Streamline momentum transfer

In streamline flow, E is very small and approaches zero, so that xj p determines the shear stress. In turbulent flow, E is negligible at the wall and increases very rapidly with distance from the wall. LAUFER(7), using very small hot-wire anemometers, measured the velocity fluctuations and gave a valuable account of the structure of turbulent flow. In the operations of mass, heat, and momentum transfer, the transfer has to be effected through the laminar layer near the wall, and it is here that the greatest resistance to transfer lies. 

The value AP can change in the axial direction in the hollow fiber (AP is the pressure drop in the membrane matrix due to the momentum transfer, the velocity through the membrane is u0 , where e is the membrane porosity). Kelsey etal. [11] have solved the equation system in all three cases, namely for closed-shell operation, partial ultrafiltration and complete ultrafiltration and have plotted the dimensionless axial and radial velocities as well as the flow streamlines. Typical axial and radial velocity profiles are shown in the hollow-fiber membrane bioreactor at several axial positions in Figure 14.8 plotted by Kelsey etal. [ 11]. This figure illustrates clearly the change of the relative values of both the axial and the radial velocity [V=vL/(u0Ro), U=u/u0 where uc is the inlet centerline axial velocity]. 

The turbulent mechanism that carries motion, heat, or matter from one part of the fluid to another is absent in laminar flow. The agency of momentum transfer is the shear stress arising from the variations in velocity, that is, the viscosity. Similarly, heat and matter can only be transferred across streamlines on a molecular scale, heat by conduction and matter by diffusion. These mechanisms that are present but less important in turbulent flow are comparatively slow. Velocity, temperature, and concentration gradients are, therefore, much higher than in turbulent flow. 

A general study of the streamlines for a circular cylinder in simple shear flow can he found in the following papers C. R. Robertson and A. Acrivos, Low Reynolds number shear flow past a rotating circular cylinder, Part I, Momentum Transfer, J. Fluid Mech. 40, 685-704 (1970) R. G. Cox, I. Y. Z. Zia, and S. G. Mason, Particle motions in sheared suspensions, 15. Streamlines around cylinders and spheres, J. Colloid Interface Sci. 27, 7-18 (1968). 

In die streamline boundary layer the only forces acting within the fluid are pure viscous forces and no transfer of momentum takes place by eddy motion. 

The highly ordered fluid motion characterized by smooth streamlines is called laminar. The highly disordered fluid motion that typically occurs at high velocities is characterized by velocity fluctuations is called turbulent. The random and rapid fluctuations of groups of fluid panicles, called eddies, provide an additional mechanism for momentum and heat transfer. 

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