In the theory of capillarity, Bosanquet equation is an improved modification of the simpler
Lucas–Washburn theory for the motion of a liquid in a thin
capillary
A capillary is a small blood vessel from 5 to 10 micrometres (μm) in diameter. Capillaries are composed of only the tunica intima, consisting of a thin wall of simple squamous endothelial cells. They are the smallest blood vessels in the bod ...
tube or a
porous material
A porous medium or a porous material is a material containing pores (voids). The skeletal portion of the material is often called the "matrix" or "frame". The pores are typically filled with a fluid ( liquid or gas). The skeletal material is us ...
that can be approximated as a large collection of capillaries. In the Lucas–Washburn model, the
inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
of the fluid is ignored, leading to the assumption that flow is continuous under constant viscous
laminar Poiseuille flow conditions without considering the effects of mass transport undergoing acceleration occurring at the start of flow and at points of changing internal capillary geometry. The Bosanquet equation is a differential equation that is second-order in the time derivative, similar to
Newton's Second Law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
, and therefore takes into account the fluid inertia. Equations of motion, like the Washburn's equation, that attempt to explain a velocity (instead of acceleration) as proportional to a driving force are often described with the term ''
Aristotelian mechanics
Aristotelian physics is the form of natural science described in the works of the Greek philosopher Aristotle (384–322 BC). In his work ''Physics'', Aristotle intended to establish general principles of change that govern all natural bodies, b ...
''.
[Arthur Stinner, "The story of force: from Aristotle to Einstein", Phys. Educ. 29. (1994)]
Definition
When using the notation
for dynamic viscosity,
for the liquid-solid contact angle,
for
surface tension ,
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
[Joachim Schoelkopf , Patrick A. C. Gane, Cathy J. Ridgway, OMYA AG, Oftringen, Switzerland and G. Peter Matthews, "Influence of Inertia on Liquid Absorption into Paper Coating Structures", University of Plymouth, UK]
:
assuming that the motion is completely driven by surface tension, with no applied pressure at either end of the capillary tube.
Solution
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
: