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Let us consider the motion of a viscous incompressible fluid past a rotating rigid body in 3D, where the translational and angular velocities of the body are prescribed but time-dependent. In a reference frame attached to the body, we have the Navier-Stokes system with the drift and (one half of the) Coriolis terms in a fixed exterior domain. The existence of the evolution operator $T(t,s)$ in the space $L^q$ generated by the linearized non-autonomous system was proved by Hansel and Rhandi [26] and the large time behavior of $T(t,s)f$ in $L^r$ for $(t-s)\to\infty$ was then developed by the present author [33] when $f$ is taken from $L^q$ with $q\leq r$. The contribution of the present paper concerns such $L^q$-$L^r$ decay estimates of $\nabla T(t,s)$ with optimal rates, which must be useful for the study of stability/attainability of the Navier-Stokes flow in several physically relevant situations. Our main theorem completely recovers the $L^q$-$L^r$ estimates for the autonomous case (Stokes and Oseen semigroups, those semigroups with rotating effect) in 3D exterior domains, which were established by [37], [42], [39], [36] and [44]., Comment: 45pages

The diffusion of cells in a viscous incompressible fluid (e.g. water) may be viewed like movement in a porous medium and there is a bidirectorial oxygen exchange between water and their surrounding air in thin fluid layers near the air–water contact surface. This leads to the following chemotaxis-Navier–Stokes system with nonlinear diffusion: [Formula: see text] endowed with the inhomogeneous boundary conditions [Formula: see text] and the initial data [Formula: see text] in [Formula: see text], where the incoming oxygen [Formula: see text] is non-negative, and the outgoing oxygen molecule is modeled by [Formula: see text] with positive coefficient [Formula: see text]. In this paper, we investigate the asymptotic dynamics of the above system in a bounded domain [Formula: see text] with the smooth boundary [Formula: see text]. We will show that arbitrary porous medium diffusion mechanism [Formula: see text] can inhibit the singularity formation. In the incoming oxygen-free case, we further prove that the solution will stabilize to the unique mass-preserving spatial equilibrium [Formula: see text] in the sense that as [Formula: see text], [Formula: see text] hold uniformly with respect to [Formula: see text], where [Formula: see text].

We study unsteady shear flows realized in a half-plane with viscous incompressible fluid, where the law of motion of the boundary oscillating along itself is given. Either the longitudinal velocity of the boundary or the shear stress on it can be specified. The statement of the linearized problem with respect to small initial perturbations imposed on the kinematics in the entire half-plane is presented. For a flat picture of perturbations, the statement consists of a single biparabolic equation with variable coefficients with respect to the complex-valued stream function that generalizes the Orr-Sommerfeld equation to the nonstationary case and of four homogeneous boundary conditions. Using the method of integral relations, we derive exponential estimates for the decay of perturbations. The result is compared with the three-dimensional picture of variations.

We studied the non-isothermal flow of an incompressible viscous fluid through a porous tube. Motivated by filtration problems, Darcy’s law was incorporated on the walls of the tube and the flow was pressure driven. The main goal was to investigate the thermodynamic part of the system, assuming that the hydrodynamic part is known. In view of the applications we wanted to model, the fluid inside the tube was supposed to be cooled (or heated) by the surrounding medium. Using asymptotic analysis with respect to the small parameter (being the ratio between the tube’s thickness and its length), we constructed the explicit second-order approximation for the temperature distribution of the fluid. Numerical examples are provided to compare the obtained solution with the one derived for a rigid tube and also to show the corrections due to higher-order terms.

We consider the Stokes-transport system, a model for the evolution of an incompressible viscous fluid with inhomogeneous density. This equation was already known to be globally well-posed for any L 1 ∩ L ∞ initial density with finite first moment in R 3 . We show that similar results hold on different domain types. We prove that the system is globally well-posed for L ∞ initial data in bounded domains of R 2 and R 3 as well as in the infinite strip R × ( 0 , 1 ) . These results contrast with the ill-posedness of a similar problem, the incompressible porous medium equation, for which uniqueness is known to fail for such a density regularity.

The paper considers a mathematical model for natural vibrations of a periodic layered medium. The medium consists of a viscoelastic Kelvin-Voigt material and a viscous incompressible fluid. For the given model, two homogenized models are derived. They correspond to the cases of transverse and longitudinal vibrations of the layered medium. It is shown that the spectrum of each homogenized model is the union of roots of the corresponding quadratic equations.

