Navier-Stokes equation for incompressible flow

When the continmty equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the time-averaged velocities and pressures are obtained which appear identical to the original equations (6-18 through 6-28), except for the appearance of additional terms in the Navier-Stokes equations. Called Reynolds stress terms, they result from the nonlinear effects of momentum transport by the velocity fluctuations. In each i-component (i = X, y, z) Navier-Stokes equation, the following additional terms appear on the right-hand side ... 

The model is composed of two parts. In the first part, steady state two-dimensional Navier-Stokes equations for incompressible flow are used to relate local velocity w and pressure p ... 

A vorticity-transport equation can be derived by taking the taking the vector curl of the full Navier-Stokes equations. For incompressible flows with constant viscosity, the vorticity-transport equation can be expressed in a form that is quite similar to the other transport equations. Begin with the full Navier-Stokes equations, which for constant viscosity can be written in compact vector form as (Eq. 3.61)... 

When the continuity equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the time-averaged... 

This is known as the Navier-Stokes equation for incompressible flow. 

Computational fluid dynamics (CFD) emerged in the 1980s as a significant tool for fluid dynamics both in research and in practice, enabled by rapid development in computer hardware and software. Commercial CFD software is widely available. Computational fluid dynamics is the numerical solution of the equations or continuity and momentum (Navier-Stokes equations for incompressible Newtonian fluids) along with additional conseiwation equations for energy and material species in order to solve problems of nonisothermal flow, mixing, and chemical reaction. 

Runnels and Eyman [41] report a tribological analysis of CMP in which a fluid-flow-induced stress distribution across the entire wafer surface is examined. Fundamentally, the model seeks to determine if hydroplaning of the wafer occurs by consideration of the fluid film between wafer and pad, in this case on a wafer scale. The thickness of the (slurry) fluid film is a key parameter, and depends on wafer curvature, slurry viscosity, and rotation speed. The traditional Preston equation R = KPV, where R is removal rate, P is pressure, and V is relative velocity, is modified to R = k ar, where a and T are the magnitudes of normal and shear stress, respectively. Fluid mechanic calculations are undertaken to determine contributions to these stresses based on how the slurry flows macroscopically, and how pressure is distributed across the entire wafer. Navier-Stokes equations for incompressible Newtonian flow (constant viscosity) are solved on a three-dimensional mesh ... 

The starting point is a Mathematical Model, i.e. the set of equations and boundary conditions, which covers the physics of the flow most suitable. For some problems the governing equations are known accurately (e.g. the Navier-Stokes equations for incompressible Newtonian fluids). However for many phenomena (e.g. turbulence or multiphase flow) and especially for the description of ceramic materials or wall slip phenomena the exact equations are either not available or a numerical solution is not feasible. 

The numerical simulation of single-phase fluid flow in fibrous porous medium is considered in this paper. The object of the study is the square domain which includes 4 cylinders (see Fig. 1). Planar flow that perpendicular to the axes of cylinders is considered in this paper. This model is based on the numerical solution of the Navier-Stokes equations for incompressible fluid flow ... 

The computer simulation model for studying the hydro-dynamically developing flow-field and developing heat and mass transport phenomena in the gas flow channel is given based on the incompressible Navier-Stokes equation for fluid flow and heat and mass transport equations as described below. 

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 ... 

The principle can be illustrated by examining the Navier-Stokes equation for two-dimensional incompressible flow. The x-component of the equation is... 

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,... 

The application of Navier-Stokes equations to turbulent flows are discribed in sect 1.2.7. The Reynolds averaged equations for incompressible flows are normally adopted deriving the transfer coefficients for heat and mass. 

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]... 

In the equation of motion for the endolymph fluid, the force Tf dA acts on the fluid at the fluid-otoconial layer interface (Figure 64.2). This shear stress tf is responsible for driving the fluid flow. The linear Navier-Stokes equation for an incompressible fluid are used to describe this endolymph flow. Expressions for the pressure gradient, the flow velocity of the fluid measured with respect to an inertial reference frame, and the force due to gravity (body force) are substituted into the Navier-Stokes equation for flow in the x-direction yielding... 

The velocity, density, pressure, and other properties of the fluid are all defined at every point in space and time and the solutions of fluid fiow in the continuum regime are obtained by using the Navier-Stokes equations. For an incompressible flow with constant properties, we have... 

The Navier-Stokes equation for the incompressible, two-phase unsteady flow takes the form... 

In the following three sections, we present solutions to Navier-Stokes equations for classical laminar flows. These solutions are specific to an incompressible fluid of homogeneous mass density. The system to be solved is then reduced to the three Navier-Stokes equations, plus incompressibility. 

For the simulations we use our in-house program package Free Surface 3D (FS3D). FS3D is a direct numerical simulation code for multiphase flows with sharp interfaces between the immiscible phases and has been developed and improved at the ITLR for the last 20 years. FS3D is a finite volume code which solves the incompressible Navier-Stokes equations for volume and momentum conservation ... 

Due to its great importance in reactor simulations, a brief survey of the main steps involved in Stokes solution of the Navier-Stokes equation for creeping motion about a smooth immersed rigid sphere is provided. The details of the derivation is not repeated in this book as this task is explained very well in many textbooks [14, 15, 88, 140]. Consider an incompressible creeping motion of a uniform flow of speed 1/ = v vj ej approaching a solid sphere of radius rp, as illustrated in Fig. 5.1. To describe the surface drag force in the direction of motion it is convenient... 

The application of Navier-Stokes equations to turbulent flows are discribed in Sect. 1.27. The Reynolds averaged equations for incompressible flows are normally adopted deriving the transfer coefficients for heat and mass. For convenience the governing Reynolds averaged equations valid for fully developed turbulent boundary layer flow are listed below [111, 140]. In these equations several simplifications are made, in particular = 0 and the dominant flow is in the x-direction so = 0. Moreover, the diffusion in the x- and z-directions are neglected. 

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