In the limit of large the function and hence eq. (11), gives the cross-over time as a atc f(atc)1, 2/3rd power law in the equilibrium penetration length, i.e. The cross-over length, as a tcxe2/3. xc, fraction of the equilibrium length, can be written using the equilibrium length6, xe, (13) 2.3. Non-dimensional analysis of the different regimes To identify the different regimes, the relationship between and can be used, and in order t/x2 x to generalize the obtained solutions the data can be plotted in a non-dimensional form as T/X2 against where and is a characteristic time motivated by the visco-gravitational X=x/xe, T=t/ solution (eq. (4)) and defined as =(a (14) b/g2sin2φ)= axe/gsin In this form, the first of the limiting regimes, in increasing value of is the inertial regime, x, which from eq. (6) is, (15) This crosses over into the Bosanquet solution (eq. (3)). The second regime, the Lucas-Washburn solution, is given from eq. (7) as, (16) and the regime, the visco-gravitational solution, is given from eq. (4) as, (17) In this form, the from Bosanquet to visco-gravitational is given from eq. Xc=xc/xe, (13) by, (18) and the defining differential equation is, (19)
In the limit of large atc the function f(atc)1, and hence eq. (11), gives the cross-over time as a 2/3rd power law in the equilibrium penetration length, i.e. tcxe2/3. The cross-over length, xc, as a fraction of the equilibrium length, xe, can be written using the equilibrium length6, (13) 2.3. Non-dimensional analysis of the different regimes To identify the different regimes, the relationship between t/x2 and x can be used, and in order to generalize the obtained solutions the data can be plotted in a non-dimensional form as T/X2 against X=x/xe, where T=t/ and is a characteristic time motivated by the visco-gravitational solution (eq. (4)) and defined as =(a b/g2sin2φ)= axe/gsin (14) In this form, the first of the limiting regimes, in increasing value of x, is the inertial regime, which from eq. (6) is, (15) This crosses over into the Bosanquet solution (eq. (3)). The second regime, the Lucas-Washburn solution, is given from eq. (7) as, (16) and the regime, the visco-gravitational solution, is given from eq. (4) as, (17) In this form, the Xc=xc/xe, from Bosanquet to visco-gravitational is given from eq. (13) by, (18) and the defining differential equation is, (19)
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