Inhomogeneous Incompressible Fluid Research Articles

Classical Solutions of Rayleigh–Taylor instability for inhomogeneous incompressible viscous fluids in bounded domains

Classical Solutions of Rayleigh–Taylor instability for inhomogeneous incompressible viscous fluids in bounded domains

Uniform regularity and zero capillarity-viscosity limit for an inhomogeneous incompressible fluid model of Korteweg type in half-space

In this paper, we study the uniform regularity and zero capillarity-viscosity limit for an inhomogeneous incompressible fluid model of Korteweg type in the half-space R+3 . We consider the Navier-slip boundary condition for velocity and the Dirichlet boundary condition for the gradient of density. By establishing the conormal energy estimates, we prove that there exists a unique strong solution of the model in a finite time interval [0,T0] , where T 0 is independent of the capillary and viscosity coefficients, and the solution is uniformly bounded in a conormal Sobolev space. Based on the aforementioned uniform estimates, we further show that there exists a constant 0<T1⩽T0 , such that the solutions of this model converge to the solution of the inhomogeneous incompressible Euler equations with the rates of convergence in L∞(0,T1;L2(R+3)) and L∞(0,T1;H1(R+3)) , as the capillary and viscosity coefficients tend to zero simultaneously.

Применение методов Монте-Карло для решения регулярной и вырожденной задачи двухфазной фильтрации

One of popular mathematical models of filtration is the classical elastic regime model describing the nonstationary equilibrium filtration. It is also called the Muskat-Leverett model. Solving filtration problems by Monte Carlo methods makes it possible to find the solution of the problem at an individual point of the domain and to estimate derivatives of the solution. This paper is devoted to applying algorithms of the Monte Carlo method to problems of filtration. The Monte Carlo algorithms of random walk by spheres and on boundaries are used for solving the stationary problem of filtration of two immiscible inhomogeneous incompressible fluids in a porous medium and for estimating the solution and the derivatives of the solution of this problem.

Устойчивость сферического движения твердого тела с неоднородной жидкостью, совершающей однородное вихревое движение

In this paper, the equations of spherical motion of a solid body with a rotating inhomogeneous incompressible fluid that fills a completely ellipsoidal cavity are obtained and investigated. The stability of rotation of a solid with an inhomogeneous fluid having a linear density distribution is considered. Arbitrary fields of density and velocity of fluid particles are represented as a power series by spatial variables with coefficients that depend only on time. Sufficient conditions for the stability of the rotation of a solid body with a fluid around the vertical axis of dynamic symmetry are presented. The obtained equations of motion make it possible to study the stability of stationary motions of the system under consideration in order to assess the effect of fluid separation on the dynamics of the body. By analogy with the motion of a solid body, it is stated that the obtained conditions are also necessary and sufficient conditions for stationary rotations of an inhomogeneous fluid in an ellipsoidal cavity.

The Flow of An Inhomogeneous Fluid Inside a Sphere

The research deals with the stationary flow of an inhomogeneous incompressible fluid inside a spherical vessel under the influence of a potential mass force. Using the methods of four-dimensional analysis, the solution to the problem is constructed in an explicit analytical form. Exact solutions of the Euler equations for a homogeneous fluid are obtained only for some of the simplest problems. Researchers usually prove the existence and uniqueness of solutions to various initial - boundary value problems for Euler equations using the methods of a priori estimation. After that, the problem is usually solved by numerical methods. For an inhomogeneous fluid, when the unknown density is a variable, even obtaining a priori estimates becomes much more complicated, not to mention finding exact solutions. Nevertheless, in recent years, new methods of four-dimensional mathematics have been developed, giving previously unknown approaches to the study of nonlinear problems. In this paper, an exact analytical solution of the Euler equations describing the flow of an ideal inhomogeneous fluid inside a sphere is obtained. At the same time, the authors demonstrate new methods of four-dimensional analysis.

Global well-posedness for 3D incompressible inhomogeneous asymmetric fluids with density-dependent viscosity

In this paper, we investigate the 3D inhomogeneous incompressible asymmetric fluids system with density-dependent viscosity. By the assumption of the smallness of initial velocity in the critical Besov space, B˙p,13/p−1 for 1<p<6 and the initial density in the critical Besov space and bounded away from vacuum, the local and global well-posedness of 3D inhomogeneous incompressible asymmetric fluids is obtained. By giving the different estimates for pressure in the system for 1<p<32 and 32≤p<6, it shows the a priori estimate for the corresponding linearized equation. This not only improves the previous results for 3D inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity, but also obtains the new results on the micropolar system without smallness for density in critical Besov spaces.

