Momentum incompressible

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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. 

HEM for Two-Phase Pipe Discharge With a pipe present, the backpressure experienced by the orifice is no longer qg, but rather an intermediate pressure ratio qi. Thus qi replaces T o iri ihe orifice solution for mass flux G. ri Eq. (26-95). Correspondingly, the momentum balance is integrated between qi and T o lo give the pipe flow solution for G,p. The solutions for orifice and pipe now must be solved simultaneously to make G. ri = G,p and to find qi and T o- This can be done explicitly for the simple case of incompressible single-phase (hquid) inclined or horizontal pipe flow The solution is implicit for compressible regimes. 

In fluid mechanics the principles of conservation of mass, conservation of momentum, the first and second laws of thermodynamics, and empirically developed correlations are used to predict the behavior of gases and liquids at rest or in motion. The field is generally divided into fluid statics and fluid dynamics and further subdivided on the basis of compressibility. Liquids can usually be considered as incompressible, while gases are usually assumed to be compressible. 

A detailed study of the influence of viscous heating on the temperature field in micro-channels of different geometries (rectangular, trapezoidal, double-trapezoidal) has been performed by Morini (2005). The momentum and energy conservation equations for flow of an incompressible Newtonian fluid were used to estimate... 

If we ignore inertia force and follow the conventional assumptions in liquid lubrication, motion equation (here we consider the incompressible fluids in the absence of volume force and volume momentum) is ... 

The principles of conservation of mass and momentum must be applied to each phase to determine the pressure drop and holdup in two phase systems. The differential equations used to model these principles have been solved only for laminar flows of incompressible, Newtonian fluids, with constant holdups. For this case, the momentum equations become... 

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-... 

The vapor-layer model developed in Section IV.A.2 is based on the continuum assumption of the vapor flow. This assumption, however, needs to be modified by considering the kinetic slip at the boundary when the Knudsen number of the vapor is larger than 0.01 (Bird, 1976). With the assumption that the thickness of the vapor layer is much smaller than the radius of the droplet, the reduced continuity and momentum equations for incompressible vapor flows in the symmetrical coordinates ( ,2) are given as Eqs. (43) and (47). When the Knudsen number of the vapor flow is between 0.01 and 0.1, the flow is in the slip regime. In this regime, the flow can still be considered as a continuum at several mean free paths distance from the boundary, but an effective slip velocity needs to be used to describe the molecular interaction between the gas molecules and the boundary. Based on the simple kinetic analysis of vapor molecules near the interface (Harvie and Fletcher, 2001c), the boundary conditions of the vapor flow at the solid surface can be given by... 

As the vapor flows in the direction along the spherical surface of the particle, a boundary layer coordinate ( , X, co) given in Fig. 21 is employed to describe the vapor-layer equation. In this coordinate, the continuity and momentum equations for incompressible vapor flows with gravitation terms neglected... 

Such an approach is conceptually different from the continuum description of momentum transport in a fluid in terms of the NS equations. It can be demonstrated, however, that, with a proper choice of the lattice (viz. its symmetry properties), with the collision rules, and with the proper redistribution of particle mass over the (discrete) velocity directions, the NS equations are obeyed at least in the incompressible limit. It is all about translating the above characteristic LB features into the physical concepts momentum, density, and viscosity. The collision rules can be translated into the common variable viscosity, since colliding particles lead to viscous behavior indeed. The reader interested in more details is referred to Succi (2001). 

In the last term of equation 1.23, the averages are taken over the fixed volume V of the section. This term is simply the rate of change of the momentum of the fluid instantaneously contained in the section. It is clear that accumulation of momentum may occur with unsteady flow even if the flow is incompressible. In general, the mass flow rates M and M2 into and out of the section need not be equal but, by continuity, they must be equal for incompressible or steady compressible flow. 

In which directions do the forces arising from the change of fluid momentum act for steady incompressible flow in the pipe-work shown in Figure 1.3 ... 

