Dual solutions on three-dimensional nanofluid flow and heat transfer over a permeable non-linearly shrinking surface with second-order velocity slips

PurposeThis paper aims to present the steady dual solutions on three-dimensional flow and heat transfer of nanofluid over a permeable non-linearly shrinking surface with two-order velocity slips conditions. Boundary layer assumption is considered in the mathematical modelling. Authors comprehend from previous studies and papers that the shrinking surfaces are extremely important in current engineering and environmental systems. Design/methodology/approachUsing appropriate similarity variables, the full partial differential equations (PDF) are modified into a specific set of ordinary (similar) differential equations (ODE). The resulting non-linear ordinary differential system is then solved both analytically for some particular cases and numerically for the general case using the function bvp4c from MATLAB for characteristic values of the parameters which govern the equations. The transformed mathematical model is analysed using the bvp4c procedure. Based on the given assumptions, this study is able to produce multiple solutions of the problem. FindingsThe ordinary (similarity) differential equations have two branches solutions, upper and lower branch solutions, given some interval of shrinking and velocity slip parameters. The authors consider here a temporal stability analysis, as they want to establish which of the solutions are stable and which are not. In a distinct paragraph, the authors discuss in detail and present in a graphical manner the effects of shrinking and second-order slip flow model on the skin friction coefficient, surface wall heat flux and dimensionless velocity and temperature profiles. The analysis reveals that the second order slip has a big influence on the flow and heat transfer characteristics. Originality/valueThe present discoveries are unique and truly new for the research of the three-dimensional stretching/shrinking forced convection flow and heat transfer nanofluids. The nanofluid is a water-based nanofluid (H2O), which contains one type of nanoparticles, namely, copper (Cu). Of course, the analysis can be further extended considering other types of nanoparticles such as alumina (Al2O3). The authors assume that the thermal equilibrium is reached for the base fluid together with the suspended nanoparticles and that the nanoparticles are uniform in dimension and form.