Flow of Incompressible Fluids in Pipes

Flow in circular conduits is important not only in its own right as an engineering operation but as an example of the quantitative relationships involving fluid flow in general. It is therefore discussed in some detail in this chapter. This discussion is restricted to steady flow. 

SHEAR-STRESS DISTRIBUTION IN A CYLINDRICAL TUBE. Consider the steady flow of a viscous fluid at constant density in fully developed flow through a horizontal tube. Visualize a disk-shaped element of fluid, concentric with the axis of the tube, of radius r and length dL, as shown in Fig. 5.1. Assume the element is isolated as a free body. Let the fluid pressure on the upstream and downstream faces of the disk be p and p + dp, respectively. Since the fluid possesses a viscosity, a shear force opposing flow will exist on the rim of the element. Apply the momentum equation (4.14) between the two faces of the disk. Since the flow is fully developed, j8j, = and Fj, = F , so that E F = 0. The quantities for substitution in Eq. (4.15) are 

The shear force acting on the rim of the element is the product of the shear stress and the cylindrical area, or 2nrdL)T. [Force F, equals F in Eq. (4.15).] Since the channel is horizontal, F is zero. Substituting these quantities into Eq. (4.15) gives 

In steady flow, either laminar or turbulent, the pressure at any given cross section of a stream tube is constant, so that dp/dL is independent of r. Equation (5.1) can be written %r the entire cross section of the tube by taking t = t, and r = r, , where is the shear stress at the wall of the conduit and r, is the radius of the tube. Equation (5.1) then becomes 

Fluid element in steady flow through pipe. 

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