This paper deals with a two-chemical reaction type Keller-Segel system coupled with incompressible viscous fluid equations which models the dynamics of cells in fluid in a two dimensional bounded domain $\varOmega $ $$ \left\{ \textstyle\begin{array}{l} \partial _{t}n+\mathbf {u}\cdot \nabla n = \Delta n - \nabla \cdot \bigl(n\chi(n,v,w,x)\nabla v\bigr), \\ \partial _{t}v+\mathbf {u}\cdot \nabla v= \Delta v -v + w, \\ \partial _{t}w+\mathbf {u}\cdot \nabla w = \Delta w -w + n, \\ \partial _{t}\mathbf {u}+ \nabla P = \Delta \mathbf {u}+ n\nabla \psi, \quad \nabla \cdot \mathbf {u}= 0. \end{array}\displaystyle \right. $$ Here, $\chi(n,v,w,x)$ represents the saturated sensitivity. Our result suggests that suitable saturation can prevent the blow-up arising from the classical Keller-Segel type signal production mechanism without any smallness condition on initial mass.

Рассмотрены плоские нестационарные течения вязкой несжимаемой жидкости в потенциальном поле внешних сил. Получено уравнение в частных производных эллиптического типа, каждое решение которого является функцией тока вихревого течения, описываемого некоторым точным решением уравнений Навье--Стокса. Полученные решения обобщают течения Бельтрами--Тркала и Беллаба. Даны примеры таких новых решений. Они предназначены для верификации численных алгоритмов и компьютерных программ.

This article examines the influence of isothermal and isoflux heating/cooling with heat source/sink on unsteady hydromagnetic-free convective flow of a viscous incompressible fluid in an annulus. The flow is induced by buoyancy forces due to temperature differences as a result of the isothermal heating and constant heat flux applied on the outer surface of the inner cylinder. Our treatment of the governing energy and momentum equation is based on a two-step approach. First, the nonlinear partial differential equations are transformed to the Laplace domain using the Laplace transform method and solved analytically in the Laplace domain. The Laplace domain solution is then inverted back to time domain using a numerical scheme known as Riemann-Sum Approximation (RSA). The derived numerical values by the method of RSA exhibit an excellent conformity with PDEPE and steady-state solutions at large time. The influence of the various dimensionless controlling parameters on the flow formation is depicted graphically and in tabular form.

In this paper, we provide stability analysis for a stationary Stokes hemivariational inequality where along the tangential direction of the slip boundary, an inclusion relation involving the generalized subdifferential of a superpotential is specified. With viscous incompressible fluid flows as application background, stability is analyzed for solutions with respect to perturbations in the superpotential and the density of external forces. We also present a result on the existence of a solution to an optimal control problem for the stationary Stokes hemivariational inequality.

The paper presents a new class of exact solutions for the Navier–Stokes equations. These solutions describe unsteady three-dimensional in velocities and two-dimensional in coordinates for a viscous incompressible fluid flow. The procedure for constructing an exact solution generalizes Trkal's method proposed for studying screw flows. The new class of exact solutions allows to describe non-hecical flows (the velocity vector forms a nonzero angle with the vorticity vector) and fluid flows existing in a finite time.

The recurrent system of second order partial differential equations serving for asymptotic approximation to the Stokes equation has been derived by the asymptotic method. The zeroth order and basic approximation is the Prandtl equation used to study the non-steady-state boundary layer. The exact analytical solutions of the Prandtl equation with a pressure gradient have been obtained.

This paper considers the steady and unsteady swirling axisymmetric flows of a homogeneous viscous incompressible fluid. The possibility of the existence of helical vortex lines on the surface of revolution homeomorphic to a torus is investigated. An example of unsteady flow in which there are helical vortex lines is given. It is proved that the existence of helical vortex lines lying on the surface of revolution homeomorphic to a torus is impossible in a steady axisymmetric flow of a viscous incompressible fluid.

This paper describes an overdetermined system of equations that describe three-dimensional layered unsteady flows of a. viscous incompressible fluid at a. constant pressure. Studying the compatibility of this system makes it possible to reduce it to coupled quasilinear parabolic equations for velocity components. The reduced equations allow constructing several classes of exact solutions. In particular, polynomial and spatially localized self-similar solutions of the equations of motion are obtained. The transition to the ideal fluid limit is investigated.

