Multiple exact solutions in tri-hybrid nanofluid flow: a study of elastic surface effects

Purpose The purpose of this study is to investigate the simultaneous effects of normal wall transpiration, stretching strength parameter, velocity slip and nanoparticles on the flow of a ternary hybrid nanofluid through an elastic surface. The goal is to understand the behavior of the flow field, temperature distribution, skin friction and temperature gradient under these conditions, and to explore the existence and nature of solutions under varying parameter values. Design/methodology/approach The analysis involves expressing the flow field, power-law temperature field, skin friction and temperature gradient in closed-form formulas. The study examines both stretching and shrinking surfaces, distinguishing between unique and dual solutions. The methodology includes deriving exact solutions for exponential and algebraic temperature and temperature rate formulas analytically by deriving the system of governing equations into ordinary differential equations. Findings The study reveals that for a stretching sheet, the solution is unique, whereas dual solutions are observed for a shrinking surface. Special solutions are provided for various parametric values, showing the behavior of the exponential and algebraic temperature and temperature rate, with a focus on identifying turning points that demarcate the existence and non-existence of single or multiple solutions. The solutions are represented through graphs and tables to facilitate a comprehensive qualitative analysis. The research identifies turning points that determine the presence or absence of single or multiple solutions, uncovering multiple solutions for different parameter sets. These findings are displayed graphically and in tabular form, highlighting the complex interplay between the parameters and the resulting flow behavior. Originality/value This analysis contributes to the field by providing new insights into the multiple solution phenomena in ternary hybrid nanofluid flows, particularly under the combined effects of normal wall transpiration, stretching strength, velocity slip and nanoparticle presence. The identification of turning points and the exact solutions for various temperature profiles are of significant value, offering a deeper understanding of the factors influencing the flow and thermal characteristics in such systems. The study’s findings have potential applications in optimizing fluid flow in engineering systems where such conditions are prevalent.