Diffusion body-force

Affinity at reaction stage r Acceleration Activity of species a Left Cauchy-Green tensor Body force per unit mass Diffusive body force Right Cauchy-Green tensor Mass-molar concentration Mass concentration of species a Mass-energy flux of the system... 

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. 

Electrokinetics. The first mathematical description of electrophoresis balanced the electrical body force on the charge in the diffuse layer with the viscous forces in the diffuse layer that work against motion (6). Using this force balance, an equation for the velocity, U, of a particle in an electric field... 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

There are three basic distinct types of phenomena that may be responsible for intrinsic instabilities of premixed flames with one-step chemistry body-force effects, hydrodynamic effects and diffusive-thermal effects. Cellular flames—flames that spontaneously take on a nonplanar shape—often have structures affected most strongly by diffusive-thermal... 

To find in equations (18) and (19) we must use the boundary-layer approximation to the y component of equation (1-7). Since the body-force and pressure-gradient terms are negligible in the boundary-layer approximation for the y component, the diffusion equation becomes... 

The product in equation (11) may be related to the binary diffusion coefficients by considering the limiting case of a constant-pressure process in a two-component system with no body forces, for which equation (11) reduces to... 

Consider a microchannel filled with an aqueous solution. There is an eleetrieal doubly layer field near the interface of the channel wall and the liquid. If an electric field is applied along the length of the channel, an electrical body force is exerted on the ions in the diffuse layer. In the diffuse layer of the double layer field, the net charge density, pe is not zero. The net transport of ions is the excess counterions. If the solid surface is negatively eharged, the counterions are the positive ions. These excess counterions will move under the influenee of the... 

The body force acting on a unit of fluid volume is equal to pg and changes from point to point in the solution. It is natural to consider that most of the change in concentration occurs in a very thin layer and most of the change in temperature occurs in a larger, but still thin layer. The changes in temperature and in concentration are the causes of fluid motion. Therefore, one can safely assume that fluid motion also occurs in this layer. Thus, the theories for the hydrodynamic boxmdary layer can also be applied to fluid motion in thermal free convection. In this case, the hydrod3mamic boundary layer coincides with the thermal diffusion layer. 

The potential work term, denotes the rate at which work is done on each of the individual species c in the fluid per unit volume by the individual species body forces, gc, due to the diffusion of the various components in external fields such as an applied electro-magnetic potential. The term can be formulated as a surface integral, and as before converted to a volume integral by use of Gauss theorem (App. A) ... 

The physical meaning of the terms in the enthalpy equation can be identified from the above modeling analysis. The term on the LHS denotes the rate of accumulation of enthalpy within the control volume per unit volume the first term on the RHS denotes the net rate of increase of enthalpy by convection per unit volume the second term on the RHS, that is already known from the foregoing discussion, denotes the rate of increase of enthalpy by the heat flow (e.g., conduction, inter-diffusion effects, Dufour effects and radiation) per unit volume the third term on the RHS denotes the rate of work done by the pressure, which is induced by the surrounding fluid motion, acting on the mixture within the control volume per unit volume the last two terms are already known from the foregoing discussion, nevertheless the fourth term on the RHS denotes the irreversible rate of increase of enthalpy by viscous dissipation per unit volume the fifth term on the RHS denotes the rate of work done by external body forces acting on the mixture within 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. 

We note that just as with our analytic solution for the Eshelby inclusion, the equilibrium equations within the inclusion will have a source term (i.e. an effective body force field) associated with the eigenstrain describing that inclusion. In addition, we require continuity of both displacements, Uj = Uout, and tractions, tj = tout, at the interface between the inclusion and the surrounding matrix. The point of contact between the elastic problem and the diffusion problem is the observation that the interfacial concentration depends upon the instantaneous elastic fields. These interfacial concentrations, in turn, serve as boundary conditions for our treatment of the concentration fields which permits the update of our particle geometries in a way that will be shown below. The concentration at the interface between the inclusion and the matrix may be written as... 

When the particle relative motion is driven by a body force or by the thermocapiUary migration (rather than by self-diffusion). Equation 5.322 is no longer vahd. Instead, in Equation 5.321 we have to formally substitute the following expression for (see Rogers and Davis ) ... 

When an electrical field of intensity E is applied in parallel to a charged flat interface, the excess of counterions in the diffuse layer gives rise to a body force exerted on the liquid. The liquid starts moving with local velocity varying from zero in the plane of shear (x = x ) to some maximal value. 

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