Question 6.9: The capillary rise h of a fluid of density ρ and surface ten...
The capillary rise h of a fluid of density ρ and surface tension σ in a tube of diameter D depends upon the contact angle \phi and acceleration due to gravity g. Find an expression for h in terms of dimensionless variables by Rayleighs indicial method.
Capillary rise h is the dependent variable of the problem and can be expressed in terms of the independent variables as,
h=A \rho^{a} \sigma^{b} D^{c} g^{d} \phi (6.52)
where A is a dimensionless constant.
(\phi is not raised to any exponent, since it is a dimensionless variable and hence an independent π term).
Expressing the variables in terms of their fundamental dimensions in above equation we get,
L = A \left( ML ^{-3}\right)^{a}\left( MT ^{-2}\right)^{b} L ^{c}\left( LT ^{-2}\right)^{d}Equating the exponents of M, L and T in LHS and RHS of the equation, we have
a + b = 0
-3a + c + d = 1
-2b 2d = 0
Solving these three equations in terms of a, we get
b = -a
c = 1 + 2a
d = a
Substituting these values in Eq. (6.52), we get
h=A D\left(\frac{\rho g D^{2}}{\sigma}\right)^{a} \phior \frac{h}{D}=A\left(\frac{\rho g D^{2}}{\sigma}\right)^{a} \phi
This is the required expression.