Viscous flow between parallel plates

Schwiebert MK, Leong WH. Underfill flow as viscous flow between parallel plates driven by capillary action. Proc. IEEE Trans Components Packages Mfg. Technol, Part C Apr. 1996 19(3) 133-7. 

Schwiebert, M. K., and Leong, W. H., Underfill Flow as Viscous Flow Between Parallel Plates Driven by Capillary Action, Proc. IEEE Transactions on Components, Packages, andMfg. Technology, Part C, 19(3) 133-137 (Apr. 1996)... 

There are two types of fluid flows that can be important in microfluidic devices. One type of flow is called Couette flow, which is the steady viscous flow between parallel plates, one of which is moving relative to the other, as shown in Figure 6.1. The velocity of the fluid varies linearly from zero at the stationary plate up to the velocity V at the moving plate. Another type of fluid flow is Poiseuille flow. This is a pressure-driven flow between stationary parallel plates, as shown in Figure 6.3. It has a parabolic variation of the pressure with the maximum flow velocity in the middle of the plates and zero flow velocity at the walls. 

Viscous heating temperature rise due to a combined drag-pressure flow between two parallel plates. Using RFM, Estrada [4] computed the temperature rise of a fluid subjected to a combination of drag and pressure flow between parallel plates and compared the results of an anlytical and a boundary element dual reciprocity (BEM-DRM) solutions presented by Davis et al. [3], Figure 11.2 presents a schematic of the problem with dimensions, physical properties and boundary conditions. 

Heat Transfer in Parallel Plate Microchannels Similarly, Nu correlations were developed for flow between two parallel plates for both constant wall heat flux and uniform wall temperature with and without viscous dissipation. For constant heat flux on one heat wall without viscous dissipation between parallel plates, the Nu expression is [32]... 

Squeeze flow between parallel plates was analyzed in Section 6.3 as an elementary model of compression molding. In that treatment we were able to obtain an analytical solution to the creeping flow equations for isothermal Newtonian fluids by making the kinematical assumption that the axial velocity is independent of radial position (or, equivalently, that material surfaces that are initially parallel to the plates remain parallel). In this section we show a finite element solution for non-isothermal squeeze flow of a Newtonian hquid. The geometry is shown schematically in Figure 8.16. We retain the inertial terms in the Navier-Stokes equations, thus including the velocity transient, and we solve the full transient equation for the temperature, including the viscous dissipation terms. The computational details. 

Consider the isothermal, incompressible, viscous flow between two closely spaced parallel plates that are separated by a height H (Fig. 5.13). There is a uniform injection velocity V from the lower boundary and a uniform exit velocity V0 from the upper plate. Flow enters the channel from the left with a mean velocity U. The net injection velocity is given as... 

To understand how gravity force capillary viscometers work, consider the layers of fluid that we previously drew between parallel plates (Figure 6.2) now to be concentric layers in a cylinder as sketched in Figure 6.3(b). If each layer has radius r, thickness dr, and the no slip condition holds at the walls, then the velocity of each layer is zero at the outer wall and increases toward the center. The viscous force on the shell is given by the force per unit area (t]dv/dr) times the area (2jtrl) so that in a steady flow condition ... 

We have already discussed confinement effects in the channel flow of colloidal glasses. Such effects are also seen in hard-sphere colloidal crystals sheared between parallel plates. Cohen et al. [103] found that when the plate separation was smaller than 11 particle diameters, commensurability effects became dominant, with the emergence of new crystalline orderings. In particular, the colloids organise into z-buckled" layers which show up in xy slices as one, two or three particle strips separated by fluid bands see Fig. 15. By comparing osmotic pressure and viscous stresses in the different particle configurations, tlie cross-over from buckled to non-buckled states could be accurately predicted. 

Viscous flow between two infinitely wide parallel plates with a very narrow gap separating the two walls. When one wall is catalytically active, the following solntion is apphcable (Solbrig and Gidaspow, 1967) ... 

Aspect Ratio Required to Achieve Viscous Flow between Two Parallel Plates with a Narrow Gap... 

According to the Cubic Law, fracture is formed by the smooth, flat, infinite long parallel plates without filling medium, and the water flow between the plates is viscous incompressible flow which is permanent laminar flow. Thus, according to the basic principle of fluid mechanics, the discharge per unit width through the crack surface i.e. q can be calculated by the follow formula ... 

The flow along the fiber direction can be modeled as a channel flow (the viscous flow between two parallel plates). Loss and Springer (29) validated thermomechanical models, including the chemorheological behavior of resin and resin flow models. 

Fordham, E. J., Bittleston, S. H., and Ahmadi Tehran , M., Viscoplastic flow in centered annuli, pipes and slots, Ind. Eng. Chem. Res. iO(3) 517- 524 (1991). Worth, R. A., Accuracy of the parallel-plate analogy for representation of viscous flow between coaxial cylinders, J. Appl. Polym. Sci. 24 319-328 (1979). McKelvey, J. M., Polymer Processing, Wiley, New York, 1962. 

Worth R A (1979), Accuracy of parallel - plate analogy for representation of viscous flow between cyUnders ,/oMwa/ of Applied Polymer Sciences, 24,319-328. 

Thus, the problem of flow of a viscoelastic fluid between two flat parallel plates one of which is moving in a direction transverse to the main flow is reduced to a solution of simplified system (7) at boundary conditions (1). Analysis of relationships (7) for specific boundary conditions indicates that the problem is reduced to the case of a non-Newtonian viscous fluid. In other words, the velocity profile v(y) is determined only by viscous characteristics of the media and the effect of high-elasticity properties of the melt upon velocity (flow rate) characteristics of the flow can be neglected. 