The above example shows the effect of a change in pipe diameter, and therefore flow area, on the momentum flow rate. It is clear that for steady, fully developed, incompressible flow in a pipe of constant diameter, the fluid s momentum must remain constant. However, it is possible for the fluid s momentum to change even in a straight pipe of constant diameter. If the (incompressible) flow were accelerating, as during the starting of flow, the momentum flow rates into and out of the section would be equal but there would be an accumulation of momentum within the section. (The mass of fluid in the section would remain constant but its velocity would be increasing.) Consequently, a force must act on the fluid in the direction of flow. 

Determine the magnitude and direction of the reaction on the bend shown in Figure 1.8 arising from changes in the fluid s momentum. The pipe is horizontal and the flow may be assumed to be steady and incompressible. 

It has been assumed that the flow is incompressible so that there are no fluctuations of the density. Equation 1.91 shows that the momentum flux consists of a part due to the mean flow and a part due to the velocity fluctuation. The extra momentum flux is proportional to the square of the fluctuation because the momentum is the product of the mass flow rate and the velocity, and the velocity fluctuation contributes to both. The extra momentum flux is equivalent to an extra apparent stress perpendicular to the face, ie a normal stress component. As (v x)2 is always positive it produces a compressive stress, which is positive in the negative sign convention for stress. 

It has been assumed that the density is constant in writing these equations, which are therefore strictly valid only for incompressible flow. ed is called the eddy diffusivity and eh the eddy thermal diffusivity. Although s can be interpreted as the eddy diffusivity of momentum, it is usually called the eddy viscosity and sometimes by the better name eddy kinematic viscosity. 

In principle, this is the same as for single-phase flow. For example in steady, fully developed, isothermal flow of an incompressible fluid in a straight pipe of constant cross section, friction has to be overcome as does the static head unless the pipe is horizontal, however there is no change of momentum and consequently the accelerative term is zero. In the case of compressible flow, the gas expands as it flows from high pressure to low pressure and, by continuity, it must accelerate. In Chapter 6 this was noted as an increase in the kinetic energy. 

For the case in which the Schmidt number is equal to 1, it can be shown [7] that the conservation equations [in terms of Cl see Eq. (6.17)] can be transposed into the form used for the momentum equation for the boundary layer. Indeed, the transformations are of the same form as the incompressible boundary layer equations developed and solved by Blasius [30], The important difference... 

REDIFEM—This fire model has applications including steady state releases of compressible gas/vapor, incompressible liquid and transient release from a gas vessel, Gaussian Plume models, continuous free momentum, BLEVE, and confined and unconfined vapor cloud explosions. REDIEEM is reported to have internal validation with ISO 9001 and checked against PHAST and ERED. 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

This natural circulation occurs by a direct transfer of momentum across the interface, and the presence of a monolayer at the interface will affect it in two ways. Firstly, the surface viscosity of the monolayer may cause a dissipation of energy and momentum at the surface, so that the drop behaves rather more as a solid than as a liquid, i.e., the internal circulation is reduced. Secondly, momentum transfer across the surface is reduced by the incompressibility of the film, which the moving stream of gas will tend to sweep to the rear of the drop (Fig. 14b) whence, by its back-spreading pressure n, it resists further compression and so damps the movement of the surface and hence the transfer of momentum into the drop. This is discussed quantitatively below, where Eq. (32) should apply equally well to drops of liquid in a gas. 

Because the no-slip condition requires that the velocities at the wall vanish, the axial momentum equation at the wall has a significantly reduced form. Stated in terms of vor-ticity, the incompressible Navier-Stokes equations can be written as... 

In incompressible problems (i.e., p = constant) neither the pressure or the density appears in the continuity equation. Nevertheless, the coupled continuity-momentum system is still third order. The pressure is still a dependent variable and the pressure gradients are retained in the momentum equations. 

