Navier Stokes Type Equations Research Articles

On a non‐stationary fluid flow problem in an infinite periodic pipe

We study linearized, non‐stationary Navier–Stokes type equations with the given flux in an infinite pipe periodic of period length L with respect to . The existence and uniqueness of the solution is proved. Moreover, the convergence of the solution in a finite pipe of length to the L‐periodic solution as is investigated.

On numerical schemes for phase‐field models for electrowetting with electrolyte solutions

AbstractWe present an energy‐stable, decoupled discrete scheme for a recent model (see [1]) supposed to describe electrokinetic phenomena in two‐phase flow with general mass densities. This model couples momentum and Cahn–Hilliard type phase‐field equations with Nernst–Planck equations for ion density evolution and an elliptic transmission problem for the electrostatic potential.The transport velocities in our scheme are based on the old velocity field updated via a discrete time integration of the force densities. This allows to split the equations into three blocks which can be treated sequentially: The phase‐field equation, the equations for ion transport and electrostatic potential, and the Navier–Stokes type equations. By establishing a discrete counterpart of the continuous energy estimate, we are able to prove the stability of the scheme. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Compressible Navier–Stokes limit of binary mixture of gas particles

Abstract In this paper we study the compressible Navier–Stokes limit of binary mixture of gas particles in which a species is dense and the other is sparse. Their collisions are decided by Grad's hard potentials. When the Knudsen number of dense species of the Boltzmann system goes to zero, we show that the hydrodynamic variables satisfy compressible Navier–Stokes type equations. It turns out that the macro fluid variables corresponding to the dense species satisfy the standard compressible Navier–Stokes equations. But the fluid equations for sparse species contain influence terms of dense species. Like single species gas, we employed Enskog–Chapman and moment methods up to the first order.

To the problem of rod heating

The construction of control boundary conditions is considered in the paper for the problem of heating a one-dimensional rod up to specified temperature. Two modifications of the method proposed by A. V. Fursikov are presented, these modifications allow us to take into account restrictions posed onto the structure of solution and the control. Results of calculations are presented for heating elements located both outside and inside of the rod. The obtained algorithms admit natural generalizations to a wide class of equations including nonlinear Navier-Stokes type equations and problems of stabilization by initial data and the right-hand side.

Numerical investigation of water-entry of flatted-bottom seafloor mining tool in ocean waves

A numerical wave load model based on two-phase (water-air) Reynolds-Averaged Navier Stokes (RANS) type equations is used to evaluate hydrodynamic forces exerted on flatted-bottom seafloor mining tool during its entering ocean waves of deploying process. The discretization of the RANS equations is achieved by a finite volume approach (FV). The volume of fluid method (VOF) is employed to track the complicated free surface. A numerical wave tank is built to generate the ocean waves which are suitable for deploying seafloor mining tool. A typical deploying condition is employed to reflect the process of flatted-bottom body impacting with waves, and the pressure distribution of bottom is also presented. Four different lowering velocities are applied to obtain the time histories of maximum pressure of bottom, and it can be concluded that the pressure coefficient decreases with water velocity increasing, which is similar with ordinary water entry case. The numerical results clearly demonstrate the characteristics of flatted-bottom body entering ocean waves.

Regularity Results for a Cahn-Hilliard-Navier-Stokes System with Shear Dependent Viscosity

We analyze a diffuse interface model describing the behavior of a mixture of two incompressible fluids. More precisely, we consider Navier-Stokes type equations with power-law like shear dependent viscosity. Such equations are nonlinearly coupled with a convective Cahn-Hilliard equation for the order parameter. The resulting system is endowed with no-slip and no-flux boundary conditions. We prove some regularity properties of weak solutions under rather general conditions. This is a generalization of previous results already proven for single fluids. Some consequences of such results are also addressed.

On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems

In this paper, we consider the well-known Fattorini's criterion for approximate controllability of infinite dimensional linear systems of type $y'=A y+Bu$. We precise the result proved by H. O. Fattorini in \cite{Fattorini1966} for bounded input $B$, in the case where $B$ can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini's criterion is satisfied and if the set of geometric multiplicities of $A$ is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in using the Fattorini's criterion to obtain the feedback stabilizability of linear and nonlinear parabolic systems with feedback controls in a finite dimensional space. In particular, for systems described by partial differential equations, such a criterion reduces to a unique continuation theorem for a stationary system. We illustrate such a method by tackling some coupled Navier-Stokes type equations (MHD system and micropolar fluid system) and we sketch a systematic procedure relying on Fattorini's criterion for checking stabilizability of such nonlinear systems. In that case, the unique continuation theorems rely on local Carleman inequalities for stationary Stokes type systems.

