Magnetohydrodynamic nanofluid radiative thermal behavior by means of Darcy law inside a porous media

Radiative nanomaterial thermal behavior within a permeable closed zone with elliptic hot source is simulated. Darcy law is selected for simulating permeable media in existence of magnetic forces. Contour plots for various buoyancy, Hartmann numbers and radiation parameter were illustrated. Carrier fluid is Al2O3-water with different shapes. Outputs prove that conduction mode augments with enhance of Ha. Nu augments with considering radiation source term.

Radiation is often encountered in frequent engineering problems. Keeping in view its applications Sheikholeslami et al. [19][20][21][22][23] presented the application of nanomaterial in various domains. Some recent publications about heat transfer can be found in [24][25][26][27][28][29][30][31][32] . To preserve the conduction of about fluid low, nano liquids have been recommended in past ages. Influence electric field on ferrofluid inside a tank with dual adaptable surfaces was demonstrated by Sheikholeslami et al. 33 . The investigation of nanofluid with magnetic forces with physical effects and applications can studied from [34][35][36] . Turbulator effect on swirling nanofluid flow was examined by Sheikholeslami et al. 37 . Utilizing such tools make the flow more complex. New model was introduced by Yadav et al. 38 for thermal instability. Furthermore, instability of thermal treatment of nanomaterial within a penetrable zone was exemplified by Yadav et al. 39 . They considered variation of nanomaterial viscosity in their simulation. Viscous heating effect on nanomaterial radiative behavior in existence of electric field was scrutinized by Daniel et al. 40 . In addition, they considered double stratification with magnetic field. Nanomaterial free convection with double-diffusive was scrutinized by Yadav et al. 41 involving rotation system. Permeable plate with considering radiative impact was modeled by Daniel et al. 42 . They imposed Lorentz forces and utilized HAM to solve the problem. Nanomaterial exergy loss with implementation of innovative approach was established by Sheikholeslami 43 . He is expert in this field and shows the approach applications in appearance of magnetic field. Entropy production during transient nanomaterial MHD flow was demonstrated by Daniel et al. 44 . They derived governing equations with considering electric field effect. Developments on numerical approach for simulating treatment of nanomaterial were presented in different publications [45][46][47][48][49][50][51] .
In current study, effects magnetic force and radiation on migration of nanofluid inside a porous medium was illustrated. CVFEM is considered as tool for showing roles of Rd, Ra, & Ha on performance.

Problem Explanation
The shape of enclosure and its boundary conditions have been demonstrated in Fig. 1   www.nature.com/scientificreports www.nature.com/scientificreports/ Characteristics of nanofluid have following formulas:    Table 2. Impact of "m" on Nu ave when Ra = 600, φ = 0.04 Rd = 0.8. www.nature.com/scientificreports www.nature.com/scientificreports/ nf f p To get the properties of carrier fluid, we utilized alike model used in 52 . To estimate temperature dependent properties, Rokni et al. 53,54 provide new formulation.
The following non dimensional variables by using of the stream function and, can be gained: www.nature.com/scientificreports www.nature.com/scientificreports/ Simulation technique, grid and verification. Combine of two influential approaches has been assembled in CVFEM. As explained in ref. 33 and shown in Fig. 1(b), such grid is applied in CVFEM. Final equations have attainment to values of θ, Ψ by using of Gauss-Seidel technique. Table 1

Outcome and Discussion
Radiative nanofluid heat transmission through a penetrable enclosure by means of Darcy law was displayed. Effects of Brownian motion and shape factor on nanomaterial behavior were examined. CVFEM was applied to display the variations of Rayleigh number (R a = 100 to 600), radiation (Rd = 0 → 0.08), Concentration of Alumina (φ = 0 to 0.04) and magnetic forces (Ha = 0 to 20). Deviations of Nu respect to m are represented in Table 2. Higher value of Nu is described for Platelet shape. Thus, it is designated for more simulations. Role of scattering Al 2 O 3 in H 2 O have exemplified in Fig. 3. It is observed that ψ max and Nu enhances by diffusing Al 2 O 3 . Since Lorentz force acting, the impact of φ on isotherms is not important. Impacts of substantial parameters on isotherms and streamlines are displayed in Figs 4, 5 and 6. ψ max rises with increase of buoyancy effect while it diminishes with escalation of Ha. Simulations for higher Ra leads to complex shape of isotherm with imposing greater buoyancy forces and thermal plume appears. Imposing Lorentz forces make suppress the plume and isotherms force to being parallel to each other's. For better description, below formula was derived and Fig. 7  Greater values of radiation parameter and Ra lead to thinner boundary layer which indicates greater Nu ave . Slender thickness of boundary layer was seen with reduce of Hartmann number which proves reduction effect of Hartmann number on Nu ave .

Conclusions
Imposing Lorenz forces influence on nanomaterial flow by means of Darcy law inside a porous enclosure is reported. Shape factor role was involved to predict nanomaterial properties as well as Brownian motion. CVFEM modeling was done to find the variations of Lorentz and buoyancy forces and radiation parameter on nanofluid thermal characteristic were demonstrated. The concluded points are given as www.nature.com/scientificreports www.nature.com/scientificreports/ • Outputs depict that Nu improves with improve of buoyancy force but it decrease with augment of Ha.
• Higher value of Nu is described for Platelet shape.
• Nu augments with considering radiation source term.
• As Ha enhances, the velocity of working fluid decreases.