Thin Films and Fingering Problems in Complex Flows

We consider two problems in complex fluids: (i) thickening of thin films in the Landau-Levich problem [1] of dip coating and in motion of long bubbles in capillary tubes [2]. In both of these problems, thickening of thin films is observed experimentally in the presence of interfacial surfactants which has been confirmed experimentally. Considering small concentration of insoluble surfactants at the interfaces with a small variation of it along the interfaces, we theoretically prove these thickening effects using lubrication theory and boundary layer methods. In the case of Landau-Levich problem of dip coating, we obtain a formula for an upper bound in terms of the Marangoni and Capillary number which is then used to show that the upper bound in the “clean” case (without surfactant) is less than the lower-bound in the “surfactant” case. For the slowly and steadily moving long bubble in a capillary tube, the thickening of the thin film left behind the moving front by the presence of surfactant is based on first obtaining a formula of the film thickness in terms of Marangoni number and concentration of interfacial surfactant. A comparison with Bretherton’s “clean” case then shows the thickening effect of surfactant. References [3] and [4] have many details on these problems including many references. We are currently studying extension of these results for the complex fluids, specially polymeric fluids, and also studying the fingering problem in complex fluids in a Hele-Shaw cell.