Viscous shear force, solid sphere

Answer Use the postulated form of the one-dimensional velocity profile developed in part (a) and neglect the entire left side of the equation of motion for creeping flow conditions at low rotational speeds of the solid sphere. The fact that does not depend on cp, via symmetry, is consistent with the equation of continuity for an incompressible fluid. The r and 9 components of the equation of motion for incompressible Newtonian fluids reveal that dynamic pressure is independent of r and 9, respectively, when centrifugal forces are negligible. Symmetry implies that does not depend on cp, and steady state suggests no time dependence. Hence, dynamic pressure is constant, similar to a hydrostatic situation. Fluid flow is induced by rotation of the solid and the fact that viscous shear is transmitted across the solid-liquid interface. As expected, the -component of the force balance yields useful information to calculate v. The only terms that survive in the (/ -component of the equation of motion are... 

At the lower boundary its velocity would be v — RD. When a particle is present, this will no longer be the case. If there is no slip at the liquid/solid surface, the forces acting on the upper and lower hemispheres of the particle will cause it to rotate. However, this rotation will be resisted by viscous forces over the whole surface of the particle for example, there will be shears down and up, respectively, on the leading and trailing surfaces. Consequently the velocity of the liquid in contact with the top surface of the sphere will be lowered and that at the bottom increased. The velocity profile of the fluid round the perimeter of the particle will be distorted as indicated schematically in Figure 8.4(b). If the overall flow is to be kept constant, the stress must increase, i.e. the viscosity is increased by the presence of the particle. In a dispersion the total effect will be proportional to the concentration of particles and to their volume, leading to the relationship in equation (8.5). 

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