Velocity profiles, couette flow

Laminar flow-linear velocity profile (Couette flow) p 120D Vp = velocity of upper plate = spacing of plates u = shear velocity... 

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... 

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 ... 

Couette flow is shear-driven flow, as opposed to pressure-driven. In this instance, two parallel plates, separated by a distances h, are sheared relative to one another. The motion induces shear in the interstitial fluid, generating a linear velocity profile that depends on the motion of the moving surface. If we assume a linear shear rate, the shear stress is given simply by... 

Solve the Couette-flow problem to determine the velocity profile in the large annular gap that carries the primary flow. 

However, it is clear that for a general tensor Vu, trajectory analysis based on the SLLOD dynamics in Eqs. [129] will yield incorrect results. Equation [132] has an extra term in the force, which is equivalent to saying that the momenta in Eqs. [129] are not peculiar with respect to a general flow (indeed, Eqs. [129] yield peculiar velocities for the case of planar Couette flow), and therefore the flow profile produced will not be q Vu as expected. Equations [129] also lead to problems when one is considering definitions of pressure... 

The last parameter, CLi, is determined investigating inhomogeneous high Reynolds number, fully developed channel flows (i.e., these flows are sometimes referred to as 2D Couette flows). Actually, the turbulence model is applied describing the flow in regions near walls, where the logarithmic velocity profile applies. 

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. 

COUETTE FLOW. In one form of layer flow, illustrated in Fig. 5.17, the fluid is bounded between two very large, flat, parallel plates separated by distance B. The lower plate is stationary, and the upper plate is moving to the right at a constant velocity Uq. For a newtonian fluid the velocity profile is linear, and the velocity u is zero at y = 0 and equals Uo t y = B, where y is the vertical distance measured from the lower plate. The velocity gradient is constant and equals uq/B. Considering an area A of both plates, the shear force needed to maintain the motion of the top plate is, from Eqs. (3.3) and (3.4),... 

For Couette flow the shear force per unit area is constant, and since it is proportional to the local velocity gradient, it follows that the velocity profile is linear. If the no-slip boundary condition is satisfied at both plates. 

Consider now the case when the liquid moves with a non-uniform profile of velocity, which can happen, for example, during the flow in a pipe (Poiseurlle flow) or in a channel whose walls are moving with different velocities (Couette flow) [21]. Since the size of particles is very small, at distances of about several particle sizes, the non-uniform velocity profile can be regarded as linear, and the flow can be regarded as a shear flow. For simplicity, consider the flow to be two-dimensional. Inasmuch as W 1, it is possible to be limited to the consideration of one particle, as before. Denote by V = (ay, 0) the velocity of the liquid at the point X = (x(t), y(t)), and hy U = (u(f), v(t)) - the velocity of the particle. 

The fluid path in a screw pump is therefore of a complex hehcal form within the channel section. The velocity components along the channel depend on the pressure generated and the resistance at the discharge end. If there is no resistance, the velocity profile in the direction of the channel will be of the Couette type, as depicted in Figure 3.39a. With a totally closed discharge end, the net flow would be zero, but the velocity components at the walls would not be affected. As a result, the flow field necessarily would be of the form shown in Figure 3.39b. 

To solve Eqs. 12 and 13 with oscUlating velocity boundary conditions, simple models such as the Couette flow model and ID Stokes model have been used. These models ignore the finite size and edge effects, as both of them model the device as two infinitely large parallel plates with one (the proof mass) oscillating on the top of the other (substrate). The Couette model further assumes a steady flow, resulting in a linear velocity profile between the plates. As shown later, the quality factor obtained by these two models is overpredicted by a factor of two, indicating the importance of 3D effects. 

The contribution to the drag from the fluid between the resonator and the substrate (denoted as the bottom force) is predicted reasonably well by two simple ID models. It is not surprising that the two ID models are in such close agreement. The small gap (g = 2 pm) and slow motion imply that the penetration depth, defined as the distance over which the motion amplitude has dropped to 1 % of its maximum value, is 37 times larger than the gap. This indicates that the flow between the resonator and the substrate is fully developed and the velocity profile will be reasonably predicted by the Couette model. That the two ID models are so close also... 

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. 

The slip length also depends on the shear rate imposed on the fluid particle. Thompson and Troian [8] have reported the molecular dynamics simulation of Couette flow at different shear rates. At lower shear rate, the velocity profile follows the no-slip boundary condition. The slip length increases with increase in shear rate. The critical shear rate for slip is very high for simple liquids, i.e., 10 s for water, indicating that slip flow can be achieved experimentally in very small devices at very high speeds. Experiments performed with the SEA and AFM have also showed shear dependence slip in the hydrodynamic force measurements. 

Boundary Slip of Liquids, Fig. 7 Velocity profile for (a) Couette and (b) Poiseuille flow comparison between molecular dynamics simulation, no-slip boimdary condition, and partial slip boundary condition... 

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... 

To investigate the effect of viscosity ratio between the two liquids, different values of 2 are chosen. Figure 4 shows the dimensionless velocity profiles at the symmetric line when 2 has different values. The velocity profile of the nonconducting fluid is very much like the Couette flow as the nonconducting liquid is dragged by the conducting liquid fluid through the interfacial shear stress. As 2 = smaller value of P2... 

Fig. 7.1 Illustration of the profiles of flow velocities in (a) Couette flow and (b) Poiseuille flow, respectively...

In order to extract both parameters, one can investigate the flow profiles of two qualitatively different tyjres of flows -planar shear flow (the Couette flow) and pressrue-driven flow (the Poiseuille flow) - in a thin film of thickness D. Both flows are characterized by a Unear stress profile such that nonlocal viscosity, which may arise from the extended molecirlar strac-ture,i2 i2 is not expected to affect the resirlts. At the center of the channel, the flow is described by the Navier-Stokes eqrra-tion yielding a linear velocity profile with shear ratey ... 

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,...

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