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The flow and heat transfer characteristics of a thermo-micropolar hybrid nanofluid (TMPHN) over a shrinking perpendicular surface are studied and analyzed in this paper. The TMPHN contains alumina and copper nanoparticles suspended in an aqua-base fluid. The adopted approach involves taking into account the masses of the base fluid and nanoparticles instead of relying on the first and second nanoparticles’ volume fractions. This method aligns with the single-phase approach known as the Tiwari–Das model. The complicated basic PDEs are converted to nondimensional forms utilizing the process of similarity transformation. Moreover, the MATLAB software’s bvp4c routine is used for problem analysis. The computational procedure is reliable since it complies well with earlier reports. The analysis shows that the present boundary value problem can have more than one unique solution. The solutions (first and second) have been obtained for important parameters such as the second nanoparticle mass, the shrinking parameter, the micropolar heat conduction parameter and the suction parameter. A stability analysis was done to explore and separate stable and unstable solutions. In-plane wall movement, classified as “shrinking”, leads to a reduction in boundary layer thickness, increasing fluid velocity and temperature near the wall. These changes collectively contribute to a higher wall shear stress and an enhanced wall heat flux. In specific conditions, for a thermo-micropolar hybrid nanofluid with a mass of 15g for each type of nanoparticle and of 100g for the aqua-base fluid, incorporating an additional 15g of the second nanoparticle into the base fluid results in a 3.3% improvement in heat transfer. These results can be attractive in both mathematical and physical aspects.

In this analysis, the stability of MHD stagnation point flow and heat transfer of an electrically conducting viscous incompressible fluid on a horizontally stretching/shrinking permeable cylinder with volumetric heat source are investigated. The cylindrical surface is subjected to a cross flow and a heat flux. This analysis finds applications in the areas of polymeric processing unit due to stretching/shrinking of surface and biofluid flows with therapeutic effects through the arteries, vein and digestive system that have a tube-like structure. Other aspects of the analysis are: magnetic field intensity whose strength can be remotely controlled and embodied generating/absorbing thermal power. The solution of the boundary value problems (BVP) is carried out with MATLAB’s inbuilt solver bvp4c. The most significant findings are recorded as: an increase in magnetic field strength increases the skin friction at the solid surface which is consistent with progressive thinning of boundary layer. This striking result is of interest in industrial applications because it is easy to regulate the magnetic field strength by electromagnetic devices such as controlling the voltage in the electric circuit. There is a point of neutrality of thermal energy distribution during the high fluctuation of temperature irrespective of the presence of heat generation or absorption. This can be attributed to the dominating effect of stretching/shrinking of the surface. The first solution provides stability of the flow with an increase in velocity in the presence of heat sink that is commensurate with the strength.

In this study, we investigated dual solutions for the influence of chemical reaction and radiation effect on axisymmetric flow of magneto-Cross nanomaterial towards a radially shrinking disk on taking account of stagnation point. The governing expressions which describe the assumed flow are reduced to ordinary differential equations by opting suitable similarity variables. The dual solutions on the performance of dimensionless velocity, thermal, concentration gradients, skin friction, rate of heat and mass transfer with the impact of relevant parameters are studied using suitable graphs. Result outcomes reveal that, upsurge in Brownian motion parameter improves the thermal gradient in case of both the solution but, converse trend is detected in concentration gradient. The uplift of thermophoresis parameter boosts up the concentration gradient in both branch solution but reverse trend is noticed in concentration profile for inclined values of Schmidt number. Further, dual nature of solutions exists only for certain range of shrinking parameter.

This research examines the flow of 2D micropolar fluid across an inclined linear shrinking/stretching surface with suction, convective, slip and thermal radiation impact. The governing partial differential equations (PDEs) are transformed into an ordinary differential equations (ODEs) system using a linear similarity transformation, and the bvp4c technique is then used to solve the ODEs system in MATLAB. Comparisons of both solutions are presented. The effects of regulating flow parameters on dimensionless temperature profiles, angular velocity, velocity, skin friction, wall couple stress and heat transfer are shown graphically. Stable velocity profiles are inversely related to velocity slip and material parameter. For a given suction and stretching/shrinking parameter value, graphs show double solutions. The couple stress and skin friction in the first solution, drop then increase as the material parameter K increases. Critical points of the stretching/shrinking parameter, ${\lambda }_{ci}\ge \lambda$ where both solutions occur, are indicated by the symbols ${\lambda }_{ci}$ (i.e., $i=1,2,3)$, and that there is no solution when ${\lambda }_{ci}>\lambda$. A table summarizes the stability study’s results and highlights.

