Bosanquet equation

When using the notation ${\displaystyle \eta }$ for dynamic viscosity, ${\displaystyle \theta }$ for the liquid-solid contact angle, ${\displaystyle \gamma }$ for surface tension , ${\displaystyle \rho }$ for the fluid density, t for time, and r for the cross-sectional radius of the capillary and x for the distance the fluid has advanced, the Bosanquet equation of motion is[2]

${\displaystyle {\frac {d}{dt}}\left(\pi r^{2}\rho x{\frac {dx}{dt}}\right)+8\pi \eta x{\frac {dx}{dt}}=2\pi r\gamma \cos \theta ,}$

assuming that the motion is completely driven by surface tension, with no applied pressure at either end of the capillary tube.

The solution of the Bosanquet equation can be split into two timescales, firstly to account for the initial motion of the fluid by considering a solution in the limit of time approaching 0 giving the form[2]

${\displaystyle x^{2}(t)-x^{2}(0)={\frac {2b}{a}}\left[t-{\frac {1}{a}}(1-e^{-at})\right]}$

where

${\displaystyle a={\frac {8\eta }{\rho r^{2}}}}$

and

${\displaystyle b={\frac {2\gamma \cos \theta }{\rho r}}.}$

For the condition of short time this shows a meniscus front position proportional to time rather than the Lucas-Washburn square root of time, and the independence of viscosity demonstrates plug flow.

As time increases after the initial time of acceleration, the equation decays to the familiar Lucas-Washburn form dependent on viscosity and the square root of time.

• Hagen–Poiseuille equation
• Washburn's equation