Thermosolutal Marangoni instabilities in a weakly viscoelastic falling film with soluble surfactant
We examine the linear stability and nonlinear evolution of a heated falling film containing a soluble surfactant and a weakly viscoelastic liquid. The stress response is described by the short-memory Walters model, while surface tension depends on temperature and surface concentration. An Orr--Sommerfeld analysis and its long-wave limit determine how elastic memory and the two Marangoni stresses enter the instability threshold. A five-field weighted-residual model is then derived to describe finite-amplitude disturbances. The bulk and surface transport equations are written with compatible fluxes, which conserve their combined projected surfactant inventory in a periodic domain. The numerical examples show that solutal Marangoni stress suppresses disturbance growth and reduces the amplitude of the nonlinear waves. Thermal Marangoni stress increases the maximum growth rate, although its effect on the width of the unstable wavenumber band is different. Elastic memory changes both the onset and the finite-wavelength response. The nonlinear profiles develop a main hump and capillary ripples, accompanied by variations in bulk concentration, surface concentration, and interfacial temperature. These results describe the competition between thermocapillary forcing, surfactant redistribution and elastic stress within the retained long-wave model. The larger elastic parameters are interpreted as formal predictions of the truncated constitutive equations, rather than as a controlled extension to strongly elastic flow.