Kinetic Energy of Tube FlowTo get the kinetic energy of laminar flow in a tube, an average of the square of the velocity must be taken to account for the velocity profile.
The average of the square of the speed is given by  The average kinetic energy per unit volume of the flowing fluid can be expressed in terms of the fluid density ρ and the maximum flow velocity vm.  |
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Velocity Relationship, Tube FlowThe nature of viscosity is such that successive lamina in the tube exert forces on each other according to the viscous force relationship:  When a pressure gradient dP/dx drives a section of lamina of length Δx at constant velocity, the force equation takes the form: 
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Velocity Equation, Tube Flow Collecting terms gives the velocity equation in terms of radius r. 
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Velocity Profile for Tube FlowThe relationship governing the velocity as a function of distance r from the center of a tube under conditions of laminar flow is:
The general form of the solution to this differential equation is v = A + Br2 where A and B are constants which must be fit to the boundary condition of the flow: v=0 at r=R. Substituting the general solution by taking the derivatives gives: 
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Velocity Profile for Tube FlowUnder conditions of laminar flow, the nature of viscosity dictates a flow profile where the velocity increases toward the center of the tube as illustrated. 
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