Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe of diameter D or radius R = D/2 inclined at angle α (Fig. P9–105). There is no applied pressure gradient (∂P/∂x = 0). Instead, the fluid flows down the pipe due to gravity alone. We adopt the coordinate system shown, with x down the axis of the pipe. Derive  an expression for the x-component of velocity u as a function of radius r and the other parameters of the problem. Calculate the volume flow rate and average axial velocity through the pipe.

FIGURE P9–105

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Too -D- α Pipe wall Fluid: p. p R