Capillary Performance of Evaporating Flow in Micro Grooves: An Analytical Approach for Very Small Tilt Angles

The axial flow of an evaporating thin film through a V-shaped micro channel was investigated for the case of a grooved plate inclined with a very small tilt angle. For this problem, the addition of the gravity term caused by the tilt angle alters the form of the governing equation from linear to nonlinear. Because the effect of the tilt angle is very small, a perturbation method was applied to obtain a first order perturbation solution of R = R₀ + εR₁, where R₀ is the undisturbed solution caused by the tilt angle. A generalized graph of the nondimensional radius of curvature as a function of the nondimensionalized axial length was obtained. The perturbation parameter, ε, was expressed as a function of five nondimensional parameters. The results indicate that when the geometry is fixed, the effects of the Bond number and Capillary number are dominant. In addition, it was found that ε is inversely proportional to the square root of the Capillary number. This indicates that the perturbation method is applicable and accurate, even for the case of somewhat larger tilt angles (where the initial applicability of perturbation appears to be violated), is a sufficiently large heat flux is applied.