Shell Momentum Balance and Velocity Profile in Laminar Flow

9 SHELL MOMENTUM BALANCE AND VELOCITY PROFILE IN LAMINAR FLOW  

In many engineering problems a knowledge of the complete velocity profile is not needed, but a knowledge of the maximum velocity, average velocity, or the shear stress on a surface is needed. In this section we show how to obtain these quantities from the velocity profiles. 

Engineers often deal with the flow of fluids inside a circular conduit or pipe. In Fig. 2.9-1 we have a horizontal section of pipe in which an incompressible Newtonian fluid is flowing in one-dimensional, steady-state, laminar flow. The flow is fully developed i.e., it is not influenced by entrance effects and the velocity profile does not vary along the axis of flow in the x direction. 

The cylindrical control volume is a shell with an inside radius r, thickness Ar, and length Ax. At steady state the conservation of momentum, Eq. (2.8-3), becomes as follows sum of forces acting on control volume = rate of momentum out — rate of momentum into volume. The pressure forces become, from Eq. (2.8-17), 

2 Principles of Momentum Transfer and Overall Balances 

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Figure 2.9-2. Velocity and momentum flux profiles for laminar flow in a pipe. Sec. 2.9 Shell Momentum Balance and Velocity Profile in Laminar Flow...

Sec. 2.9 Shell Momentum Balance and Velocity Profile in Laminar Flow... 

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