Steady Marangoni instabilities in variable-viscosity liquid layer in the presence of insoluble surfactant

Abstract

Abstract The ability to control the complex convective flow patterns is important and one of the factors that alter the dynamics of the surface tension is surface-active agents or surfactants. Surfactants affect the surface tension at the free upper surface by producing tangential stress. This study theoretically examines the onset of steady instabilities in a thin nondeformable and deformable variable-viscosity liquid layer in the presence of insoluble surfactant. The surface tension at the free surface is assumed to be linearly dependent on temperature and concentration gradients. The dynamic viscosity is assumed to be exponentially dependent on temperature. The stability problems are governed by the systems of nonlinear partial differential equations consisting of continuity, momentum and energy equations. Linear stability analysis consisting of scaling, perturbation of infinitesimal disturbances and superposition in normal modes are used to transform the system of partial differential equations into system of ordinary differential equations. The exact analytical solutions are derived and the marginal curves are illustrated to show effects of some controlling parameters particularly the variable viscosity and elasticity parameter. Viscosity variation and elasticity parameter of surfactant act as destabilizer while the Biot number and Lewis number stabilize. For deformable surface, convection sets in at long wavelength for liquid with variable viscosity but convection occurs at short wavelength for constant viscosity liquid.