The considered physical geometry with related parameters and coordinates are shown in Fig. 6.1A. A rectangular body with height t and width H/2 is placed in the center of the enclosure, and is supposed to be isothermal at higher temperature than two vertical isothermal walls, while the top and bottom walls are insulated. In addition, it is also assumed that the uniform magnetic field ($\overrightarrow{B}=B_x\overrightarrow{e_x}+B_y\overrightarrow{e_y}$) of constant magnitude $B=\sqrt{B_x^2+B_y^2}$ is applied, where $\overrightarrow{e_x}$ and $\overrightarrow{e_y}$ are unit vectors in the Cartesian coordinate system. The orientation of the magnetic field forms an angle θ_M with horizontal axis such that θ_M = B_x/B_y. The electric current J and the electromagnetic force F are defined by $J=\sigma \left(\overrightarrow{V}\times \overrightarrow{B}\right)$ and $F=\sigma \left(\overrightarrow{V}\times \overrightarrow{B}\right)\times \overrightarrow{B}$, respectively. In this section, θ_M is equal to zero [1].

One of the novel computational fluid dynamics (CFD) methods, which is solved Boltzmann equation to simulate the flow instead of solving the Navier–Stokes equations, is called lattice Boltzmann methods (LBM) [or thermal lattice Boltzmann methods (TLBM)]. LBM has several advantages such as simple calculation procedure and efficient implementation for parallel computation, over other conventional CFD methods, because of its particulate nature and local dynamics. The thermal LB model utilizes two distribution functions, f and g, for the flow and temperature fields, respectively. It uses modeling of movement of fluid particles to capture macroscopic fluid quantities such as velocity, pressure, and temperature. In this approach, the fluid domain discretized to uniform Cartesian cells. Each cell holds a fixed number of distribution functions, which represent the number of fluid particles moving in these discrete directions. The D₂Q₉ model was used, and values of w₀ = 4/9 for |c₀| = 0 (for the static particle), w_1–4 = 1/9 for |c_1–4| = 1, and w_5–9 = 1/36 for $\left|c_{5-9}\right|=\sqrt{2}$ are assigned in this model (Fig. 6.1B). The density and distribution functions, that is, the f and g, are calculated by solving the lattice Boltzmann equation, which is a special discretization of the kinetic Boltzmann equation. After introducing BGK approximation, the general form of lattice Boltzmann equation with external force is as follows.

$f_i(x+c_i∆t,t+∆t)=f_i(x,t)+\frac{∆t}{{\tau }_v}[f_i^{eq}(x,t)-f_i(x,t)]+∆t{c}_i{F}_k$

(6.1)

$g_i(x+c_i∆t,t+∆t)=g_i(x,t)+\frac{∆t}{{\tau }_C}[g_i^{eq}(x,t)-g_i(x,t)]$

(6.2)

where ∆t denotes lattice time step, c_i is the discrete lattice velocity in direction i, F_k is the external force in direction of lattice velocity, and τ_v and τ_C denote the lattice relaxation time for the flow and temperature fields. The kinetic viscosity υ and the thermal diffusivity α are defined in terms of their respective relaxation times, that is, $\upsilon =c_s^2({\tau }_v-1/2)$ and $\alpha =c_s^2({\tau }_C-1/2)$, respectively. Note that the limitation 0.5 < τ should be satisfied for both relaxation times to ensure that viscosity and thermal diffusivity are positive. Furthermore, the local equilibrium distribution function determines the type of problem that needs to be solved. It also models the equilibrium distribution functions, which are calculated with Eqs. (6.3) and (6.4) for flow and temperature fields, respectively.

