Solutions to the Navier-Stokes Equations Describing Local Flows of Stratified and Compressible Fluids

A family of exact solutions of the Navier-Stokes equations is used to describe local flows of incompressible stratified and compressible fluids. For some of the flows, the coefficient of viscosity can depend on the temperature. An example of an incompressible stratified flow for which the analysis is applicable is the sheared swirling flow that is produced between two parallel plates that translate with different velocities and rotate with different angular velocities about different, but parallel, axes. The fluid may be stratified in the direction normal to the plates. These generalized von Karman flows are relevant to the study of strong local atmospheric disturbances, such as might be produced by the passage of a tornado. Also, when the coefficient of viscosity depends on the temperature, they can be used to analyze the flow of molten metals between surfaces that are in relative motion. An example of a compressible flow for which the analysis is applicable is that produced by a plane shock wave as it traverses a layer where the fluid is sheared in a direction normal to the shock.     

Uniaxial exchange flows of two Bingham fluids in a cylindrical duct

Buoyancy driven flows of two Bingham fluids in an inclined ductare considered, providing a simplified model for many oilfieldcementing processes. The flows studied are near-uniaxial andstratified, with the heavy fluid moving down the incline, displacingthe lighter fluid upwards. Existence and uniqueness resultsare obtained for quite general flows and for those which satisfyan axial flow rate constraint. Parametric dependence of thesolutions on the axial pressure gradient is studied. Flows whichsatisfy a zero net axial flow constraint result from an axialpressure gradient which minimizes the viscous dissipation, butnot the plastic dissipation. A regularization method is usedto compute solutions to these problems for (more or less) arbitraryfluid-fluid interfaces and duct-cross sections. Examples relatedto a number of practical applications are presented.     

On power-law fluids with the power-law index proportional to the pressure

In this short note we study special unsteady flows of a fluid whose viscosity depends on both the pressure and the shear rate. Here we consider an interesting dependence of the viscosity on the pressure and the shear rate; a power-law of the shear rate wherein the exponent depends on the pressure. The problem is important from the perspective of fluid dynamics in that we obtain solutions to a technologically relevant problem, and also from the point of view of mathematics as the analysis of the problem rests on the theory of spaces with variable exponents. We use the theory to prove the existence of solutions to generalizations of Stokes’ first and second problem.     

Global existence of solutions for flows of fluids with pressure and shear dependent viscosities

There is clear and incontrovertible evidence that the viscosity of many liquids depends on the pressure. While the density, as the pressure is increased by orders of magnitude, suffers small changes in its value, the viscosity changes dramatically. It can increase exponentially with pressure. In many fluids, there is also considerable evidence for the viscosity to depend on the rate of deformation through the symmetric part of the velocity gradient, and most fluids shear thin, i.e., viscosity decreases with an increase in the rate of shear. In this paper, we study the flow of fluids whose viscosity depends on both the pressure and the symmetric part of the velocity gradient. We find that the shear thinning nature of the fluid can be gainfully exploited to obtain global existence of solution, which would not be possible otherwise. Previous studies of fluids with pressure dependent viscosity require strong restrictions to all data, or assume forms that are clearly contrary to experiments, namely that the viscosity decreases with the pressure. We are able to establish existence of space periodic solutions that are global in time for both the two- and three-dimensional problem, without restricting ourselves to small data.     

Global well-posedness for the dynamical Q-tensor model of liquid crystals

We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions.     

Global well-posedness for the dynamical <Emphasis Type="Italic">Q</Emphasis>-tensor model of liquid crystals

We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions.     

A Class of Solutions for the Equations Governing the Steady Two-Dimensional Motion of a Viscous Incompressible Liquid

A general viscosity dependent solution for the stream function is found satisfying the simplest nondegenerate form of the steady flow Navier-Stokes equations for a viscous incompressible liquid. The solution is two-dimensional and is expressed in terms of arbitrary analytic functions in the fluid domain. This class of flows is generated by complex stream functions, and the region of definition is restricted by an inequality containing these analytic functions. A general potential flow, and degenerate Stokes or creeping flows are recovered as particular solutions in limiting cases.     

Shear flows of a new class of power-law fluids

We consider the flow of a class of incompressible fluids which are constitutively defined by the symmetric part of the velocity gradient being a function, which can be non-monotone, of the deviator of the stress tensor. These models are generalizations of the stress power-law models introduced and studied by J. Málek, V. Pr??a, K.R. Rajagopal: Generalizations of the Navier-Stokes fluid from a new perspective. Int. J. Eng. Sci. 48 (2010), 1907–1924. We discuss a potential application of the new models and then consider some simple boundary-value problems, namely steady planar Couette and Poiseuille flows with no-slip and slip boundary conditions. We show that these problems can have more than one solution and that the multiplicity of the solutions depends on the values of the model parameters as well as the choice of boundary conditions.     

