Couette-shear flow

In a simple plane Couette shearing flow, as depicted at the top of Fig. 1-5, there is only one nonzero velocity component, namely ui, and it varies only in direction 2, the direction orthogonal to the plates. Hence Vv is... 

If each of the three epsilons are zero we recover the boundary conditions of planar Couette (shear) flow. For y = 0, there are three basic elongation flow fields defined as follows if one of the epsilons is also zero, the flow field is referred to as planar elongation flow if one of the epsilons is negative while the other two are positive and equal, the flow field is referred to as biaxial stretching flow and if one of the epsilons is positive while the other two are negative and equal, the flow field is referred to as uniaxial stretching flow. 

COUETTE SHEAR FLOW BETWEEN COAXIAL CYLINDERS... 

We therefore conclude that for (d/R) 1, y (RQId), a result we might have guessed from the similarity of Couette shear flow with d/R 1 to planar Couette shear. Thus, Couette shear flow measurements performed using small cylinder spacings (typically shear stress Zg and shear rate y. 

The main characteristics of Newtonian liquids is that simple shear flow (e.g., Couette flow) generates shear stress t, which is proportional to the shear rate... 

Couette and Poiseuille flows are in a class of flows called parallel flow, which means that only one velocity component is nonzero. That velocity component, however, can have spatial variation. Couette flow is a simple shearing flow, usually set up by one flat plate moving parallel to another fixed plate. For infinitely long plates, there is only one velocity component, which is in the direction of the plate motion. In steady state, assuming constant viscosity, the velocity is found to vary linearly between the plates, with no-slip boundary conditions requiring that the fluid velocity equals the plate velocity at each plate. There... 

A graphic example of the consequences of the existence of in stress in simple steady shear flows is demonstrated by the well-known Weissenberg rod-climbing effect (5). As shown in Fig. 3.3, it involves another simple shear flow, the Couette (6) torsional concentric cylinder flow,3 where x = 6, x2 = r, x3 = z. The flow creates a shear rate y12 y, which in Newtonian fluids generates only one stress component 112-Polyisobutelene molecules in solution used in Fig. 3.3(b) become oriented in the 1 direction, giving rise to the shear stress component in addition to the normal stress component in. 

The breakup or bursting of liquid droplets suspended in liquids undergoing shear flow has been studied and observed by many researchers beginning with the classic work of G. I. Taylor in the 1930s. For low viscosity drops, two mechanisms of breakup were identified at critical capillary number values. In the first one, the pointed droplet ends release a stream of smaller droplets termed tip streaming whereas, in the second mechanism the drop breaks into two main fragments and one or more satellite droplets. Strictly inviscid droplets such as gas bubbles were found to be stable at all conditions. It must be recalled, however, that gas bubbles are compressible and soluble, and this may play a role in the relief of hydrodynamic instabilities. The relative stability of gas bubbles in shear flow was confirmed experimentally by Canedo et al. (36). They could stretch a bubble all around the cylinder in a Couette flow apparatus without any signs of breakup. Of course, in a real devolatilizer, the flow is not a steady simple shear flow and bubble breakup is more likely to take place. 

Table 3.3 Summary of the relations for Newtonian and shear thinning/shear thickening fluids in the case of a simple fully-developed shear flow (Couette flow, isothermal) [6], [14]...

Figure 3.19 Profiles of shear stress, shear rate and speed through the cross-section in the case of an isothermal, simple shear flow (Couette flow) with Newtonian, shear thinning, and shearthickening fluids...

Simple shear flow is the characteristic of Couette flow, and no pressure gradient exists in the direction of motion. Since the pressure gradient term is also zero, the governing equation reduces to... 

FIG. 15.2 Types of simple shear flow. (A) Couette flow between two coaxial cylinders (B) torsional flow between parallel plates (C) torsional flow between a cone and a plate and (D) Poisseuille flow in a cylindrical tube. After Te Nijenhuis (2007). 

A plot of yapp against 1 / h will then be a straight line with slope 2Vs. This method has been used to measure the slip velocity for polyethylene melts in a sliding plate (plane Couette) rheometer by Hatzikiriakos and Dealy (1991). Analogous methods have been applied to shearing flows of melts in capillaries and in plate-and-plate rheometers (Mooney 1931 Henson and Mackay 1995 Wang and Drda 1996). 

We wish to study the effects of planar Couette flow on a system that is in the NPT (fully flexible box) ensemble. In this section, we consider the effects of the external field alone on the dynamics of the cell. The intrinsic cell dynamics arising out of the internal stress is assumed implicitly. The constant NPT ensemble can be employed in simulations of crystalline materials, so as to perform dynamics consistent with the cell geometry. In this section, we assume that the shear field is applied to anisotropic systems such as liquid crystals, or crystalline polytetrafluoroethylene. For an anisotropic solid, we assume that the shear field is oriented in such a way that different weakly interacting planes of atoms in the solid slide past each other. The methodology presented is quite general hence it is straightforward to apply for simulations of shear flow in liquids in a cubic box, as well. 

Equation [213] can be equivalently obtained using the Mori-Zwanzig formalism. " It is also seen that, in contrast to LRT developed for shear flow in bulk fluids, the one presented here has two coefficients, (which is similar to the shear viscosity q) and, which has no parallel in bulk fluids. should be interpreted as an average location at which hydrodynamics is found to be nominally invalid. Note that although the surface may have corrugations in the X as well as the y direction, the corrugation in the x direction alone matters to the frictional force in the planar Couette geometry. 

Next we turn to the stability of Couette flow for parallel rotating cylinders. This is an important flow for various applications, and, though it is a shear flow, the stability is dominated by the centrifugal forces that arise because of centripetal acceleration. This problem is also an important contrast with the first two examples because it is a case in which the flow can actually be stabilized by viscous effects. We first consider the classic case of an inviscid fluid, which leads to the well-known criteria of Rayleigh for the stability of an inviscid fluid. We then analyze the role of viscosity for the case of a narrow gap in which analytic results can be obtained. We show that the flow is stabilized by viscous diffusion effects up to a critical value of the Reynolds number for the problem (here known as the Taylor number). 

Figure 2-9. A number of simple flow geometries, such as concentric cylinder (Couette), cone-and-plate, and parallel disk, are commonly employed as rheometers to subject a liquid to shear flows for measurement of the fluid viscosity (see, e.g., Fig. 3-5). In the present discussion, we approximately represent the flow in these devices as the flow between two plane boundaries as described in the text and sketched in this figure.

Hence, to achieve the best possible approximation to a linear shear flow, the Couette device must have a very thin gap relative to the cylinder radius. 

On the other hand, because the circular Couette flow is generally adopted as a convenient substitute for a simple shear flow, it maybe tempting to analyze the experimental data as though we exactly had simple shear between two plane boundaries. This would mean dividing rrg r=a with an estimated velocity gradient given by the velocity difference of the two walls divided by the gap width as would be exactly correct for a linear shear flow. The latter is simply... 

Problem 3-35. Temperature Distribution in a Combined Plane Couette and Poiseuille Flow with a Linearly Increasing Wall Temperature. In this problem you will examine convective heat transfer in a combined plane Couette and Poiseuille flow. Suppose we have the channel depicted in the figure. The upper wall moves with a velocity U and the lower wall is fixed. In addition to the shear flow there is a pressure-driven backflow resulting in the purely quadratic dependence of velocity on y (e.g., the shear rate at the lower wall is zero). 

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