Continuity equation, incompressible fluid

In general, fluid velocity is given by the Navier-Stokes and continuity equations. For fluids that are Newtonian (shear stress linearly related to fluid shear rate) and incompressible, the Navier-Stokes equation is written as... 

The general equation of continuity for incompressible fluid flow (Equation 7-1) eliminates the velocity variable from the set of equations ... 

APPENDIX 8.A EQUATIONS OF CONTINUITY FOR INCOMPRESSIBLE FLUIDS UNDER ISOTHERMAL FLOW CONDITIONS... 

As already explained the necessity to satisfy the BB stability condition restricts the types of available elements in the modelling of incompressible flow problems by the U-V P method. To eliminate this restriction the continuity equation representing the incompressible flow is replaced by an equation corresponding to slightly compressible fluids, given as... 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

Wc now obtain the integral of the continuity equation for incompressible fluids with respect to the local gap height hr this flow domain... 

Simplified forms of Eq. (6-8) apply to special cases frequently found in prac tice. For a control volume fixed in space with one inlet of area Ai through which an incompressible fluid enters the control volume at an average velocity Vi, and one outlet of area Ao through which fluid leaves at an average velocity V9, as shown in Fig. 6-4, the continuity equation becomes... 

This is the microscopic (local) continuity equation and must be satisfied at all points within any flowing fluid. If the fluid is incompressible (i.e., constant p), Eq. (5-7) reduces to... 

Example 5-3 Diffuser. A diffuser is a section in a conduit over which the flow area increases gradually from upstream to downstream, as illustrated in Fig. 5-3. If the inlet and outlet areas (Ai and A2) are known, and the upstream pressure and velocity (Pi and V ) are given, we would like to find the downstream pressure and velocity (P2 and If ). If the fluid is incompressible, the continuity equation gives V2 ... 

Note that there are 11 dependent variables, or unknowns in these equations (three u s, six r,y s, P, and p), all of which may depend on space and time. (For an incompressible fluid, p is constant so there are only 10 unknowns. ) There are four conservation equations involving these unknowns (the three momentum equations plus the conservation of mass or continuity equation), which means that we still need six more equations (seven, if the fluid is compressible). These additional equations are the con-... 

For an incompressible fluid, the term in parentheses is zero as a result of the conservation of mass (e.g., the microscopic continuity equation). Equation (13-25) can be generalized to three dimensions as... 

An important application of Bernoulli s equation is in flow measurement, discussed in Chapter 8. When an incompressible fluid flows through a constriction such as the throat of the Venturi meter shown in Figure 8.5, by continuity the fluid velocity must increase and by Bernoulli s equation the pressure must fall. By measuring this change in pressure, the change in velocity can be determined and the volumetric flow rate calculated. 

Some of the simplifications that may be possible are illustrated by the case of steady, fully-developed, laminar, incompressible flow of a Newtonian fluid in a horizontal pipe. The flow is assumed to be axisymmetric with no swirl component of velocity so that derivatives wrt 6 vanish and vg = 0. For fully-developed flow, derivatives wrt z are zero. With these simplifications and noting that the flow is incompressible, the continuity equation (equation A. 11) reduces to... 

If we substitute Eqs. (2.12) and (2.13) into Eq. (2.2) and perform the averaging operation indicated by the overbar, using the equation of continuity for an incompressible fluid,... 

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... 

This may not seem like much help, because we have expanded three terms into six. However, if the flow is assumed to be incompressible, a derivation given in fluid mechanics texts (the continuity equation) is... 

Regardless of how an incompressible element of fluid changes shape, its volume cannot change. Therefore, for an incompressible fluid, it is apparent that volumetric dilatation must be zero. Thus it must be the case that V-V = 0 for incompressible flows. The fact that V V = 0 for an incompressible fluid is also apparent from the mass-continuity equation, Eq. 2.35. 

