Momentum stresses

The Ricci tensor that represents the geometry of space is next equated with the so-called energy-momentum (stress) tensor of the matter field that defines the influence of matter and field energy... 

Rate of change Shear Bouyancy force of momentum stress per unit element... 

For this reason the surface tension force can be interpreted as a correction to the momentum stress tensor, i.e., it represents a continuum surface stress (CSS) tensor. 

In the second model the viscous fluid flow inside the fracture is taken into account. In this case the propagation model is unsteady. The process unsteadiness is taken into account by the fluid-flow continuity equation. Meanwhile all other equations describing momentum balance, elastic equilibrium, and material rapture are stationary. The dynamics of the propagation process is represented by the static conditions of flow momentum, stress field, and elastic media displacements in various moments of time. 

The equation of momentum conservation, along with the linear transport law due to Newton, which relates the dissipative stress tensor to the rate of strain tensor = 1 (y. 4, and which introduces two... 

For simplicity, we define T - and T (A iooTe/At). As explained by Luo and Tanner (1989), the decoupled method requires a suitable variable transfonna-tion in the governing equations (3.20) and (3.21). This is to ensure that the discrete momentum equations always contain the real viscous term required to recover the Newtonian velocity-pressure formulation when Ws approaches zero. This is achieved by decomposing the extra stress T as... 

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

As velocity continues to rise, the thicknesses of the laminar sublayer and buffer layers decrease, almost in inverse proportion to the velocity. The shear stress becomes almost proportional to the momentum flux (pk ) and is only a modest function of fluid viscosity. Heat and mass transfer (qv) to the wall, which formerly were limited by diffusion throughout the pipe, now are limited mostly by the thin layers at the wall. Both the heat- and mass-transfer rates are increased by the onset of turbulence and continue to rise almost in proportion to the velocity. 

The creation terms embody the changes in momentum arising from external forces in accordance with Newton s second law (F = ma). The body forces arise from gravitational, electrostatic, and magnetic fields. The surface forces are the shear and normal forces acting on the fluid diffusion of momentum, as manifested in viscosity, is included in these terms. In practice the vector equation is usually resolved into its Cartesian components and the normal stresses are set equal to the pressures over those surfaces through which fluid is flowing. 

Stressing by Impact. Size reduction is achieved by the impact of a particle against a soHd surface (Fig. 3c) or another particle (Fig. 3d). The particle can be accelerated to impact against the surface, or the surface can be accelerated to impact the particle, as in an impact mill. The momentum transferred is limited by the mass of the particle and the achievable impact velocity. 

For smooth pipe, the friction factor is a function only of the Reynolds number. In rough pipe, the relative roughness /D also affects the friction factor. Figure 6-9 plots/as a function of Re and /D. Values of for various materials are given in Table 6-1. The Fanning friction factor should not be confused with the Darcy friction fac tor used by Moody Trans. ASME, 66, 671 [1944]), which is four times greater. Using the momentum equation, the stress at the wall of the pipe may be expressed in terms of the friction factor ... 

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 ... 

In Eulerian coordinates x and t, the mass and momentum conservation laws and material constitutive equation are given by (u = = particle velocity,, = longitudinal stress, and p = material density)... 

Where momentum is transfeiTed by shearing stresses, in which the transfer is perpendicular to the direction of flow. This category includes the rotating disc and cone agitators. 

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 ... 

Conservation relations are used to derive mechanical stress-volume states from observed wave profiles. Once these states have been characterized through experiment or theory they may, in turn, predict wave profiles for the material in question. For the case of a well-defined shock front propagating at constant speed L/ to a constant pressure P and particle velocity level u, into a medium at rest at atmospheric pressure, with initial density, p, the conservation of momentum, mass, and energy leads to the following relations ... 

While the general form of the generalized Euler s equation (equation 9.9) allows for dissipation (through the term Hifc) expression for the momentum flux density as yet contains no explicit terms describing dissipation. Viscous stress forces may be added to our system of equations by appending to a (momentarily unspecified) tensor [Pg.467]

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