(Solved): Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe ...

Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe of diameter \\( \\mathrm{D} \\) or radius \\( \\mathrm{R}=\\mathrm{D} / 2 \\) inclined at angle \\( \\alpha \\). There is no applied pressure gradient \\( \\left(\\frac{d P}{d x}=0\\right. \\) ). Instead, the fluid flows down the pipe due to gravity alone. We adopt the coordinate system shown, with \\( \\mathrm{x} \\) down the axis of the pipe. Derive an expression for the \\( \\mathrm{x} \\)-component of velocity \\( \\mathrm{u} \\) as a function of radius \\( \\mathrm{r} \\) and the other parameters of the problem.