The base flid is assumed Newtonian and incompressible. The fluid flow is laminar in the laminar regime, set in motion by free convection. The fluid properties are assumed constant. The density variation is treated according to Boussinesq approximation, while the viscous dissipation effects are neglected. The viscous incompressible flow and the temperature distribution inside the cavity are described by the Navier-Stokes and the energy equations, respectively. The governing equations of the present problem are as follows:

Conservation of mass:

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$(1)

Conservation of momentum:

$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{{\rho }_{nf}}\frac{\partial p}{\partial x}+\frac{{\mu }_{nf}}{{\rho }_{nf}}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)$ (2)

$\begin{array}{c}u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{{\rho }_{nf}}\frac{\partial p}{\partial y}+\frac{{\mu }_{nf}}{{\rho }_{nf}}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)\\ +\frac{1}{{\rho }_{nf}}\left[g{\beta }_{nf}{\rho }_{nf}\left(T-T_c\right)\right]-\frac{1}{{\rho }_{nf}}\left[{\sigma }_{nf}{B}_0^{2}v\right]\end{array}$ (3)

Conservation of Energy:

$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha }_{nf}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)$(4)

Properties of the nanofluid are combinations of the properties of the base fluid, denoted by the subscript “f”, and those of the nanoparticle, denoted by the subscript “s”. These properties are defined as follows:

Effective density

${\rho }_{nf}=\left(1-\varphi \right){\rho }_f+\varphi {\rho }_s$

where φ is the solid volume fraction of nanoparticles.

Heat capacitance

${\left(\rho C_p\right)}_{nf}=\left(1-\varphi \right){\left(\rho C_p\right)}_f+\varphi {\left(\rho C_p\right)}_s$

Thermal expansion coefficient

${\beta }_{nf}=\left(1-\varphi \right){\beta }_f+\varphi {\beta }_s$

Thermal diffusivity

${\alpha }_{nf}=k_{nf}/{\left(\rho C_p\right)}_{nf}$

Electrical conductivity

${\sigma }_{nf}={\sigma }_f\left[1+\frac{3\phi \left(\frac{{\sigma }_s}{{\sigma }_f}-1\right)}{\left\{\left(\frac{{\sigma }_s}{{\sigma }_f}\right)+2\right\}-\left\{\left(\frac{{\sigma }_s}{{\sigma }_f}\right)-1\right\}\phi }\right]$ [23]

Thermal conductivity

$k_{nf}=k_f\left[\frac{\left(k_s+2k_f\right)-2\phi \left(k_f-k_s\right)}{\left(k_s+2k_f\right)+\phi \left(k_f-k_s\right)}\right]$ [24]

Viscosity

${\mu }_{nf}={\mu }_f\left(1+39.11\phi +533.9{\phi }^2\right)$ [25]

At the bottom wall:

$u\left(x,0\right)=0,v\left(x,0\right)=0,T=T_h,\forall y=0,0\le x\le L$

At the left wall:

$u\left(0,y\right)=0,v\left(0,y\right)=0,\partial T/\partial x=0,\forall x=0,0\le y\le L$

At the right wall:

$u\left(0,y\right)=0,v\left(0,y\right)=0,\partial T/\partial x=0,\forall x=L,0\le y\le L$

At the top wall:

$u\left(x,L\right)=0,v\left(x,L\right)=0,T=T_c,\forall y=L,0\le x\le L$

To obtain non-dimensional governing equations, we incorporate the following dimensionless dependent and independent variables:

$X=\frac{x}{L}$, $Y=\frac{y}{L}$, $U=\frac{uL}{{\alpha }_f}$, $V=\frac{vL}{{\alpha }_f}$, $P=\frac{p{L}^2}{{\rho }_{nf}{\alpha }_f^2}$ and $\theta =\frac{T-T_c}{T_h-T_c}$

where X and Y are the coordinates varying along horizontal and vertical directions, respectively, U and V are the velocity components in the X and Y directions, respectively, θ is the dimensionless temperature, P is the dimensionless pressure, and α is the thermal diffusivity of the fluid.

After applying the above dimensionless variable, Equations (1)-(4) are transformed into dimensionless form as the following:

$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$ (5)

$U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{{\nu }_{nf}}{{\alpha }_f}\left(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right)$(6)

$\begin{array}{c}U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial Y}+\frac{{\nu }_{nf}}{{\alpha }_f}\left(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right)+\left(\frac{{\beta }_{nf}}{{\beta }_f}\right)RaPr\theta \\ -\left(\frac{{\rho }_f}{{\rho }_{nf}}\right)\left(\frac{{\sigma }_{nf}}{{\sigma }_f}\right)H{a}^{2}PrV\end{array}$(7)

$U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{{\alpha }_{nf}}{{\alpha }_f}\left(\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right)$(8)

Here the dimensionless parameters are defined as follows:

Prandtl number, $Pr=\frac{\nu }{\alpha }$ (ratio of viscous to thermal diffusion rates, which indicates the ratio or dominance of heat transfer mode-convection over conduction)

Hartmann number, $Ha=B_{0}L\sqrt{\frac{\sigma }{\mu }}$ (ratio of electromagnetic force to the viscous force)

Grashof number, $Gr=\frac{g\beta L^3\left(T_h-T_c\right)}{{\nu }^2}$ (ratio of the buoyancy to viscous force)

Rayleigh number, $Ra=\frac{g\beta L^3\left(T_h-T_c\right)Pr}{{\nu }^2}$ (product of Gr and Pr. It also indicates the ratio or dominance of heat transfer mode-convection over conduction, but incorporates the buoyancy force).