The fact that V2p = 0 indicates a clear elliptic behavior of the pressure field, notwithstanding the first derivatives in the momentum equations themselves. For an incompressible fluid, pressure communicates among all the boundaries and within the interior instantly (i.e., infinite sound speed). 

Steady parallel flow can be realized in ducts of essentially arbitrary cross section. A linear elliptic partial differential equation must be solved to determine the velocity field and the shear stresses on the walls. For an incompressible, constant-viscosity fluid, the axial momentum equation states that... 

Consider a long circular duct in which an incompressible, constant-property fluid is initially at rest. Suddenly a constant pressure gradient is imposed. The axial momentum equation that describes the transient response of the velocity profile for this situation is... 

Consider a long cylindrical shell whose interior is filled with an incompressible fluid. If the fluid is initially at rest when the cylinder begins to rotate, a boundary layer develops as the momentum diffuses inward toward the center of the cylinder. The fluid s circumferential velocity vu comes to the cylinder-wall velocity immediately, owing to the no-slip condition. At very early time, however, the interior fluid will be only weakly affected by the rotation, with the influence increasing as the boundary layer diffuses inward. If the shell continues to rotate at a constant angular velocity, the fluid inside will eventually come to rotate as a solid body. 

The fully developed, steady, incompressible, constant-viscosity flow in the annulus is described by a momentum equation in the form... 

Beginning with the constant-viscosity, incompressible Navier Stokes equations, write a reduced form of the radial and circumferential momentum equations that is appropriate to represent the fully developed flow in the circular channel. 

I. Assuming steady, incompressible, isothermal, axisymmetric flow, write out the full system of equations that describe the conservation of mass, momentum, and species... 

Consider the steady-state, fully developed, incompressible flow between parallel disks, such as illustrated in Fig. 5.9. In concert with the Jeffery-Hamel assumptions that were made in the previous configurations, one can assume that only the radial velocity is nonzero. As a consequence the continuity and momentum equations reduce to the following ... 

Consider the flow of an incompressible fluid in the entry region of a circular duct. Assuming the inlet velocity profile is flat, determine the length needed to achieve the parabolic Hagen-Poiseuille profile. Recast the momentum equation in nondimensional form, where the Reynolds number is based on channel diameter and inlet velocity emerges as a parameter. Based on solutions at different Reynolds numbers, develop a correlation for the entry length as a function of inlet Reynolds number. 

The continuity and momentum equations can be obtained for the fluid mixture as a single fluid using Chapman-Enskog expansion procedure in the nearly incompressible limit 43... 

The fuel flow in the gas channels is modeled by applying the equation of state and the principles of mass and momentum conservation. From Equation (3.27), considering that no reaction takes place within the gas channel (at the anode side only humidified hydrogen is provided), and that the fluid flow is regarded as incompressible (assumption (3)), the mass conservation equation becomes ... 

An example of a flow element would be a pipe with a particular pressure drop. The change in mass flow rate with respect to time will depend on a given upstream and downstream pressure. For incompressible flow (Mach numbers below 0.3), the lumped momentum differential equation used to calculate mass flow rate for a pipe is simplified to... 

The use of the compressibility term can be described as follows. The greater the stiffness a system model has, the more quickly the flow reacts to a change in pressure, and vice versa. For instance, if all fluids in the system are incompressible, and quasi-steady assumptions are used, then a step change to a valve should result in an instantaneous equilibrium of flows and pressures throughout the entire system. This makes for a stiff numerical solution, and is thus computationally intense. This pressure-flow solution technique allows for some compressibility to relax the problem. The equilibrium time of a quasi-steady model can be modified by changing this parameter, for instance this term could be set such that equilibrium occurs after 2 to 3 seconds for the entire model. However, quantitative results less than this timescale would then potentially not be captured accurately. As a final note, this technique can also incorporate flow elements that use the momentum equation (non-quasi-steady), but its strength is more suited by quasi-steady flow assumptions. 

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. 

The z-momentum equation for a Newtonian, incompressible flow (Navier-Stokes equations)... 

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