$f_i^{eq}=w_i\rho \left[1+\frac{c_i.u}{c_s^2}+\frac{1}{2}\frac{{(c_i.u)}^2}{c_s^4}-\frac{1}{2}\frac{u^2}{c_s^2}\right]$

(6.3)

$g_i^{eq}=w_{i}T\left[1+\frac{c_i.u}{c_s^2}\right]$

(6.4)

$\begin{array}{l}F=F_x+F_y\\ F_x=3w_i\rho \left[A\left(v\sin \left({\Theta }_{\text{M}}\right)\cos \left({\Theta }_{\text{M}}\right)\right)-\left(u{\sin }^2\left({\Theta }_{\text{M}}\right)\right)\right],\\ F_y=3w_i\rho \left[g_y\beta \left(T-T_m\right)+A\left(u\sin \left({\Theta }_{\text{M}}\right)\cos \left({\Theta }_{\text{M}}\right)\right)-\left(v{\cos }^2\left({\Theta }_{\text{M}}\right)\right)\right]\end{array}$

(6.5)

where A is $A=\frac{H{a}^2\mu }{L^2}$, $Ha=L{B}_0\sqrt{\frac{\sigma }{\mu }}$ is Hartmann number, and θ_M is the direction of the magnetic field. For natural convection, the Boussinesq approximation is applied and radiation heat transfer is negligible. To ensure that the code works in near incompressible regime, the characteristic velocity of the flow for natural $(V_{\text{natural}}\equiv \sqrt{\beta g_y∆TH})$ regime must be small compared with the fluid speed of sound. In this chapter, the characteristic velocity selected as 0.1 of sound speed.

$\begin{array}{l}\text{Flow density}:\rho =\sum _i{f}_i\\ \text{Momentum}:\rho \text{u}=\sum _i{\text{c}}_i{f}_i\\ \text{Temperature}:T=\sum _i{g}_i\end{array}$

(6.6)

$S_{\text{gen}}^{•}=HTI+FFI$

(6.7)

$\begin{array}{l}HTI=\frac{k\left(\nabla T.\nabla T\right)}{T^2}\\ FFI=\frac{\mu \phi }{T}\end{array}$

(6.8)

$Be=\frac{HTI}{HTI+FFI}$

(6.9)

$Ns=\frac{{\left(\frac{H}{\Omega }\right)}^2{S}_{\text{gen}}^{•}}{k}$

(6.10)

$Ns=\frac{{\left(\frac{\partial \Theta }{\partial x}\right)}^2+{\left(\frac{\partial \Theta }{\partial y}\right)}^2}{{\Omega }^2{\left({\Omega }^{-1}+\Theta \right)}^2}+\frac{Ge\varphi }{Ra{\Omega }^2\left({\Omega }^{-1}+\Theta \right)}$

(6.11)

$\Omega =\frac{T_{\text{h}}-T_{\text{c}}}{T_{\text{c}}}$

(6.12)

$\varphi =2{\left(\frac{\partial u}{\partial x}\right)}^2+2{\left(\frac{\partial v}{\partial x}\right)}^2+{\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial x}\right)}^2$

(6.13)

$Ge=\frac{g\beta H}{C_{\text{p}}}$

(6.14)

$<Ns>=\underset{A}{\int }\frac{Ns}{A}dA$

(6.15)

$<S_{\text{gen}}^{•}>=\frac{q^{″}}{H}\left(\frac{1}{T_{\text{c}}}-\frac{1}{T_{\text{H}}}\right)$

(6.16)

$<S_{\text{gen}}^{•}>=4Nu\frac{k}{H^2}\frac{{\Omega }^2}{1+\Omega }$

(6.17)

$<S_{\text{gen}}^{•}>\approx \frac{4Nu{\Omega }^2\left(1-\Omega \right)k}{H^2}$

(6.18)

$<Ns>=\frac{4Nu}{1+\Omega }$

(6.19)

${\rho }_{\text{n}\text{f}}={\rho }_{\text{f}}(1-\phi )+{\rho }_{\text{s}}\phi \text{,}$

(6.20)

${\left(\rho C_{\text{p}}\right)}_{\text{n}\text{f}}={\left(\rho C_{\text{p}}\right)}_{\text{f}}(1-\phi )+{\left(\rho C_{\text{p}}\right)}_{\text{s}}\phi$

(6.21)

${\left(\rho \beta \right)}_{\text{n}\text{f}}={\left(\rho \beta \right)}_{\text{f}}(1-\phi )+{\left(\rho \beta \right)}_{\text{s}}\phi$

(6.22)

