Motion Of Viscous Incompressible Fluid Research Articles

An inverse problem for generalized Kelvin–Voigt equation with p-Laplacian and damping term

In this paper, we consider the nonlinear inverse problem for generalized Kelvin–Voigt equations with the p-Laplace diffusion and damping term, describing the motion of incompressible viscous fluids. We assume that the damping term in the momentum equation depends on whether its signal is positive or negative, which may realizes the presence of a source or a sink within the system. The investigated inverse problem consists of finding a coefficient f(t) of the right-hand side of the momentum equation, a vector of velocity field v, and a pressure π. An additional information on a solution of the inverse problem is given as integral overdetermination condition. Under several assumptions on the exponents p, m, the coefficients μ, κ, γ, the dimension of the space d, and specified initial data, we prove the existence and uniqueness of the weak solution of the problem.

First-order necessary conditions of optimality for the optimal control of two-dimensional convective Brinkman–Forchheimer equations with state constraints

This work is concerned about some distributed optimal control problems governed by two-dimensional convective Brinkman–Forchheimer (CBF) equations, which describe the motion of incompressible viscous fluid through a rigid, homogeneous, isotropic, porous medium. We address a distributed optimal control problem of minimizing a suitable cost functional subjected to 2D CBF equations with state constraints. We establish the existence of an optimal solution and obtain the first-order necessary conditions of optimality for such optimal control problems.

The Time Optimal Control of Two Dimensional Convective Brinkman–Forchheimer Equations

In this work, we discuss the time optimal control of two dimensional convective Brinkman–Forchheimer (2D CBF) equations, which describe the motion of incompressible viscous fluid through a rigid, homogeneous, isotropic, porous medium. We establish Pontryagin’s maximum principle for the time optimal control of the 2D CBF equations.

Moderate deviation principle for the 2D stochastic convective Brinkman–Forchheimer equations

ABSTRACT This work is concerned about some asymptotic behaviour of the two-dimensional stochastic convective Brinkman–Forchheimer (SCBF) equations, which describes the motion of incompressible viscous fluid through a rigid, homogeneous, isotropic, porous medium. We establish a central limit theorem and a moderate deviation principle for the SCBF equations using a variational method (based on weak convergence approach) developed by Budhiraja and Dupuis.

Speed–direction description of turbulent flows

In this note, we introduce speed and direction variables to describe the motion of incompressible viscous fluid. Fluid velocity u is decomposed into u = ur, with u = |u| and r = u/|u|. We consider a directional split of the Navier–Stokes equations into a coupled system of equations for u and for r. The equation for u is particularly simple but solely maintains the energy balance of the system. Under the assumption of a weak correlation between fluctuations in speed and direction in a developed turbulent flow, we further illustrate the application of u–r variables to describe mean statistics of a shear turbulence. The standard (full) Reynolds stress tensor does not appear in a resulting equation for the mean flow profile.

Модель руху крові по артеріальному руслу з урахуванням біоактивності стінки судин

The modification of a two-dimensional model of incompressible viscous fluid motion along a deformed thick-walled tube from viscoelastic bioactive material is proposed in connection to the modeling of blood flow along the arterial bed is proposed. The motion of a viscous incompressible fluid is described by a system of equations including the Navier-Stokes equations and the continuity equation. The behavior of the tube wall material is described by a 5-element rheological model with one active element. The solution of the problem is solved setting boundary conditions on the interface of the two media, the outer surface of the tube is considered as non-moving. At the end of the tube, a zero-dimensional Frank model with regulation is considered, as a model of the microcirculatory bed. The dispersion equation for the propagation of wave velocity is obtained for the case of active properties of tube, the amplitudes of fluid velocities, wall displacements, and fluid and tube pressures. Numerical computations have been carried out for the model parameters corresponded to the normal and pathological arterial wall.

