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Low Reynolds number flow in slowly varying axisymmetric tubes. J. Fluid Mech. 49, 451. , and R. L. Laurence. (1979). Influence of slip velocity at a membrane surface on ultrafiltration performancemlI. Tube flow system. Int. J. Heat Mass Transfer 12, 731. This Page Intentionally Left Blank 2 Laminar Flow through a Tube with a Thin Annular Adherent Liquid Film l n thischapter we return to flow within a uniform tube, but we look at some problems that arise when there is an annular layer of liquid adjacent to the inside wall of the tube, which in turn surrounds an inner or "core" flow of a second fluid.

25 numerically. However, one simple result falls out of inspection of Eq. 25, and it reveals a serious limitation of the technique used in this analysis. First we nondimensionalize Eq. 25, and then rewrite it slightly in the form ~R 3 APex - 8~tO - _ 679 Reo 1575 (RolSd R Jz l) ~R! 26) 1-4 An Integral Technique for Solution of the Lubrication Equations 21 If we now understand that the excess pressure drop is to be calculated over a length z 2 - z 1 that is symmetricabout z = 0, we see that the first integral in the expression above vanishes if the shape function is symmetric about z = 0.

Influence of slip velocity at a membrane surface on ultrafiltration performancemlI. Tube flow system. Int. J. Heat Mass Transfer 12, 731. This Page Intentionally Left Blank 2 Laminar Flow through a Tube with a Thin Annular Adherent Liquid Film l n thischapter we return to flow within a uniform tube, but we look at some problems that arise when there is an annular layer of liquid adjacent to the inside wall of the tube, which in turn surrounds an inner or "core" flow of a second fluid. On the assumption that the annular liquid film is stable, we find that this layer has the ability to "lubricate" the tube in the sense that its presence can reduce the pressure drop required to maintain a specified volumetric flow rate of the inner fluid.