The Kinematic Condition

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. 

Let us denote the bulk-phase densities on the two sides of the interface as p and p and the fluid velocities as u and u. The orientation of surface S is specified in terms of a unit normal n. In general, the surface S is not a material surface. For example, if there is a phase transition occurring between the two bulk phases (e.g., a solid phase is melting or a liquid phase is evaporating), mass will be transferred across S. However, the surface S is not a source or sink for mass, and thus mass conservation requires that the net flux of mass to (or from) the surface must be zero. 

It is probably useful to think of two specific situations. In one, there is no phase transformation occurring, and in this case S is in fact a material surface separating a viscous fluid and a second medium that may either be solid or fluid. Hence, in the absence of phase change at S, the normal component of velocity must be continuous across it and equal to the normal velocity of the surface  

If the second phase is a solid wall, then u = L som, which is assumed to be known. In a frame of reference fixed to a solid wall, u n = 0, and in this frame of reference 

A generalization of the condition (2-112) is required if there is an active phase transformation occurring at S, i.e., if the liquid is vaporizing or the solid is melting. In this case, we must distinguish between the bulk fluid velocities in the limit as we approach the interface, and the velocity of the interface itself, u1 n (where the interface is specified still by the criteria of zero excess mass discussed earlier). The condition of conservation of mass then requires that 

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The kinematic condition requires that no fluid can transverse the interface, i.e. the local flow velocity Wj relative to the velocity of the interface should be zero... 

Via Eq. (136) the kinematic condition Eq. (131) is fulfilled automatically. Furthermore, a conservative discretization of the transport equation such as achieved with the FVM method guarantees local mass conservation for the two phases separately. With a description based on the volume fraction fimction, the two fluids can be regarded as a single fluid with spatially varying density and viscosity, according to... 

Fig. 10. (a) He time-of-flight spectrum taken from a LiF(001) surface along the < 100) azimuth at an incident angle Si = 64.2°. The primary beam energy was 19.2 meV. (After Ref 25.). (b) Measured Rayleigh phonon dispersion curve of LiFfOOl) < 100), including a scan curve (dashed) for the kinematical conditions in (a). (After Ref. 25.)... 

For high order reflections with a large g, the rapid increase in the excitation error away from the Bragg condition results a rapid decrease in diffraction intensity. Under the kinematical condition, the maximum intensity occurs at the Bragg condition, which appears as a straight line within a small convergence angle. 

Taking good HREM images is a critical step of any structure determination. The thinnest parts of the crystals should be used, to avoid strong multiple scattering. Only then are we close to the kinematical condition, where the relation between the amplitudes and phases extracted... 

From the analysis of the kinematic conditions at a moving boundary 36), with Eq. (4.26) taken into account, we obtain ... 

The flow rate can be related to the interface position through the kinematic condition. 

Here K=x/(prga), y=l-(dpv/dx)/(pLg)=l-4aK/dh and dh is the hydraulic diameter of the part of channel cross-section occupied by gas phase. The same symbols for dimensionless variables been used again and p is dimensionless pressure in the liquid. With account for (3.4) the kinematic condition on the interface (3.3) and (3.2) gives ... 

Both velocity components at the solid-liquid interface y = ( t,x) must satisfy u I. = 0 and v = 0. The kinematic condition at the liquid-air surface y = t,x) is dt+ dx-. , . At the liquid-air surface the shear stress must vanish ... 

In the absence of an active phase-transformation process, both sides of (2.114a) are zero, i.e. u n = u1 n and u n = u1 n, and the condition (2-114a) reduces to (2 112). It is important to emphasize that the kinematic condition is a direct consequence of mass conservation at S, and must always be satisfied, regardless of the specific fluid properties or any details of the flow. 

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. 

This is the most general form of the kinematic condition. Obviously, in view of (2-112), it can be written in terms of either u or u. The reader should note that if the shape of the... 

We have assumed that the fluid properties are independent of temperature, and hence no natural convection will occur. However, the kinematic condition, (2.114b), at an interface involving a phase transformation from liquid to solid requires that there be a relative velocity in the liquid relative to the velocity of the interface ... 

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... 

However, for the distortion mode, (4—298), the kinematic condition, (4-299), requires... 

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... 

As usual, we can also derive the same result by first solving for w by means of the continuity equation, (6-216b), plus the impermeability condition, (6-218a), atz = 0, and then applying the 0(8) contribution to the kinematic condition, (6-244b). The result for w is... 