$\frac{{\sigma }_{\text{nf}}}{{\sigma }_{\text{f}}}=1+\frac{3\left(\frac{{\sigma }_{\text{s}}}{{\sigma }_{\text{f}}}-1\right)\phi }{\left(\frac{{\sigma }_{\text{s}}}{{\sigma }_{\text{f}}}+2\right)-\left(\frac{{\sigma }_{\text{s}}}{{\sigma }_{\text{f}}}-1\right)\phi }$

(6.23)

$\begin{array}{l}{k}_{\text{nf}}=1+\frac{3\left(\frac{k_{\text{p}}}{k_{\text{f}}}-1\right)\phi }{\left(\frac{k_{\text{p}}}{k_{\text{f}}}+2\right)-\left(\frac{k_{\text{p}}}{k_{\text{f}}}-1\right)\phi }+5\times {10}^4{g}^{\prime }(\phi ,T,d_{\text{p}})\phi {\rho }_{\text{f}}{c}_{\text{p,f}}\sqrt{\frac{{\kappa }_{\text{b}}T}{{\rho }_{\text{p}}{d}_{\text{p}}}}\\ g^{\prime }\left(\phi ,T,d_{\text{p}}\right)=\left(a_6+a_{7}Ln\left(d_{\text{p}}\right)+a_{8}Ln\left(\phi \right)+a_{9}Ln\left(\phi \right)\ln \left(d_{\text{p}}\right)+a_{10}Ln{\left(d_{\text{p}}\right)}^2\right)\\ +Ln\left(T\right)\left(a_1+a_{2}Ln\left(d_{\text{p}}\right)+a_{3}Ln\left(\phi \right)+a_{4}Ln\left(\phi \right)\ln \left(d_{\text{p}}\right)+a_{5}Ln{\left(d_{\text{p}}\right)}^2\right)\\ R_{\text{f}}+d_{\text{p}}/k_{\text{p}}=d_{\text{p}}/k_{\text{p,eff}},R_{\text{f}}=4\times {10}^{-8}\text{k}{\text{m}}^2/\text{W}\end{array}$

(6.24)

${\mu }_{\text{nf}}=\frac{{\mu }_{\text{f}}}{{\left(1-\phi \right)}^{2.5}}+\frac{k_{\text{Brownian}}}{k_{\text{f}}}\times \frac{{\mu }_{\text{f}}}{\mathrm{Pr}}$

(6.25)

$N{u}_{\text{ave}}=\frac{k_{\text{n}\text{f}}}{k_{\text{f}}}\underset{0}{\overset{1}{\int }}\frac{1}{4}\left({\left.\frac{\partial T}{\partial x}\right|}_{X=0}+{\left.\frac{\partial T}{\partial x}\right|}_{X=1}\right)dy$

(6.26)

6.1.2. Effects of active parameters

Fig. 6.5 shows the effects of the nanoparticle volume fraction, Rayleigh number, and Hartmann number for Cu–water nanofluids on average Nusselt number ratio $\left(N{u}^{*}=\left({\left.N{u}_{\text{ave}}\right|}_{\phi =0\text{.04}}\right)/\left({\left.N{u}_{\text{ave}}\right|}_{\phi =0}\right)\right)$. It can be found that the effect of nanoparticles is more pronounced at low Rayleigh numbers than at high Rayleigh numbers because of greater values of average Nusselt number ratios. This observation can be explained by noting that at low Rayleigh numbers the heat transfer is dominant by conduction. Therefore, the addition of high thermal conductivity nanoparticles will increase the conduction and therefore make the enhancement more effective. When Ha < 75 minimum values of average Nusselt number ratio are obtained for Ra = 10⁵ but when Ha > 75 minimum values of this ratio occurs for Ra = 10⁴.

Effects of Hartmann number, the nanoparticle volume fraction, Rayleigh number, and dimensionless temperature difference for CuO–water on dimensionless entropy generation number are shown in Fig. 6.6. The dimensionless entropy generation number increases with the increase of nanoparticle volume fraction and Rayleigh number. As shown in Fig. 6.6, <Ns> decreases with the increase of Ω. This fact is in line with the predictions of our Eq. (6.11). This also makes physical sense because, as Ω = T_h/T_c − 1, higher values of Ω imply a greater temperature difference (leading to higher heat transfer rates) and consequently boosted HTI values. So decrease in <Ns>, according to $S_{\text{gen}}^{•}=Ns.k{\left(\Omega /L\right)}^2$, leads to increase in the total entropy generation. Also, these figures indicate that increasing Hartmann number leads to decrease in dimensionless entropy generation number.

