Shear flow, laminar

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

Shear stresses are developed in a fluid when a layer of fluid moves faster or slower than a nearby layer of fluid or a solid surface. In laminar flow, the shear stress is equal to the product of fluid viscosity and velocity gradient or rate of shear. Under laminar-flow conditions, shear forces are larger than inertial forces in the fluid. 

Equation 5.2, with the modified parameter X used in place of X, may be used for laminar flow of shear-thinning fluids whose behaviour can be described by the power-taw model. 

Figure 7.1. Schematic representation of the thinning of fluid elements due to laminar shear flow...

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

In a laminar shear flow, the maximum viscous shearing stress, Xn,ax> on the surface of freely suspended spherical particle, such a unicellular micro-organism, is given as ... 

Assuming a parabolic velocity profile for laminar conditions, the flow averaged shear stress is calculated as ... 

Soule et al. [141] constructed a sparged, concentric cylinder bioreactor for the cultivation of suspensions of Pirus malus. Growth was reduced under all rotational conditions. Sun and Linden [106] employed a rotating wall vessel (Rotary Cell Culture System, Synthecon, Houston, TX, USA) to cultivate suspensions of Taxus cuspidata under laminar flow conditions. Shear rates were... 

Another electrical measurement useful in detecting flocculation in aniso-metric particles is the response of dielectric constant to shear. The alignment along streamlines of flow which results from laminar shear in a viscometer decreases the dielectric constant of the system if the dipole moment lies along the long axis of the particle. Another way in which this phenomenon can be meas-... 

Deviation from laminar shear flow [88,89],by calculating the material functions r =f( y),x12=f( Y),x11-x22=f( y),is assumed to be of a laminar type and this assumption is applied to Newtonian as well as viscoelastic fluids. Deviations from laminar flow conditions are often described as turbulent, as flow irregularities or flow instabilities. However, deviation from laminar flow conditions in cone-and-plate geometries have been observed and analysed for Newtonian and viscoelastic liquids in numerous investigations [90-95]. Theories have been derived for predicting the onset of the deviation of laminar flow between a cone and plate for Newtonian liquids [91-93] and in experiments reasonable agreements were found [95]. 

NAD(P)H oxidase can also be activated by fluid shear stresses. This is one reason why branched and curved arteries tend to develop atherosclerotic plaques earlier than straight arteries. Using a spin probe to detect 02 , Hwang et al. have demonstrated that monolayer cultures of EC exposed to oscillatory (but not laminar) shear stresses produce the radical using NAD(P)H oxidase.292 A subsequent study showed that XO also responds to oscillatory shear stress.293 Other workers, using BMPO, have detected the flow-induced production of 02 by mitochondria.294... 

For fully developed laminar flow, the shear stress at the wall of a circular duct is... 

The State of Stress in Laminar Shear Flow and its Relation to Flow... 

For convenience, a mathematically simple arrangement is considered. It consists of a fluid layer of finite constant thickness, confined by two rigid parallel planes of infinite extension. Steady laminar shear flow is created in this layer by fixing one plane in space and moving the other one with constant speed in a direction parallel to both planes. In this way, a truly uniform and time independent shear rate q is created in the liquid. The magnitude of this shear rate is simply given by the ratio of the said speed to the mutual distance of the planes. Experimentally such an arrangement is approximated e.g. by the use of two coaxial cylinders. When the gap between the inner surfaces of these cylinders is made small compared with their radii, the above mentioned situation can be realized to a sufficient extent. 

For the special case of slow laminar shear flow in the direction of the 1-axis eq. (2.35) appears to reduce to ... 

This expression is by the factor (rf/i/rf) smaller than the one originally given by Peterlin (76). However, since the coil expansion is a qualitative measure of the state of deformation of coil molecules in laminar shear flow and, moreover, the first relaxation time is, in general, by far the largest one, the original equation of Peterlin can be used unchanged, if desired. 

This means that for such particles the contribution by rotary diffusion is predominant but not exclusive. In fact, as a consequence of its rigidity, a rigid rod can only follow the rotary component of laminar shear flow There remains some radial flow of solvent along the particle which causes the hydrodynamic contribution to the intrinsic viscosity. 

It seems, therefore, useful to take the numerically simple laminar shear flow as a starting point for the description. As recently published (Bouldin), an easily manageable model has been derived which enables an ad hoc prediction to be made of the critical sfteanate at which mechanical chain scission takes place (cf. Fig. 34). 

To describe the mechanical degradation in turbulent flow, two limiting cases must be introduced the laminar shear flow and the elongational flow. 

For a non-Newtonian fluid, the viscosity is not constant even for the laminar flow. Therefore, shear rate is easier to estimate than shear stress. 

Tubular flow reactors (TFR) deviate from the idealized PFR, since the applied pressure drop creates with viscous fluids a laminar shear flow field. As discussed in Section 7.1, shear flow leads to mixing. This is shown schematically in Fig. 11.9(a) and 11.9(b). In the former, we show laminar distributive mixing whereby a thin disk of a miscible reactive component is deformed and distributed (somewhat) over the volume whereas, in the latter we show laminar dispersive mixing whereby a thin disk of immiscible fluid, subsequent to being deformed and stretched, breaks up into droplets. In either case, diffusion mixing is superimposed on convective distributive mixing. Figure 11.9(c) shows schematically the... 

Table 3.3 provides a summary of the relations for simple drag flow (laminar, stationary). Here, even if the fluids show different rheological behaviors, the speed profiles do not change (see Fig. 3.19). The shear rate is also not dependent on rheological properties of the fluid. 

In viscoelastic fluids at steady-state laminar flow, besides shear stress t = o21 = p y, normal stresses are observed in all three directions ... 

Plastic processing is primarily the flow and shaping of viscous liquids. The scientific study of this flow is called rheology. Assuming laminar shear flow, viscosity is defined as the ratio of shear stress to shear rate. 

Swift DL, Friedlander SK. The coagulation of hydrosols by Brownian motion and laminar shear flow. J Colloid Sd 1964 19 621-647. 

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