Flow Between Moving Flat Plates

FIGURE 8.5 Drag flow between parallel plates with the upper plate in motion and no axial pressure drop. 

The plates are separated by distance H, and the y-coordinate starts at the bottom plate. The velocity profile is linear  

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. 

---

Step 6 The streamlines in Figure 10.11 indicate the extent to which fully developed flow between two flat plates occurs as one moves from left to right. There is also some velocity rearrangement at each inlet because the velocity has been assumed to be +1 at the inlet, which is inconsistent with the no-slip boundary condition on each wall. This problem can easily be fixed. Instead of using v = 1 on the boundary, use Eq. (10.18), which has the... 

For simplicity, consider flow between two flat plates as shown in Fig. 8.11. The flow is laminar and the velocity prohle is quadratic. Now suppose the entering fluid has a second chemical with the concentration shown, 1.0 in the top half and 0 in the bottom half As this fluid moves downstream (to the right), the velocity is highest in the center, so the second chemical moves fast there. However, it can also diffuse sideways, and the chemical that does so first then moves more slowly down the flow channel. At a point not on the centerline, the concentration... 

Rheology is the study of the deformation and flow of matter under the influence of an applied stress. Commonly a sample such as a polymer or gel is placed between two flat plates or between two concentric cylinders and the stress is applied by moving one plate or cylinder relative to the other while the resulting strain is measured using a pressure transducer. We can define a number of categories of rheological behaviour as a... 

Several important dimensionless numbers in combined heat and momentum transfer in fluids can be derived by considering the simple flow of a Newtonian fluid between two flat plates, one stationary and one moving at velocity, v see Fig. 5.1. 

Because liquid crystals are fluids, they also show anisotropy in their flow behaviom. This can easily be imderstood by imagining measuring the viscosity of a liquid crystal by placing it between two flat plates and measuring the force necessary to move one plate past the other at a certain velocity. In Figure 2.1, the plates lie in the xy plane and are separated by a distance d. The bottom plate is fixed and the force acting on the top plate is in the x-direction, F. The velocity of the top plate is also in the x-direction, v. ... 

Development of an understanding of turbulence requires consideration of the details of turbulent motion. Much of our intuitive sense of fluid flow is based on what we can observe with the naked eye, and much of this intuitive sense can be applied to an understanding of turbulence, if we proceed with some care. We begin with the classical definition of simple shear flow, as shown in Figure 2-6. In this figure a Newtonian fluid is placed between two flat plates. The top plate moves with velocity Vx, requiring a force per unit area of plate surface (F/A) to maintain the motion. The force required is in proportion to the fluid viscosity. 

In convective heat flow, heat transfer takes effect by means of two mechanisms the direct conduction through the fluid, and the motion of the fluid itself Figure 2.2 illustrates convective heat transfer between a flat plate and a moving fluid. 

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... 

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. 

We shall consider first the flow of solution over a flat plate. In such systems, two forces will be acting upon the fluid. The first is the cause of the flow (that generated by a pump or a solution head) and is known as the inertial force. The second opposes the flow and results from viscous forces at the interface between the plate and the solution. Suppose we assume that the solution may be divided into elements, then the viscous force will initially cause that element next to the stationary plate to be retarded and later each element will be slowed down by that closer to the plate. Hence as the solution flows over the plate, a boundary layer of more slowly moving solution will develop, as shown in Fig. 1.11. In an electrolytic cell, the flat plate would normally be the electrode and therefore the formation of such boundary layers has particular importance. The electrode reaction takes place in the boundary layer in the presence of velocity gradients. 

For a study of liquid crystals, flat plane capillaries with transparent plates are very convenient, because in this case we can create and observe a proper orientation of the director. There can be distinguished a simple shear flow and the Poiseuille flow, both shown in Fig. 9.5. As discussed above, the shear flow occurs when the upper plate is moving with constant velocity Vq and the lower plate is fixed. Then, for small Vo, the profile of velocity of isotropic liquid is linear (the dash line in Fig. 9.5a). The Poiseuille flow occurs when the liquid is moved between two immobile plates under an external pressure, as discussed in the previous paragraph. In this case the profile has a form of parabola (the dash line in Fig. 9.5b). In both... 

In many cases a fluid is flowing over completely immersed bodies such as spheres, tubes, plates, and so on, and heat transfer is occurring between the fluid and the solid only. Many of these shapes are of practical interest in process engineering. The sphere, cylinder, and flat plate are perhaps of greatest importance with heat transfer between these surfaces and a moving fluid frequently encountered. 

Figure 7.105 illustrates that the difference between the cylindrical and flat plate system increases when the screw root radius reduces. When the root radius is 80% of the barrel radius, the flow rate in the flat plate system with moving barrel is about 4% lower than the cylindrical system. However, the flow rate in the flat plate system with moving screw is about 30% lower. The flat plate system with moving screw results in significant errors and should not be used in serious analysis of screw extruders. 

In many electrolysis cells it is the solution rather than the electrode which moves, and as an example of such systems, we shall consider briefly the flow of solution over a flat plate. As the solution flows across the plate, two forces act upon it the first is the cause of the flow and is known as the inertial force (i.e. that generated by the pump or solution head), while the second opposes flow and results from viscous forces between the plate and the solution. Hence as the solution flows over the plate, the layer adjacent to the surface will continuously be slowed down, and the boundary layer, where the rate of flow is less than that in the bulk, will expand into the solution. This is illustrated in Fig. 4.6. The shape of the flow contours and the thickness of the boundary layer will depend on the relative importance of the forces leading to solution flow and those leading to the retardation of flow at the plate/solution interface. Because of the importance of this ratio of inertial/viscous forces, it is given a name, the Reynolds number. Re, This is a dimensionless parameter defined by... 

A power-law fluid is conflned between two parallel, flat plates in simple shearing flow. The lower plate is fixed and the upper plate moves with a velocity V. The plates are separated by a distance S. Calculate and sketch the velocity profiles for n = 0.5, = 1, and = 1.5. 

Consider the case of a Newtonian fluid undergoing laminar, pressure-driven flow between two parallel, infinite flat plates separated by a distance B (Figure 1.10). The bottom plate is stationary and the top plate moves at a constant velocity Fup. For a constant dynamic pressure gradient, AP/Ax, P = p - g -r,-we wish to calculate the resulting velocity profile. 

Fig. 1 a, b. Schematic diagram of a flow of fluid under combined shear conditions a — between flatly parallel plates under the action of pressure difference AP = P -P2 (the upper plane moves in the direction transverse to the main flow) b — between two coaxial cylinders rotating towards one another at angular velocities flj and fi2... 

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),... 

Plate and frame Flat porous plates covered with polymeric membrane material are assembled together with alternate hollow spacers to produce a cross-flow system where feed moves through the annular spaces between adjacent membrane surfaces. Although now largely superseded by other designs, the plate and frame system is still available with membrane areas up to 80 m. ... 

---