$Y=H-\left\{a\left(H+\sin \left(\pi x-\pi /2\right)\right)\right\}$

(6.27)

where a is the dimensionless amplitude of the sinusoidal wall. It is also assumed that the uniform magnetic field ($\overrightarrow{B}=B_x\overrightarrow{e_x}+B_y\overrightarrow{e_y}$) of constant magnitude $B=\sqrt{B_x^2+B_y^2}$ is applied, where $\overrightarrow{e_x}$ and $\overrightarrow{e_y}$ are unit vectors in the Cartesian coordinate system. The orientation of the magnetic field forms an angle γ with horizontal axis such that γ = B_x/B_y. The electric current J and the electromagnetic force F are defined by $J=\sigma \left(\overrightarrow{V}\times \overrightarrow{B}\right)$ and $F=\sigma \left(\overrightarrow{V}\times \overrightarrow{B}\right)\times \overrightarrow{B}$, respectively.

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

(6.28)

$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{{\rho }_{\text{nf}}}\frac{\partial P}{\partial x}+{\upsilon }_{\text{nf}}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)+\frac{{\sigma }_{\text{nf}}{B}^2}{{\rho }_{\text{nf}}}\left(v\sin \lambda \cos \lambda -u{\sin }^2\lambda \right)$

(6.29)

$\begin{array}{l}u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{{\rho }_{\text{nf}}}\frac{\partial P}{\partial y}+{\upsilon }_{\text{nf}}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)+{\beta }_{\text{nf}}g\left(T-T_{\text{c}}\right)\\ +\frac{{\sigma }_{\text{nf}}{B}^2}{{\rho }_{\text{nf}}}\left(u\sin \lambda \cos \lambda -v{\cos }^2\lambda \right)\end{array}$

(6.30)

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

(6.31)

${\rho }_{\text{nf}}={\rho }_{\text{f}}(1-\phi )+{\rho }_{\text{s}}\phi$

(6.32)

${\left(\rho C_{\text{p}}\right)}_{\text{nf}}={\left(\rho C_{\text{p}}\right)}_{\text{f}}(1-\phi )+{\left(\rho C_{\text{p}}\right)}_{\text{s}}\phi$

(6.33)

${\alpha }_{\text{nf}}=\frac{k_{\text{nf}}}{{\left(\rho C_{\text{p}}\right)}_{\text{nf}}}$

(6.34)

${\beta }_{\text{nf}}={\beta }_{\text{f}}(1-\phi )+{\beta }_{\text{s}}\phi$

(6.35)

${\mu }_{\text{nf}}=\frac{{\mu }_{\text{f}}}{{(1-\phi )}^{2.5}}$

(6.36)

$\frac{k_{\text{n}\text{f}}}{k_{\text{f}}}=\frac{k_{\text{s}}+2k_{\text{f}}-2\phi (k_{\text{f}}-k_{\text{s}})}{k_{\text{s}}+2k_{\text{f}}+\phi (k_{\text{f}}-k_{\text{s}})}$

(6.37)

$\frac{{\sigma }_{\text{nf}}}{{\sigma }_{\text{f}}}=1+\frac{3\left({\sigma }_{\text{s}}/{\sigma }_{\text{f}}-1\right)\phi }{\left({\sigma }_{\text{s}}/{\sigma }_{\text{f}}+2\right)-\left({\sigma }_{\text{s}}/{\sigma }_{\text{f}}-1\right)\phi }$

(6.38)

$u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x},\omega =\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$

(6.39)

$\begin{array}{l}\frac{\partial \psi }{\partial y}\frac{\partial \omega }{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial \omega }{\partial y}={\upsilon }_{\text{nf}}\left(\frac{{\partial }^2\omega }{\partial x^2}+\frac{{\partial }^2\omega }{\partial y^2}\right)+{\beta }_{\text{nf}}g\left(\frac{\partial T}{\partial x}\right)\\ +\frac{{\sigma }_{\text{nf}}{B}^2}{{\rho }_{\text{nf}}}\left(-\frac{\delta v}{\delta y}\sin \lambda \cos \lambda +\frac{\delta u}{\delta y}{\sin }^2\lambda +\frac{\delta u}{\delta x}\sin \lambda \cos \lambda -\frac{\delta v}{\delta x}{\cos }^2\lambda \right)\end{array}$

