Navier-Stokes equation incompressible liquid

In Chapter 5 the general equation for the flow of an incompressible, Newtonian viscous liquid was obtained the Navier-Stokes equation. For liquid flow in the x direction only the equation is ... 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

The Navier-Stokes equation [Eq. (1)] provides a framework for the description of both liquid and gas flows. Unlike gases, liquids are incompressible to a good approximation. For incompressible flow, i.e. a constant density p, the Navier-Stokes equation and the corresponding mass conservation equation simplify to... 

To compute the motion of two immiscible and incompressible fluids such as a gas liquid bubble column and gas-droplets flow, the fluid-velocity distributions outside and inside the interface can be obtained by solving the incompressible Navier-Stokes equation using level-set methods as given by Sussman et al. (1994) ... 

When considering flow of a liquid in contact with a solid surface, a basic understanding of the hydrodynamic behavior at the interface is required. This begins with the Navier-Stokes equation for constant-viscosity, incompressible fluid flow, such that Sp/Sf = 0,... 

Next we turn to the Navier-Stokes equation, which for creeping flow of incompressible liquids, neglecting inertial and gravitational forces, reduces to... 

We limit our discussion here to laminar flows governed by the steady or unsteady, incompressible Navier-Stokes equations. In addition, we restrict ourselves to flows where the solution to the energy or the concentration equation does not influence the flow field, a circumstance not uncommon to isothermal constant viscosity liquid flows of relevance for many electrochemical systems. The incompressible, constant-property, Navier-Stokes equations are given below, with summation over repeated indices ... 

The starting point for the description of these complex phenomena is the set of hydrodynamic equations for the hquid crystal and Maxwell s equation for the propagation of the light. The relevant physical variables that these equations contain are the director field n(r, t), the flow of the liquid v(r, t) and the electric field of the light E/jg/jt(r, t). (We assume an incompressible fluid and neglect temperature differences within the medium.) The Navier-Stokes equation for the velocity v can be written as [5]... 

A most important disadvantage of this equation, though simple for the solution, was that the equation considers the incompressible nature of the medium, which results in significant deviations in the predictions of the collapse conditions from the realistic values. More recently, the gas dynamics inside collapsing bubbles has been studied considering the compressibility of liquid in Navier-Stokes equations (Moss et ah, 1999, Storey and Szeri, 1999). The equation reported by Tomita and Shima (1986) also considers the compressibility of the liquid medium. The... 

Consider a fluid domain consisting of a liquid/gas interface. The Navier-Stokes equations (1.1) and the continuing equations (1.2) for an incompressible Newtonian flow describing such a system can be written as ... 

To analyze the fundamentals of a droplet motion actuated by ctMitmuous electrowetting principles, for illustration, one may consider two infinite parallel plates separated by a distance H, with an intervening liquid. For a steady, fully developed incompressible flow along the x-direction, the Navier-Stokes equation assumes the following simplified form ... 

The flow of incompressible liquids in continuum state can be described by the Navier-Stokes equations ... 

The above equation, together with a condition related to the incompressibility of the liquid (Vi7 = 0), constitutes what is known as the Navier-Stokes equation. It is normally extremely difficult to solve for three reasons ... 

Abstract. In this paper, the motion model of the two-component incompressible viscous fluid with variable viscosity and density is considered for modeling the process of the surface wave propagation. The model consists of the non-stationary Navier-Stokes equations with variable viscosity and density, the convection-diffusion equation and equations for determining the viscosity and density depending on the concentration of the components. Thus we model the two-component medium, one of the components being more dense and viscous liquid. The results of calculations for two-dimensional and three-dimensional problems are presented. 

Ghaini et al. [47] observed different aqueous-organic two-phase systems in capillaries of 1mm internal diameter. The authors carried out CFD simulations using the VOF model [48] based on the incompressible Navier-Stokes equation with appropriate boundary conditions between the two phases. The authors demonstrated the significance of the wall film in the hydrodynamics and mass transfer liquid-liquid slug flow. 

As a very simple example of the use of the Navier-Stokes equation under creeping flow conditions, consider the flow of a Newtonian liquid in a parallel-sided, semi-infinite channel in the absence of body forces. Let the boundaries be planes located at y = /z, each lying in an xz plane and let the liquid flow be in the z direction. The motion is assumed to be so slow that a creeping solution is obtained, the liquid is assumed to be incompressible and r is treated as a constant. From the symmetry of the problem the liquid flow velocity is a function of y only, so that the three components of the Navier-Stokes equation are ... 

In this chapter, we discuss the principles of how to calculate fluid flow. As we shall see, hydrodynamics is governed by a partial differential equation, the Navier -Stokes equation. It can be solved analytically only for a few simple cases. A systematic introduction into hydrodynamics is beyond the scope of this book. For an instructive introduction, we recommend Refs [625, 626]. New methods for the calculation of hydrodynamic interactions in dispersions are described in Ref. [627]. As one important example, we derive the hydrodynamic force between a rigid sphere and a plane in an incompressible liquid. Finally, hydrodynamic interactions between fluid boundaries are discussed. 

Pulsatile flow in an elastic vessel is very complex, since the tube is able to undergo local deformations in both longitudinal and circumferential directions. The unsteady component of the pulsatile flow is assumed to be induced by propagation of small waves in a pressurized elastic tube. The mathematical approach is based on the classical model for the fluid-structure interaction problem, which describes the dynamic equilibrium between the fluid and the tube thin wall (Womersley, 1955b Atabek and Lew, 1966). The dynamic equilibrium is expressed by the hydrodynamic equations (Navier-Stokes) for the incompressible fluid flow and the equations of motion for the wall of an elastic tube, which are coupled together by the boundary conditions at the fluid-wall interface. The motion of the liquid is described in a fixed laboratory coordinate system (f , 6, f), and the dynamic... 

The third term on the left hand side of (18.1) is an artificial compression term which is active only in the interface region. U ei is a velocity field suitable to compress the interface. The gas-liquid flow is governed by the incompressible Navier—Stokes (N-S) equations in which the parameters about physical properties such as density ip) and dynamic viscosity (//) are calculated as weighted averages by the linear interpolation of the volume fraction. The continuum surface force (CSF) method is employed to calculate the surface tension force [2]. Therefore, the N-S equatimis can be expressed as follows ... 

The CFD model simulates particle-droplet interaction in the surrounding gas by solving the incompressible Navier—Stokes (N—S) equations coupled with the Volume of Fluid (VOF) method, six Degrees of Freedom (6-DOF) method, and dynamic mesh technique [42]. A solid particle is represented as a rigid body with 6-DoF motion, and the gas—liquid interface is described by the VOF method. A body-attached mesh, which follows the body motion, is used for rigid body motion simulation. The CFD model has been developed in the open-source CFD code OpenFOAM. 

---