Marangoni instability in a heated viscoelastic liquid film: Long-wave versus short-wave perturbations

We investigate the Marangoni instability in a thin layer of viscoelastic fluid, confined between its deformable free surface and a substrate of low thermal conductivity. Following a theoretical analysis, we study the stability of the present system for the case when the fluid layer is subjected to heating from below. Here, we use the Maxwell model to depict the rheology of the viscoelastic fluid. Linear stability analysis of the quiescent base state reveals that, in addition to the conventional short-wave mode, a long-wave instability can also emerge in this system. We demonstrate the appearance of both the long-wave monotonic and oscillatory instabilities in such a system. We study this long-wave mode analytically using the scaling k similar to root Bi (k is the wave number and Bi is the Biot number), whereas the short-wave mode is examined numerically. The influential role of elasticity of the fluid and the other involved parameters on the stability of the system is aptly discussed, and their ranges are identified within which a particular instability mode gets critical.