Nonlinear dynamics of micropolar two-phase fluids: Multiple exact solutions

Dual and triple solutions induced by a flexible planar surface for a micropolar two-phase fluid model are studied. The two-phase behavior in the micropolar fluid model occurs due to phase transitions between the fluid phases, influenced by interfacial stresses and heat transfer. The physical implications of these transitions are significant in understanding flow behavior under different mechanical and thermal conditions. This study examines the critical parameters and conditions that lead to these phase transitions, resulting in dual or triple solutions in the flow dynamics. The flow and thermal fields are exact solutions of the steady, two-dimensional two-phase micropolar fluid equations in the form of similarity solution. It is shown that dual and triple exact solutions exist for a highly nonlinear system. Triple solutions exist for the skin friction and temperature gradient identified by the critical numbers $a_c$ and ${\mu }_c.$ It is noted that for sufficiently small values of stretching strength parameter the dual branches for two of the triple solutions exist only in the regions $\mu \ge {\mu }_{c_3},$ and $\mu \le {\mu }_{c_4},$ where ${\mu }_{c_3}=-5.23$ and ${\mu }_{c_4}=-7.72.$ Numerical results are also provided, validating the model and offering insights into its accuracy and behavior of the model.