Shear stress potential flow

This index describes the potential of the tablet to cap and laminate, and is related to the ability to relieve shear stresses within the compact via material flow. In Equation (1), TS equals the tensile strength of the normal tablet and TSo is measured from a tablet with a hole in its center, which simulates a defect. 

The relation between the frequency v of local jumps, shear stress r, temperature T and macroscopic rate of creep e was well established by Eyring s reaction rate theory [41]. Let us consider that a number of vl0 thermally activated structural units attempt per unit time to cross a potential barrier Ur, the net flow v, of units that will succeed is then given by ... 

Figure 11.18 Predictions of the tumbling parameter A as a function of reduced concentration C/C2 from the Smoluchowski equation for hard rods with the Onsager potential. The exact result from the spherical-harmonic expansion is shown, compared to approximate results from an analytic formula and from the perturbation expansion of Kuzuu and Doi. The open circles (O) are estimates from the periods of shear stress oscillations in transient shearing flows for PEG solutions (see Walker et al. 1995), and the closed circle ( ) is from a direct conoscopic measurement of Muller et al. (1994).

Of course the modulus of a block copolymer with ordered spherical microdomains is much lower than that of a crystalline solid. Near the disordering transition, the potential energy holding each domain or atom in place is of order ksT, and the modulus is roughly vksT, where v is the number of domains or atoms. This gives an elastic modulus 10 -10 dyn/cm for typical block copolymers with spherical domains, as opposed to 10 -10 dyn/cm for atomic crystals. Ordered spherical diblock copolymers are therefore soft solids. They deflect under an imposed shear stress, but do not flow continuously unless that stress exceeds a critical value, the yield stress (Watanabe and Kotaka 1984). 

This index describes the potential of the tablet to cap and laminate, and is related to the ability to relieve shear stresses within the compact via material flow. 

The analyses of Hunt, Liebovich and Richards, 1988 [287] and of Finnigan and Belcher, 2004 [189] divide the flow in the canopy and in the free boundary layer above into a series of layers with essentially different dynamics. The dominant terms in the momentum balance in each layer are determined by a scale analysis and the eventual solution to the flow held is achieved by asymptotically matching solutions for the flow in each layer. The model apphes in the limit that H/L 1. By adopting this limit, Hunt, Liebovich and Richards [287] were able to make the important simplification of calculating the leading order perturbation to the pressure held using potential how theory. This perturbation to the mean pressure, A p x, z), can then be taken to drive the leading order (i.e. 0(II/I.) ]) velocity and shear stress perturbations over the hill. 

Hartree18 also obtained a family of solutions for f3 between 0 and —0.1988 that were physically acceptable in the sense that 1 from below as i] —> oo. Several such profiles are sketched in Fig. 10-7. These correspond to the boundary layer downstream of the corner in Fig. 10-6(b) (assuming that the upstream surface is either a slip surface or is short enough that one can neglect any boundary layer that forms on this surface). It should be noted that solutions of the Falkner-Skan equation exist for (l < -0.1988, but these are unacceptable on the physical ground that f —> 1 from above as r] —> oo, and this would correspond to velocities within the boundary layer that exceed the outer potential-flow value at the same streamwise position, x. It may be noted from Fig. 10-7 that the shear stress at the surface (r] = 0) decreases monotonically as (l is decreased from 0. Finally, at /3 = -0.1988, the shear stress is exactly equal to zero, i.e., /"(0) = 0. It will be noted from (10-113) that the pressure gradient... 

Clearly, in the limit Re y> 1, the leading-order approximation for the solution to this problem is identical to the inviscid flow problem for a solid sphere. Although the no-slip boundary condition has been replaced in the present problem with the zero-shear-stress condition, (10-197), this has no influence on the leading-order inviscid flow approximation because the potential-flow solution can, in any case, only satisfy the kinematic condition u n = 0 at r = 1. Hence the first approximation in the outer part of the domain where the bubble radius is an appropriate characteristic length scale is precisely the same as for the noslip sphere, namely, (10-155) and (10-156). However, this solution does not satisfy the zero-shear-stress condition (10-197) at the bubble surface, and thus it is clear that the inviscid flow equations do not provide a uniformly valid approximation to the Navier-Stokes... 

