External body force

In this balance p is the static pressure, xtj is the stress tensor, and pgt is the gravitational body force. Ft is an external body forces component it can include forces from interaction between phases, centrifugal forces, Coriolis forces, and... 

In these equations pt is the mass density (g. cm.-3) of the fth chemical species, fc is the rate of production of the fth chemical species by chemical reaction (g. cm.-3 sec.-1), and Fi is the external body force per unit mass acting on the ith species. The velocity v is the local mass average velocity (that velocity measured by a Pitot tube), p is the over-all density of the fluid, and U is the local thermodynamic internal energy (per unit mass) of the mixture. The j, are the fluxes of the various chemical species in g. cm.-2 sec.-1 with respect to the local mass average velocity, v. It should be noted that 2j, = 0, 2/c,- = 0, and = p these relations are used in deriving the over-all equation of continuity [Eq. (4)] by adding up the individual equations of continuity given in Eq. (24). 

This relation, called the mechanical energy equation, describes the rate of increase of kinetic energy in a fluid element as a result of the action of external body forces, pressure, and reversible stress work. 

Equations (7.91) and (7.92) are valid for mixtures at mechanical equilibrium, containing no external body forces, and with negligible surface effects. Also, mass-average velocity is small even under an initially large concentration gradient. For a ternary mixture, Eqs. (7.91) and (7.92) become... 

In these equations, tt is the molecular flux of momentum and g and F are gravitational acceleration and external body forces, respectively. The physical interpretation of the various terms appearing in these equations again follows similar lines the first term is the rate of increase in momentum per unit volume the second term represents... 

It is sometimes convenient to express Eqs. 2.3.10 and 2.3.11 in terms of the external body force exerted per mole of / the corresponding equations are... 

The external body force per mole acting on species Z, is given by (see Newman, 1991)... 

The physical meaning of the terms in the momentum equation (1.78) is inferred from the above modeling analysis. The term on the LHS denotes the rate of accumulation of momentum within the control volume per unit volume the first term on the RHS denotes the net rate of momentum increase by convection per unit volume the second term on the RHS denotes the pressure force acting on the control volume per unit volume the third term on the RHS denotes the viscous force acting on the control volume per unit volume and the fourth term on the RHS denotes the external body forces acting on the control volume per unit volume. 

As will be shown shortly, this term occurs with opposite sign in the internal energy equation the sixth term on the RHS denotes the rate of work done by external body forces on the fluid within the control volume. 

For each balance law, the values of -0, J and 4> defines the transported quantity, the diffusion flux and the source term, respectively, v denotes the velocity vector, T the total stress tensor, gc the net external body force per unit of mass, e the internal energy per unit of mass, q the heat flux, s the entropy per unit mass, h the enthalpy per unit mass, u>s the mass fraction of species s, and T the temperature. 

Absence of external body forces (such as gravity) or body torques (such as magnetic fields). 

The momentum equation, as represented by the Navier-Stokes equation, is not restricted to a single-component fluid but is valid for a multicomponent solution or mixture so long as the external body force is such that each species is acted upon by the same external force (per unit mass), as in the case with gravity. In the following section we consider external forces associated with an applied external field, which differ for different species. The reason for there being no distinction between the various contributions to the stress tensor associated with diffusive transport is that the phenomenological relation for the stress is unaltered by the presence of concentration gradients. This is seen from the fact that the stress tensor must be related to the spatial variations in fluid... 

Up to this point we have considered distributed dilute dispersions of colloidal size particles and macromolecules in continuous liquid media. Where the particles are uncharged and of finite size, they are always separated by a fluid layer irrespective of the nature of the hydrodynamic interactions that take place. In the absence of external body forces such as gravity or a centrifugal field or some type of pressure filtration process, the uncharged particles therefore remain essentially uniformly distributed throughout the solution sample. We have also considered the repulsive electrostatic forces that act between the dispersed particles in those instances where the particles are charged. These repulsive forces will tend to maintain the particles in a uniform distribution. The extent to which a dispersion remains uniformly distributed in the absence of applied external forces, such as those noted above, is described in colloid science by the term stability, whereas colloidal systems in which the dispersed material is virtually insoluble in the solvent are termed lyophobic colloids. 

In Eq. (1) it is assumed that the only external body forces acting on the fluid are either due to gravity or are otherwise derivable from a potential. As such, these forces have been absorbed into the pressure term. 

Momentum Rate Processes Due to External Body Forces (i.e., 5)... 

If a magnetic field with components 0, H ) is applied, the external body force... 

Let the cholesteric film be bounded between the planes z = 0 and z = h, and let there be a temperature gradient along the screw axis z. The components of the director in a right-handed cartesian coordinate system are cos0(z, ), sin0(z, ),0. We assume that there are no heat sources within the liquid crystal, no external body forces and that the velocity vector is zero. Hence T = T(z),ff = G, = rfy = Wy = 0. Thus from (4.3.4)... 

The thermokinetic process (3.133)-(3.135) in thermoelastic material (3.125) fulfilling mass balance generates the admissible thermodynamic process. Indeed, for chosen values of F, T and g = (Gradr)F at X (or place x) and t (see (3.13)) the fields of responses (3.115) follow by (3.125) the symmetric response T fulfils the balance of moment of momentum. Mass balance is satisfied by (3.135) and the balance of momentum (3.78) and energy (3.107) are satisfied by the appropriate choice of external body force b(Y, t) (or/and inertial force i(Y, r)) and volume heating Q(V,x) because (3.116) are controlled Ifom the outside. 

A slip boundary model in microflows accounting for external body forces was derived by To et al. [9]. The botmdary condition based on Maxwell s collision theory between gas molecules is... 

Here, S = 5 gj is the surface stress, f r the external body force per unit mass of material surface, Sa- the specific internal surface energy, q . = q a the surface heat flux vector, rjg. the surface entropy density, So- = i a the surface entropy flux vector, and h /d the surface entropy production. 

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