Creeping flow around a sphere

Figure 10. Velocity field of creeping flow around a sphere [24], Each arrow represents the velocity at the origin of the depicted vector. The length of the arrow corresponding with the free velocity v would equal the radius of the sphere...

The values of n and m vary depending on the system and the geometry, but typical values are n = m = 0.5 for creeping flow around a sphere in a gas-liquid system, and n = m = 0.33 for creeping flow around a sphere in liquid-solid systems. 

When fluid flows around the outside of an object, an additional loss occurs separately from the frictional energy loss. This loss, called form drag, arises from Bernoulli s effect pressure changes across the finite body and would occur even in the absence of viscosity. In the simple case of very slow or creeping flow around a sphere, it is possible to compute this form drag force theoretically. In all other cases of practical interest, however, this is essentially impossible because of the difficulty of the differential equations involved. 

The time required for a ball to fall a given distance in a fluid is probably the simplest and certainly one of the oldest viscosity tests (Stokes, 1851). Unfortunately, creeping flow around a sphere is very complex. Thus the falling ball is really an index test and requires a constitutive equation for complete analysis. Analyses of the flow have been made for inelastic (Gottlieb, 1979 Beris et al., 1985) and viscoelastic fluids (Hassager and Bisgaard, 1983 Graham et al., 1989). Usually an apparent viscosity based on the Newtonian analysis is reported. 

Creeping flow around a sphere. From Taneda (1979). 

Chhabra, R. R, P. H. J. Uhlherr, and D. V Boger, The Influence of Fluid Elasticity on the Drag Coefficient for Creeping Flow Around a Sphere, J. Non-Newtonian Fluid Mech., 6, 187-199, 1980. 

Now, the denominator of the final term in (11-24) can be evaluated explicitly via exact fluid dynamics solutions for creeping flow around a solid sphere, and for creeping and potential flow around a gas bubble. In the creeping or laminar flow regimes, the momentum boundary layer is not thin. Hence, the following claim ... 

For creeping flow around a stationary solid sphere. 

Unlike creeping flow about a solid sphere, the r9 component of the rate-of-strain tensor vanishes at the gas-liquid interface, as expected for zero shear, but the simple velocity gradient (dvg/dr)r R is not zero. The fluid dynamics boundary conditions require that [(Sy/dt)rg]r=R = 0- The leading term in the polynomial expansion for vg, given by (11-126), is most important for flow around a bubble, but this term vanishes for a no-slip interface when the solid sphere is stationary. For creeping flow around a gas bubble, the tangential velocity component within the mass transfer boundary layer is approximated as... 

In flow around a sphere, for example, the fluid changes velocity and direction in a complex manner. If the inertia effects in this case were important, it would be necessary to keep all the terms in the three Navier-Stokes equations. Experiments show that at a Reynolds number below about 1, the inertia effects are small and can be omitted. Hence, the equations of motion, Eqs. (3.7-36)-(3.7-39) for creeping flow of an incompressible fluid, become... 

The interphase mass transfer coefficient of reactant A (i.e., a,mtc), in the gas-phase boundary layer external to porous solid pellets, scales as Sc for flow adjacent to high-shear no-slip interfaces, where the Schmidt number (i.e., Sc) is based on ordinary molecular diffusion. In the creeping flow regime, / a,mtc is calculated from the following Sherwood number correlation for interphase mass transfer around solid spheres (see equation 11-121 and Table 12-1) ... 

A rigorous solution exists for fj for the limiting condition of very low fluid flow rates around a sphere - in which the fluid streamlines follow the contours of the sphere, with no separation at the upper surface (the so called creeping flow regime). This may be regarded to occur at particle Reynolds numbers Re below about 0.1 ... 

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the drag coefficient ... 

The curvature correction factor in parentheses in (11-29) is calculated explicitly for creeping flow of an incompressible Newtonian fluid around a solid sphere, where... 

There is no contribution from (3vr/30),=i to the r-9 component of the rate-of-sfrain tensor at the solid-liquid interface because the solid is nondeformable. Creeping flow of an incompressible Newtonian fluid around a stationary solid sphere produces the following expressions for the tangential velocity component ... 

Figure 11-1 Thickness of the mass transfer boundary layer around a solid sphere, primarily in the creeping flow regime. This graph in polar coordinates illustrates 8c 9) divided by the sphere diameter vs. polar angle 9, and the fluid approaches the solid sphere horizontally from the right. No data are plotted at the stagnation point, where 9=0.

Effect of Flow Regime on the Dimensionless Mass Transfer Correlation. For creeping flow of an incompressible Newtonian fluid around a stationary solid sphere, the tangential velocity gradient at the interface [i.e., g 9) = sin6>] is independent of (he Reynolds number. This is reasonable because contributions from accumulation and convective momentum transport on the left side of the equation of motion are neglected to obtain creeping flow solutions in the limit where Re 0. Under these conditions. 

Compare mass transfer boundary layer thicknesses for creeping flow of identical fluids around (a) a sohd sphere, and (b) a gas bubble at the same value of the Reynolds number. In which case is the boundary layer thickness greater ... 

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