Couette and Poiseuille Flow

Here the scale pressure pmax is some estimate of the maximum pressure in the system. For low-speed flows the nondimensionalization above may not be appropriate and the following nondimensional pressure can be used  

The dynamics of the incompressible fluid flow depend on small changes in the pressure through the flowfield. These changes are negligible compared to the absolute value of the thermodynamic pressure. The reference value can then be taken as some pressure at a fixed point and time in the flow. Changes in pressure result from fluid dynamic effects and an appropriate pressure scale is where Vmax is a measure of the maximum velocity in 

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 

Poiseuille flow also occurs in a simple shearing situation, but it presumes that there is a pressure gradient that drives the flow and that the solid boundaries are fixed. Flow in a pipe or tube is an example of Poiseuille flow. It is a straightforward matter to combine these Couette and Poiseuille effects, and still find an exact analytic solution. 

For long rod-guide systems it is reasonable to assume that the only nonzero velocity component is u, the axial velocity. Inasmuch as the rod and guide may have different axial velocities, it is clear that the fluid velocity must be permitted to vary radially. Given that the radial and circumferential velocities v and w are zero, the mass-continuity equation, Eq. 6.3, requires that 

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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). 

Berdichevski, V., Fridlyand, A., and Sutyrin V., Prediction of turbulent velocity profile in Couette and Poiseuille flows from first principles, Phys. Rev. Letters, Vol. 76, No. 21, pp. 3967-3970, 1996. 

Barrat and Bocquet [6] carried out the molecular dynamics simulation of Couette and Poiseuille flows. In Couette flow, the upper wall is moved with a cmistant velocity, and in Poiseuille flow an external force drives the flow. Sample results from molecular dynamics simulation are reproduced in Fig. 7. The application of no-slip boundary condition leads to the expected linear and parabolic profiles, respectively, for Couette and Poiseuille flows. However, the velocity profile obtained from molecular dynamics simulation shows a sudden change of velocity in the near-waU region indicating the slip flow. The velocity profile for Couette flow away from the solid surface is linear with different slope than that of the no-slip case. The velocity for slip flow case is higher than that observed in the no-slip case for Poiseuille flow. For both Couette and Poiseuille flows, the partial slip boundary condition at the wall predict similar bulk flow as that observed by molecular dynamics simulation. Some discrepancy in the velocity profile is observed in the near-wall region. 

Barrat and Bocquet [6] reported slip in Couette and Poiseuille flows using molecular dynamics simulation (Fig. 7). Tretheway and Meinhart [4] reported micron-resolution velocity profile in hydrophilic and hydrophobic microchannels of cross section 30 x 300 pm using the p-PIV technique (Fig. 5a). Their results showed significant fluid velocity near a hydrophobic (octadecyltrichlorosilane or... 

For partial Brownian motion of small rods, there is, as yet, no satisfactory solution. In any case, the viscosity will decrease with the flow gradient and increase with increasing Brownian motion (i.e. with kT), Because of the variable gradient in a capillary viscosimeter, different results are to be expected for Couette and Poiseuille flow. 

Figure 9 Molecular dynamics simulation of a Lennard-Jones, bead-spring model, (a) Slip length, 8, as a function of the strength, [ an, of attraction between a hard, corrugated substrate and liquid for temperature, kgT/ [ =. 2. The solid line with circles is obtained from the Couette and Poiseuille profiles (NEMO) according toeqn [37], whereas the dashed line with squares, from the Green-Kubo (GK) relation, eqn [39]. The curve marks the behavior 1/ [ jii in accord with eqn [40]. The inset illustrates the velocity profiles of the Couette and Poiseuille flows, from which the slip length has been estimated for [mii = 0.6, measured in units of the Lennard-Jones parameter, [. Adapted from Servantie, J. Muller, M. Phys. Rev. Lett. 2008, 101,...

Couette and Poiseuille flow of polyacrylamide and polyisobutylene solutions... 

As in wide gap Couette and Poiseuille flow (Chapter 6), shear rate is not constant. Thus we must use a derivative to relate shear stress to total torque. The resulting equations are given below and then derived in the remainder of this section. 

Huang, P. Y., Feng, J., Hu, H. H., Joseph, D. D. (1997), Direct simulation of the motion of solid particles in Couette and Poiseuille flows of viscoelastic fluids , J Fluid Mech, 343, 73-94. 

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