizing the wave equation so that it is obtained fluid flow model. The resulting mathematical model is a Laplace equation with boundary conditions in the fluid.

Hölder continuity of solutions for unsteady generalized Navier–Stokes equations with p(x,t)-power law in 2D

We prove Hölder continuity of gradient of a unique weak solution for unsteady generalized Navier–Stokes equations with p(x,t)-power law with Dirichlet type boundary condition under 2<p(x,t)<6 in 2D. This is a new result even for p=const in the sense that the upper bound p=const<4 is improved to 6, but also is a generalization of the results concerning a constant exponent p to a non-constant exponent p(x,t).

Generalized Navier–Stokes Equations with Non-Homogeneous Boundary Conditions

We study the generalized unsteady Navier–Stokes equations with a memory integral term under non-homogeneous Dirichlet boundary conditions. Using a suitable fractional Sobolev space for the boundary data, we introduce the concept of strong solutions. The global-in-time existence and uniqueness of a small-data strong solution is proved. For the proof of this result, we propose a new approach. Our approach is based on the operator treatment of the problem with the consequent application of a theorem on the local unique solvability of an operator equation involving an isomorphism between Banach spaces with continuously Fréchet differentiable perturbations.

Global Existence of Strong Solutions for the Generalized Navier-Stokes Equations with Damping

Global Existence of Strong Solutions for the Generalized Navier-Stokes Equations with Damping

Extensions to the Navier–Stokes equations

Historically, the mass conservation and the classical Navier–Stokes equations were derived in the co-moving reference frame. It is shown that the mass conservation and Navier–Stokes equations are Galilean invariant—they are valid in any arbitrary inertial reference frame. From the mass conservation and Navier–Stokes equations, we can derive a wave equation, which contains the speed of pressure wave as its parameter. This parameter is independent of the speed of the source—the fluid element velocity. The speed of pressure wave is determined from the thermodynamic equation of state of the fluid, which is reference frame independent. It is well known that Lorentz transformation ensures wave speed invariant in all inertial frames, and the Lorentz invariance holds for different inertial observers. Based on these arguments, general Navier–Stokes equations (conservation law for the energy–momentum) can be written in any arbitrary inertial reference frame, they are transformed from one reference frame into another with the help of the Lorentz transformation. The key issue is that the Lorentz factor is parametrized by the local Mach number. In the instantaneous co-moving reference frame, these equations will degrade to the classical Navier–Stokes equations—the limit of the non-relativistic ones. These extended equations contain a square of the Lorentz factor. When the local Mach number is equal to one (the Lorentz factor approaches infinity), the extended Navier–Stokes equations will embody an intrinsic singularity, meaning that the transitions from the subsonic flow to the supersonic flow will happen. For the subsonic flow, the square of the Lorentz factor is positive, while for the supersonic flow, the square of the Lorentz factor becomes a negative number, which represents that the speed of sound cannot travel upstream faster than the flow velocity.

Existence of global weak solutions for unsteady motions of incompressible chemically reacting generalized Newtonian fluids

We study a system of nonlinear partial differential equations describing the unsteady motions of incompressible chemically reacting non-Newtonian fluids. The system under consideration consists of the generalized Navier–Stokes equations with a power-law-type stress-strain relation, where the power-law index depends on the concentration of a chemical, coupled to a convection-diffusion equation for the concentration. This system of equations arises in the rheology of the synovial fluid found in the cavities of synovial joints. We prove the existence of global weak solutions of the non-stationary model by using the Galerkin method combined with monotone operator theory and parabolic De Giorgi–Nash–Moser theory. As the governing equations involve a nonlinearity with a variable power-law index, our proofs exploit the framework of generalized Sobolev spaces with variable-integrability exponent.

Statistical Analysis of Hydrodynamic Forces in Power-Law Fluid Flow in a Channel: Circular Versus Semi-Circular Cylinder

This numerical study is about the steady incompressible non-Newtonian fluid flow in a channel with static obstacles. The flow field is governed by the Generalized Navier-Stokes equations incorporating the constitutive relation of power-law fluids. Three cases are considered: 1) circular obstacle(C1), 2) semicircular obstacle(C2), and 3) both circular and semicircular obstacles. A range of values of the power-law index0.3≤n≤1.7are considered atRe=20to check the impact of shear-thinning and shear-thickening viscosity on the drag and lift coefficients. The correlation between drag and lift coefficients is calculated against the power-law index. The simulated results of velocity and pressure are investigated at different sections of the channel. Benchmark results of drag and lift for the Newtonian fluid are reproduced as a special case. A strong positive correlation is observed between drag and lift coefficients in the case of a single obstacle, while in the case of dual obstacles and inverse correlation, drag and lift coefficients have been found.

Hydro–Hygro–Thermo–Magneto–Electro​ elastic wave propagation of axially moving nano-cylindrical shells conveying various magnetic-nano-fluids resting on the electromagnetic-visco-Pasternak medium

The present study addresses hydro-magneto-electro wave propagation of the axially moving circular cylindrical nanoshells conveying magnetic nanofluid and embedded on an electromagnetic-visco-Pasternak foundation under three types of the longitudinal, circumferential, and simultaneous longitudinal–peripheral thermal and hygrothermal forces fields. It is assumed that the visco-Pasternak elastic medium consists of springs in the form of airy coils, so the application of an electric field to the system will also create a magnetic field and thus, the nano-system will become a magnetic–electric–elastic (MEE) state. The magnetic field will affect both the nanostructure as a Lorentz force and the passing nanofluid. The effect of magnetic nano-fluid on phonon dispersion is considered using Knudsen and Hartmann numbers. The equations governing the nano-fluid–structure interaction (Nano-FSI) problem under thermal, moisture, electrical, magnetic, and shear forces are derived using the proposed generalized theory of high-order shear deformation in cylindrical coordinates considering sinusoidal parameters and non-local elasticity, which we call CSN-HSDT for short. The Hamilton principle and the generalized Navier–Stokes equations are used for this purpose. In addition, the densities variations of nanostructures and different nanofluids (liquid or gas) due to thermal and thermal humidity fields, as well as the effects of the velocity of the nanostructure and the passing nanofluid on the system’s natural frequency are investigated. The results demonstrate that the axial velocity of the nanoshell will have a greater effect on the wave scattering and increasing the natural frequency of the system than the velocity of the transient fluid and this effect is decreased by increasing the wavenumber. Moreover, increasing the nanosystem stability and shifting the magnitude values of wave frequencies to higher amounts occurs by applying the fields of thermal and resulting humidity at room-temperatures, and using the negative voltage electric field, Lorentz force, and shear modulus. While the different results are obtained by utilizing the heat and humidity forces at high-temperatures, a positive voltage electric field, passing the magnetic nanofluid, and Winkler modulus. Here, the simultaneous effect of the mentioned force fields on the manner and speed of energy dispersion on a dynamic nanosystem under the influence of electromagnetic substrate and passing magnetic nanofluid is discussed for the first time. The results of the simultaneous application of these force fields in this study can be profitably exploited in the design and manufacturing of new smart structures used in structural health monitoring, energy harvesting, and drug delivery vessels in nanoscale.