Poiseuille’s Formula and It’s Derivation by Dimensional Analysis | Physics Grade 11 Notes

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Poiseuille’s Formula and It’s Derivation by Dimensional Analysis | Physics Grade 11 Notes

Poiseuille’s Formula and It’s Derivation by Dimensional Analysis

Poiseuille’s Formula

Poiseuille's Formula DiagramPoiseuille studied the streamline flow of a liquid in capillary tubes as shown in the figure. He concluded that the volume, V of the liquid flowing per second through a capillary tube is

  1. Directly proportional to the difference of pressure, P between the two ends of the tube. i.e; V α P
  2. Directly proportional to the fourth power of the radius, r of the capillary tube. i.e; V α r4
  3. Inversely proportional to the coefficient of viscosity, η of the liquid. i.e; V α 1/η
  4. Inversely proportional to the length, l of the capillary tube. i.e;V α 1/l
Combining these factors, we get
V α P r/ η l
or, V = K P r/ η l
 
Where k = π/8, a constant of proportionality.
.: V= π P r/ 8 η l
This equation is called Poiseuille’s equation. 
 
Derivation of Poiseuille’s Formula by Dimensional Analysis
Poiseuille found that the volume of a liquid flowing through a capillary tube per second depends upon:
  1. The pressure gradient (P/l) (i.e. the rate of change of pressure with length)
  2. Radius of the capillary tube, r
  3. Coefficient of the viscosity of liquid, η
That is, the volume of liquid flowing per second V α (P/l)arbηc, where a, b and c are constants to be determined
.: V = K (P/l)arbηc …… (i)
Where K is proportionality constant.
  • The dimension of V per second = [LT-1]
  • The dimension of P/l = [M L-1 T-2] / [L] = [M L-2 T-2]
  • The dimension of r = [L]
  • The dimension of η = [M L-1 T-1]
Putting the dimensions of the quantities in Equation (i), we get
[L3 T-1] = [M L-2 T-2]a [L]b [M L-1 T-1]c
or, [M0 L3 T-1] = [Ma+c L-2a+b-c T2a-c]
 
Applying the principle of homogeneity of dimensions, we get
Powers of M, a+c = 0 ….... (ii)
Powers of L, -2a+b-c = 3 ….... (iii)
Powers of T, -2a-c = -1 …..... (iv)
 
Solving these three equations, we get a = 1, b = 4 and c = -1 and substituting the values of a, b and c in Equation (i), we get
V = K (P/l)1 (r)4η-1
  = K P r4/ η l
 
The value of K is found experimentally to be π/8 and the above equation changes to
V= π P r/ 8 η l

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