Straining shear flow

Solid particles. In the case of axisymmetric straining shear flow, the boundary conditions (2.5.1) remote from the particle have the form... 

Arbitrary three-dimensional straining shear flow. Such flow is characterized by the boundary condition (2.5.1) remote from the drop with symmetric matrix of shear coefficients, Gkj = Gjk The solution of the problem on an arbitrary three-dimensional straining shear flow past a drop leads to the following expressions for the velocities outside and inside the drop [26,475] ... 

Arbitrary three-dimensional straining shear flows past a porous particle were considered in [524], The flow outside the particle was described by using the Stokes equations (2.1.1). It was assumed that the percolation of the outer liquid into the particle obeys Darcy s law (2.2.24). The boundary conditions (2.5.1) remote from the particles and the conditions at the boundary of the particle described in Section 2.2 were satisfied. An exact closed solution for the fluid velocities and pressure inside and outside the porous particle was obtained. 

The mean Sherwood number for spherical solid particles, drops, and bubbles in a linear straining shear flow (Gkm = 0 for k m) at low Reynolds numbers and high Peclet numbers... 

Linear Straining Shear Flow. High Peclet Numbers... 

For a solid spherical particle in an arbitrary linear straining shear flow, the following interpolation formula was suggested in [27] for the mean Sherwood number ... 

For an arbitrary straining shear flow (Gkm = Gmk), the mean Sherwood number for a solid sphere can be expressed by the similar formula... 

In the case of an arbitrary straining shear flow past a spherical drop for 0 < PeM 200, the expressions (4.8.5) and (4.8.7) must be substituted into the expression (4.7.3) of the mean Sherwood number. 

The generalization of this equation to an arbitrary straining shear flow has the form... 

Let us consider mass transfer for a translational flow past a solid spherical particle, where the flow field remote from the particle is the superposition of a translational flow with velocity U and an axisymmetric straining shear flow, the translational flow being directed along the axis of the straining flow. The dimensional fluid velocity components in the Cartesian coordinates relative to the center of the particle have the form... 

Formula (5.6.4) is valid for an arbitrary laminar flow without closed streamlines for particles and drops of an arbitrary shape. The quantity Sh(l,Pe) corresponds to the asymptotic solution of the linear problem (5.6.1) at Pe > 1. For spherical particles, drops, and bubbles in a translational or linear straining shear flow, the values of Sh(l, Pe) are shown in the fourth column in Table 4.7. 

Non-Newtonian Fluids Die Swell and Melt Fracture. Eor many fluids the Newtonian constitutive relation involving only a single, constant viscosity is inappHcable. Either stress depends in a more complex way on strain, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known coUectively as non-Newtonian and are usually subdivided further on the basis of behavior in simple shear flow. 

Viscoelasticity illustrates materials that exhibit both viscous and elastic characteristics. Viscous materials tike honey resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain instantaneously when stretched and just as quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both of these properties and, as such, exhibit time-dependent strain. Viscoelasticity is the result of the diffusion of atoms or molecules inside an amorphous material. Rubber is highly elastic, but yet a viscous material. This property can be defined by the term viscoelasticity. Viscoelasticity is a combination of two separate mechanisms occurring at the same time in mbber. A spring represents the elastic portion, and a dashpot represents the viscous component (Figure 28.7). 

Illustration Optimum strain per period in shear flows with periodic reorientation. Many practical mixing flows (e.g., single screw extruder with mixing... 

With the gel equation, we can conveniently compute the consequences of the self-similar spectrum and later compare to experimental observations. The material behaves somehow in between a liquid and a solid. It does not qualify as solid since it cannot sustain a constant stress in the absence of motion. However, it is not acceptable as a liquid either, since it cannot reach a constant stress in shear flow at constant rate. We will examine the properties of the gel equation by modeling two selected shear flow examples. In shear flow, the Finger strain tensor reduces to a simple matrix with a shear component... 

