Marangoni instability in the linear Jeffreys fluid with a deformable surface

Viscoelastic polymer films are present in numerous industrial applications such as in paint coatings, lubricants, and optoelectronics which involve a large variation in temperatures. The present study deals with the general linear stability analysis (GLSA) of a Jeffreys viscoelastic fluid layer having a deformable free surface deposited on a perfectly conducting slippery plate. Variation in temperature results in variation in surface tension thereby inducing Marangoni stresses leading, under certain conditions, to the emergence of instability of the quiescent, heat-conducting base state which may be either stationary or oscillatory. Accounting for the deformability of the free surface of the layer, the GLSA reveals the existence of the long-wavelength Marangoni instability absent in the previous studies that assumed a nondeformable free surface. The analysis reveals a strong effect of variation in the capillary number, Ca, on the stationary mode, whereas the oscillatory mode is moderately affected. The critical parameters for the stationary mode remain unaffected by variation in the Prandtl number, Pr, and the dimensionless relaxation L1 and retardation L2 parameters; however, their impact on the oscillatory modes is significant. Variation in the Biot number, Bi, has a stabilizing effect on the system. Variation in the Bond number, Bo, is found to strongly influence the long-wave regime of the stationary mode, but it does not influence the short-wave regime of the stationary mode and oscillatory modes. Variation in the slip coefficient leads to a moderate increase in Mac in the case of the stationary mode, whereas in the case of the oscillatory mode variation of both Mac and kc is nonmonotonic. The energy balance analysis suggests that viscoelasticity of the fluid is responsible for the emergence of oscillatory instability. Our analysis reveals the existence of a wide parameter range where oscillatory modes determine the stability properties of the system. This result emphasizes that neglecting viscoelasticity of the fluid may lead to significant errors when the dynamics of a viscoelastic fluid is investigated. The present study further demonstrates that even for a highly elastic fluid, the interface deformability prevails and promotes the long-wave stationary instability at the expense of the oscillatory instability.