Question 6.11: The capillary rise h, of a fluid of density ρ, and surface t...
The capillary rise h , of a fluid of density \rho , and surface tension \sigma , 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 Rayleigh’s 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 (6.50)
where A is a dimensionless constant.
( \phi is not raised to any exponent, since it is a dimensionless variable and hence an independent \pi term).
Expressing the variables in terms of their fundamental dimensions in the above equation, we get
Equating the exponents of M, L and T in LHS and RHS of the equation, we have
a+b =0
-3 a+c+d =1
-2 b-2 d =0
Solving these three equations in terms of a, we get
b=-a
c=1+2 a
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.