Simple Derivations of Some Important Relationships in Capillary Flow

Abstract

Use is made of geometrical constructions to demonstrate the conditions under which a plot of V/πR³ against ½pR gives a unique curve independent of the value of R, and also to show how account can be taken of discrepancies due to modified flow near the wall of the tube. In a similar way, the reasoning from which the velocity gradient G_W at the wall of the tube can be deduced from experimental figures for V, p and R has been set out in a geometrical form, which should be helpful to those to whom a pictorial representation makes a ready appeal. The deductions, though simple, involve no loss of generality. The data of Farrow, Lowe and Neale for two percent starch paste are considered by way of example, and it is shown that their form $$G_W{\mathrm{=(V/\pi R}}^3\text{)(N+3)\hspace{0.17em}\hspace{0.17em}}\mathrm{where}\mathrm{\hspace{0.17em}\hspace{0.17em}N=d\; (}\log \mathrm{V)/d\hspace{0.17em}(}\log \mathrm{p)}$$ of the equation for the velocity gradient at the wall has special advantages. Later work, by disclosing a wider basis, has shown that N need not be constant as they supposed, and also that, where modified flow occurs near the wall of the tube, V/πR³ becomes V_β/πR³, the limiting value for large radii.