Kinematic boundary condition

It is seen that we are comparing kinematic viscosity, thermal diffusivity, and diffu-sivity of the medium for both air and water. In air, these numbers are all of the same order of magnitude, meaning that air provides a similar resistance to the transport of momentum, heat, and mass. In fact, there are two dimensionless numbers that will tell us these ratios the Prandtl number (Pr = pCpv/kj = v/a) and the Schmidt number (Sc = v/D). The Prandtl number for air at 20°C is 0.7. The Schmidt number for air is between 0.2 and 2 for helium and hexane, respectively. The magnitude of both of these numbers are on the order of 1, meaning that whether it is momentum transport, heat transport, or mass transport that we are concerned with, the results will be on the same order once the boundary conditions have been made dimensionless. 

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. 

Ya.B. assumes that 1) a fast dynamo based on this mechanism can be generated by a time-periodic three-dimensional fluid flow, but that 2) a fast kinematic dynamo (in three-dimensional space with steady flow and periodic boundary conditions) is impossible. 

Equations (8.10)—(8.12), tensorial ranks and boundary conditions (8.14)-(8.15) notwithstanding, embody a structure similar in format and symbolism to their counterparts for the transport of passive scalars, e.g., the material transport of the scalar probability density P (Brenner, 1980b Brenner and Adler, 1982), at least in the absence of convective transport. As such, by analogy to the case of nonconvective material transport, the effective kinematic viscosity viJkl of the suspension may be obtained by matching the total spatial moments of the probability density Pu to those of an equivalent coarse-grained dyadic probability density P j, valid on the suspension scale, using a scheme (Brenner and Adler, 1982) identical in conception to that used to determine the effective diffusivity for material transport at the Darcy scale from the analogous scalar material probability density P. In particular, the second-order total moment M(2) (sM, ) of the probability density P, defined as... 

Other kinematic regions require a complete description of the collision, which may be facilitated by including the boundary condition for the three charged particles in the final state. This is nontrivial because there is no separation distance at which the Coulomb forces in the three-body system are strictly negligible. The pioneering experiments of Ehrhardt et al. (1969) are of this type. 

With kinematic boundary conditions at the surface and bottom, the following continuity equation integrated over the depth of the water column is obtained ... 

The order of accuracy of the upwind scheme can be improved by using a higher-order accurate scheme such as QUICK (quadratic upwind interpolation for convective kinematics).The concentration at an interface is interpolated by means of a parabola instead of a straight line. The use of QUICK or similar methods may, however, complicate implementation of boundary conditions or lessen the convergence rate of the solution algorithm. 

Turning now to the question of boundary conditions, the solution of (2 108) (2 110) requires both thermal boundary conditions relating the temperature or its derivatives and the velocity and its derivatives on the two sides of S. We begin with the so-called kinematic boundary condition, which derives from the principle of mass conservation at any boundary of the flow domain. 

The third type of boundary condition at the surface S involving the bulk-phase velocities is known as the dynamic condition. It specifies a relationship between the tangential components of velocity, [u - (u n)n] and [u - (u n)n]. However, unlike the kinematic and thermal boundary conditions, there is no fundamental macroscopic principle on which to base this relationship. The most common assumption is that the tangential velocities are continuous across S, i.e.,... 

In summary, we have so far seen that there are two types of boundary conditions that apply at any solid surface or fluid interface the kinematic condition, (2-117), deriving from mass conservation and the dynamic boundary condition, normally in the form of (2-122), but sometimes also in the form of a Navier-slip condition, (2-124) or (2-125). When the boundary surface is a solid wall, then u is known and the conditions (2-117) and (2-122) provide a sufficient number of boundary conditions, along with conditions at other boundaries, to completely determine a solution to the equations of motion and continuity when the fluid can be treated as Newtonian. 

When a bounding surface is a fluid fluid interface instead of the surface of a solid, the kinematic and dynamic boundary conditions can be seen, from (2 112) and (2-122), to provide either two (or three) independent relationships between the unknown velocity vectors, u and u. However, there are a total of either four or six unknown components of u and u (the number depending on whether the flow is 2D or frilly 3D), and thus additional conditions must be imposed at an interface to completely specify the solutions of the Navier-Stokes and continuity equations. In this section, we assume that there is no phase change at the interface. 

For motions of a single fluid involving sohd boundaries, we have already noted that the no-slip and kinematic boundary conditions are sufficient to determine completely a solution of the equations of motion, provided the motion of the boundaries is specified. In problems involving two fluids separated by an interface, however, these conditions are not sufficient because they provide relationships only between the velocity components in the fluids and the interface shape, all of which are unknowns. The additional conditions necessary to completely determine the velocity fields and the interface shape come from a force equilibrium condition on the interface. In particular, because the interface is viewed as a surface of zero thickness, the volume associated with any arbitrary segment of the interface is zero, and the sum of all forces acting on this interface segment must be identically zero (to avoid infinite acceleration). 

Here, we consider only the simpler situation in which the surfactant is assumed to be relatively dilute so that it is mobile on the interface and contributes a change only in the interfacial tension, without any more complex dynamical or rheological effects. In this case, the boundary conditions derived for a fluid interface still apply. Specifically, the dynamic and kinematic boundary conditions, in the form (2 122) and (2-129), respectively, and the stress balance, in the form (2 134), can still be used. However, the interfacial tension, which appears in the stress balance, now depends on the local concentration of surfactant. We shall discuss how this concentration is defined shortly. First, however, we note that flows involving an interface with surfactant are qualitatively similar to thermocapillary flows. The primary difference is that the concentration distribution of surfactant on the interface is almost always dominated by convection and diffusion within the interface, whereas the... 

The kinematic boundary condition, (2-118), is satisfied identically because the velocity component normal to the walls is identically equal to zero. Although this problem is extremely simple to solve, the solution is temporarily postponed. 

The time-dependent fiinction Hit ) is determined by the rate of increase or decrease in the bubble volume. The governing equations and boundary conditions that remain to be satisfied are (1) the radial component of the Navier Stokes equation (2) the kinematic condition, in the form of Eq. (2 129), at the bubble surface and (3) the normal-stress balance, (2 135), at the bubble surface with = 0. Generally, for a gas bubble, the zero-shear-stress condition also must be satisfied at the bubble surface, but xrti = 0, for a purely radial velocity field of the form (4-193), and this condition thus provides no usefirl information for the present problem. 

The relationship between Hit) and the bubble radius R(t) is determined from the kinematic boundary condition. In particular, for a bubble containing only an insoluble gas, the kinematic condition takes the form... 

To avoid this, we use domain perturbation theory (see Section E) to transform from the exact boundary conditions applied at rs = R + sf to asymptotically equivalent boundary conditions applied at the spherical surface rs = R(t). For example, instead of a condition on ur at r = R(t) + ef from the kinematic condition, we can obtain an asymptotically equivalent condition at r = R by means of the Taylor series approximation... 

Then the kinematic boundary condition, (4-294), takes the form... 

The boundary conditions at the cylinder surfaces are just the kinematic and no-slip conditions, namely,... 

The primary new feature is that the boundary conditions (6-124a) are now replaced with boundary conditions for an interface. We assume that the fluid above the interface is air (or some other gas). Hence these boundary conditions can be adopted from Eqs. (6-9) (6-21). The kinematic condition, (6-9), (for a steady interface shape) becomes,... 

The kinematic condition remains in the form (6-19) because it does not involve a or 0. However, the thermal boundary condition, (6-199) is now... 

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