Kinematic conditions

Table II. Kinematic Conditions for the Formation of Tertiary Ions by Various Collision Mechanisms...

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 ... 

One of the early hopes of heavy ion induced transfer reactions was that new states in nuclei would be preferentially populated The fact that this hope diminished was due primarily to insufficient understanding of the reaction mechanism, but also to the generally poorer energy resolution obtained with heavy ions as compared to light ions. In this paper, I hope to demonstrate that with the proper kinematical conditions there is a remarkable selectivity which can be. obtained with a proper. choice of the reaction and that these reactions can be valuable spectroscopic tools The data in this talk have been taken using beams from the Brookhaven National Laboratory double MP tandem facility with particles identified in the focal plane of a QDDD spectrometer ... 

C. Rauwendaal, T.A. Osswald, G. Tellez, and PJ. Gramann. Flow analysis in screw extruders -effect of kinematic conditions. International Polymer Processing, 13(4) 327-333, 1998. 

Figure 9. Phase difference for the transmitted beam in the undisturbed crystal comparing the phase under kinematical conditions and for (lower curve) 0002 or (upper curve) 11-20 two-beam conditions as a function of the thickness.

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

In developing the kinematical theory, we made use in Section 3.7 of the column approximation. In view of the preceding discussion, it is important to estimate the crystal thickness t for which the column approximation is valid. If kinematical conditions are to apply within a single column of unit cells of width x, and adjacent columns are to diffract independently of each other, then 2 t < x. Thus, for x 0.5 nm, the crystal thickness must be less than about 25 nm. 

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,... 

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