Finally, let us examine the flow of viscous fluid between two parallel plates in relative motion [Fig. E2.4(c)]. Because of intermolecular forces, the fluid layer next to the bottom plate will start moving. This layer will then transmit, by viscous drag, momentum to the layer above it, and so on. The velocity gradient for vx(0) > vx(b) is positive and given by... 

In this example, we consider the viscous, isothermal, incompressible flow of a Newtonian fluid between two infinite parallel plates in relative motion, as shown in Fig. E2.5a. As is evident from the figure, we have already chosen the most appropriate coordinate system for the problem at hand, namely, the rectangular coordinate system with spatial variables x, y, z. 

According to the lubrication approximation, we can quite accurately assume that locally the flow takes place between two parallel plates at H x,z) apart in relative motion. The assumptions on which the theory of lubrication rests are as follows (a) the flow is laminar, (b) the flow is steady in time, (c) the flow is isothermal, (d) the fluid is incompressible, (e) the fluid is Newtonian, (f) there is no slip at the wall, (g) the inertial forces due to fluid acceleration are negligible compared to the viscous shear forces, and (h) any motion of fluid in a direction normal to the surfaces can be neglected in comparison with motion parallel to them. 

Example 6.14 Squeezing Flow between Two Parallel Disks This flow characterizes compression molding it is used in certain hydrodynamic lubricating systems and in rheological testing of asphalt, rubber, and other very viscous liquids.14 We solve the flow problem for a Power Law model fluid as suggested by Scott (48) and presented by Leider and Bird (49). We assume a quasi-steady-state slow flow15 and invoke the lubrication approximation. We use a cylindrical coordinate system placed at the center and midway between the plates as shown in Fig. E6.14a. 

A constant property fluid flows between two horizontal, semiinfinite, parallel plates, kept at a distance 2m apart. The upper plate is at a constant temperature Ti and the lower plate is at a constant temperature T2. Consider the fully developed velocity and temperature profiles region for laminar flow. Include viscous dissipation. Find the heat flux to each of the plates. 

Consider constant-property, fully developed laminar flow between two large parallel plates, i.e., in a wide plane duct. One plate is adiabatic and the other is isothermal and the velocity is high enough for viscous dissipation effects to be significant. Determine the temperature distribution in the flow. 

For laminar flow, the characteristic time of the fluid phase Tf can be deflned as the ratio between a characteristic velocity Uf and a characteristic dimension L. For example, in the case of channel flows confined within two parallel plates, L can be taken equal to the distance between the plates, whereas Uf can be the friction velocity. Another common choice is to base this calculation on the viscous scale, by dividing the kinematic viscosity of the fluid phase by the friction velocity squared. For turbulent flow, Tf is usually assumed to be the Kolmogorov time scale in the fluid phase. The dusty-gas model can be applied only when the particle relaxation time tends to zero (i.e. Stp 1). Under these conditions, Eq. (5.105) yields fluid flow. This typically happens when particles are very small and/or the continuous phase is highly viscous and/or the disperse-to-primary-phase density ratio is very small. The dusty-gas model assumes that there is only one particle velocity field, which is identical to that of the fluid. With this approach, preferential accumulation and segregation effects are clearly not predicted since particles are transported as scalars in the continuous phase. If the system is very dilute (one-way coupling), the properties of the continuous phase (i.e. density and viscosity) are assumed to be equal to those of the fluid. If the solid-particle concentration starts to have an influence on the fluid phase (two-way coupling), a modified density and viscosity for the continuous phase are generally introduced in Eq. (4.92). 

The hydrodynamic resistance of dispersion medium in the gap between particles against flowing out is one of the kinetic stability factors. The decrease in thickness of fluid layer between the particles during coagulation is related to viscous flow of liquid out of a narrow gap between the particles. For solid particles the liquid flow velocity is zero at the interface and highest in the center of a gap. The rate with which the gap between two circular plane-parallel plates of radius r (Fig. VII-7) shrinks, dh/dt, is related to the volume of liquid that flows per second across the side surface of cylindrical gap, dV/dt, via the following relationship ... 

Let us first recall briefly the classical Benard-Rayleigh problem of thermal convection in an isotropic liquid. When a horizontal layer of isotropic liquid bounded between two plane parallel plates spaced d apart is heated from below, a steady convective flow is observed when the temperature difference between the plates exceeds a critical value A 7. The flow has a stationary cellular character with a spatial periodicity of about 2d. The mechanism for the onset of convection may be looked upon as follows. A fluctuation T in temperature creates warmer and cooler regions, and due lO buoyancy effects the former tends to move upwards and the latter downwards. When AT < AT, the fluctuation dies out in time because of viscous effects and heat loss due to conductivity. At the threshold the energy loss is balanced exactly and beyond it instability develops. Assuming a one-dimensional model in which T and the velocity (normal to the layer) vary as exp (i j,y) with x ji/rf, the threshold is given by the dimensionless Rayleigh number... 

Viscosity describes the tendency of fluids to resist reciprocal laminar displacement of two adjacent layers the so-called "inner friction". Viscosity is therefore a measure of fluid flow resistance. 1 % solutions of sodium hyaluronate (e.g., Hea-lon or Proviso ) show a viscosity 400,000 times higher than aqueous fluid. They are dependent on concentration, molecular weight, addition of solvent, and temperature. Viscosity can usually be determined by placing a certain amount of viscous fluid between two parallel plates of equal size at a predetermined distance from one another and sliding them in the same direction, at different speeds (Fig. 10). 

The steady-state heat convection between two parallel plates and in circular, rectangular, and annular channels with viscous heat generation for both thermally developing and fully developed conditions is solved. Both constant wall temperature and constant heat flux boundary conditions are crmsidered. The velocity and the temperature distributions are derived from the momentum and energy equations, and the proper slip-flow boundary conditions are considered. 

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