Global strong solution to the 2D inhomogeneous incompressible magnetohydrodynamic fluids with density-dependent viscosity and vacuum

In this paper, we investigate an initial boundary value problem for two-dimensional inhomogeneous incompressible MHD system with density-dependent viscosity. First, we establish a blow-up criterion for strong solutions with vacuum. Precisely, the strong solution exists globally if |nabla mu (rho )|_{L^{infty }(0, T; L^{p})} is bounded. Second, we prove the strong solution exists globally (in time) only if |nabla mu (rho _{0})|_{L^{p}} is suitably small, even the presence of vacuum is permitted.

The global regularity for 3D inhomogeneous incompressible fluids with vacuum

In this paper, we investigate the global regularity for 3D inhomogeneous incompressible Navier–Stokes equations with vacuum. More precisely, we establish several Prodi–Serrin type regularity criteria in the Besov space in terms of only velocity or gradient of velocity which allow the initial density to contain vacuum. By using the embedding theory of Besov space, it is shown that our results improve the Prodi–Serrin type regularity condition which is scaling invariant in the time–space Lebesgue space.

On classical solutions of Rayleigh\u2013Taylor instability in inhomogeneous viscoelastic fluids

We study the nonlinear Rayleigh–Taylor (RT) instability of an inhomogeneous incompressible viscoelastic fluid in a bounded domain. It is well known that there exist strong solutions of RT instability in H^{2}-norm in inhomogeneous incompressible viscoelastic fluids, when the elasticity coefficient κ is less than some threshold kappa _{mathrm{C}}. In this paper, we prove the existence of classical solutions of RT instability in L^{1}-norm in Lagrangian coordinates based on a bootstrap instability method with finer analysis, if kappa <kappa _{mathrm{C}}. Moreover, we also get classical solutions of RT instability in L^{1}-norm in Eulerian coordinates by further applying an inverse transformation of Lagrangian coordinates.

Shallow flow of an inhomogeneous incompressible fluid down an inclined plane

In this paper, we study a model for the shallow flow of an inhomogeneous incompressible fluid down an inclined plane. The constitutive response of this class of fluids has been discussed in Malek and Rajagopal (Mech Mater 38(3):233–242, 2006). We consider a planar geometry in which the length scale ratio is small so that the lubrication approximation can be applied. We analyze two different flow regimes: in the first one, the viscous and the gravitational forces are balanced and the system is governed by a hyperbolic equation for the fluid film thickness. In the second regime, the frictional forces are balanced by the pressure gradient and the system is governed by a parabolic equation for the film thickness. For both regimes, we perform numerical simulations that show the evolution of the film thickness using some general data that are well suited for highlighting the main features of the flow.

Thin-Film Flow of an Inhomogeneous Fluid with Density-Dependent Viscosity

In this paper, we study the pressure-driven thin film flow of an inhomogeneous incompressible fluid in which its viscosity depends on the density. The constitutive response of this class of fluids can be derived using a thermodynamical framework put into place to describe the dissipative response of materials where the materials’ stored energy depends on the gradient of the density (Mechanics of Materials, 2006, 38, pp. 233–242). Assuming a small aspect ratio for the channel, we use the lubrication approximation and focus on the leading order problem. We show the mathematical problem reduce to a nonlinear first order partial differential equation (PDE) for the density in which the coefficients are integral operators. The problem is solved numerically and plots that describe the evolution of the density in the fluid domain are displayed. We also show that it was possible to determine an analytical solution of the problem when the boundary data are small perturbations of the homogeneous case. Finally, we use such an analytical solution to validate the numerical scheme.

Waves in inhomogeneous plasmas and liquid and gas flows. Analogies between electro- and gas-dynamic phenomena

Two rather closely analogous classes of phenomena — plasma and hydrodynamic — are reviewed. Topics covered include: surface waves in an inhomogeneous plasma layer and in an inhomogeneous incompressible fluid flow; the instability of the tangential discontinuity of an inhomogeneous gas flow and the negative mass instability of an electron-beam-carrying plasma; radiative Cherenkov instabilities of an electron beam in a plasma and of a gas flow in a gas; the instability of plasma flows and of incompressible fluid flows with inhomogeneous velocity profiles. Special attention is given to the collisionless damping of plasma surface waves. It is shown that no such damping occurs for surface waves in gas flows.