Some Self-Similar Two-Phase Flows of Non-Newtonian Fluids Through a Porous Medium

Pascal This paper addresses the question of the rheological effects of non-Newtonian fluids in a flow system, in which a two-phase flow zone is coupled to a single-phase flow zone by a moving fluid interface. This flow system is involved in a technique for oil displacement in a porous medium, where a non-Newtonian displacing fluid (a polymer solution) is used to displace a non-Newtonian heavy oil. The self-similar solutions of the equations governing the dynamics of the moving interface, separating the displacing and displaced fluids, are obtained for the one-dimensional and plane radial flows. The effects associated with the presence of a two-phase flow zone, behind the moving interface, on the interface movement are analyzed. The existence of a pressure front ahead of the moving interface, moving with a finite velocity, is also shown. The relevance of this result to the propagation of pressure disturbances in a non-Newtonian fluid flowing through a porous medium is discussed with regard to interpretation of the transient pressure response in an injection well for polymer-solution floods.     

精确计算Bingham流体圆管层流压降及速度分布规律的新方法

Bingham(宾汉)模型情况下,多采用通用公式进行圆管层流压降的解析计算,即将Bingham模型本构方程代入粘性流体圆管层流流动通用公式进行计算,仅能得到压降的解析解.新方法结合Bingham流体本构方程与运动方程,建立有关力学平衡方程,并运用代数方程的根式解理论对圆管层流流动时的非线性方程进行求解,可直接求得Bingham流体圆管层流压降及速度流核区半径的解析解,进一步可求得圆管层流速度解析解;Bingham流体圆管层流速度的直接影响因素为流量、塑性粘度和屈服值,研究发现速度流核宽度与屈服值成正比,与流量及塑性粘度成反比,且流核的宽度越大,流核区的速度越小.     

On stationary flows with energy dependent nonlocal viscosities

A nonlocal constitutive law for an incompressible viscous flow in which the viscosity depends on the total dissipation energy of the fluid is obtained as the limit case of very large thermal conductivity when the viscosity varies with the temperature. A rigorous analysis is illustrated within the Hilbertian framework for unidirectional stationary flows of Newtonian and Bingham fluids with heating by viscous dissipation. An extension to quasi-Newtonian fluids of power law type and with temperature dependent viscosities is obtained in the context of the heat equation with an L¹-term. The nonlocal model proposed by Ladyzhenskaya in 1966 as a modification of Navier-Stokes equations can be, in particular, obtained with this procedure. Bibliography: 14 titles.Dedicated to O. A. Ladyzhenskaya on the occasion of her 80th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 99–117.     

Exact solutions of problems of the three-dimensional flow of ideal viscous incompressible fluids near cylindrical surfaces

Three-dimensional flows of an incompressible fluid, the parameters of which depend on two coordinates and time, are considered. The stream surfaces of such flows are cylindrical. The equations of continuity and the Navier-Stokes equations can be transformed to relations, one of which is the equation for the stream function the other is the integral of the equations relating the pressure and the stream function, and the third is a linear equation for the projection of the velocity vector onto the axis parallel to the generatrix of the cylindrical surfaces. The problems of modelling the flows are considered on the basis of the exact solutions of the Navier-Stokes equations and Euler's equations using examples. Relations for the distribution of the flow parameters in the channel created by hyperbolical cylinders are derived for the case of unsteady inviscid flow. The streamlines of these flows are situated on the side surfaces of the hyperbolical cylinders and intercept the generatrices of the cylinders at certain indirect angles. The flow around a circular cylinder and the flow of fluid inside an elliptic cylinder are considered in the case of steady inviscid flow. The streamlines on the circular cylinder are arranged transverse to the cylinder (the projection of the velocity vector onto the coordinate axis, parallel to the generatrix of the cylinder, is equal to zero). Far from the cylinder the streamlines are also situated on a cylindrical surfaces, but not transverse to the cylinder, making certain indirect angles with the generatrix. Viscous three-dimensional flows, possessing a certain symmetry, are considered. In the case of radial symmetry the streamlines are helical lines. The non-planar Couette flow between parallel moving planes is characterized by the fact that the velocity vectors, being situated in the same plane, are collinear, while the velocity vectors in parallel planes are not collinear. Relations for viscous steady three-dimensional flows, using well-known relations, obtained for the stream function of two-dimensional flows, are given.     

A note on the flow of a fluid with pressure-dependent viscosity in the annulus of two infinitely long coaxial cylinders

The dependence of the viscosity of fluids on pressure has been well established by experiments and it needs to be taken into consideration in problems where there is a large variation of pressure in the flow domain. In this paper we consider the flow of a fluid in the annulus between two cylinders whose viscosity depends on the pressure. First we consider the steady flow in the annulus due to the rotation of one cylinder with respect to the other. Then we study the problem of flow in the annular region due to torsional and longitudinal oscillations of one cylinder with respect to the other. In both the problems considered the flow is found to be markedly different from that for the incompressible Navier–Stokes fluid with constant viscosity.     