Numerical simulation of wave interaction with porous structures using an improved smoothed particle hydrodynamic method

A smoothed particle hydrodynamic (SPH) model is developed to simulate wave interaction with porous structures. The mean flow outside the porous structures is obtained by solving Reynolds Averaged Navier–Stokes (RANS) equations and the turbulence field is calculated by a large eddy simulation (LES) model. The porous flow is described by the spatially averaged Navier–Stokes type equations with the resistance effect of the porous media being represented by an empirical frictional source term. The interface boundaries between the porous flow and the outside flow are modeled by means of specifying a transition zone along the interface. The model is validated against other available numerical results and experimental data for wave damping over porous seabed with different levels of permeability. The validated model is then employed to investigate wave breaking over a submerged porous breakwater and good agreements between the SPH model results and the experimental data are obtained in terms of free surface displacement. In addition the predicted velocity, vorticity and pressure fields near the porous breakwater and in the breaking wave zone are also analyzed.

Numerical Investigation of Wave Slamming of Flat Bottom Body during Water Entry Process

A numerical wave load model based on two-phase (water-air) Reynolds-averaged Navier-Stokes (RANS) type equations is used to evaluate hydrodynamic forces exerted on flat bottom body while entering ocean waves of deploying process. The discretization of the RANS equations is achieved by a finite volume (FV) approach. The volume of fluid (VOF) method is employed to track the complicated free surface. A numerical wave tank is built to generate the ocean waves which are suitable for deploying offshore structures. A typical deploying condition is employed to reflect the process of flat bottom body impacting waves, and the pressure distribution of bottom is also presented. Four different lowering velocities are applied to obtain the relationship between slamming force and wave parameters. The numerical results clearly demonstrated the characteristics of flat bottom body impacting ocean waves.

Unifying binary fluid diffuse-interface models in the sharp-interface limit

AbstractRecent results published by Gugenberger et al. on surface diffusion (Phys. Rev. E, vol. 78, 2008, 016703), show that the sharp-interface limit of the phase field models often adopted in the literature fails to produce the appropriate boundary conditions. With this knowledge, we consider the sharp-interface limit of phase field models for binary fluids, obtained carefully, where hydrodynamic equations are coupled to phase field evolution based on Cahn–Hilliard or Allen–Cahn theories, in a variety of guises, and unify and contrast their forms and behaviours in the sharp-interface limit. In particular, a tensorial mobility model is analysed, which allows the bulk fluids in the outer region to satisfy classical Navier–Stokes type equations to all orders in the Cahn number.

Incompressible viscous Newtonian flow in a fissured medium of general deterministic type

We study homogenization for nonstationary Navier-Stokes systems in a fissured medium of general deterministic type. Assuming that the blocks of the porous medium consist of deterministically distributed inclusions and the elasticity tensors satisfy general deterministic hypotheses, we prove that the macroscopic problem is a Navier-Stokes type equation for Newtonian fluid in a fixed domain. Our setting includes the classical periodic framework, the weakly almost periodic one, and some others.

Analysis of a Cahn–Hilliard–Ladyzhenskaya system with singular potential

The evolution of a mixture of two incompressible and (partially) immiscible fluids is here described by Ladyzhenskaya–Navier–Stokes type equations for the (average) fluid velocity coupled with a convective Cahn–Hilliard equation with a singular (e.g., logarithmic) potential. The former is endowed with no-slip boundary conditions, while the latter is subject to no-flux boundary conditions so that the total mass is conserved. Here we first prove the existence of a weak solution in three-dimensions and some regularity properties. Then we establish the existence of a weak trajectory attractor for a sufficiently general time-dependent external force. Finally, taking advantage of the validity of the energy identity, we show that the trajectory attractor actually attracts with respect to the strong topology.

Longtime behavior of a diffuse interface model for binary fluid mixtures with shear dependent viscosity

We consider a system which describes the behavior of a binary mixture of immiscible incompressible fluids with shear dependent viscosity by means of the diffuse interface approach. This system consists of Navier–Stokes type equations, characterized by a nonlinear stress-strain law, which are nonlinearly coupled with a convective Cahn–Hilliard equation for the order parameter. We analyze the corresponding dynamical system and, by means of the short trajectory method, we prove the existence of global and exponential attractors. We also discuss the dependence of an upper bound of the fractal dimension on the physical parameters of the system.

Examples of equations of Navier-Stokes type not strongly solvable in the large

In the space of functions with values in Hilbert space, we consider the Cauchy problem u′t + Au + B(u, u) = f(t), u(0) = 0, 0 ≤ t ≤ T. We construct examples of a self-adjoint operator A ≥ E and a bilinear transformation B satisfying the condition 〈B(u, v), v〉 = 0 such that the Cauchy problem is not strongly solvable.

