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 φ, and acceleration due to gravity g. Find an expression for h in terms of dimensionless variables by Rayleigh’s indicial method.

Question

The capillary rise [latex] h [/latex], of a fluid of density [latex] \rho [/latex], and surface tension [latex] \sigma [/latex], in a tube of diameter D, depends upon the contact angle [latex] \phi [/latex], and acceleration due to gravity [latex] g[/latex] . Find an expression for [latex] h [/latex] in terms of dimensionless variables by Rayleigh’s indicial method.

Accepted Answer

Capillary rise [latex] h [/latex], is the dependent variable of the problem and can be expressed in terms of the independent variables as

[latex] h=A \rho^{a} \sigma^{b} D^{c} g^{d} \phi [/latex] (6.50)

where [latex] A [/latex] is a dimensionless constant.

([latex] \phi [/latex] is not raised to any exponent, since it is a dimensionless variable and hence an independent [latex] \pi [/latex] term).
Expressing the variables in terms of their fundamental dimensions in the above equation, we get

[latex] \mathrm{L}=\mathrm{A}\left(\mathrm{ML}^{-3}\right)^{a}\left(\mathrm{MT}^{-2}\right)^{b} \mathrm{~L}^{c}\left(\mathrm{LT}^{-2}\right)^{d} [/latex]

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

[latex] h=A D\left(\frac{\rho g D^{2}}{\sigma}\right)^{a} \phi [/latex]

Or [latex] \frac{h}{D}=A\left(\frac{\rho g D^{2}}{\sigma}\right)^{a} \phi [/latex]

This is the required expression.

Question 6.11: The capillary rise h, of a fluid of density ρ, and surface t...

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