The aim of this paper is to examine the combined effects of unsteadiness and thermal radiation on the flow of a mixture nanoliquid due to a shrinking disk. Investigation of flow near a stagnation-point naturally arises in several industrial sectors and is useful owing to its significance in thermal enhancement. The motion is persuaded by a radially stretched/shrinked exterior of the disk in adding to the flow near a stagnation point. As a result, this research examines the thermophysical belongings of the unsteady flow near a stagnation-point past a stretching/shrinking disk by using twin-kind nanoelements, viz. a mixture nanoliquid. Cu and Al₂O₃ nanoparticles have been considered in water to produce hybrid nanofluid. The numerous uses like electronic cooling system, heat exchangers and radiators are available. Using similarity transformations, self-similar equations have been obtained which are then solved numerically using MATHEMATICA software. Comparisons have been made with the available data in the literature and found a complete agreement for some special case of the problem. Double solutions subsist for equally stretched and shrinked disks, whereas a unique solution exists for the static disk. For flow past a stretching or a shrinking disk, a 27.79% of increase in warmth transport speed is noted for mixture nanofluid, contrasted to the nanoliquid. Hence, this study will be of interest to several scientists and engineers, owing its capability to augment the speed of heat transport in contrast to the regular nanofluids.

The aim of this study is to present forced convective nanofluid flow over a moving plate embedded in an absorbent medium. Following Darcy law’s for porous medium, the flow analysis is explored in attendance of warmth basis/drop. The main objective of this study is to explore the effects of Brownian motion and thermophoresis. The plate is considered to move in both directions: in the way of movement of fluid and in the opposite route of fluid movement. Similarity alterations have been applied to alter the leading partial differential equations (PDEs) to ordinary differential equations (ODEs). Numerical solutions have been obtained with the help of MATHEMATICA software. Dual solutions have been obtained when the plate and liquid go in reverse ways. Wall shear stress, Nusselt and Sherwood numbers all are found to rise with the rising permeability parameter of absorbent medium. For Nusselt and Sherwood numbers, ranges of dual solutions diminish by the mounting values of permeability parameter K. The critical values for porosity parameter $K=0.01$, 0.02, 0.03 are $R_{c_1}=1.872909$, $R_{c_2}=1.927211$, $R_{c_3}=1.9824284$, respectively. For decreasing values of s, range of dual solutions decreases. For $s=-0.45$, dual solutions exist in the range $(1.19,1.20)$.

Purpose: A key step in the propagation of microparticles across temperature gradients, thermophoresis deposition of particles is important for electrical and aerosol technologies. Thus, the impact of thermophoresis deposition of particles is encountered in a mixed convective flow of Williamson hybrid nanofluids (HNFs) across a stretching/shrinking sheet to monitor the fluctuation of mass deposition.

Design/methodology/approach: The transformation is used to transmute the PDEs into ODEs and then the bvp4c approach is applied to solve the modified governing equations of the problem. Graphs are used to illustrate the key variables that influence the heat, mass and flow profiles.

Findings: The results suggest that the mass transfer rate (MTR) decreases for both solutions due to the higher consequences of the thermophoretic parameter. In addition, the developed impact of the radiation parameter and the posited hybrid nanoparticles markedly raise the heat transfer rate (HTR) for both solutions. Hybrid nanoparticles are effective for generating maximum energy. They play a crucial role in providing a high rate of acceleration in the flow.

Practical implications: The stable outcomes of this examination could be very helpful in improving the energy efficiency of thermal systems.

Originality/value: The double-diffusive non-Newtonian Williamson fluid by incorporating nanofluids across a vertical stretching/shrinking sheet with thermal radiation has not been considered yet. The obtained numerical results have been validated with the published work to validate the numerical method.