Abstract:
This note deals with the influence of heat and mass transfer on peristaltic flow of an viscous fluid with wall slip condition. The flow is investigated in a wave frame of reference moving with the velocity of the wave. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitude and phase. The momentum and energy equations have been linearized under the assumption of long-wavelength approximation. The arising equations are solved by perturbation technique and the expressions for Temperature, Concentration, Velocity and Stream function are constructed. Graphical results are sketched for various embedded parameters and discussed in detail.

Abstract:
We study in this paper the movement of a rigid solid inside an incompressible Navier-Stokes flow, within a bounded domain. We consider the case where slip is allowed at the fluid/solid interface, through a Navier condition. Taking into account slip at the interface is very natural within this model, as classical no-slip conditions lead to unrealistic collisional behavior between the solid and the domain boundary. We prove for this model existence of weak solutions of Leray type, up to collision, in three dimensions. The key point is that, due to the slip condition, the velocity field is discontinuous across the fluid/solid interface. This prevents from obtaining global H1 bounds on the velocity, which makes many aspects of the theory of weak solutions for Dirichlet conditions unadapted.

Abstract:
We study a nonlinear, unsteady, moving boundary, fluid-structure interaction (FSI) problem arising in modeling blood flow through elastic and viscoelastic arteries. The fluid flow, which is driven by the time-dependent pressure data, is governed by 2D incompressible Navier-Stokes equations, while the elastodynamics of the cylindrical wall is modeled by the 1D cylindrical Koiter shell model. Two cases are considered: the linearly viscoelastic and the linearly elastic Koiter shell. The fluid and structure are fully coupled (2-way coupling) via the kinematic and dynamic lateral boundary conditions describing continuity of velocity (the no-slip condition), and balance of contact forces at the fluid-structure interface. We prove existence of weak solutions to the two FSI problems (the viscoelastic and the elastic case) as long as the cylinder radius is greater than zero. The proof is based on a novel semi-discrete, operator splitting numerical scheme, known as the kinematically coupled scheme, introduced in \cite{GioSun} to solve the underlying FSI problems. The backbone of the kinematically coupled scheme is the well-known Marchuk-Yanenko scheme, also known as the Lie splitting scheme. We effectively prove convergence of that numerical scheme to a solution of the corresponding FSI problem.

Abstract:
We study a nonlinear, moving boundary fluid-structure interaction problem between an incompressible, viscous Newtonian fluid, modeled by the 2D Navier-Stokes equations, and an elastic structure modeled by the shell or plate equations. The fluid and structure are coupled via the {\em Navier slip boundary condition} and balance of contact forces at the fluid-structure interface. The slip boundary condition is more realistic than the classical no-slip boundary condition in situations, e.g., when the structure is "rough", and in modeling dynamics near, or at a contact. Cardiovascular tissue and cell-seeded tissue constructs, which consist of grooves in tissue scaffolds that are lined with cells, are examples of "rough" elastic interfaces interacting with and incompressible, viscous fluid. The problem of heart valve closure is an example of a fluid-structure interaction problem with a contact. We prove the existence of a weak solution to this class of problems by designing a constructive proof based on the time discretization via operator splitting. This is the first existence result for fluid-structure interaction problems involving elastic structures satisfying the Navier slip boundary condition

Abstract:
We consider the 3-D full Navier-Stokes equations whose the viscosity coefficients and the thermal conductivity coefficient depend on the density and the temperature. We prove the local existence and uniqueness of the strong solution in a domain $\Omega\subset\mathbb{R}^3$. The initial density may vanish in an open set and $\Omega$ could be a bounded or unbounded domain. We also prove a blow-up criterion for the solution. Finally, we show the blow-up of the smooth solution to the compressible Navier-Stokes equations in $\mathbb{R}^n$ ($n\geq1$) when the initial density has compactly support and the initial total momentum is nonzero.

Abstract:
In this paper we study a coupled system modeling the movement of a deformable solid immersed in a fluid. For the solid we consider a given deformation that has to obey several physical constraints. The motion of the fluid is modeled by the incompressible Navier-Stokes equations in a time-dependent bounded domain of $\R^3$, and the solid satisfies the Newton's laws. Our contribution consists in adapting and completing some results of ARMA 2008 in dimension 3, in a framework where the regularity of the deformation of the solid is limited. We rewrite the main system in domains which do not depend on time, by using a new means of defining a change of variables, and a suitable change of unknowns. We study the corresponding linearized system before setting a local-in-time existence result. Global existence is obtained for small data, and in particular for deformations of the solid which are arbitrarily close to the identity.

Abstract:
The present numerically study investigates the influence of the Hall current and constant heat flux on the Magneto hydrodynamic (MHD) natural convection boundary layer viscous incompressible fluid flow in the manifestation of transverse magnetic field near an inclined vertical permeable flat plate. It is assumed that the induced magnetic field is negligible compared with the imposed magnetic field. The governing boundary layer equations have been transferred into non-similar model by implementing similarity approaches. The physical dimensionless parameter has been set up into the model as Prandtl number, Eckert number, Magnetic parameter, Schmidt number, local Grashof number and local modified Grashof number. The numerical method of Nactsheim-Swigert shooting iteration technique together with Runge-Kutta six order iteration scheme has been used to solve the system of governing non-similar equations. The physical effects of the various parameters on dimensionless primary velocity profile, secondary velocity profile, and temperature and concentration profile are discussed graphically. Moreover, the local skin friction coefficient, the local Nusselt number and Sherwood number are shown in tabular form for various values of the parameters.

Abstract:
We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain $\Omega = \Omega_0 \times (0,L) \in \mathbb{R}^3$. We show existence of a solution $(v,\rho) \in W^2_p(\Omega) \times W^1_p(\Omega)$, where $v$ is the velocity of the fluid and $\rho$ is the density, that is a small perturbation of a constant flow $(\bar v \equiv [1,0,0], \bar \rho \equiv 1)$. We also show that this solution is unique in a class of small perturbations of $(\bar v,\bar \rho)$. The term $u \cdot \nabla w$ in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence $(v^n,\rho^n)$ that is bounded in $W^2_p(\Omega) \times W^1_p(\Omega)$ and satisfies the Cauchy condition in a larger space $L_{\infty}(0,L;L_2(\Omega_0))$ what enables us to deduce that the weak limit of a subsequence of $(v^n,\rho^n)$ is in fact a strong solution to our problem.

Abstract:
We consider a boundary value problem for the system of equations describing the stationary motion of a viscous nonhomogeneous asymmetric fluid in a bounded planar domain having a $C^2$ boundary. We use a stream-function formulation after the manner of N. N. Frolov [Math. Notes, \textbf{53}(5-6), 650--656, 1993] in which the fluid density depends on the stream-function by means of another function determined by the boundary conditions. This allows for dropping some of the equations, most notably the continuity equation. With a fixed point argument we show the existence of solutions to the resulting system.