It is clear that sound, meaning pressure waves, travels at finite speed. Thus some of the hyperbolic—wavelike-characteristics associated with pressure are in accord with everyday experience. As a fluid becomes more incompressible (e.g., water relative to air), the sound speed increases. In a truly incompressible fluid, pressure travels at infinite speed. When the wave speed is infinite, the pressure effects become parabolic or elliptic, rather than hyperbolic. The pressure terms in the Navier-Stokes equations do not change in the transition from hyperbolic to elliptic. Instead, the equation of state changes. That is, the relationship between pressure and density change and the time derivative is lost from the continuity equation. Therefore the situation does not permit a simple characterization by inspection of first and second derivatives. 

For the steady flow of an incompressible fluid, state the appropriate mass-continuity equation in spherical coordinates. What can be inferred from the reduced continuity equation about the functional form of of the circumferential velocity v 2... 

For an incompressible fluid, the continuity equation is written in general vector form as... 

Write out the mass-continuity equation that describes the flow of an incompressible fluid in the toroidal channel. 

The theorem is easily generalized also to the case when the fluid density is p = p(z) or p = p(r). Indeed, in this case, the continuity equation implies that for vz = 0 we have div v = 0, i.e., the problem reduces to that already considered. We note that in an ideally conducting medium, for arbitrary density, the quantity Hz/p is conserved (instead of Hz in the case of an incompressible fluid), since instead of (4) we have in this case the equation... 

When 7 —> 00, the fluid is considered incompressible and eqns. (9.92) and (9.93) are fully satisfied. Replacing the continuity equation (9.94) in the momentum equations we get... 

For the solution of this problem, the momentum and continuity equations for the steady-state flow of an incompressible viscoelastic fluid are given by... 

Example 1 Force Exerted on a Reducing Bend An incompressible fluid flows through a reducing elbow (Fig. 6-5) situated in a horizontal plane. The inlet velocity Vl is given and the pressures pl and p. are measured. Selecting the inlet and outlet surfaces 1 and 2 as shown, the continuity equation Eq. (6-9) can be used to find the exil velocity Vz = VjA/A. The mass flow rate is obtained by m = p V. A. ... 

The LGA is a variant of a cellular automaton, introduced as an alternative numerical approximation to the partial differential equation of Navier-Stokes and the continuity equations, whose analytical solution leads to the macroscopic approach of fluid dynamics. The microscopic behavior of the LGA has been shown to be very close to the Navier-Stokes (N-S) equations for incompressible fluids at the macroscopic level. 

Another commonly used model is based on the general differential balance of mass and momentum [Burgers, 1948]. Consider a steady, incompressible, and axially symmetric flow in which the body forces are negligible. In cylindrical coordinates, the equation of continuity of the fluid can be given as... 

The equation of continuity and the Bernoulli theorem together show, for a stream of incompressible fluid, that (a) where the cross-sectional area is large and the streamlines are widely spaced, the velocity is low and the pressure is high, and (6) where the cross-sectional area is small and the streamlines are crowded together, the velocity is high and the pressure is low Hence a flow net gives a picture not only of the velocity field but also of the pressure field. 

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... 

Solution In a pipe flow we have, in principle, three velocity components vz, vg, and vr. The equation of continuity in cylindrical coordinates is given in Table 2.1. For an incompressible fluid, this equation reduces to... 

Many hydrodynamic systems have been studied theoretically7-11. The solution to (5.45) proceeds through analysis of the velocity profile, derived from the momentum continuity equation and which is, for an incompressible fluid,... 

In Eqs. (6.20) and (6.21), the terms on the left-hand side are the inertia forces, the first term on the right-hand side is the body force, the second term is the pressure force, and the last term within brackets is the viscous forces acting on the fluid element. With known body forces Bx and By, the continuity equation (6.5) and the two momentum equations (6.20) and (6.21) are three independent equations for the solution of the three unknown quantities u, v and p for the steady, two-dimensional flow of an incompressible fluid. The solution of these equations are not... 

For incompressible flow of a fluid with constant properties, the particles are all moving in the direction parallel to the plates so that the velocity component v normal to the plates must be zero. By putting v = 0 in the continuity equation,... 

Let us look at an incompressible, constant-property fluid flowing laminarly inside a circular tube in regions away from the inlet where the velocity profile is fully developed. The continuity equation gives... 

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