Unimodal solutions of the generalized Constantin–Lax–Majda equation with viscosity

Steady-states of the generalized Constantin–Lax–Majda equation with the viscosity and an external force are computed numerically by the spectral method. This equation is regarded as a model for two-dimensional turbulent motion of incompressible viscous fluid. We demonstrate numerically that the equation admits unimodal solutions—solutions with one and only one peak and bottom, if the Reynolds number is sufficiently large. We also report some interesting properties of the spectra of unimodal solutions.

Well-posedness of steady state Navier–Stokes equations with slip boundary conditions

ABSTRACTWe study the motion of a viscous incompressible fluid in a bounded region of containing finitely many rigid solid particles of small size. On the boundary of the particles, we prescribe the slip boundary condition of the form developed in Fujita H. [A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. Mathematical fluid mechanics and modeling (Kyoto, 1994). Srikaisekikenkysho Kkyroku No. 888; 1994. p. 199–216] and Le Roux C, Tani A. [Steady solutions of the Navier–Stokes equations with threshold slip boundary conditions. Math Methods Appl Sci. 2007;30(5):595–624]. We derive a weak formulation of the problem and obtain an equivalent variational inequality formulation. Sufficient conditions for existence, uniqueness, and continuous dependence on data are obtained.

A class of new exact solutions of the system of PDE for the plane motion of viscous incompressible fluids in the presence of body force

The purpose of this paper is to indicate a class of exact solutions of the system of partial differential equations governing the steady, plane motion of incompressible fluid of variable viscosity with body force term to the right-hand side of Navier-Stokes equations. The class consists of the stream function characterized by the equation  in polar coordinates  and  where  and  are continuously differentiable functions and the function  is such that  where a non-zero constant is  and overhead prime represents derivative with respect to . When  or  we show exact solutions for given one component of the body force for both the cases when the function  is arbitrary and when it is not. For the arbitrary function case,  appears in the coefficient of a linear second order ordinary differential equation showing a large numbers of solutions of this equation. This in turn establishes an infinite set of exact solutions to the problem concerned however; we show three examples of such exact solutions. The alternate case fixes  and provides viscosity as derivative of temperature function for  and . Anyhow, we find an infinite set of streamlines, the velocity components, viscosity function, generalized energy function and temperature distribution.

Numerical Investigation of Tungsten Reduction Process

The subject of this research was a chemical reactor for the producing tungsten coverings by means of chemical vapor deposition. The physical and mathematical model of viscous incompressible fluid motion and heat and mass transfer in a vortex chamber has been described. The velocity field, the temperature and the concentration distributions in a vortex chamber were obtained. Calculations reliability was verified by different methods. Parametric studies on effects of the Reynolds, Prandtl and Rossbi criteria on the flow characteristics were performed.

Global solvability of the Navier–Stokes equations with a free surface in the maximal Lp-Lq regularity class

We consider the motion of incompressible viscous fluids bounded above by a free surface and below by a solid surface in the N-dimensional Euclidean space for N≥2. The aim of this paper is to show the global solvability of the Navier–Stokes equations with a free surface, describing the above-mentioned motion, in the maximal Lp-Lq regularity class. Our approach is based on the maximal Lp-Lq regularity with exponential stability for the linearized equations, and also it is proved that solutions to the original nonlinear problem are exponentially stable.

Numerical estimation of various influence factors on a multipoint hydrostatic leveling system

Among various methods that allow controlling quasi-static vertical displacements of structures, the hydrostatic leveling method remains relevant. A multi-point hydrostatic leveling systems allows controlling the vertical displacement field with the required spatial resolution. It is assumed that the liquid level in the each measuring vessel has the same absolute elevation. However, it is influenced by various external factors, which are difficult to eliminate when implementing the method in real constructions. Consequently, it is necessary to assess the influence of these factors and develop methods aimed at their reduction or compensation. A mathematical model, describing the viscous incompressible fluid motion located in a multipoint level with absolutely rigid walls, is presented in the paper. The experiments performed with two point levels of different lengths, as well as analytical estimates of other authors, made it possible to estimate the degree of confidence of the model and the boundaries of applicability. The influence of non-uniform density of a liquid on the liquid level in a 4-point hydrostatic level of different topologies is numerically estimated using the model. An estimation of transient processes in the level, caused by an air pressure surge over one of the measuring vessels is carried out.

