Kinetic Energy of Tube Flow

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 v_m.

Velocity Relationship, Tube Flow

The 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:

Velocity Equation, Tube Flow

Collecting terms gives the velocity equation in terms of radius r.

Velocity Profile for Tube Flow

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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:

Velocity Profile for Tube Flow

Under conditions of laminar flow, the nature of viscosity dictates a flow profile where the velocity increases toward the center of the tube as illustrated.