Plane channel laminar flow

In microscale channels, the viscous forces dominate the inertial effect resulting in a low Reynolds numbers. Hence, laminar flow behavior is dominant and mixing occurs via diffusion. However, in a liquid-liquid system, the interfacial forces acting on the interface add complexity to the laminar flow as the relationship between interfacial forces and other forces of inertia and viscous results in a variety of interface and flow patterns. Gunther and Jensen [202] illustrated this relationship as a function of the channel dimension and velocity as shown in Figure 4.12. The most regularly shaped flow pattern is achieved when interfacial forces dominate over inertia and viscous forces at low Reynolds numbers, as represented in Figure 4.12 by the area below the yellow plane [202,203]. 

In order to. illustrate how natural convection in a vertical channel can be analyzed, attention will be given to flow through a wide rectangular channel, i.e., to laminar, two-dimensional flow in a plane channel as shown in Fig. 8.15. This type of flow is a good model of a number of flows of practical importance. 

Heat and Mass Transfer in a Laminar Flow in a Plane Channel... 

Temperature field. We shall study the heat exchange in laminar flow of a fluid with parabolic velocity profile in a plane channel of width 2h. Let us introduce rectangular coordinates X, Y with the X-axis codirected with the flow and lying at equal distances from the channel walls. We assume that on the walls (at Y = h) the temperature is constant and is equal to Ti for X < 0 and to T2 for X > 0. Since the problem is symmetric with respect to the X-axis, it suffices to consider a half of the flow region, 0 < Y < h. 

Kaganov, S. A., On steady-state laminar flow of incompressible fluid in a plane channel and in a circular cylindrical tube with regard to heat friction and dependence of viscosity on temperature, J. Appl. Mech. Techn. Phys., No. 3, 1962. 

A number of analytical results are available for fully developed Nusselt values for the laminar flow of power law fluids in rectangular channels having aspect ratios ranging from 0 (i.e., plane parallel plates) to 1.0 (i.e., a square duct). Newtonian results (n = 1) are available for the T, HI, and H2 boundary conditions for the complete range of aspect ratios. Another limiting case for which many results are available is the slug or plug flow condition, which corresponds to n = 0. At other values of n, results are available for plane parallel plates and for the square duct. 

Shear stress It is the component of stress tangent to the plane on which the forces act. Shear stress is equal to the force divided by the area sheared, yielding MPa (or psi). During processing it is the stress developed in a plastic melt when the layers in a cross section are sliding along each other or along the wall of the channel or cavity in laminar flow. 

Haldenwang, P., 2007. Laminar flow in a two-dimensional plane channel with local pressure-dependent crossflow. J. Fluid Mech. 593, 463 73. 

There have been attempts to explain the onset of melt fracture by using hnear stability theory, along the hnes discussed in the preceding chapter for the spinline. The procedure is the same in principle, but the computational problem to solve the relevant linear eigenvalue problem for channel flow with any viscoelastic constitutive equation is a very difficult one. Based on a number of successful solutions it appears that plane laminar flow of viscoelastic liquids at very low Reynolds numbers is stable to infinitesimal disturbances. 

Consider fully developed pressure-driven laminar flow in a plane channel, with Vx = Vx(y), Vy — v = 0. Following the development in Section 3.2.1, which is valid for any fluid, we And that the pressure gradient is a constant and the shear stress varies linearly across the channel, passing through zero at the center plane ... 

More detail of the cross-plane (y - z) geometry is shown in Figure 9.2 (also not to scale). The high aspect ratio of a unit cell suggests the following dimensional reduction That the x direction transport is dominated by the gas channel and coolant flow. Furthermore, since the flow is slow enough to be laminar, these flows can be described simply with average quantities in the x direction. This leads to a 2 -I-1 D model, in which the cross-plane (y-z) problem can be solved for each x and connected to 1-D models for the channel flow. 

Figure 4.8 Three-dimensional velocity field of of-planecomponentofthevelocity,z-component stereo- X-PIV measurements in the mixing zone is displayed colour coded. Only the lower half of of a T-shaped micromixer at Re = 120. The flow is the 3D scan from z = 22 to lOOpm in the center of laminar and station a 7. The in-plane velocity the channel ofthe 800 x 200 Xm cross-section is distribution is presented as vectors and the out- shown [12] (by courtesy of Springer-Verlag).

Where H is the charmel height (the smaller dimension in a rectangular channel), tw,av the average wall shear stress, V the kinematic viscosity, and p the density of the fluid. In internal flows, the laminar to turbulent transition in abrupt entrance rectangular ducts was found to occur at a transition Reynolds number Ret = 2200 for an aspect ratio ac = I (square ducts), to Ret = 2500 for flow between parallel planes with = 0 [4]. For intermediate channel aspect ratios, a linear interpolation is recommended. For circular tubes. Ret = 2300 is suggested. These transition Reynolds number values are obtained from experimental observations in smooth channels in macroscale applications of 3 mm or larger hydraulic diameters. Their applicability to microchannel flows is still an open question. 

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