Steady-state creeping motion

Substituting Eq. (8) into Eq. (7) results in Stokes law for the unidirectional steady-state creeping motion of a sphere in a viscous fluid ... 

At higher stresses, thermally activated dislocation motion such as dislocation climb out of its slip plane limits the rate of steady state creep. When a lattice vacancy exchanges position with an atom in a dislocation core, a jog is formed and in the process an element of the dislocation moves to a parallel slip plane one atom distance away. Dislocations can get around obstacles by repeated climb events. The direction of climb is controlled by the applied stress state, but, of course, the rate increases with stress as the material acts to reduce its free-energy state. The rate of climb under constant stress is controlled by the rate of diffusion. The theory of dislocation creep mechanisms begins with the well-known Johnston-Gilman equation... 

The sample must have reached steady state before cessation of the test or the application of a second step. Steady state in a creep test is seen as a constant slope in the strain curve. A constant slope in the stress curve may also be seen in a stress relaxation test, but often the signal is lost in the noise. A material that is liquid-like in real time will need a test period of 5 to 10 min. A stress relaxation test is likely to be somewhat shorter than a creep test since the signal inevitably decays into the noise at some point. A creep test will last indefinitely but will probably reach steady state within an hour. For a material that is a solid in real time, all experiments should be longer as molecular motion is, by definition, slower. Viscoelastic materials will lie in between these extremes. Polymer melts can take 1 hr or more to respond in a creep test, but somewhat less time in a stress relaxation test. 

However, this is just the Reynolds number that, according to the creeping-motion assumption, is arbitrarily small. In addition, in this case, S = 1 and Re/S = Re 1. It follows that the velocity and pressure fields adjust instantaneously relative to the rate at which the geometry of the flow domain changes and therefore always appear to be at steady state with respect to the present configuration. Thus time appears in a creeping-flow solution only as a parameter that characterizes the instantaneous boundary velocity, or boundary geometry, either of which may depend on time. 

Equation (9 16) is known as the steady-state heat conduction equation and is completely analogous to the creeping-motion equation of Chaps. 7 and 8. It can be seen that convection plays no role in the heat transfer process described by (9 16) and (9 17). Thus the form of the velocity field is not relevant, and in spite of the initial assumption (9 15), there is no dependence of 0o on the Reynolds number of the flow. The solution of (9 16) and (9-17) depends on only the geometry of the body surface, represented in (9 17) by S. 

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... 

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