The flow of fluids over the boundaries of a rotating disc has many practical uses, including boundary-layer control and separation. Therefore, the aim of this study is to discuss the impact of unsteady magnetohydrodynamics (MHD) hybrid ferrofluid flow over a stretching/shrinking rotating disk. The time-dependent mathematical model is transformed into a set of ordinary differential equations (ODE’s) by using similarity variables. The bvp4c method in the MATLAB platform is utilised in order to solve the present model. Since the occurrence of more than one solution is presentable, an analysis of solution stabilities is conducted. Both solutions were surprisingly found to be stable. Meanwhile, the skin friction coefficient, heat transfer rate—in cooperation with velocity—and temperature profile distributions are examined for the progressing parameters. The findings reveal that the unsteadiness parameter causes the boundary layer thickness of the velocity and temperature distribution profile to decrease. A higher value of magnetic and mass flux parameter lowers the skin friction coefficient. In contrast, the addition of the unsteadiness parameter yields a supportive effect on the heat transfer rate. An increment of the magnetic parameter up to 30% reduces the skin friction coefficient by 15.98% and enhances the heat transfer rate approximately up to 1.88%, significantly. In contrast, the heat transfer is rapidly enhanced by improving the mass flux parameter by almost 20%.

This numerical study was devoted to examining the occurrence of non-unique solutions in boundary layer flow due to deformable surfaces (cylinder and flat plate) with the imposition of prescribed surface heat flux. The hybrid Al${}_2$O${}_3$-Cu/water nanofluid was formulated using the single phase model with respective correlations of hybrid nanofluids. The governing model was simplified by adopting a similarity transformation. The transformed differential equations were then numerically computed using the efficient bvp4c solver with the ranges of the control parameters $0.5\%\le {\varphi }_1,{\varphi }_2\le 1.5\%$ (Al${}_2$O${}_3$ and Cu volumetric concentration), $0\le K\le 0.2$ (curvature parameter), $2.6<S\le 3.2$ (suction parameter) and $-2.5<\lambda \le 0.5$ (stretching/shrinking parameter). Dual steady solutions are presentable for both a cylinder $(K>0)$ and a flat plate $(K=0)$ with the inclusion of only the suction (transpiration) parameter. The real and stable solutions were mathematically validated through the stability analysis. The Al${}_2$O${}_3$-Cu/water nanofluid with ${\varphi }_1=0.5\%$ (alumina) and ${\varphi }_2=1.5\%$ (copper) has the highest skin friction coefficient and heat transfer rate, followed by the hybrid nanofluids with volumetric concentrations $({\varphi }_1=1\%,{\varphi }_2=1\%)$ and $({\varphi }_1=1.5\%,{\varphi }_2=0.5\%)$, respectively. Surprisingly, the flat plate surface abates the separation of boundary layer while it enhances the heat transfer process.

Nanofluids are a novel class of heat transfer fluid that plays a vital role in industries. In mathematical investigations, these fluids are modeled in terms of traditional integer-order partial differential equations (PDEs). It is recognized that traditional PDEs cannot decode the complex behavior of physical flow parameters and memory effects. Therefore, this article intends to study the mixed convection heat transfer in nanofluid over an inclined vertical plate via fractional derivatives approach. The problem in hand is modeled in connection with Atangana–Baleanu fractional derivatives without singular and local kernel with a strong memory. Human blood is considered as base fluid and carbon nanotube (CNTs) (single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs)) are dispersed into it to form blood-CNTs nanofluid. The nanofluid is considered to flow in a saturated porous medium under the influence of an applied magnetic field. The exact analytical expressions for velocity and temperature profiles are acquired using the Laplace transform technique and plotted in various graphs. The empirical results indicate that the memory effect decreases with increasing fractional parameters in the case of both temperature and velocity profiles. Moreover, the temperature profile is higher for blood SWCNTs because of higher thermal conductivity whereas this trend is found opposite in the case of velocity profile due to densities difference.

The Casson model is a fascinating model, which is genuinely recommended for use with fluids of a non-Newtonian type. The conventional model is not capable to represent the Casson model with the suspension of foreign bodies (dust particles). Due to this, the two-phase model for the mixture of Casson model fluid and dust particles is formulated. This study examines the emerging role of dust particles in changing the behavior of Casson model. In particular, two-phase flow of dusty Casson model with modified magnetic field and buoyancy effect under Newtonian heating boundary condition along a vertically stretching sheet is considered. The equations that govern under Casson model, together with dust particles, are reduced to a system of nonlinear ordinary differential equations by employing the suitable similarity variables. These transformed equations are then solved numerically by implementing the Runge–Kutta–Fehlberg (RKF45) method. The numerical results of skin friction coefficient plus Nusselt number are displayed graphically. The results revealed the fluid’s velocity tends to deteriorate due to the existence of dust particles, whilst its temperature is increased. The two-phase flow is one of the mathematical modeling techniques for multiphase flow, where the relationship between the fluid and solid is examined more closely. It is expected that the present findings can contribute to the understanding of the theory of two-phase flow mathematically, which will continue to produce significant research in this field.

Most of the fluid used in industrial application (i.e. Oils and gas industry, food manufacturing, lubrication and biomedical) do not conform to the Newtonian postulate. In contrast to the Newtonian fluid, the viscosity of the fluid can change when under force to either more liquid or more solid and dependent on shear rate history. This behaviour of fluids is commonly known as non-Newtonian fluid. The non-Newtonian fluid is so widespread in nature and technology resulting in very high interest of investigating among scientist. The Reiner-Philippoff fluid is one of the types of non-Newtonian fluid models that exhibiting the dilatant, pseudoplastic and Newtonian behaviors. Hence, this study is devoted to analyze the flow and heat transfer of Reiner-Philippoff fluid with the presence of first and second order velocity slip together with the temperature jump effects over a stretching sheet. Partial differential equations of continuity, momentum and energy equations were transformed into the similarity equations. The obtained equations were then solved via bvp4c function in MATLAB software. For the validation purpose, the present model and its numerical solution were compared with previous established solutions under limiting case where the present model is condensed to be identical with the reported model and turn to be in very strong agreement. The consequences of pertinent parameters on fluid’s characteristics are analyzed in details through the plotted graphic visuals and tabular form.

This significant study is designed to analyze the axisymmetric hybrid nanofluid flow with heat transfer on a convectively heated stretching/shrinking disk. The combination of metal (Cu) and metal oxide (Al2O3) nanoparticles with water (H2O) as the base fluid is used for the analysis. Similarity transformation is adopted to reduce the complexity of the PDEs into a system of ODEs. The utilization of suction in maintaining the steady flow solution for the shrinking disk case discloses the presence of dual solutions. Besides, an upsurge of Biot number and suction’s strength enhances the heat transfer operation. The application of Cu-Al2O3/water nanofluid can extend the range of solutions’ existence and consequently, decelerate the separation of laminar flow.