The impact of deformable interfaces and Poiseuille flow on the thermocapillary instability of three immiscible phases confined in a channel

The majority of studies considering thermocapillary convection in multilayer systems assume the interface is rigid (non-deformable) and the fluids are stationary. Motivated by the potential of using thermally generated convection patterns in moving droplets for mixing, this paper re-examines the Marangoni-Benard instability for a three-layer system with deformable interfaces undergoing Poiseuille flow. Taking into account the deformability of the interface reveals new physics. Linear stability analysis shows that at small wave numbers a deformable interface is of the orders of magnitude less stable than a non-deformable interface. At large wave numbers, both a rigid and a deformable interface have the same stability. Furthermore, a base planar Poiseuille flow affects the linear stability of the system and the type of instability when the interface is allowed to deform. Flow stabilizes an already unstable system and works to destabilize a stable system. Lastly, the dependence of the linear stability of the system on viscosity ratio, depth ratio, and Prandtl number, Pr, experimentally adjustable parameters, is discussed. Whereas Pr has little effect on the stability of the system, we show that a small viscosity ratio and a large depth ratio are advantageous in making the system unstable.