Two-phase flow of dusty Carreau nanofluid with dual-Cattaneo–Christov heat flux and non-uniform heat source over a disk

The main aim of the present work is to examine a mathematical model for the flow and heat transfer characteristics of a Carreau nanofluid with suspended dust particles over a rotating disk. It explores the influence of dual Cattaneo–Christov heat flux models, which combine relaxation and thermal retardation effects, on the temperature distribution in the fluid. By integrating theoretical models with real-world engineering problems, this study advances our understanding of expertise in optimizing systems such as dusty flows and nanofluids. The BVP4C built-in solver in MATrix LABoratory (MATLAB) software is used to solve the governing equations using the shooting technique. A graphic representation of the relationship between temperature, velocity, concentration, skin friction coefficients, heat, and mass transfer rates over the rotating disk is displayed for a range of values of the pertinent parameters, such as interaction parameter $$( {{\beta }_v = 1.0,\,2.0,\,3.0} )$$, Weissenberg number $$( {We = 0.1,\,1.0,\,2.0} )$$, mass concentration $$(l = 0.5,\,1.5,\,2.5)$$ Hartmann number $$(Ha = 0.0,\,1.0,\,2.0)$$, Thermal radiation $$( {Rd = 0.0,\,1.0,\,2.0} )$$, Thermophoresis $$( {Nt = 0.1,\,0.2,\,0.3} )$$, Brownian motion $$( {Nb = 0.3,\,0.5,\,0.7} )$$, fluid temperature $$( {{\beta }_t = 1.0,\,2.0,\,3.0} )$$, Space-dependent parameter $$( {{A}_1 = 0.0,\,2.0,\,4.0} )$$, the specific heat ratio $$( {\gamma = 0.5,\,1.0,\,1.5} )$$. The Graphs and tables illustrate the impacts of active parameters on the fluid's transport properties. An increase in Hartmann's number decelerates the radial and azimuthal velocities for fluid and dusty flow cases. For the radiation parameter (Rd = 0.5), the Nusselt number is 4.8. When the Radiation parameter increases to 1.5, the Nusselt number improves to 6.3, representing a 31.25% enhancement in heat transfer efficiency. Also, it was observed that the temperature profile increases as the Nusselt number rises, indicating better thermal transport due to thermophoretic forces.