Stability of the one-dimensional motions of a viscous gas with a linear dependence of the velocity on the coordinates

The stability of a new solution of the equations of one-dimensional gas dynamics is investigated. This solution is a generalization of the solutions of Sedov /1, 2/ to the case of a viscous, thermally conducting ideal gas with an exponential dependence of the coefficient of viscosity and thermal conductivity on temperature. The linearized equations for small perturbations (the effects of thermal conductivity are not allowed for in the equations for the perturbations), which contain functions of time and the radial coordinate in the coefficients, can be solved by separation of the variables. The conditions under which instability arises are determined from an analysis of the time parts of the solutions. The stability of the solutions /1/ has been considered in /3–5/.     

Somk remarks on the navier-stokes equations with a pressure-dependent viscosity

We discuss the Navier-Stokes equations for an incompressible fluid wit a viscosity that is allowed to depend on the pressure. Ellipticity and the complementing condition of Agmon, Douglis and Nirenberg [l] are discussec It is found that the pressure dependence of viscosity leads to the possibilit of a change of type. It is shown that the Dirichlet initial-boundary valu problem is well posed as long as the equations do not change type     

Unsteady flows of fluids with pressure dependent viscosity in unbounded domains

In order to describe the behavior of various liquid-like materials at high pressures, incompressible fluid models with pressure dependent viscosity seem to be a suitable choice. In the context of implicit constitutive relations involving the Cauchy stress and the velocity gradient these models are consistent with standard procedures of continuum mechanics. Understanding the mathematical properties of the governing equations is connected with various types of idealizations, some of them lead to studies in unbounded domains. In this paper, we first bring up several characteristic features concerning fluids with pressure dependent viscosity. Then we study the three-dimensional flows of a class of fluids with the viscosity depending on the pressure and the shear rate. By means of higher differentiability methods we establish the large data existence of a weak solution for the Cauchy problem. This seems to be a first result that analyzes flows of considered fluids in unbounded domains. Even in the context of purely shear rate dependent fluids of a power-law type the result presented here improves some of the earlier works.     

Unsteady motion of a spherical bubble in a complex fluid: Mathematical modelling and simulation

The nonlinear response of an oscillatory bubble in a complex fluid is studied. The bubble is immersed in a Newtonian liquid, which may have a dilute volume fraction of anisotropic additives such as fibers or few ppm of macromolecules. The constitutive equation for the fluid is based on a Maxwell model with an extensional viscosity for the viscous contribution. The model is considered new in the study of bubble dynamics in complex fluids. The numerical computation solves a system of three first order ordinary differential equations, including the one associated with the solution of the convolution integral, using a fifth order Runge–Kutta scheme with appropriated time steps. Asymptotic solutions of governing equation are developed for small values of the pressure forcing amplitude and for small values of the elastic parameter. A study of the bubble collapse radius is also presented. We compare the results predicted by our model with other model in the literature and a good agreement is observed. The calculated asymptotic solutions are also used to test the results of the numerical simulations. In addition, the orientation of the additives is considered. The angular probability density function is assumed to be a normal distribution. The results show that the model based on the fully aligned additives with the radial direction overestimates the tendency of the additives to stabilize the bubble motion, since the effect of extensional viscosity occurs due to the particle resistance to the movement throughout its longitudinal direction.     

Unsteady Couette flows in a second grade fluid with variable material properties

Fluid solid mixtures are generally considered as second grade fluids and are modeled as fluids with variable physical parameters. Thus, an analysis is performed for a second grade fluid with space dependent viscosity, elasticity and density. Two types of time-dependent flows are investigated. An eigen function expansion method is used to find the velocity distribution. The obtained solutions satisfy the boundary and initial conditions and the governing equation. Remarkably some exact analytic solutions are possible for flows involving second grade fluid with variable material properties in terms of trigonometric and Chebyshev functions.     

Remarks on Vanishing Viscosity Limits for the 3D Navier-Stokes Equations with a Slip Boundary Condition

The authors study vanishing viscosity limits of solutions to the 3-dimensional incompressible Navier-Stokes system in general smooth domains with curved boundaries for a class of slip boundary conditions. In contrast to the case of flat boundaries, where the uniform convergence in super-norm can be obtained, the asymptotic behavior of viscous solutions for small viscosity depends on the curvature of the boundary in general. It is shown, in particular, that the viscous solution converges to that of the ideal Euler equations in C([0, T];H1(Ω)) provided that the initial vorticity vanishes on the boundary of the domain.     

An exact solution for flow in a porous pipe

Summary An exact solution of the Navier-Stokes equations for flow in a porous pipe is presented. This solution allows the suction or injection at the wall to vary with axial distance and will provide new insight into flows through porous pipes.

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Resumé Une solution exacte d'équation de Navier-Stokes est présentée pour l'écoulement d'un liquide visqueux dans un tube perméable. Ce liquide est aspiré ou injecté avec une vélocité variable et la solution donne une nouvelle optique quant aux tubes poreuses.