A Splitting Method for Numerical Simulation of Free Surface Flows of Incompressible Fluids with Surface Tension

Abstract The paper studies a splitting method for the numerical time-integration of the system of partial differential equations describing the motion of viscous incompressible fluid with free boundary subject to surface tension forces. The method splits one time step into a semi-Lagrangian treatment of the surface advection and fluid inertia, an implicit update of viscous terms and the projection of velocity into the subspace of divergence-free functions. We derive several conservation properties of the method and a suitable energy estimate for numerical solutions. Under certain assumptions on the smoothness of the free surface and its evolution, this leads to a stability result for the numerical method. Efficient computations of free surface flows of incompressible viscous fluids need several other ingredients, such as dynamically adapted meshes, surface reconstruction and level set function re-initialization. These enabling techniques are discussed in the paper as well. The properties of the method are illustrated with a few numerical examples. These examples include analytical tests and the oscillating droplet benchmark problem.

Mathematical modeling of Black Sea level change for the last 20000 years

A mathematical model of the Black Sea level change for the last 20,000 years is presented in this paper, based on the equations of viscous incompressible fluid motion according to Navier–Stokes, Sen-Venans, and the equation of water balance in the sea. The solution of the differential equations shows that the Black Sea level has been generally rising due to glacier melting and increase of freshwater balance in the late Pleistocene. Water which was flowing into the Black Sea did not have time to fully flow into the Marmara Sea because water was flowing through the long and very narrow canyon, which the Bosphorus strait was then. Water accumulating in the Black Sea increased the sea level, and increased the slope of the surface of river in the strait towards the Marmara Sea. When the level of the world ocean rose above river level, the strait began to form. Increasing depth caused excessive water to flow from the Black Sea. As a result of regression, the Black Sea level approached the Marmara Sea level, and the global oceanic transgression of the two seas began, continuing to the present.When the depth of the strait was 12–15 m, the penetration of the Marmara Sea's dense water into the Black Sea occurred. Water discharge in the counterflow was increasing when the ocean level was rising.The calculations showed that secondary fluctuations of the Black and Marmara Seas levels greater than 50 cm were theoretically impossible during the Holocene, as the deep strait allowed the possibility of intensive flow toward either the Black or Marmara Sea. When the value of the freshwater balance is positive, the regression of the Black Sea level below Marmara Sea level is theoretically impossible.Vertical movement of Earth crust can modify the shape of the eustatic sea level curves. A method allows calculation of the eustatic sea level course using the known local curves. The eustatic Mediterranean Sea level change and world ocean level change are calculated using nine local Mediterranean Sea level change curves. Tectonic components of the nine local curves were calculated. An assessment of the method's accuracy was made.

A note on the pipe flow with a pressure-dependent viscosity

We study a stationary motion of incompressible viscous fluid with a pressure-dependent viscosity in a thin (or long) straight pipe with variable cross-section. We assume that the flow is governed by the prescribed pressure drop between pipe’s ends. Under very general assumption on the viscosity law satisfied by the Barus formula and other empiric laws, the effective behavior of the flow is found via rigorous asymptotic analysis with respect to the pipe’s thickness.

