Fluid momentum equation

Electric analog models are a class of lumped models and are often used to simulate flow through Ae network of blood vessels. These models are useful in assessmg Ae overall performance of a system or a subsystem. Integration of Ae fluid momentum equation (longitudinal direction, in cy-... 

Inspection of the K-H stability condition indicates that the structure of Equation 18 is invariant with the specific modelling of the wall and interfacial shear stresses and evolves essentially from the continuity equations and the left hand side of the momentum equations. On the other hand. Equation 19 for is directly related to the quasi-steady models adopted for the various shear stresses terms (the rhs of the two-fluid momentum Equation 8). In this sense, the form of 18 is general and is affected by the specific modelling of shear stresses only indirectly through the value. Thus, given different correlations for the shear stresses, the general form of 19 provides the corresponding values for... 

It is further of interest to note that in the particular case of thin laminar liquid layers in channel flow, the liquid layer thickness (as appears in Fr, Equation 27), can be obtained explicitly by solving the steady-state two-fluid momentum equations. In this case, the transitional criterion for wind generated waves, > 1 reduces to a critical superficial gas phase Reynolds number, Re > 1.113 x 10 [103]. 

Upon introduction of particles to a fluid phase, interaction terms must be incorporated into the single fluid momentum equations. These interaction terms are expressed as drag forces. The equations of motion in three directions can be written in the cylindrical system for an incompressible, Newtonian fluid ... 

The fourth and fifth terms on the RHS of (4.299) vanish because the particle velocity, is independent of the single particle species, and the single particle molecular temperature, Tt, by definition. Likewise, the first part of the last term vanishes as the single particle velocity c, (t) is independent of the single particle mass. It follows that the solid phase fluid momentum equation is reduced to the form ... 

Note that the second term on the right-hand side of the fluid momentum equation (8.24), -e(dpldz), comprises the sum of two terms the buoyant interaction with the particles in the control volume, (1 — e)dpjdz, and the net surface force across the control volume boundaries, —dpidz. 

The stability of the homogeneously fluidized state is analysed in terms of the full set of system equations, eqns (8.21)-(8.24), in Chapter 11. For the case of gas fluidization, however, where pp pf, terms in the fluid momentum equation that are proportional to fluid density will be negligible compared to the drag and pressure gradient terms, which are involved in supporting the fluidized particles. Equation (8.24) then reduces to ... 

Here g is the gravity vector and tu is the force per unit area exerted by the surroundings on the fluid in the control volume. The integrand of the area integr on the left-hand side of Eq. (6-10) is nonzero only on the entrance and exit portions of the control volume boundary. For the special case of steady flow at a mass flow rate m through a control volume fixed in space with one inlet and one outlet, (Fig. 6-4) with the inlet and outlet velocity vectors perpendicular to planar inlet and outlet surfaces, giving average velocity vectors Vi and V9, the momentum equation becomes... 

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... 

Fluid statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, Vp = pg. Letting z be directed vertically upward, so that g, = —g where g is the gravitational acceleration (9.806 mVs), the pressure field is given by... 

Example 2 Simplified Ejector Figure 6-6 shows a very simplified sketch of an ejector, a device that uses a high velocity primary fluid to pump another (secondary) fluid. The continuity and momentum equations may he... 

Application of the momentum equation to ejectors of other types is discussed in Lapple (Fluid and Paiticle Dynamics, University of Delaware, Newark, 1951) and in Sec. 10 of the Handbook. 

The momentum equation is a mathematieal formulation of the law of eonservation of momentum. It states that the rate of ehange in linear momentum of a volume moving with a fluid is equal to the surfaee forees and body forees aeting on a fluid. Figure 3-2 shows the veloeity eomponents in a generalized turbomaehine. The veloeity veetors as shown are resolved into three mutually perpendieular eomponents the axial eomponent (FJ, the tangential eomponent (Fg), and the radial eomponent (F ). 

It is possible to determine the x-component of the momentum equation by setting the rate of change of x-momentum of the fluid particle equal to the total force in the x-direction on the element due to surface stresses plus the rate of increase of x-momentum due to sources, which gives ... 

Wallis (1969) has presented the general one-dimensional momentum equation for the partiele phase of a fluid-partiele eontinuum as... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

While the modified energy equation provides for calculation of the flowrates and pressure drops in piping systems, the impulse-momenlum equation is required in order to calculate the reaction forces on curved pipe sections. I he impulse-momentum equation relates the force acting on the solid boundary to the change in fluid momentum. Because force and momentum are both vector quantities, it is most convenient to write the equations in terms of the scalar components in the three orthogonal directions. 

