The maximum-velocity occurs at centre where r = 0. Put in Equation

U_max  = -1/4μ  ∂p / ∂x R²

Mean or Average velocity is obtained by dividing the discharge of the fluid across the corss sectional area of pipe (πr²).

Te discharge (Q) across the section is obtained by considering the flow through a circular ring element of radius ‘r’ and thickness ‘dr’ as shown in.

dQ  = Velocity at a radius r x Area of ring element

       = u x 2 πr dr = - 1/4μ  ∂p / ∂x [ R² - r² ] x 2 πr dr

        R                                                R
Q  =  ∫  dQ = - 1/4μ  (∂p / ∂x ) x 2π  ∫  (R² - r²) r dr
        0                                                0
                                      R
    = 1/4μ ( - ∂p / ∂x )2π ∫ ( R² r - r³) dr
                                      0

    = 1/4μ (- ∂p / ∂x )2π (R⁴ / 2 - R⁴ / 4) = 1/4μ (- ∂p / ∂x )2π  R⁴ / 4

Q  = π / 8μ (- ∂p / ∂x ) R⁴

Average velocity,
-
U_ave  =  Q / A  = π / 8μ (- ∂p / ∂x ) R⁴ / πR²

-
U_ave  = 1 / 8μ (- ∂p / ∂x )R²

Ratio of maximum and average velocity

U_max  = - 1 / 4μ (∂p / ∂x )R² / 1 / 8μ (- ∂p / ∂x )R²

U_max  = 8/4 = 2
-
U

Ratio of maximum velocity to average velocity is 2.

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