(6.40)

$\frac{\partial \psi }{\partial y}\frac{\partial T}{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial T}{\partial y}={\alpha }_{\text{nf}}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)$

(6.41)

$\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}=-\omega$

(6.42)

$X=\frac{x}{L},Y=\frac{y}{L},\Omega =\frac{\omega L^2}{{\alpha }_{\text{f}}},\Psi =\frac{\psi }{{\alpha }_{\text{f}}},\Theta =\frac{T-T_{\text{c}}}{T_{\text{h}}-T_{\text{c}}},U=\frac{uL}{{\alpha }_{\text{f}}},V=\frac{vL}{{\alpha }_{\text{f}}}$

(6.43)

where in Eq. (6.43) $L=r_{\text{out}}-r_{\text{in}}=r_{\text{in}}$. Using the dimensionless parameters, the equations now become:

$\begin{array}{l}\frac{\partial \Psi }{\partial Y}\frac{\partial \Omega }{\partial X}-\frac{\partial \Psi }{\partial X}\frac{\partial \Omega }{\partial Y}=\left[\frac{P{r}_{\text{f}}}{{\left(1-\phi \right)}^{2.5}\left(\left(1-\phi \right)+\phi \frac{{\rho }_{\text{s}}}{{\rho }_{\text{f}}}\right)}\right]\left(\frac{{\partial }^2\Omega }{\partial X^2}+\frac{{\partial }^2\Omega }{\partial Y^2}\right)\\ +R{a}_{\text{f}}P{r}_{\text{f}}\left[\left(1-\phi \right)+\phi \frac{{\beta }_{\text{s}}}{{\beta }_{\text{f}}}\right]\left(\frac{\partial \Theta }{\partial X}\right)\\ +H{a}^{2}P{r}_{\text{f}}\left(1+\frac{3\left(\frac{{\sigma }_{\text{s}}}{{\sigma }_{\text{f}}}-1\right)\phi }{\left(\frac{{\sigma }_{\text{s}}}{{\sigma }_{\text{f}}}+2\right)-\left(\frac{{\sigma }_{\text{s}}}{{\sigma }_{\text{f}}}-1\right)\phi }\right)\left(\frac{1}{\left(1-\phi \right)+\phi \frac{{\rho }_{\text{s}}}{{\rho }_{\text{f}}}}\right)\left(-\frac{\delta V}{\delta Y}\tan \lambda +\right.\\ \left.\frac{\delta U}{\delta Y}{\tan }^2\lambda +\frac{\delta U}{\delta X}\tan \lambda -\frac{\delta V}{\delta X}\right)\end{array}$

(6.44)

$\frac{\partial \Psi }{\partial Y}\frac{\partial \Theta }{\partial X}-\frac{\partial \Psi }{\partial X}\frac{\partial \Theta }{\partial Y}=\left[\frac{\frac{k_{\text{nf}}}{k_{\text{f}}}}{\left(\left(1-\phi \right)+\phi \frac{{\left(\rho Cp\right)}_{\text{s}}}{{\left(\rho Cp\right)}_{\text{f}}}\right)}\right]\left(\frac{{\partial }^2\Theta }{\partial X^2}+\frac{{\partial }^2\Theta }{\partial Y^2}\right)$

(6.45)

$\frac{{\partial }^2\Psi }{\partial X^2}+\frac{{\partial }^2\Psi }{\partial Y^2}=-\Omega$

(6.46)

where Ra_f = gβ_fL³(T_h − T_c)/(α_fυ_f) is the Rayleigh number for the base fluid, $Ha=L{B}_x\sqrt{{\sigma }_{\text{f}}/{\mu }_{\text{f}}}$ is the Hartmann number, and Pr_f = υ_f/α_f is the Prandtl number for the base fluid. The boundary conditions as shown in Fig. 6.7 are as follows:

$\begin{array}{l}\Theta =1.0\text{on}\text{the}\text{hot}\text{wall}\\ \Theta =0.0\text{on}\text{the}\text{cold}\text{wall}\\ \partial \Theta /\partial n=0.0\text{on}\text{the}\text{two}\text{other}\text{insulation}\text{boundaries}\\ \Psi =0.0\text{on}\text{all}\text{solid}\text{boundaries}\end{array}$

(6.47)

$N{u}_{\text{loc}}=\frac{k_{\text{nf}}}{k_{\text{f}}}{\left.\frac{\partial \Theta }{\partial n}\right|}_{\text{hot}\text{wall}}$

(6.48)

$N{u}_{\text{ave}}=\frac{1}{2}\underset{0}{\overset{2}{\int }}N{u}_{\text{loc}}dS$

(6.49)

6.2.2. Effects of active parameters

$E=\frac{Nu\left(\phi =0\text{.06}\right)-Nu\left(\text{base fluid}\right)}{Nu\left(\text{base fluid}\right)}\times 100$

(6.50)

$\varepsilon =\sqrt{a^2-b^2}/a\text{or}b=\sqrt{1-{\varepsilon }^2}.a$

(6.51)

Also, it is also assumed that the uniform magnetic field ($\overrightarrow{B}=B_x\overrightarrow{e_x}+B_y\overrightarrow{e_y}$) of constant magnitude $B=\sqrt{B_x^2+B_y^2}$ is applied, where $\overrightarrow{e_x}$ and $\overrightarrow{e_x}$ are unit vectors in the Cartesian coordinate system. The orientation of the magnetic field forms an angle λ with horizontal axis such that λ = B_x/B_y. The electric current J and the electromagnetic force F are defined by $J=\sigma \left(\overrightarrow{V}\times \overrightarrow{B}\right)$ and $F=\sigma \left(\overrightarrow{V}\times \overrightarrow{B}\right)\times \overrightarrow{B}$, respectively. In this chapter, λ is equal to zero.

$N{u}_{\text{loc}}=\left(\frac{k_{\text{nf}}}{k_{\text{f}}}\right)\frac{\partial \Theta }{\partial r}$

(6.52)

$N{u}_{\text{ave}}=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}N{u}_{\text{loc}}\left(\zeta \right)d\zeta$

(6.53)

6.3.2. Effects of active parameters

$\begin{array}{ll}\hfill & Nu=a_{13}+a_{23}{Y}_1+a_{33}{Y}_2+a_{43}{Y_1}^2+a_{53}{Y_2}^2+a_{63}{Y}_1{Y}_2\hfill \\ \hfill & Y_1=a_{11}+a_{21}R{a}^{*}+a_{31}H{a}^{*}+a_{41}R{a}^{{*}^2}+a_{51}H{a}^{{*}^2}+a_{61}H{a}^{*}R{a}^{*}\hfill \\ \hfill & Y_2=a_{12}+a_{22}\gamma +a_{32}\phi +a_{42}{\gamma }^2+a_{52}{\phi }^2+a_{62}\gamma \phi \hfill \end{array}$

(6.54)

${\left(\frac{X}{a}\right)}^{2\widehat{n}}+{\left(\frac{Y}{b}\right)}^{2\widehat{n}}=1$

(6.55)

When a = b and $\widehat{n}=1$, the geometry becomes a circle. As $\widehat{n}$ increases from 1, the geometry would approach a rectangle for a # b and square for a = b. It is also assumed that the uniform magnetic field ($\overrightarrow{B}=B_x\overrightarrow{e_x}+B_y\overrightarrow{e_y}$) of constant magnitude $B=\sqrt{B_x^2+B_y^2}$ is applied, where $\overrightarrow{e_x}$ and $\overrightarrow{e_y}$ are unit vectors in the Cartesian coordinate system. The orientation of the magnetic field forms an angle λ with horizontal axis such that λ = B_x/B_y. In this chapter, λ equals to zero. The electric current J and the electromagnetic force F are defined by $J=\sigma \left(\overrightarrow{V}\times \overrightarrow{B}\right)$ and $F=\sigma \left(\overrightarrow{V}\times \overrightarrow{B}\right)\times \overrightarrow{B}$, respectively.