The procedure adopted here consists of taking a momentum balance on an element of fluid. The resulting Momentum Equation involves no assumptions concerning the nature of the flow. However, it includes an integral, the evaluation of which requires a knowledge of the velocity profile ux = f(y). At this stage assumptions must be made concerning the nature of the flow in order to obtain realistic expressions for the velocity profile. 

Derive the momentum equation for the flow of a fluid over a plane surface for conditions where the pressure gradient along the surface is negligible. By assuming a sine function for the variation of velocity with distance from the surface (within the boundary layer) for streamline flow, obtain an expression for the boundary layer thickness as a function of distance from the leading edge of the surface. 

Derive the momentum equation for the flow of a viscous fluid over a small plane surface. 

Derive the momentum equation for the flow of a viscous fluid over a small plane surface. Show that the velocity profile in the neighbourhood of the surface may be expressed as a sine function which satisfies the boundary conditions at the surface and at the outer edge of the boundary layer. 

Following usual conventions, repeated indices indicate summation and fy denotes df/dXj. The permutation S5mibol is used to present the vector cross product in indicial notation. Due to the anisotropic nature, traction and body couples can exist, and thus the angular momentum equation must be considered. For purely viscous fluids this equation says simply that the deviatoric stresses are symmetric. 

Thermally-Driven Buoyancy Flow. Thermal gradients can Induce appreciable flow velocities in fluids, as cool material is pulled downward by gravity while warmer fluid rises. This effect is Important in the solidification of crystals being grown for semiconductor applications, and might arise in some polymeric applications as well. To illustrate how easily such an effect can be added to the flow code, a body force term of pa(T-T ) has been added to the y-coraponent of the momentum equation, where here a is a coefficient of volumetric thermal expansion. 

Momentum balance equations are of importance in problems involving the flow of fluids. Momentum is defined as the product of mass and velocity and as stated by Newton s second law of motion, force which is defined as mass times acceleration is also equal to the rate of change of momentum. The general balance equation for momentum transfer is expressed by... 

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... 

Example 5-8 Turbine Blade. Consider a fluid stream impinging on a turbine blade that is moving with a velocity Vs. We would like to know what the velocity of the impinging stream should be in order to transfer the maximum amount of energy to the blade. The system is the fluid in contact with the blade, which is moving at velocity Vs. The impinging stream velocity is V, and the stream leaves the blade at velocity Va. Since V0 = Vro + Vs and V = Vri + Vs, the system velocity cancels out of the momentum equation ... 

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 foregoing procedure can be used to solve a variety of steady, fully developed laminar flow problems, such as flow in a tube or in a slit between parallel walls, for Newtonian or non-Newtonian fluids. However, if the flow is turbulent, the turbulent eddies transport momentum in three dimensions within the flow field, which contributes additional momentum flux components to the shear stress terms in the momentum equation. The resulting equations cannot be solved exactly for such flows, and methods for treating turbulent flows will be considered in Chapter 6. 

For steady, uniform, fully developed flow in a pipe (or any conduit), the conservation of mass, energy, and momentum equations can be arranged in specific forms that are most useful for the analysis of such problems. These general expressions are valid for both Newtonian and non-Newtonian fluids in either laminar or turbulent flow. 

Show how the Hagen-Poiseuille equation for the steady laminar flow of a Newtonian fluid in a uniform cylindrical tube can be derived starting from the general microscopic equations of motion (e.g., the continuity and momentum equations). 

By either integrating the microscopic momentum equations (see Example 5-9) or applying a momentum balance to a slug of fluid in the center... 

From the continuity and momentum equations for the fluid and solid phases along with the boundary conditions, the following groups of independent dimensionless parameters are found to control the hydrodynamics, noting our assumption that the particle-particle forces are only dependent on hydrodynamic parameters,... 

It is noted that the virtual body force Fp depends not only on the unsteady fluid velocity, but also on the velocity and location of the particle surface, which is also a function of time. There are several ways to specify this boundary force, such as the feedback forcing scheme (Goldstein et al., 1993) and direct forcing scheme (Fadlun et al., 2000). In 3-D simulation, the direct forcing scheme can give higher stability and efficiency of calculation. In this scheme, the discretized momentum equation for the computational volume on the boundary is given as... 

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