Navier-Stokes equations incompressible

T able 17.1. Navier-Stokes equations, incompressibility, and velocity curl expressed in a cylindrical coordinate system with axis Oz... 

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... 

These are the two components of the Navier-Stokes equation including fluctuations s., which obey the fluctuation dissipation theorem, valid for incompressible, classical fluids ... 

Brooks, A. N, and Hughes, T. J.R., 1982. Streamline-upwind/Petrov Galerldn formulations for convection dominated hows with particular emphasis on the incompressible Navier -Stokes equations. Cornput. Methods Appl Meek Eng. 32, 199-259. 

Gresho, P. M., Lee, R. L. and Sani, R. L., 1980. On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions. In Recent Advances in Numerical Methods in fluids, Ch. 2, Pineridge Press, Swansea, pp. 27-75. 

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

Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fliiid mechanics texts (e.g., Slatteiy [ibid.] Denu Whitaker and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) lead to the Navier-Stokes equations, whose three Cartesian components are... 

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 ... 

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. 

The incompressible, time-averaged continuity and the Navier-Stokes equations can be written as... 

The steady, laminar, incompressible fluid flow in cyclone collectors is governed by the Navier-Stokes equations ... 

For steady, incompressible fluid flow in a cyclone separator, the governing Navier-Stokes equations of motion are given, in a Cartesian coordinate system, by ... 

The Navier-Stokes equations are the fundatnenta equations describing incompressible, fluid flow see Chapter 9. 

The incompressible Navier-Stokes equations are obtained by substituting the above form for into the generalized Euler equation (equation 9.9) and by using the incompressibility condition (5 ) dvijdxi = 0 equation 9.4) and Euler s equation dvijdt = -Y k Vkidvi/dxk) - dp/dxi) equation 9.7) ... 

If we set ( = (2/3)/i,t the last two terms cancel and the incompressible Navier-Stokes equations take their conventional form [trittSS] ... 

Next, we substitute these dimensionless variables into the incompressible Navier-Stokes equations (equation 9.16). In Cartesian coordinates, the T component of the first equation reads... 

So what does the list of basic properties of a LG necessary to yield an emergent behavior consistent with the incompressible Navier-Stokes equations look like Such... 

Incompressible Limit In order to obtain the more familiar form of the Navier-Stokes equations (9.16), we take the low-velocity (i,e. low Mach number M = u I /cs) limit of equation 9,104, We also take a cue from the continuous case, where, if the incompressible Navier-Stokes equations are derived via a Mach-number expansion of the full compressible equations, density variations become negligible everywhere except in the pressure term [frisch87]. Thus setting p = peq + p and allowing density fluctuations only in the pressure term, the low-velocity limit of equation 9,104 becomes... 

These values can be compared to predicted values for numerical solutions of the incompressible Navier-Stokes equations. For d = 2, for example, we have the lower bounds Sa=2 (TZ/M) and Wd=2 TZ /M for a LG and the bounds S num, d=2 and d=2 where the bounds for the numerical solutions... 

Wu and Cheng (2003) measured the friction factor of laminar flow of de-ionized water in smooth silicon micro-channels of trapezoidal cross-section with hydraulic diameters in the range of 25.9 to 291.0 pm. The experimental data were found to be in agreement within 11% with an existing theoretical solution for an incompressible, fully developed, laminar flow in trapezoidal channels under the no-slip boundary condition. It is confirmed that Navier-Stokes equations are still valid for the laminar flow of de-ionized water in smooth micro-channels having hydraulic diameter as small as 25.9 pm. For smooth channels with larger hydraulic diameters of 103.4-103.4-291.0pm, transition from laminar to turbulent flow occurred at Re = 1,500-2,000. 

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... 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

The situation is different for incompressible flow. In that case, no equation of motion for the pressure field exists and via the mass conservation equation Eq. (17) a dynamic constraint on the velocity field is defined. The pressure field entering the incompressible Navier-Stokes equation can be regarded as a parameter field to be adjusted such that the divergence of the velocity field vanishes. 

The Navier-Stokes equation and the enthalpy equation are coupled in a complex way even in the case of incompressible fluids, since in general the viscosity is a function of temperature. There are, however, many situations in which such interdependencies can be neglected. As an example, the temperature variation in a microfluidic system might be so small that the viscosity can be assumed to be constant. In such cases the velocity field can be determined independently from the temperature field. When inserting the computed velocity field into Eq. (77) and expressing the energy density e by the temperature T, a linear equahon in T is... 

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) ... 

Vuik, C., Fast iterative solvers for the discretized incompressible Navier-Stokes equations , Delft University of Technology, TMI TR93-98 (1993). 

However, one difference exists with classical theory in this latter case, the Navier-Stokes equation (443) and the incompressibility condition (444) are assumed to be valid for all distances rict. In this case, it is an easy matter to calculate explicitly the higher-order terms in Eq. (445), and the boundary condition at the B-particle (assumed to be spherical) imposes the condition... 

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 hydrodynamic equation of motion (Navier-Stokes equation) for the stationary axial velocity, vfr), of an incompressible fluid in a cylindrical pore under the influence of a pressure gradient, dP /dz, and an axial electric field, E is... 

Equations (if.4) and (ff.S) are solved, along with the continuity equation (which does not change upon nondimensionalization), in a Cartesian coordinate system using the Fourier-Galerkin (spectral) technique under periodic boundary conditions in all three space dimensions. The scheme is similar to that used by Orszag [8] for direct solution of the incompressible Navier-Stokes equations. More details can be found in [9] and [7], and the scheme may be considered to be pseudospectral. ... 

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 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 ... 

Application of Newton s second law of motion to an infinitesimal element of an incompressible Newtonian fluid of density p and constant viscosity p, acted upon by gravity as the only body force, leads to the Navier-Stokes equation of motion ... 

With this simplification, the equations governing incompressible fluid motion are Eq. (1-33) and the continuity equation, Eq. (1-9). Several important consequences follow from inspection of these equations. The fluid density does not appear in either equation. Both equations are reversible in the sense that they are still satisfied if u is replaced by — u, whereas the nonlinearity of the Navier-Stokes equations prevents such reversibility. If we take the divergence of Eq. (1-33) and apply Eq. (1-9), we obtain... 

These are called the incompressible Navier-Stokes equations in Cartesian coordinates. 

The viscosity of the medium is t, and 1 is the unit tensor. (The Oseen tensor is the Green s function for the Navier-Stokes equation under the conditions that the fluid is incompressible, convective effects can be neglected, and inertial effects coming from the time derivative can be neglected.)... 

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