In this equation, is the equilibrium free energy of capillary flow. An imbalance of the three interfacial tensions near

the three-phase contact line, solid–liquid (σ sl), solid-vapor (σ sa), and liquid–vapor (σ), results in the out-of-equilibrium interfacial energy (σ(cos θ 0 − cos θ)) which changes the total free energy of capillary flow. The frequency of the three-phase contact line motion in forward direction (+) and backward direction (−) is [26]: (5) where n is the number of adsorption sites per unit area on solid surface. The net frequency of contact line motion is then as follows [26]: (6) For small arguments of sinh, Equations 3 and 6 result in linear MKT [31]: (7) where is in units of Pa s and is termed as the coefficient of friction at the three-phase contact line. It is noted

that this selleck products equation is identical Evofosfamide solubility dmso to equation twenty-two of [33] for U = 0 and σ cos(θ 0) = σ sa − σ sl (Young’s equation). Left hand side (LHS) of Equation 7 is the out-of-equilibrium interfacial energy which is the driving force of capillary flow. Right hand side (RHS) of Equation 7 only includes dissipation of the free energy due to the contact line friction. De Ruijter et al. [30] showed that the corresponding dissipation function (TΣ l ) is: (8) In the next section, the wedge film viscous dissipation is calculated and added to Equation 8 to form the total dissipation function from which the total drag force is calculated. The total drag force is then equated to the LHS of Equation 7 to form the complete equation of the three-phase contact line motion. Hydrodynamic theory To calculate Casein kinase 1 the wedge film

viscous dissipation (TΣ W ), Navier–Stokes equation of motion is solved in the wedge film region. From Figure 4 for the film thickness (H) much smaller than the radial distance ρ (H ≪ ρ) and for capillary number Ca ≪ 1, lubrication theory is used: (9) where p is the pressure and u is the Bindarit in vitro velocity distribution at distance x inside the wedge film. For no stress boundary condition at the free fluid-air interface and no slip boundary condition at the solid surface, solution to Equation 9 gives: (10) where η n is replaced by its expression in Equation 1. The average cross-sectional fluid velocity in the wedge film ( ) is equal to the three-phase contact line velocity ( ). This results in: (11) The viscous dissipation in the wedge film can be obtained as follows [5]: (12) where τ is the shear stress (= η n  ∂ u/∂ z), and x m is the cutoff length similar to slip length in HDT [27, 28]. Without consideration of x m , dissipation of energy at the wedge film grows infinitely close to the three-phase contact line.