The dimensionless boundary conditions become:

$U=V=0,\theta =1$ at bottom wall and heated conical body (at higher constant temperature)

$U=V=0,\theta =0$ at top wall (at lower constant temperature)

$U=V=0,\frac{\partial \theta }{\partial N}=0$ at side walls (thermally insulated)

$P=0$ Fluid pressure at the inside and on the walls of the cavity

where X and Y are dimensionless coordinates varying along horizontal and vertical directions, respectively; U and V are dimensionless velocity components in X and Y directions, respectively; $\theta$ is the dimensionless temperature.

The model is subdivided into many smaller discrete elements, and the set of equations are solved for each element. The mesh structure of finite elements for the present physical model displays in Figure 3 .

Grid independence test was performed with different number of elements and the average Nusselt number was assumed to be the test variable. The results are shown in Table 1 and Figure 4 .

It is seen that after about 8000 elements, further refinement does not have any appreciable impact on the average Nusselt number. Therefore, grid independence is attained around 8000. The present model used 8039 elements for the computations. Table 2 shows the thermo physical properties of the base fluid and the nano particle.

Figure 5 shows the effect of Ra on the streamlines with no nanoparticles (φ = 0) and no magnetic field (Ha = 0), as a baseline case. The streamlines represent the fluid flow pathways, while the colors represent the velocity. The velocity magnitude is indicated by the color legend shown at the right side of each figure.

At Ra = 10³, two vortices are seen on both side of the cone. The velocity varies from 0.05 to 0.5. At Ra = 10⁴, four additional major and two minor vortices appear, and the velocities increase by an order of magnitude. Further increase of Ra to 10⁵, total number of vortices remains the same but their shapes changed. Velocity also increased noticeably. However, the flow patterns remain symmetric around the cone for all cases.

Next, Ra is kept constant at 10⁴, and the values of φ and Ha are increased. The reason for keeping Ra fixed at a high value is that, the effects of Ha become more pronounced at higher Ra’s, providing better understanding of the phenomenon being studied. The results are shown in Figure 6 .

It is seen that with increasing φ, the additional vortices disappear, and the velocities reduce significantly. However, for a given φ, increasing Ha does not have much effect. With increasing φ, the effect of Ha becomes less and less noticeable. It implies that the nano-particles have a more dominating effect over the magnetic field to influence the fluid movement. The physical explanation to this behavior is that, adding nano particles makes the nano-fluid heavier, thus slowing down the motion. This dominated over magnetic field because the nano particles considered in this study has very low electrical conductivity and magnetic susceptibility.

Isotherms: Figure 7 show the effect of Ra on temperature distribution in the form of isotherms, with φ = 0, and Ha = 0. The labels next to the curves show the values of θ, the dimensionless temperature whose value range is from 0 to 1. The coloring also indicate temperature gradually changing from hotter to cooler (from red to green).

It is noted that the distortion of the isotherms becomes more and more prominent with increasing Ra. The isotherm corresponding to 0.95 moved closer to the bottom wall with increasing Ra, indicating that the larger portion of the cavity is cooler.

Next, Ra is kept constant at 10⁵, and the values of φ and Ha are increased. The reason for doing this is mentioned in the previous section. The results are shown in Figure 8 . It is noted that, the nano-particles have a greater influence on the temperature distribution as well, compared to that of the magnetic field. Similar to the influence on fluid movement as seen in the previous section ( Figure 6 ).

Heat flux: It is seen that in the previous figures that, adding more nanoparticle retarded flow. Increasing Hartmann number also retarded flow. Adding more nanoparticle also affected temperature distribution by making the isotherms smoother. Increasing Hartmann number also had similar effect but to a lesser degree. However, these observations are not sufficient to say conclusively whether heat transfer was enhanced or not. Therefore, Heat Flux was calculated for different values of Ra, Ha, and φ. The results are presented next in Figure 9 .

Figure 9(a) shows the case of base fluid only, with no nano particles added. It is seen that the effect of Ha is quite noticeable at higher values of Ra. It also shows that, increasing Ha decreased the heat flux. Figures 9(b)-(d) are the cases with increasing concentration of nano particles. It is noted that while heat flux increased with Ra, the difference in heat flux due to different values of Ha almost vanished. In other words, the nano-particles just overwhelmed the heat transfer process, making the Magnetic field almost irrelevant.

Figure 10(a) compares the effects of nano particles when no magnetic field is applied. It shows that each increment of φ has a more prominent effect on the heat flux. The greater the φ, the higher is the heat flux. Figure 10(b) shows the same, with the application of a strong magnetic field. The behavior of heat flux is almost same as in Figure 9(a) . However, there is hardly any effect due to the magnetic field. Therefore, both Figure 9 and Figure 10 lead to same conclusion, that φ has the dominating effect on convection. This phenomenon can be explained by the nature of the thermal conductivity of nanofluid used in this work. For the base fluid and the nano particle considered, the thermal conductivity of the solid nano particles is significantly higher than that of the base fluid. Therefore, the nanoparticles carry significant amount of heat along with the base fluid. Thus, the overall heat transfer of the system is increased by adding more nanoparticles. Moreover, the nanoparticle has very low electrical conductivity and magnetic susceptibility, thus it is indifferent to the magnetic field applied. This finding has practical implications. It is possible to design efficient heat exchangers just by adding nano particles without worrying about magnetic interference from any source. However, it is possible that there is a cut-off limit for adding φ, beyond which the fluid will become too “thick” or too “heavy” for easy circulation, thus retarding heat transfer. This point can be a potential ground for future research.