International audience; In this paper, we consider a system modeling the interaction between a viscous incompressible fluid and an elastic structure. The fluid motion is represented by the classical Navier-Stokes equations while the elastic displacement is described by the linearized elasticity equation. The elastic structure is immersed in the fluid and the whole system is confined into a general bounded smooth domain of R3. Our main result is the local in time existence and uniqueness of a strong solution of the corresponding system.

The three-dimensional flow of a viscous incompressible fluid in a cylindrical duct with two serial diaphragms (constrictions) is studied by the numerical solution of non–stationary Navier–Stokes equations. The solution algorithm is based on the finite volume method using difference schemes second-order accurate in both space and time. The TVD form of a central-difference scheme with a flow limiter is used for the interpolation of convec tive terms. The combined computation of the velocity and pressure fields is carried out, by using the PISO procedure. It is shown that, in a certain range of Reynolds numbers, the fluid flow in the region between the diaphragms is non-stationary and is characterized by the presence of an unstable shear layer formed by the boundary layer that breaks off from the surface of the first diaphragm. In the cavity between the diaphragms, a circulating motion of the medium is formed, which can be interpreted as a hydrodynamic feedback channel that creates conditions for the occurrence of self-sustained oscillations in the system. A sequential series of ring vortices is for med in the shear layer that cause self-oscillations of the velocity and pressure fields in a vicinity of the orifice of the second diaphragm, as well as pressure oscillations in the whole medium between the diaphragms. These self-sustained oscillations may serve as an acoustic source in the duct. The obtained results are compared with the model of an axisymmetric flow in a cylindrical duct with two diaphragms. The structure of the three–dimensional flow has an azimuthal asymmetry that substantially affects the local features of the flow. There is an asymmetry of the circulating motion of the medium in the cavity between the diaphragms and of the ring vortices in the shear layer. However, the oscillation periods of the velocity and pressure fields coincide with those in the model of axisymmetric flow. Thus, the asymmetry of the flow practically does not affect its integral characteristics.

The kinetics of flow of a viscous incompressible fluid between two closely lying parabola-shaped inserts (plates) rotating with the same angular velocity around a common axis is studied. The flow, in the inter-plate gap of the separator drum, of a weakly concentrated liquid + solid type of two-phase fluid is modeled. Based on the postulates of the theory of fluid flows in “sliding” mode, the internal problem of hydrodynamics along the flow in the chosen working space of the centrifugal equipment is proposed to be solved in linear formulation. For an isolated small spherical particle moving in the flow, the critical diameter of the particle (settling in the working space of the separator), which determines the separation efficiency, is calculated by solving the external problem of hydrodynamics.

The problem of the interaction between two oppositely swirled jets of viscous incompressible fluid issuing from parallel tubes into the submerged space is considered. The scenario of the confluence of the two jets into a single jet is determined on the basis of a numerical solution. It is shown that far from the sources the flow is described by the self-similar axisymmetric Landau solution for a single swirled jet. The longitudinal vorticity field far from the source is also self-similar, with a complex decay rate. The asymptotic structure of the far field of the jet is discussed.

In this paper, we study the normal mode solutions of 3D incompressible viscous fluid flow models. The obtained theoretical results are then applied to analyze several time-stepping schemes for the numerical solutions of the 3D incompressible fluid flow models.

Several problems of motion of a viscous incompressible fluid with a point source in the flow region are considered. The corresponding initial-boundary-value problems for the Navier-Stokes equations have no solutions in the standard class of functions because the flow velocity field contains an infinite Dirichlet integral. Problem regularization allows one to prove its solvability under certain constraints on the initial data.

In this paper, we study the inhomogeneous boundary value problem for the steady MHD system of a viscous incompressible fluid in an arbitrary bounded multiply connected domain. We prove the existence of generalized solutions of the steady MHD system with some smallness conditions on the boundary values.

Three complex variable circle theorems for studying the two-dimensional Stokes flows interior to a circular cylinder are presented. These theorems are formulated in terms of the complex velocities of the fundamental singularities in an unbounded incompressible viscous fluid. Illustrative examples are given to demonstrate their usefulness.