Tangential Slip

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... 

While the fluid dynamics of the actual film-flow process is dauntingly complex, a very approximate interim flow model may be based on Nusselt s (1916) treatment of the flow of a condensate film. This model assumes that there is no shear at the gas-liquid interface, that the film is ripple-free and that there is no tangential slip... 

We can write the tangential momentum flux on a surface s located near the wall as equal to Here, is the number density of molecules crossing surface Y m is the molecular mass is the tangential (slip) velocity on the surface and Vg is the mean thermal speed of the molecule. 

The basis for the familiar non-slip boundary condition is a kinetic theory argument originally presented by Maxwell [23]. For a pure gas Maxwell showed that the tangential velocity v and its derivative nornial to a plane solid surface should be related by... 

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

Supposing constant rotational speeds, no slip, and an axial inlet, the velocity triangles are as shown in Figure 6-10. For the radial vane, the absolute tangential fluid velocity at the impeller exit is constant—even if the flow rate is increased or decreased. 

In Eq. (9.90), C2 is the tangential component of the absolute velocity at the exit if the flow is exactly in the blade direction. Since the slip factor is ieSs than 1, the total pressure increase will decrease according to Eq. ( 9.72) for the same impeller and isentropic flow. 

To obtain physically meaningful solutions, a set of appropriate boundary conditions must also be specified. One obvious requirement is that no fluid should pass through the boundary (i.e. wall) itself. Thus, if we choose a reference frame in which the boundaries are at rest, we require that v fi = 0, where fi is the unit normal to the surface. Another condition, the so-called no-slip condition ([trittSS], [feyn64]), is the requirement that the fluid s tangential velocity vanishes at the surface v x n = 0. 

When the fractions of molecules reflected specularly and diffusively are known, the slip length can be determined, as shovm by Maxwell. Maxwell introduced a tangential momentum accommodation coefficient defined as... 

All four processes have the same origin, since they are all based on the phenomenon of slip of the hquid along the surface of the other phase when a tangential electric field is present, or conversely, on the phenomenon that an electric field will arise during slip of the liquid. 

Figure 18 illustrates the difference between normal hydrodynamic flow and slip flow when a gas sample is confined between two surfaces in motion relative to each other. In each case, the top surface moves with speed ua relative to the bottom surface. The circles represent gas molecules, and the length of an arrow is proportional to the drift velocity for that molecule. The drift velocity variation with distance is illustrated by the plots on the right. When the ratio of the mean free path to the separation distance between surfaces is much less than unity (Fig. 18a), collisions between gas molecules are much more frequent than collisions of the gas molecules with the surfaces. Here, we have classical fluid flow or viscous flow. If the flow were flow in tubes, Poiseuille s law would be obeyed. The velocity of gas molecules at the surface is the same as the velocity of the surface, and in the case of the stationary surface the mean tangential velocity of the gas at the surface is zero. 

Figure 18 The contrast between (a) normal viscous flow in a fluid and (b) slip flow in a dilute gas. (a) Lid 1 (b) Lid > 0.1. u mean tangential speed in the x direction.

In the spiral or tangential cyclone inlet duct, the particles are accelerated to a velocity which is related to the inlet gas velocity Ue. Neglecting as a first approximation the slip between solids and gas, the kinetic energy of the solids which is transported per unit time into the cyclone is approximately given by... 

It is assumed here that there is no slip between the gas and the particle in the tangential direction. 

The condition on the tangential velocity at the interface is not as obvious as that on the normal velocity. There is now ample experimental evidence that the fluid velocity at the surface of a rigid or noncirculating particle is zero relative to the particle, provided that the fluid can be considered a continuum. This leads to the so-called no-slip condition, which for a fluid particle takes the form... 

The boundary conditions are the same as for steady motion considered in Chapters 1, 3, and 4, i.e., uniform flow remote from the particle, no slip and no normal flow at the particle boundary, and, for fluid particles, continuity of tangential stress at the interface. For a sphere the normal stress condition at the interface is again formally redundant, but indicates whether a fluid particle will remain spherical. 

When there is no tangential force (or no transmission of angular momentum) across the liquid—sphere interface, the sphere slips within the liquid. Within the hydrodynamic theory, the rotational relaxation time is negligible for an inertialess slippery sphere. A variable coefficient of slip (or stick), j3, may be introduced. As 3 tends from 0 to 00 the rotational relaxation time increases from 0 to r)V/kT [221, 222]. 

Using water, the heat transfer coefficient on the 50-cm-diameter disc regularly exhibited a minimum value at a radius of about 17 cm. On the other hand, with the use of a water/60% monopropylene glycol mixture, no minimum was observed and the absolute performance was much poorer than that obtained with water. This behavior is attributed to the tangential fluid slip generated as the feed liquid is brought up to the rotor s angular velocity. This slip appears to be more marked with low-viscosity liquids, which seems intuitively reasonable. 

A similar relation governs the tangential displacements in the y-direction. For the no-slip region, sx becomes zero so that... 

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