Creeping flow wall effects

Here Kjj is obtained from Fig. 9.5. Equation (9-27) and the equations of Chapter 5 can be used to determine the decrease in Sh for a rigid sphere with fixed settling on the axis of a cylindrical tube. For example, for a settling sphere with 2 = 0.4 and = 200, Uj/Uj = 0.76 and UJUj = 0.85. Since the Sherwood number is roughly proportional to the square root of Re, the Sherwood number for the settling particle is reduced only 8%, while its terminal velocity is reduced 24%. As in creeping flow, the effect of container walls on mass and heat transfer is much smaller than on terminal velocity. 

BTU/hr. sq.ft. over a wide range of viscosities and rotational speeds. This is equivalent to the thermal resistance of a fluid film equal to about 1/2 the clearance between the helical agitator and the vessel wall. This represents Reynolds numbers in the range of 10 to 10. This is the region of creeping flow where, with no inertial effects, there is little displacement of the fluid adjacent to the wall. 

The wall effect for particles settling in non-Newtonian fluids appears to be significantly smaller than for Newtonian fluids. For power law fluids, the wall correction factor in creeping flow, as well as for very high Reynolds... 

The results in Table 9.2 apply when no end effects are present. Sutterby (S7) determined simultaneous wall and end correction factors for the creeping flow range. His correlations are shown in Fig. 9.3 where the cylindrical column has closed ends a distance apart and the center of the spherical particle is distance Z from one end of the tube. The curve for D/L = 1.0 and Z/L = 1/2 is... 

Sutterby (S7) gave a useful tabulation of the viseosity ratio, defined in Eq. (9-8), for relatively low Re and a. These values, intended primarily to correct for departures from Stokes law in falling sphere viscometry, are shown in Fig. 9.6. Reynolds number is defined using the measured Uj and defined in Eq. (9-9). The curve for a = 0 accounts for departures from the creeping flow approximations in an unbounded fluid, and the relative displacement of the other curves indicates the wall effect. 

The stream function expressions of Haberman and Sayre (HI) for creeping flow permit the calculation of the effect of cylindrical containing walls on the... 

Unnikrishnan and Chhabra (1990) Cylinders (axial) lA L/d) 2 Power law Experimental results on wall effects and drag coefficient in creeping flow region... 

Now, if the Reynolds number of the flow is sufficiently small for the creeping-motion approximation to apply, it can be shown by the arguments of Subsection B.3 in Chap. 7 that no lateral motion of the drop is possible unless the drop deforms. In other words, Us = Useiin this case, though, of course, Us is not generally equal to the undisturbed velocity of the fluid evaluated at the X3 position of the drop center. The drop may either lag or lead (in principle) because of a combination of interaction with the walls and the hydrodynamic effect of the quadratic form of the undisturbed velocity profile - see Faxen s law. Because the drop deforms, however, lateral migration can occur even in the complete absence of inertia (or non-Newtonian) effects. In this problem, our goal is to formulate two... 

Thermal creep is the phenomenon in which we are able to start rarefied gasfiows because of tangential temperature gradients along the channel walls, where the fluid starts creeping in the direction from cold toward hot (see Figure 3.11). Equilibrium condition requires no flow in the channel for thick channel (A h). If channel thickness /i A (mean free path), rarefied gas effects have to be taken into account. Here, the local equilibrium mechanism is very complex, and interaction of gas molecules with the walls must be considered. 

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