Fluid mechanics, equations boundary condition

At the scales involved, electrodynamics, chemistry, and fluid mechanics are inextricably intertwined electric fields can create fluid flow, and fluid flow can create electric fields, with the surface chemistry driving the degree of coupling. The flow coupling effect can be described by electrostatic source terms in the Navier-Stokes equations or particle transport equations. Boundary conditions become an issue in microsystems, due to high surface area-volume ratios. Boundary conditions that are taken for granted at the macroscale (e.g., the no-sUp condition) can often fail in these systems. 

At Che opposite limit, where Che density Is high enough for mean free paths to be short con ared with pore diameters, the problem can be treated by continuum mechanics. In the simplest ease of a straight tube of circular cross-section, the fluid velocity field can easily be obtained by Integrating Che Nsvler-Stokes equations If an appropriate boundary condition at Che... 

Boiling at a heated surface, as has been shown, is a very complicated process, and it is consequently not possible to write and solve the usual differential equations of motion and energy with their appropriate boundary conditions. No adequate description of the fluid dynamics and thermal processes that occur during such a process is available, and more than two mechanisms are responsible for the high... 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

In the broadest sense, I found the analogy with fluid mechanics to be very helpful. Just as kinematics provides the geometrical framework of fluid mechanics by exploring the motions that are possible, so also stoicheiometry defines the possible reactions and the restrictions on them without saying whether or at what rate they may take place. When dynamic laws are imposed on kinematic principles, we arrive at equations of motion so, also, when chemical kinetics is added to stoicheiometry, we can speak about reaction rates. In fluid mechanics different materials are distinguished by their constitutive relations and allow equations for the density and velocity to be formulated thence, various flow situations are examined by adding appropriate boundary conditions. Similarly, the chemical kinetics of the reaction system allow the rates of reaction to be expressed in terms of concentrations, and the reactor is brought into the picture as these rates are incorporated into appropriate equations and their boundary conditions. 

Apropos of the analogy noted by Batchelor between vortex velocity and a magnetic field, it should be noted that for the realization of truly steady turbulence a supply of mechanical energy is necessary. The supply of energy comes about either through nonpotential volume forces or through the motion of the surfaces bounding the fluid. With these factors taken into consideration, the set of equations and boundary conditions for a vortex does... 

Clearly, the capability of relatively rapid and inexpensive numerical solutions of a proposed set of governing equations or boundary conditions that can be used in an interactive way with experimental observation represents a profound new opportunity that should greatly facilitate the development of a theoretical basis for fluid mechanics and transport phenomena for complex or multiphase fluids that are the particular concern of chemical engineers. 

Ions can be transported through an electrochemical solution by three mechanisms. These are migration, diffusion, and convection. Electroneutrality must be maintained. The movement of ions in a solution gives rise to the flow of charge, or an ionic current. Migration is the movement of ions under the influence of an electric field. Diffusion is the movement of ions as driven by a concentration gradient, and convection is the movement due to fluid flow. In combination these terms produce differential equations with nonlinear boundary conditions (1). 

The first boundary condition is equivalent to the well-known Levich approach (ca=1 for according to which, it is supposed that the concentration values vary only within a very thin concentration layer while it is supposed to keep its bulk value elsewhere [9], Eq. (3b) has been proposed by Coutelieris et al. [8] in order to ensure the continuity of the concentration upon the outer boundary of the cell for any Peclet number. Furthermore, eq. (3c) and (3d) express the axial symmetry that has been assumed for the problem. The boundary condition (3e) can be considered as a significant improvement of Levich approach, where instantaneous adsorption on the solid-fluid interface cA(ri=Tia,ff)=0) is also assumed for any angular position 0. In particular, eq. (3e) describes a typical adsorption, order reaction and desorption mechanism for the component A upon the solid surface [12,16] where ks is the rate of the heterogeneous reaction upon the surface and the concentration of component A upon the solid surface, c,is, is calculated by solving the non linear equation... 

In this case, the first fluid motion equation (3.33) and the boundary conditions (3.34), (3.35) are equivalent to the conjugation boundary-value problem (3.60) if A = Ah and Pr = 1, so that the profiles of U(z) and T(z) coincide. This is the so-called Reynolds analogy between fluid mechanics and heat (mass) fields. 

FEMLAB has many options for boundary conditions. The finite element method is based on integrating the equations by parts and applying the divergence theorem. Thus, the allowable boundary conditions are really determined by the equations. The basic conditions of fluid mechanics are that either the velocity or forces must be specified. In a three-dimensional problem, you would need to specify three velocities, for example, at each boundary point, or some combination of velocities and forces. The finite element method provides the following boundary conditions (Finlayson, 1992) ... 

It is more difficult to explain the motion of particles that are larger than the mean free path. The explanation Is based on the tangential slip velocity that develops at the surface of a particle in a temperature gradient (Kennard, 1938). This creep velocity is directed toward the high-temperature side, propelling the particle in the direction of lower temperature. An expression for the thermophoretic velocity based on the continuum equations of fluid mechanics with slip-corrected boundary conditions was derived by Brock (1962). Talbot et al. (1980) proposed an interpolation formula for the thermophoretic velocity... 

We are now in a position to begin to consider the solution of heat transfer and fluid mechanics problems by using the equations of motion, continuity, and thermal energy, plus the boundary conditions that were given in the preceding chapter. Before embarking on this task, it is worthwhile to examine the nature of the mathematical problems that are inherent in these equations. For this purpose, it is sufficient to consider the case of an incompressible Newtonian fluid, in which the equations simplify to the forms (2 20), (2-88) with the last term set equal to zero, and (2-93). 

The question then is whether methods exist to achieve approximate solutions for such problems. In fluid mechanics and in convective transport problems there are three possible approaches to obtaining approximate results from the nonlinear Navier Stokes equations and boundary conditions. 

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