Couette-flow

It is now clear that as the amplitude a of the oscillation at the upper plate approaches the critical value Uc defined by equation (5.185) then T can become arbitrarily large and that near Uc we have 

Other cases for different values of e, for example 1, have been considered by Clark et al. [46]. Further, homeotropic boundary conditions rather than the above planar boundary conditions have also been examined in a similar style in [46]. 

A great deal of experimental and theoretical work has been carried out recently on oscillatory shear problems in nematics. Readers will find much of interest in the experimental work of Borzsonyi, Buka, Krekhov and Kramer [20] and Mullin and Peacock [208]. More theoretical work can be found in the articles by Krekhov and Kramer [156] and Hogan, Mullin and Woodford [128] (who also consider experimental results) and the references cited therein. 

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If a fluid is placed between two concentric cylinders, and the inner cylinder rotated, a complex fluid dynamical motion known as Taylor-Couette flow is established. Mass transport is then by exchange between eddy vortices which can, under some conditions, be imagmed as a substantially enlranced diflfiisivity (typically with effective diflfiision coefficients several orders of magnitude above molecular difhision coefficients) that can be altered by varying the rotation rate, and with all species having the same diffusivity. Studies of the BZ and CIMA/CDIMA systems in such a Couette reactor [45] have revealed bifiircation tlirough a complex sequence of front patterns, see figure A3.14.16. 

Evans D J 1983 Computer experiment for nonlinear thermodynamics of Couette flow J. Chem. Phys. 78 3297-302... 

Simulation of the Couette flow of silicon rubber - generalized Newtonian model... 

Figure 5.10 (a,b) Comparison of the simulated die swell in a Couette flow for the power-... 

Rotating cone viscometers are among the most commonly used rheometry devices. These instruments essentially consist of a steel cone which rotates in a chamber filled with the fluid generating a Couette flow regime. Based on the same fundamental concept various types of single and double cone devices are developed. The schematic diagram of a double cone viscometer is shown in... 

In the Couette flow inside a cone-and-plate viscometer the circumferential velocity at any given radial position is approximately a linear function of the vertical coordinate. Therefore the shear rate corresponding to this component is almost constant. The heat generation term in Equation (5.25) is hence nearly constant. Furthermore, in uniform Couette regime the convection term is also zero and all of the heat transfer is due to conduction. For very large conductivity coefficients the heat conduction will be very fast and the temperature profile will... 

Cottrell equation Cottrell unit Couchman equation Couette flow Couette viscometers Cough drops Coughlozenges... 

In packed beds of particles possessing small pores, dilute aqueous solutions of hydroly2ed polyacrylamide will sometimes exhibit dilatant behavior iastead of the usual shear thinning behavior seen ia simple shear or Couette flow. In elongational flow, such as flow through porous sandstone, flow resistance can iacrease with flow rate due to iacreases ia elongational viscosity and normal stress differences. The iacrease ia normal stress differences with shear rate is typical of isotropic polymer solutions. Normal stress differences of anisotropic polymers, such as xanthan ia water, are shear rate iadependent (25,26). 

Fig. 15. Flow pattern in rotating Couette flow where and Q2 represent the outer and inner rotational speeds.

For extremely narrow openings (cracks) with deep flow paths (such as mortar joints and tight-fitting components) the flow is laminar and the flow rate, Q (mVs), can be described by the Couette flow equation - ... 

Under steady-state conditions, as in the Couette flow, the strain rate is constant over the reaction volume for a long period of time (several hours) and the system of Eq. (87) could be solved exactly with the matrix technique developed by Basedow et al. [153], Transient elongational flow, on the other hand, has two distinctive features, i.e. a short residence time (a few ps) and a non-uniform flow field, which must be incorporated into the kinetics equations. In transient elongational flow, each rate constant is a strongfunction of the strain-rate which varies with time in the Lagrangian frame moving with the center of mass of the macromolecule the local value of the strain rate for each spatial coordinate must be known before Eq. (87) can be solved. 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

Velocity profiles of Plane Couette flow — Continuum - - - First-otder slip Boltzmann Eq... 

Fig. 4—Comparison of velocity profiles of plane Couette flow between different models.

It should be pointed out that the flow rate in the case of the Couette flow is independent of the inverse Knudsen number, and is the same as the prediction of the continuum model, although the velocity profiles predicted by the different flow models are different as shown in Fig. 4. The flow velocity in the case of the plane Couette flow is given as follows (i) Continuum model ... 

Figure 3 illustrates some additional capability of the flow code. Here no pressure gradient is Imposed (this is then drag or "Couette flow only), but we also compute the temperatures resulting from Internal viscous dissipation. The shear rate in this case is just 7 — 3u/3y — U/H. The associated stress is.r — 177 = i/CU/H), and the thermal dissipation is then Q - r7 - i/CU/H). Figure 3 also shows the temperature profile which is obtained if the upper boundary exhibits a convective rather than fixed condition. The convective heat transfer coefficient h was set to unity this corresponds to a "Nusselt Number" Nu - (hH/k) - 1. 

Figure 3. Finite element simulation of plane Couette flow with thermal dissipation and conductive heat transfer. (f) — fixed temperature condition (c) — convective boundary condition.

The attractive feature of LADM Is that once the fluid structure Is known (e.g., by solution of the YBG equations given In the previous section or by a computer simulation) then theoretical or empirical formulas for the transport coefficients of homogeneous fluids can be used to predict flow and transport In Inhomogeneous fluid. For diffusion and Couette flow In planar pores LADM turns out to be a surprisingly good approximation, as will be shown In a later section. 

For the steady, planar Couette flow to be examined In a later section, the momentum balance equation yields... 

Couette Flow Simulation. MD typically simulate systems at thermodynamic equilibrium. For the simulation of systems undergoing flow various methods of nonequilibrium MD have been developed (Ifl iZ.). In all of these methods the viscosity Is calculated directly from the constitutive equation. 

The nonequilibrium MD method we employed ( ) is the reservoir method (iff.) which simulates plane Couette flow. The effective viscosity Is calculated from the constitutive relation... 

The shear stress Is uniform throughout the main liquid slab for Couette flow ( ). Therefore, two Independent methods for the calculation of the shear stress are available It can be calculated either from the y component of the force exerted by the particles of the liquid slab upon each reservoir or from the volume average of the shear stress developed Inside the liquid slab from the Irving-Kirkwood formula (JA). For reasons explained In Reference (5) the simpler version of this formula can be used In both our systems although this version does not apply In general to structured systems. The Irvlng-Klrkwood expression for the xy component of the stress tensor used In our simulation Is... 

Flow systems. In this subsection we present the results of our Couette flow simulations. Most of these results were first presented In Reference ( ). 

Although only one density profile Is shown In each of Figures 7 and 8 the density profiles of the two systems both at equilibrium and In the presence of flow that have been determined. A conclusion of great importance that is suggested by the Couette flow simulations is that the density profiles of the two systems in the presence of flow coincide with the equilibrium density profiles, even at the extremely high shear rates employed in our simulation. A detailed statistical analysis that Justifies this point was presented In Reference ( ). 

The simulation value for the effective viscosity Is almost half the viscosity of the bulk fluid. According to the LADM the effective viscosity for plane Couette flow can be Identified as... 

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