Fig. 13. Shear stress t12 and first normal stress difference N1 during start-up of shear flow at constant rate, y0 = 0.5 s 1, for PDMS near the gel point [71]. The broken line with a slope of one is predicted by the gel equation for finite strain. The critical strain for network rupture is reached at the point at which the shear stress attains its maximum value...

Near LST, the relaxation times become very long, and steady shear flow cannot be reached in the relatively short transient experiment. Large strains are the consequence for most reported data. 

A straightforward estimate of the maximum hardness increment can be made in terms of the strain associated with mixing Br and Cl ions. The fractional difference in the interionic distances in KC1 vs. KBr is about five percent (Pauling, 1960). The elastic constants of the pure crystals are similar, and average values are Cu = 37.5 GPa, C12 = 6 GPa, and C44 = 5.6 GPa. On the glide plane (110) the appropriate shear constant is C = (Cu - C12)/2 = 15.8 GPa. The increment in hardness shown in Figure 9.5 is 14 GPa. This corresponds to a shear flow stress of about 2.3 GPa. which is about 17 percent of the shear modulus, or about C l2n. 

Regarding elongation-induced structure development, Figure 9.12(B) shows the Hencky strain rate dependence of the up-rising Hencky strain (r.t) ) — f.0 x ft) recorded for PLANC at 170 °C. The r.t) increases systematically with the eo- The lower the value of s0, the smaller the value ofs,lE. This tendency probably corresponds to the rheopexy of PLANC under slow shear flow. 

Thus, only the normal Reynolds stresses (i = j) are directly dissipated in a high-Reynolds-number turbulent flow. The shear stresses (i / j), on the other hand, are dissipated indirectly, i.e., the pressure-rate-of-strain tensor first transfers their energy to the normal stresses, where it can be dissipated directly. Without this redistribution of energy, the shear stresses would grow unbounded in a simple shear flow due to the unbalanced production term Vu given by (2.108). This fact is just one illustration of the key role played by 7 ., -in the Reynolds stress balance equation. 

Now the characteristic time for shear flow is the reciprocal of the shear rate. This is the time taken for a cubic element of material to be transformed to a parallelogram with angles of 45° (i.e. the time for unit strain to be applied) as shown in Figure 1.6. The Peclet number can now be written ... 

The phase angle changes with frequency and this is shown in Figure 4.7. As the frequency increases the sample becomes more elastic. Thus the phase difference between the stress and the strain reduces. There is an important feature that we can obtain from the dynamic response of a viscoelastic model and that is the dynamic viscosity. In oscillatory flow there is an analogue to the viscosity measured in continuous shear flow. We can illustrate this by considering the relationship between the stress and the strain. This defines the complex modulus ... 

It became clear in the early development of the tube model that it provided a means of calculating the response of entangled polymers to large deformations as well as small ones [2]. Some predictions, especially in steady shear flow, lead to strange anomaUes as we shall see, but others met with surprising success. In particular the same step-strain experiment used to determine G(t) directly in shear is straightforward to extend to large shear strains y. In many cases of such experiments on polymer melts both Hnear and branched, monodisperse and polydisperse,the experimental strain-dependent relaxation function G(t,Y) may be written... 

Note 5 The Finger strain tensor for simple shear flow is... 

A subsequent analysis [66] also employed this model, with the inclusion of results for the shear strain. The dependence of the viscous effects on initial foam orientation was also noted. Further work [67] on monodisperse wet foams, where is between 0.9069 and 0.9466, demonstrated that, under shear flow, the foam viscosity increased with increasing < > (decreasing liquid content). In contrast, for small deformations, the viscous contribution to the overall stress was found to be independent of liquid content. 

Lodge and Meissner have recently examined in detail the stresses during start-up of steady elongational and shear flows (374). The Lodge equation [Eq.(6.15)], with its flow-independent memory function, described the build-up of stress rather well (even for rapid deformations) until a critical strain was reached. Beyond the critical strain (which differed somewhat in shear and... 

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