An obvious question that may occur to the reader is why the very simple method of integrating the viscous dissipation function has not been used earlier for calculation of the force on a solid body. The answer is that the method provides no real advantage except for the motion of a shear-stress-free bubble because the easily attained inviscid or potential-flow solution does not generally yield a correct first approximation to the dissipation. For the bubble, Vu T=0(l) everywhere to leading order, including the viscous boundary layer where the deviation from the inviscid solution yields only a correction of 0(Re x 2). For bodies with no-slip boundaries, on the other hand, Vu T is still 0(1) outside the boundary layer, but inside the boundary layer Vu T = O(Re). When integrated over the boundary layer, which is G(Re k2) in radial thickness, this produces an ()( / Re) contribution to the total dissipation,... 

Because of this relationship, it does not always follow that catalysts with higher titania contents must deliver the highest MI potential (see footnote 16 on page 294). There is an interaction with activation temperature as well. For example, Figure 107 is a plot of the fluidity of polymers obtained from cogelled silica-titania catalysts [548], Like MI, fluidity is an indication of how easily the molten polymer flows at low shear rates. Whereas MI measures the flow (shear) rate under constant shear stress, fluidity measures the shear stress at constant shear rate (0.1 s-1). Fluidity is the inverse of melt viscosity, and like MI, it varies inversely with MW. 

The remaining sections employ the deformed lattice and quantum picture of plastic flow to account for shear band formation as a means of achieving fte energy localization and hot spot temperatures necessary for initiation of crystalline explosives by shock or impact. Also briefly examined will be the role of the deformed lattice potential in causing particle size effects and its effect on the plastic deformation and energy dissipation rates. Finally, the dependence of the energy dissipation rate on shear stress will be shown to imply that reaction initiation will be dependent on the shape of the shock wave or impact stimulus. These predictions will be compared with experiment. 

The above expressions illustrate the central role that the tunneling probability has in determining the quantum mechanical elements of plastic flow. The tunneling probability, T(t,U), contains all of the dependencies of the applied shear stress, temperature and lattice potential that effect plastic deformation. 

The recent AFM experimental data concerning plastic flow place severe restrictions on possible theoretical accounts of plastic deformation in crystalline solids due to shock or impact. The high spatial resolution of the AFM, = 2 x lO " m, reveals substantial plastic deformation in shocked or impacted crystal lattices and molecules. Understanding how this occurs and its effect on plastic flow requires a quantum mechanical description. The semi-permanent lattice deformation has necessitated the development of a deformed lattice potential which, when combined with a quantum mechanical theory of plastic deformation, makes it possible to describe many of the features found in the AFM records. Both theory and the AFM observations indicate that shock and impact are similar shear driven processes that occur at different shear stress levels and time durations. The role of pressure is to provide an applied shear stress sufficient to cause initiation. 

The behavior of a flowing fluid depends strongly on whether or not the fluid is under the influence of solid boundaries. In the region where the influence of the wall is small, the shear stress may be negligible and the fluid behavior may approach that of an ideal fluid, one that is incompressible and has zero viscosity. The flow of such an ideal fluid is called potential flow and is completely described by the principles of newtonian mechanics and conservation of mass. The mathematical theory of potential flow is highly developed but is outside the scope of this book. Potential flow has two important characteristics (1) neither circulations nor eddies can form within the stream, so that potential flow is also called irrotational flow, and (2) friction cannot develop, so that there is no dissipation of mechanical energy into heat. 

Assume that the resistance to the cylinder motion is due to the shear stress associated with the electroosmotic flow that is generated, so that the Navier-Stokes equation reduces to a balance between viscous and electrical forces. Show that the solution for the electrophoretic velocity of the cylinder is the same as that for a sphere of the same zero potential with the Debye length small. 

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