Pressure velocity gradient

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

The procedure of Mason and Evans has the electrical analog shown in Figure 2.2, where voltages correspond to pressure gradients and currents to fluxes. As the argument stands there is no real justification for this procedure indeed, it seems improbable that the two mechanisms for diffusive momentum transfer will combine additively, without any interactive modification of their separate values. It is equally difficult to see why the effect of viscous velocity gradients can be accounted for simply by adding... 

Application of the weighted residual method to the solution of incompressible non-Newtonian equations of continuity and motion can be based on a variety of different schemes. Tn what follows general outlines and the formulation of the working equations of these schemes are explained. In these formulations Cauchy s equation of motion, which includes the extra stress derivatives (Equation (1.4)), is used to preseiwe the generality of the derivations. However, velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Cauchy s equation of motion is either substituted in terms of velocity gradients or calculated via a viscoelastic constitutive equation in a separate step. 

Figure 9.5a shows a portion of a cylindrical capillary of radius R and length 1. We measure the general distance from the center axis of the liquid in the capillary in terms of the variable r and consider specifically the cylindrical shell of thickness dr designated by the broken line in Fig. 9.5a. In general, gravitational, pressure, and viscous forces act on such a volume element, with the viscous forces depending on the velocity gradient in the liquid. Our first task, then, is to examine how the velocity of flow in a cylindrical shell such as this varies with the radius of the shell.

As more and more of the filtrate is removed, the slurry graduaUy thickens and may become thixotropic. The soHds content of the thickened slurry may be higher than that obtained with conventional pressure filtration, by as much as 10 or 20%. A range of velocity gradients from 70 to 500 L/s has been suggested as necessary to prevent cake formation and to keep the thickening slurry ia a fluid state (27). 

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

The emulsification process in principle consists of the break-up of large droplets into smaller ones due to shear forces (10). The simplest form of shear is experienced in lamellar flow, and the droplet break-up may be visualized according to Figure 4. The phenomenon is governed by two forces, ie, the Laplace pressure, which preserves the droplet, and the stress from the velocity gradient, which causes the deformation. The ratio between the two is called the Weber number. We, where Tj is the viscosity of the continuous phase, G the velocity gradient, r the droplet radius, and y the interfacial tension. 

Viscosity is defined as the shear stress per unit area at any point in a confined fluid divided by the velocity gradient in the direc tiou perpendicular to the direction of flow. If this ratio is constant with time at a given temperature and pressure for any species, the fluid is caUed a Newtonian fluid. This section is limited to Newtonian fluids, which include all gases and most uoupolymeric liquids and their mixtures. Most polymers, pastes, slurries, waxy oils, and some silicate esters are examples of uou-Newtouiau fluids. 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

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... 

Number of vanes. The greater the number of vanes, the lower the vane loading, and the closer the fluid follows the vanes. With higher vane loadings, the flow tends to group up on the pressure surfaces and introduces a velocity gradient at the exit. 

Diffusion-blading loss. This loss develops because of negative velocity gradients in the boundary layer. Deceleration of the flow increases the boundary layer and gives rise to separation of the flow. The adverse pressure gradient that a compressor normally works against increases the chances of separation and causes significant loss. 

This velocity gradient arises from the pressure gradient necessary to create mobile-phase flow. The pressure drops as we travel from the column inlet to the... 

These equations, with six farther equations for the other components of a20 and au, when solved by the Burnett iteration procedure, yield the Navier-Stokes equations when solved simultaneously, however, there is no longer the simple dependence of the pressure tensor upon the velocity gradients, and of the heat flow upon the temperature gradient, but, rather, an interdependence of these relations. 

It is the full wetted perimeter that determines the flow regime and the velocity gradients in a channel. So, in this book, de determined using the full wetted perimeter will be used for both pressure drop and heat transfer calculations. The actual area through which the heat is transferred should, of course, be used to determine the rate of heat transfer equation 12.1. 

We have stated that the only stress that can exist in a fluid at rest is pressure, because the shear stresses (which resist motion) are zero when the fluid is at rest. This also applies to fluids in motion provided there is no relative motion within the fluid (because the shear stresses are determined by the velocity gradients, e.g., the shear rate). However, if the motion involves an acceleration, this can contribute an additional component to the pressure, as illustrated by the examples in this section. 

In general, the velocity profile will be curved but as equation 1.33 contains only the local velocity gradient it can be applied in these cases also. An example is shown in Figure 1.13. Clearly, as the velocity profile is curved, the velocity gradient is different at different values of y and by equation 1.32 the shear stress r must vary withy. Flows generated by the application of a pressure difference, for example over the length of a pipe, have curved velocity profiles. In the case of flow in a pipe or tube it is natural to use a cylindrical coordinate system as shown in Figure 1.14. 

Sleiched278 has indicated that this expression is not valid for pipe flows. In pipe flows, droplet breakup is governed by surface tension forces, velocity fluctuations, pressure fluctuations, and steep velocity gradients. Sevik and Park 279 modified the hypothesis of Kolmogorov, 280 and Hinze, 270 and suggested that resonance may cause droplet breakup in turbulent flows if the characteristic turbulence frequency equals to the lowest or natural frequency mode of an... 

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