Incompressible flow relative

Vorticity The relative motion between two points in a fluid can be decomposed into three components rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described bv a tensor (Oy = dvj/dxj — dvj/dxi. The vector of vorticity given by one-half the... 

Pressure drop measurements. For the majority of experiments the instrumentation was relatively similar. Due to limitations associated with the small size of the channels, pressures were not measured directly inside the micro-channels. To obtain the channel entrance and exit pressures, measurements were taken in a plenum or supply line prior to entering the channel. It is insufficient to assume that the friction factor for laminar compressible flow can be determined by means of analytical predictions for incompressible flow. 

As a simplification, the term in Eq. (10) that accounts for the kinetic energy of the gas jets emerging from the gas distributor is based on the expression ( 9goVl/2, which is valid for incompressible flow. Experimental investigations show [27], that for relatively low gas velocities it is possible to represent the empirically determined loss coefficients q as accurately with this simplification as by the use of expressions for compressible flow. 

In this example, we consider the viscous, isothermal, incompressible flow of a Newtonian fluid between two infinite parallel plates in relative motion, as shown in Fig. E2.5a. As is evident from the figure, we have already chosen the most appropriate coordinate system for the problem at hand, namely, the rectangular coordinate system with spatial variables x, y, z. 

First of all, the density and all the thermodynamic coefficients are constants. Secondly, when the density and the transport properties are constants, the continuity and momentum equations are decoupled from the energy equation. This result is important, as it means that we may solve for the three velocities and the pressure without regard for the energy equation or the temperature. Third, for incompressible flows the pressure is determined by the momentum equation. The pressure thus plays the role of a mechanical force and not a thermodynamic variable. Fourth, another important fact about incompressible flow is that only two parameters, the Reynolds number and the Froude number occur in the equations. The Froude number, Fr, expresses the importance of buoyancy compared to the other terms in the equation. The Reynolds number indicates the size of the viscous force term relative to the other terms. It is mentioned that compressible flows are often high Re flows, thus they are often computed using the inviscid Euler (momentum) equations. 

Many important applications of fluid dynamics require that density variations be taken into account. The complete field of compressible fluid flow has become very large, and it covers wide ranges of pressure, temperature, and velocity. Chemical engineering practice involves a relatively small area from this field. For incompressible flow the basic parameter is the Reynolds number, a parameter also important in some apphcations of compressible flow. In compressible flow at ordinary densities and high velocities a more basic parameter is the Mach mnnber. At very low densities, where the mean free path of the molecules is appreciable in comparison with the size of the equipment or solid bodies in contact with the gas, other factors must be considered. This type of flow is not treated in this text. 

Low Mach number equations similar to incompressible flow equations do not describe acoustic phenomena the only compressibility effect retained is the local thermal gas expansion. To obtain a well-known Boussinesq approximation, one should additionally require a low relative temperature (density) variation. 

Scope, 52 Basis, 52 Compressible Flow Vapors and Gases, 54 Factors of Safety for Design Basis, 56 Pipe, Fittings, and Valves, 56 Pipe, 56 Usual Industry Pipe Sizes and Classes Practice, 59 Total Line Pressure Drop, 64 Background Information, 64 Reynolds Number, R,. (Sometimes used Nr ), 67 Friction Factor, f, 68 Pipe—Relative Roughness, 68 Pressure Drop in Fittings, Valves, Connections Incompressible Fluid, 71 Common Denominator for Use of K Factors in a System of Varying Sizes of Internal Dimensions, 72 Validity of K Values,... 

The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

Simple pressure/drag flow. Here we treat an idealization of the down-channel flow in a melt extruder, in which an incompressible viscous fluid constrained between two boundaries of infinite lateral extent (2). A positive pressure gradient is applied in the X-direction, and the upper boundary surface at y - H is displaced to the right at a velocity of u(H) - U this velocity is that of the barrel relative to the screw. This simple problem was solved by a 10x3 mesh of 4-node quadrilateral elements, as shown in Figure 1. 

The slopes of the peaks in the dynamic adsorption experiment is influenced by dispersion. The 1% acidified brine and the surfactant (dissolved in that brine) are miscible. Use of a core sample that is much longer than its diameter is intended to minimize the relative length of the transition zone produced by dispersion because excessive dispersion would make it more difficult to measure peak parameters accurately. Also, the underlying assumption of a simple theory is that adsorption occurs instantly on contact with the rock. The fraction that is classified as "permanent" in the above calculation depends on the flow rate of the experiment. It is the fraction that is not desorbed in the time available. The rest of the adsorption occurs reversibly and equilibrium is effectively maintained with the surfactant in the solution which is in contact with the pore walls. The inlet flow rate is the same as the outlet rate, since the brine and the surfactant are incompressible. Therefore, it can be clearly seen that the dynamic adsorption depends on the concentration, the flow rate, and the rock. The two parameters... 

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 ... 

In two-dimensional, incompressible, steady flows, there is a relatively simple relationship between the vorticity and the stream function. Consider the axisymmetric flow as might occur in a channel, Fig. 3.12. Beginning with the axisymmetric stream function as discussed in Section 3.1.2, substitute the stream-function definition into the definition of the circumferential vorticity u>q ... 

Fluent is a commercially available CFD code which utilises the finite volume formulation to carry out coupled or segregated calculations (with reference to the conservation of mass, momentum and energy equations). It is ideally suited for incompressible to mildly compressible flows. The conservation of mass, momentum and energy in fluid flows are expressed in terms of non-linear partial differential equations which defy solution by analytical means. The solution of these equations has been made possible by the advent of powerful workstations, opening avenues towards the calculation of complicated flow fields with relative ease. 

Strictly speaking, most of the equations that are presented in the preceding part of this chapter apply only to incompressible fluids but practically, they may be used for all liquids and even for gases and vapors where the pressure differential is small relative to the total pressure. As in the case of incompressible fluids, equations may be derived for ideal frictionless flow and then a coefficient introduced to obtain a correct result. The ideal conditions that will be imposed for a compressible fluid are that it is frictionless and that there is to be no transfer of heat that is, the flow is adiabatic. This last is practically true for metering devices, as the time for the fluid to pass through is so short that very little heat transfer can take place. Because of the variation in density with both pressure and temperature, it is necessary to express rate of discharge in terms of weight rather than volume. Also, the continuity equation must now be... 

Example 2.7 Nonisothermal Parallel Plate Drag Flow with Constant Thermophysical Properties Consider an incompressible Newtonian fluid between two infinite parallel plates at temperatures T(0) = T and T(H) — T2, in relative motion at a steady state, as shown in Fig. E2.7 The upper plate moves at velocity Vo (a) Derive the temperature profile between the plates, and (b) determine the heat fluxes at the plates. 

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