Effective leak conditions across a membrane

The problem of effective boundary conditions for the flow of a viscous fluid across a type of permeable membranes is considered. Threshold leak conditions of subgradient type, introduced by [H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, Res. Inst. Math. Sci. Kokyuroku 888 (1994), pp. 199–216; H. Fujita, A coherent analysis of Stokes flows under boundary conditions of friction type, J. Comput. Appl. Math. 149(1) (2002), pp. 56–79] are considered on the periodic solid part of the membrane: the normal velocity is zero unless the jump in the normal stress across the membrane reaches a yield limit. The effective conditions are of subgradient type with an effective yield limit, in the case of a densely distributed solid part, or of Navier type, in the case of dilute solid part; in the intermediate case the tangential slip cancels, whereas the normal velocity and stress are continuous. Unlike in the case of perforated walls [Sanchez-Palencia, Un problème de l'écoulement lent d'un fluide visqueux incompressible au travers d'une paroi finement perforée, in Les Méthodes de l'Homogénèisation: Théorie et Applications en Physique, Collection de la Direction des Études et Recherches d'Électricité de France, Vol. 57, Eyrolles, Paris, 1985, pp. 371–400; E. Sanchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 127, Springer-Verlag, Berlin, 1980], no stress concentrations are present.

On Leray’s problem under Navier boundary conditions

We consider the stationary motions of a viscous incompressible fluid, of kinematical viscosity ν, in domains with cylindrical outlets to infinity of section &Sigma, and axis e 1 (distorted cylinders). We require that the fluid satisfies the classical boundary Navier condition u·n=0, [&gamma u+T (u,p)·n] &tau =0 (&gamma&ge 0) and tends at infinity to the laminar flow u N in the cylinder with section &Sigma having an assigned flux through &Sigma (Leray problem). We prove that there is a computable positive constant c 0 depending only on &Sigma, such that if ||u N || L &infin (&Sigma) 0 &nu, then the problem has at least a weak solution. Moreover, for &gamma=0 we can alternatively require that u tends at infinity to an assigned constant vector &mu e 1 , &mu&ne 0. In contrast with the most part of the known existence theorems for the steady--state Navier--Stokes problem we show that a solution (u,p) of such a problem exists for every &mu and &nu, is unique for large &nu and converges to &mu e 1 at infinity according to u(x)=&mu e 1 +O(e -&zeta |x 1 | ) for some positive constant &zeta .

Global solvability of the problem on the motion of two fluids without surface tension

Unsteady motion of viscous incompressible fluids is considered in a bounded domain. The liquids are separated by an unknown interface on which the surface tension is neglected. This motion is governed by an interface problem for the Navier-Stokes system. First, a local existence theorem is established for the problem in Holder classes of functions. The proof is based on the solvability of a model problem for the Stokes system with a plane interface, which was obtained earlier. Next, for a small initial velocity vector field and small mass forces, we prove the existence of a unique smooth solution to the problem on an infinite time interval. Bibliography: 7 titles.

The homogenized model of small oscillations of complex fluids

We consider the system of equations that describes small non-stationary motions of viscous incompressible fluid with a large number of small rigid interacting particles. This system is a microscopic mathematical model of complex fluids such as colloidal suspensions, polymer solutions etc. We suppose that the system of particles depends on a small parameter $\varepsilon$ in such a way that the sizes of particles are of order $\varepsilon^{3}$, the distances between the nearest particles are of order $\varepsilon$, and the stiffness of the interaction force is of order $\varepsilon^{2}$. We study the asymptotic behavior of the microscopic model as $\varepsilon\rightarrow 0$ and obtain the homogenized equations that can be considered as a macroscopic model of diluted solutions of interacting colloidal particles.

BEM for the two-dimensional plane compressible fluid dynamics

In this study, the boundary element method, which has been established for the viscous incompressible fluid motion, is modified and extended to capture the compressible fluid state. The velocity–vorticity formulation of the time dependent set of equations is employed, where the kinematics is given with the Poisson velocity vector equation, while the kinetics is represented with the vorticity transport equation, and the pressure field function is governed by the Poisson pressure scalar equation. The method is applied to consider buoyancy driven flow in closed cavity, differentially heated under large temperature gradients. The ideal gas law is used and viscosity is given by Sutherland law.