WAVE STRUCTURE INTERACTION: ROLE OF ENTRAPPED AIR ON WAVE IMPACT AND UPLIFT FORCES

In present paper, a numerical wave load model based on compressible two-phase Navier Stokes type equations is used to evaluate hydrodynamic forces exerted on I-10 bridge across Mobil Bay which was extensively damaged during Hurricane Katrina. The volume of fluid method (VOF) is used in the model to describe dynamic free surface which is capable of simulating complex discontinuous free surface associated with wave-deck interactions. Special emphasis was put on investigating the role of entrapped air on hydrodynamic forces exerted on bridge superstructure. Numerical simulation results indicate that air entrapment can significantly amplify uplift forces applied to the bridge superstructure. To mitigate hydrodynamic forces, effectiveness of airvents is investigated. It has been shown that airvents can significantly damp out wave energy and can effectively reduce uplift forces.

On a class of three dimensional Navier-Stokes equations with bounded delay

In this paper we consider a three dimensional Navier-Stokes type equations with delay terms. We discuss the existence ofweak and strong solutions and we study the asymptotic behavior of the strong solutions. Moreover, under suitableassumptions, we show the exponential stability of stationary solutions.

Molecular flows in narrow pores at small initial perturbations

The problem of the correctness of setting the initial conditions upon the generation of a molecular flow in narrow pores is discussed. The effect of intensity of the initial perturbation of the equilibrium state of the vapor and liquid in a 15-nm-wide slitlike pore with different potential of interaction between the molecules and pore walls is studied. The microhydrodynamic approach is used to calculate the flows of dense gas and liquid in narrow pores, which makes it possible to investigate the effect of wall potential on the parameters of molecular flows. The calculations are made on the basis of the Navier-Stokes type equations in which the transport coefficients and the equation of state for a component are calculated using a simple molecular model such as the lattice-gas model. The transport coefficients and the equation of state depend on the local values of the vapor and liquid densities and on temperature. The dynamic pattern of vapor flow is shown to qualitatively remain the same at a decreasing intensity of the initial unimoment perturbation: it is of oscillating nature, just as at strong initial perturbations. Weak attraction of the molecules to the walls enhances oscillations at the initial stage of gas and liquid flow along the pore axis. Strong attraction between low-density molecules and the pore walls forms a laminar flow, and the oscillations are noticeably smoothed. The perturbation intensity changes the absolute values of the flow parameters, including the average rates of the molecular flows formed. The average rate of hydrodynamic flow in wide pores depends weakly on the energy of interaction between the molecules and the pore walls.

Global well-posedness for a Boussinesq–Navier–Stokes system with critical dissipation

In this paper we study a fractional diffusion Boussinesq model which couples a Navier–Stokes type equation with fractional diffusion for the velocity and a transport equation for the temperature. We establish global well-posedness results with rough initial data.

Stationary Stokes, Oseen and Navier–Stokes Equations with Singular Data

The concept of very weak solution introduced by Giga (Math Z 178:287–329, 1981) for the Stokes equations has hardly been studied in recent years for either the Navier–Stokes equations or the Navier–Stokes type equations. We treat the stationary Stokes, Oseen and Navier–Stokes systems in the case of a bounded open set, connected of class $${\mathcal{C}^{1,1}}$$ of $${\mathbb{R}^3}$$ . Taking up once again the duality method introduced by Lions and Magenes (Problemes aus limites non-homogenes et applications, vols. 1 & 2, Dunod, Paris, 1968) and Giga (Math Z 178:287–329, 1981) for open sets of class $${\mathcal{C}^{\infty}}$$ [see also chapter 4 of Necas (Les methodes directes en theorie des equations elliptiques. (French) Masson et Cie, Ed., Paris; Academia, Editeurs, Prague, 1967), which considers the Hilbertian case p = 2 for general elliptic operators], we give a simpler proof of the existence of a very weak solution for stationary Oseen and Navier–Stokes equations when data are not regular enough, based on density arguments and a functional framework adequate for defining more rigourously the traces of non-regular vector fields. In the stationary Navier–Stokes case, the results will be valid for external forces not necessarily small, which lets us extend the uniqueness class of solutions for these equations. Considering more regular data, regularity results in fractional Sobolev spaces will also be discussed for the three systems. All these results can be extended to other dimensions.

Motion of a vapor—liquid meniscus in narrow slitlike pores

In this work, a microhydrodynamic approach is used to investigate the motion of the meniscus on a vapor—liquid interface in narrow slitlike pores. Calculations are performed on the basis of Navier—Stokes—type equations, in which the transport coefficients and equation of state for a compound are calculated within the simplest molecular model of a gas lattice that takes into account the inherent volume of molecules and their interaction with each other and the pore wall. The dynamic modes of the flow of a liquid monatomic gas (argon) bearing a vapor bubble through nanosized pores at a given pressure difference across the pore are studied. The differences in the average rates of the liquid and vapor-bubble motion in a quasi-stationary regime caused by the intensive nonequilibrium processes of molecule exchange on the liquid—vapor and vapor—liquid boundaries, i.e., by the phase transitions on both boundaries of a bubble, are revealed.