Linear straining flow

The effect of an external straining flow on a laminar wake caused by other bodies is illustrated with linearised calculations (see Fig. 7.2b). Consider a point source of momentum in a planar linear straining flow,... 

Now, however, this can be expressed as the sum of a linear straining flow and a purely rotational flow, that is,... 

The decomposition of the tensor [Gfcm] into the symmetric and antisymmetric parts corresponds to the representation of the velocity field of a linear shear fluid flow as the superposition of linear straining flow with extension coefficients Ei, 2, 3 along the principal axes and the rotation of the fluid as a solid at the angular velocity u = ( 32,013, SI21)-... 

Arbitrary linear straining flow Gkm — Gmk 0.36 GkmGkm)1 2 (the sum is over both indices)... 

In works where the stationary flow was detected, the viscosity X = cr/x was, normally, either constant 16 20,31> or grew with increase of x 18 19-24 27> above 3r 0 (r 0 is the maximum shear viscosity) in the linear strain region. The intensive growth of viscosity X was detected, apparently, for the first time in Ref. 19). The greatest increase in viscosity X from 3r to 20ri was observed in Refs. I8 21,30>. In the same works the viscosity X started to drop after the growth (this shall be discussed in detail below). Usually the relationship x/rj (rj = a12/y is effective viscosity at stationary shear) is taken at different strain velocities, which could vary by more than an order of magnitude (see, for example, Ref.30>). The data on the existence of stationary flow and behavior of viscosity in this case, as well as descriptions of extended polymers are collected in Ref. 20>. 

The tests were also performed on computer-generated data in which additional uniform or non-uniform motion was added, to study how far the CG algorithm could be pushed beyond its original design parameters. For uniform motion, CG tracking was as successful as in the quiescent case for small drifts but failed for drifts of the order of half the particle-particle separation. For non-uniform (linear shear) flows with small strains between frames the identification worked correctly, but large non-uniform displacements caused major tracking errors. 

If we were to estimate the viscosity by using only the ratio of the measured shear stress, (3-80), to the actual velocity gradient, (3 82), at r = a (instead of the full strain rate E ) as would be valid for a linear shear flow, we would obtain... 

The closely related problem of a rigid sphere in a linear shear flow is very easy to solve now by analogy to the solution for an axisymmetric straining flow. We consider the problem in the form... 

Figure 2.10. Linear shear flow past a fixed circular cylinder (a) straining flow (Ei = 0 and Cl = 0) (b) simple shear flow (Ei = 0 and Ei = -Cl)...

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... 

The solution of hydrodynamic problems for an arbitrary straining linear shear flow (Gkm = Gmk) past a solid particle, drop, or bubble in the Stokes approximation (as Re -> 0) is given in Section 2.5. In the diffusion boundary layer approximation, the corresponding problems of convective mass transfer at high Peclet numbers were considered in [27, 164, 353]. In Table 4.4, the mean Sherwood numbers obtained in these papers are shown. 

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 a spherical drop in an arbitrary straining linear shear flow under limiting resistance of the continuous phase, one can use the interpolation formula [353] ... 

It follows from (4.11.3) that in the region -1 < fl < +1, the mean Sherwood number varies only slightly (the relative increment in the mean Sherwood number as Iflfil varies from 0 to 1 is at most 1.3%). In the special cases of purely straining (CIe = 0) and purely shear (Ifi l = 1) linear Stokes flow past a circular cylinder, formula (4.11.3) turns into those given in [342, 343]. 

Drop, bubble Arbitrary straining linear shear flow ( Gij - Gji) Hkf / 3 y/2 U = a[ GijGij, Gij are the shear matrix coefficients... 

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. 

D. C. Venerus. A critical evaluation of step strain flows of entangled linear polymer liquids. J. Rheol, 49 (2005), 277-295. 

The Maxwell model allows the approximation of elastic and creep strain under static stress as a function of time by assigning elastic compliance leading to instant elastic strain to the spring part of the model. The flow due to creep is represented by the linear slope related to the single dashpot element with linear viscous flow properties. 

The rheological properties are described by relations between stress and strain (elasticity) or between stress and rate of strain (flow). In the simplest cases these relations are a simple proportionality as in the viscosity of NEWTONfian liquids or the elasticity of HooiCEan solids (sec chapter I, 4c, p. 22, 4d, p. 28). In the following sections we shall, however, mainly be interested-in non-linear behaviour, the only exception being the NkwTONian viscosity of dilute stable suspensions. 

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