density of the liquid and its surface tension. is the angle of the tube with respect to the horizontal axis. is the contact angle of the liquid on the capillary material. Substituting these expressions leads to the first-order differential equation for the distance the fluid penetrates into the tube : Washburn's constant The Washburn constant may be included in Washburn's equation. It is calculated as follows: Fluid inertia In the derivation of Washburn's equation, the inertia of the liquid is ignored as negligible. This is apparent in the dependence of length to the square root of time, , which gives an arbitrarily large velocity dL/dt for small values of t. An improved version of Washburn's equation, called Bosanquet equation, takes the inertia of the liquid into account. Applications Inkjet printing The penetration of a liquid into the substrate flowing under its own capillary pressure can be calculated using a simplified version of Washburn's equation: where the surface tension-to-viscosity ratio represents the speed of ink penetration into the substrate. In reality, the evaporation of solvents limits the extent of liquid penetration in a porous layer and thus, for the meaningful modelling of inkjet printing physics it is appropriate to utilise models which account for evaporation effects in limited capillary penetration. Food According to physicist and Ig Nobel prize winner Len Fisher, the Washburn equation can be extremely accurate for more complex materials including biscuits. Following an informal celebration called national biscuit dunking day, some newspaper articles quoted the equation as Fisher's equation. Novel capillary pump The flow behaviour in traditional capillary follows the Washburn's equation. Recently, novel capillary pumps with a constant pumping flow rate independent of the liquid viscosity were developed, which have a significant advantage over the traditional capillary pump (of which the flow behaviour is Washburn behaviour, namely the flow rate is not constant). These new concepts of capillary pump are of great potential to improve the performance of lateral flow test. See also Bosanquet equation Mercury intrusion porosimetry (MIP) References External links Powder wettability measurement with the Washburn method Equations of fluid dynamics Porous media