Shear-stress

To derive a molenilar exyirossion for the shear str s , we begin by realizing that [see Eq. (166)] 

The shear strain then appears only in the argument of ug (r ) via 

It is then an easy matter to apply the derivation detailed in Appendix E.3.1.1 to show that 

Here we derive a molecular expression for the shear stress parallel to the force expression for the compressional stress derived in Appendix E.3.1.2. How( ver, here wc employ a slightly iliffinimt definition of the auxiliary fiuic-tions. 9i, which we introduce via 

Hciicc, wc may again apply Leibniz s rule for the differentiation of a parameter integral to obtain 

In addition to the acute effects mentioned above, shear stress also plays a role in the more chronic regulation of NOS activity. This concept is 

FIGURE 3 Original traces showing the effect of an elevation in shear stress on the intracellular pH (pHi) in cultured human umbilical vein endothelial cells. 

Bacterial lipopolysaccharides as well as proinflammatory cytokines such as tumor necrosis factor a (TNF-a) have been reported to induce the expression of a Ca -independent NOS isoform in endothelial cells (Kilbourn 

The combination of interferon-y plus TNF-a or interleukin-1 8 (IL-IjS), on the other hand, has been shown to paradoxically enhance endothelial NOS activity, despite a concurrent decrease in NOS mRNA, an effect that may be accounted for by increased endogenous tetrahydrobiopterin levels (Rosenkranz-Weiss a/., 1994). 

The shear-sensitivity of plant cells has been observed in a number of cases [17, 52-58]. For example, Takeda et al. [58] reported decreases in respiration rate, ATP content and membrane integrity of both eucalyptus and safflower cells under hydrodynamic stress in a stirred-tank reactor. 

Consider two parallel layers of a fluid having in common a surface with an area. A, in m In order to move the upper layer across the lower, it is necessary to apply a tangential force, P, in N, to produce a shear stress, t, in Pa  

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Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

Densification during liquid-phase sintering occurs in tliree stages. Initially, liquid fonns at particle intersections and redistributes tliroughout the particulate mass under the influence of the capillary action. Shear stresses due to the... 

Theoretically the apparent viscosity of generalized Newtonian fluids can be found using a simple shear flow (i.e. steady state, one-dimensional, constant shear stress). The rate of deformation tensor in a simple shear flow is given as... 

Bingham plastics are fluids which remain rigid under the application of shear stresses less than a yield stress, Ty, but flow like a. simple Newtonian fluid once the applied shear exceeds this value. Different constitutive models representing this type of fluids were developed by Herschel and Bulkley (1926), Oldroyd (1947) and Casson (1959). 

Pseudoplastic fluids have no yield stress threshold and in these fluids the ratio of shear stress to the rate of shear generally falls continuously and rapidly with increase in the shear rate. Very low and very high shear regions are the exceptions, where the flow curve is almost horizontal (Figure 1.1). 

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

G is a multiplier which is zero at locations where slip condition does not apply and is a sufficiently large number at the nodes where slip may occur. It is important to note that, when the shear stress at a wall exceeds the threshold of slip and the fluid slides over the solid surface, this may reduce the shearing to below the critical value resulting in a renewed stick. Therefore imposition of wall slip introduces a form of non-linearity into the flow model which should be handled via an iterative loop. The slip coefficient (i.e. /I in the Navier s slip condition given as Equation (3.59) is defined as... 

Figure 5,16. It is assumed that by using an exactly symmetric cone a shear rate distribution, which is very nearly uniform, within the equilibrium (i.e. steady state) flow held can be generated (Tanner, 1985). Therefore in this type of viscometry the applied torque required for the steady rotation of the cone is related to the uniform shearing stress on its surface by a simplihed theoretical equation given as...

Suppose we divide the flow segments into classes according to relaxation times and index the various states by the subscript i. Thus the relaxation time and the component of shear stress borne by the segments in class i are and Fj, respectively. The applied shear force is related to the Fj s through... 

This is the fundamental differential equation for a shear stress relaxation experiment. The solution to this differential equation is an equation which gives a as a function of time in accord with experiment. 

Shaving products Shaw process Shear breeding Shear energy Shearlings Shearometer Shear plane Shear rate Shear stresses Shear test Shear thinning behavior Shear viscosity Sheath-core fiber... 

The shear stresses are proportional to the viscosity, in accordance with experience and intuition. However, the normal stresses also have viscosity-dependent components, not an intuitively obvious result. For flow problems in which the viscosity is vanishingly small, the normal stress component is negligible, but for fluid of high viscosity, eg, polymer melts, it can be significant and even dominant. 

Flow Past Bodies. A fluid moving past a surface of a soHd exerts a drag force on the soHd. This force is usually manifested as a drop in pressure in the fluid. Locally, at the surface, the pressure loss stems from the stresses exerted by the fluid on the surface and the equal and opposite stresses exerted by the surface on the fluid. Both shear stresses and normal stresses can contribute their relative importance depends on the shape of the body and the relationship of fluid inertia to the viscous stresses, commonly expressed as a dimensionless number called the Reynolds number (R ), EHp/]1. The character of the flow affects the drag as well as the heat and mass transfer to the surface. Flows around bodies and their associated pressure changes are important. 

Flow Along Smooth Surfaces. When the flow is entirely parallel to a smooth surface, eg, in a pipe far from the entrance, only the shear stresses contribute to the drag the normal stresses are directed perpendicular to the flow (see Piping systems). The shear stress is usually expressed in terms of a dimensionless friction factor ... 

As velocity continues to rise, the thicknesses of the laminar sublayer and buffer layers decrease, almost in inverse proportion to the velocity. The shear stress becomes almost proportional to the momentum flux (pk ) and is only a modest function of fluid viscosity. Heat and mass transfer (qv) to the wall, which formerly were limited by diffusion throughout the pipe, now are limited mostly by the thin layers at the wall. Both the heat- and mass-transfer rates are increased by the onset of turbulence and continue to rise almost in proportion to the velocity. 

In configurations more complex than pipes, eg, flow around bodies or through nozzles, additional shearing stresses and velocity gradients must be accounted for. More general equations for some simple fluids in laminar flow are described in Reference 1. 

Many industrially important fluids cannot be described in simple terms. Viscoelastic fluids are prominent offenders. These fluids exhibit memory, flowing when subjected to a stress, but recovering part of their deformation when the stress is removed. Polymer melts and flour dough are typical examples. Both the shear stresses and the normal stresses depend on the history of the fluid. Even the simplest constitutive equations are complex, as exemplified by the Oldroyd expression for shear stress at low shear rates ... 

Eigure 20 compares the predictions of the k-Q, RSM, and ASM models and experimental data for the growth of the layer width 5 and the variation of the maximum turbulent kinetic energy k and turbulent shear stress normalized with respect to the friction velocity jp for a curved mixing layer... 

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