Two-dimensional thin-film flow of an incompressible inhomogeneous fluid in a channel

We study the 2D flow of an inhomogeneous incompressible fluid in a channel of small aspect ratio. The constitutive equation of the fluid consists of the classical incompressible Navier–Stokes equation plus an extra term depending on the density and its spatial gradient. This type of fluid is particularly suitable for describing the flow of incompressible granular materials. We formulate the mathematical problem and we exploit the smallness of the aspect ratio of the channel to obtain a simplified version (leading order of the lubrication expansion) that can be solved numerically. Choosing a particular set of boundary data we also determine a particular solution that can be compared to the numerical one to prove the consistency of the numerical scheme. The comparison shows good agreement. Finally we extend the model to the case of a channel with variable thickness and we compare our simulations with the ones obtained in Massoudi et al. (2012) [7].

Global strong solutions to the 3D inhomogeneous heat-conducting incompressible fluids

ABSTRACTThis paper concerns the global regularity to an initial boundary value problem for the three-dimensional (3D) inhomogeneous heat-conducting fluids with density and absolute temperature-dependent viscosity. Let be the initial velocity. Through some time-weighted a priori estimates, we establish global existence of strong solutions for positive initial density under assumption that is small. For the case when initial vacuum is allowed, we show that strong solutions globally exist provided small. It is worth pointing out that the initial temperature can be arbitrarily large.

Stability analysis of shear flows in a Hele–Shaw cell

A mathematical model describing motion of an inhomogeneous incompressible fluid in a Hele–Shaw cell is considered. Linear stability analysis of shear flow class is provided. The role of inertia, linear friction and impermeable boundaries in Kelvin–Helmholtz instability is studied. Hierarchy of simplified one-dimensional models of viscosity- and density-stratified flows is obtained in long-wave approximation. Interpretation of Saffman–Taylor instability is given in the framework of these models.

The global wellposedness of the 3D heat-conducting viscous incompressible fluids with bounded density

We consider the global wellposedness of the inhomogeneous incompressible heat-conducting viscous fluids in three dimension space. We generalize the result of Fujita & Kato for Navier–Stokes to the heat-conducting inhomogeneous incompressible viscous fluids. The key point is that we get the global wellposedness under the assumption that the initial density has positive lower and upper bound and the initial temperature can be arbitrarily large.

Analytical solutions to Stokes-type flows of inhomogeneous fluids

In this paper, we study the unsteady motion of an inhomogeneous incompressible viscous fluid, where the viscosity varies spatially according to various models. We study the Stokes-type flow for these types of fluids where in the first case the flow between two parallel plates is examined with one of the plates oscillating and in the second case when the flow is caused by a pulsatile pressure gradient. A general argument establishes the existence of oscillatory solutions to our problem. Exact solutions are obtained in terms of some special functions and comparisons are made with the cases of constant viscosity and the slow flow regimes.

Optimal boundary control of a system describing thermal convection

A problem of optimal boundary control of thermal sources for a stationary model of natural thermal convection of a high-viscosity inhomogeneous incompressible fluid in the Boussinesq approximation is investigated. Conditions for the solvability of the problem, as well as necessary and sufficient optimality conditions, are specified. Optimality conditions and the corresponding adjoint problems defining the gradient of the quality functional are written for several special cases of the functional. Computational procedures for finding an optimal control based on gradient methods are described. The results of numerical experiments are given.

Unsteady flows of inhomogeneous incompressible fluids

In this paper, we study the unsteady motion of inhomogeneous incompressible viscous fluids. We present the results corresponding to Stokes' second problem and for the flow between two parallel plates where one is oscillating.

Generalized potentiality of inhomogeneous media flows

A method for specifying a class of potential flows of inhomogeneous continuous media is developed. The general approach is based on expanding a medium material symmetry group to a special volume-preserving group, allowing us to obtain a law for the conservation of vorticity and, when there is no vorticity, to derive the unsteady Bernoulli equation. As illustrations, plane steady stationary flows of an inhomogeneous incompressible fluid and variable-entropy gas are considered. The problem of an inhomogeneous gas flow around a wedge yielding the formation of a shock wave is solved.