Here, we consider Stokes problem of uniform, streaming motion in the positive z direction, past a stationary solid sphere. The problem corresponds to the schematic representation shown in Fig. 7-11 when the body is spherical. This problem may also be viewed as that of a solid spherical particle that is translating in the negative z direction through an unbounded stationary fluid under the action of some external force. From a frame of reference whose origin is fixed at the center of the sphere, the latter problem is clearly identical with the problem pictured in Fig. 7-11. Because we have already derived the form for the stream-function under the assumption of a uniform flow at infinity, we adopt the latter frame of reference. The problem then reduces to applying boundary conditions at the surface of the sphere to determine the constants C and Dn in the general equation (7-149). The boundary conditions on the surface of a solid sphere are the kinematic condition and the no-slip condition,... 

It remains to determine the four sets of constants, An, Bn, Cn, and Dn, and the function fo that describes the 0(Ca) correction to the shape of the drop. For this, we still have the five independent boundary conditions, (7-207)-(7-210). It can be shown that the conditions (7-207), (7-208), and (7-209) are sufficient to completely determine the unknown coefficients in (7-213) and (7-215). Indeed, for any given (or prescribed) drop shape, the four conditions of no-slip, tangential-stress continuity and the kinematic condition are sufficient along with the far-field condition to completely determine the velocity and pressure fields in the two fluids. The normal-stress condition, (7-210), can then be used to determine the leading-order shape function /0. Specifically, we can use the now known solutions for the leading-order approximations for the velocity components and the pressure to evaluate the left-hand side of (7 210), which then becomes a second-order PDE for the function /(). The important point to note is that we can determine the 0(Ca) contribution to the unknown shape knowing only the 0(1) contributions to the velocities and the pressures. This illustrates a universal feature of the domain perturbation technique for this class of problems. If we solve for the 0(Cam) contributions to the velocity and pressure, we can... 

Two of these boundary conditions are straightforward, namely, continuity of velocity and the kinematic conditions. Because the drop shape is assumed to be spherical, these conditions simply reduce to... 

The rate of change of b describes the response of the interface to any rate of change of the flow. To obtain an equation for b, we apply the kinematic condition, (8-63). The result is... 

This is the key result for application of the boundary-integral technique to interface dynamics problems. Let us first consider a case in which the interfacial tension is constant, i.e., gradvi/ = 0. In this case, if the shape of the interface is specified, then V n is known, and (8-209) is an integral equation for the interface velocity, u(x,v). Hence the problem defined by (8-199)-(8-203) can be solved as follows for a specified undisturbed flow, u Uoo as x - cxc. The drop shape is initially specified (usually as a sphere). The integral equation (8-209) is then solved to obtain the interface velocity, u(xs). Then, with u(x,v) known, we can use a discretized form of the kinematic condition, (8-20 lb), to increment the drop shape forward one step in time. We then return to (8-209) with this new drop shape, and again solve for u(xv ), and so on. This process continues as long as the interface shape continues to evolve. If there is a steady-state solution, and our numerical scheme is working properly, we should find that... 

If we compare (10-12) with the full 2D Navier Stokes equation, expressed in terms of the streamfimction, we note that the latter is fourth order (the viscous terms generate V4i/f), whereas (10-12) is only second order. As a result, it is clear that the velocity field obtained from (10-12) will, at most, be able to satisfy only one of the boundary conditions of the original problem at the body surface. Intuitively, we may anticipate that the kinematic condition on the normal component of velocity,... 

The condition (10 16) is just the kinematic condition (10 13) expressed in terms of the streamfunction, whereas (10 15) requires that the velocity field approach a uniform streaming motion at large distances from the cylinder. Equation (10 14), subject to (10-15) and (10 16), is solved easily by means of separation of variables, or other standard transform methods, with the resulting solution... 

Clearly, in the limit Re y> 1, the leading-order approximation for the solution to this problem is identical to the inviscid flow problem for a solid sphere. Although the no-slip boundary condition has been replaced in the present problem with the zero-shear-stress condition, (10-197), this has no influence on the leading-order inviscid flow approximation because the potential-flow solution can, in any case, only satisfy the kinematic condition u n = 0 at r = 1. Hence the first approximation in the outer part of the domain where the bubble radius is an appropriate characteristic length scale is precisely the same as for the noslip sphere, namely, (10-155) and (10-156). However, this solution does not satisfy the zero-shear-stress condition (10-197) at the bubble surface, and thus it is clear that the inviscid flow equations do not provide a uniformly valid approximation to the Navier-Stokes... 

These forms for u satisfy the continuity condition (12-22), and thus the only remaining condition that must be satisfied is the kinematic condition (12-23). If we substitute for ur and /, this condition can be written in the form... 

Because the fluids are approximated as inviscid, neither the no-slip conditions (12-69) nor the continuity of shear-stress conditions (12-74) can be imposed. Flence the solutions of (12-75) satisfy the kinematic condition in the form (12-71) and the normal-stress condition (12-73), with the viscous-stress contribution neglected ... 

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