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

(6.56)

$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{{\rho }_{\text{nf}}}\frac{\partial P}{\partial x}+{\upsilon }_{\text{nf}}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)+\frac{{\sigma }_{\text{nf}}{B}^2}{{\rho }_{\text{nf}}}\left(v\sin \lambda \cos \lambda -u{\sin }^2\lambda \right)$

(6.57)

$\begin{array}{l}u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{{\rho }_{\text{nf}}}\frac{\partial P}{\partial y}+{\upsilon }_{\text{nf}}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)+{\beta }_{\text{nf}}g\left(T-T_c\right)\\ +\frac{{\sigma }_{\text{nf}}{B}^2}{{\rho }_{\text{nf}}}\left(u\sin \lambda \cos \lambda -v{\cos }^2\lambda \right)\end{array}$

(6.58)

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

(6.59)

$\begin{array}{l}{\rho }_{\text{nf}}={\rho }_{\text{f}}(1-\phi )+{\rho }_{\text{s}}\phi \\ {\beta }_{\text{nf}}={\beta }_{\text{f}}(1-\phi )+{\beta }_{\text{s}}\phi \\ {\left(\rho C_{\text{p}}\right)}_{\text{nf}}={\left(\rho C_{\text{p}}\right)}_{\text{f}}(1-\phi )+{\left(\rho C_{\text{p}}\right)}_{\text{s}}\phi \end{array}$

(6.60)

$\frac{{\sigma }_{\text{nf}}}{{\sigma }_{\text{f}}}=1+\frac{3\left({\sigma }_{\text{s}}/{\sigma }_{\text{f}}-1\right)\phi }{\left({\sigma }_{\text{s}}/{\sigma }_{\text{f}}+2\right)-\left({\sigma }_{\text{s}}/{\sigma }_{\text{f}}-1\right)\phi }$

(6.61)

$\begin{array}{l}{k}_{\text{nf}}=1+\frac{3\left(\frac{k_{\text{p}}}{k_{\text{f}}}-1\right)\phi }{\left(\frac{k_{\text{p}}}{k_{\text{f}}}+2\right)-\left(\frac{k_{\text{p}}}{k_{\text{f}}}-1\right)\phi }+5\times {10}^4{g}^{\prime }(\phi ,T,d_{\text{p}})\phi {\rho }_{\text{f}}{c}_{\text{p,f}}\sqrt{\frac{{\kappa }_{\text{b}}T}{{\rho }_{\text{p}}{d}_{\text{p}}}}\\ g^{\prime }\left(\phi ,T,d_{\text{p}}\right)=\left(a_6+a_{7}Ln\left(d_{\text{p}}\right)+a_{8}Ln\left(\phi \right)+a_{9}Ln\left(\phi \right)\ln \left(d_{\text{p}}\right)+a_{10}Ln{\left(d_{\text{p}}\right)}^2\right)\\ +Ln\left(T\right)\left(a_1+a_{2}Ln\left(d_{\text{p}}\right)+a_{3}Ln\left(\phi \right)+a_{4}Ln\left(\phi \right)\ln \left(d_{\text{p}}\right)+a_{5}Ln{\left(d_{\text{p}}\right)}^2\right)\\ R_{\text{f}}+d_{\text{p}}/k_{\text{p}}=d_{\text{p}}/k_{\text{p,eff}},R_{\text{f}}=4\times {10}^{-8}\text{k}{\text{m}}^2/\text{W}\end{array}$

(6.62)

${\mu }_{\text{nf}}=\frac{{\mu }_{\text{f}}}{{\left(1-\phi \right)}^{2.5}}+\frac{k_{\text{Brownian}}}{k_{\text{f}}}\times \frac{{\mu }_{\text{f}}}{\mathrm{Pr}}$

(6.63)

$u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x},\omega =\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$

(6.64)

$\begin{array}{l}\frac{\partial \psi }{\partial y}\frac{\partial \omega }{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial \omega }{\partial y}={\upsilon }_{\text{nf}}\left(\frac{{\partial }^2\omega }{\partial x^2}+\frac{{\partial }^2\omega }{\partial y^2}\right)+{\beta }_{\text{nf}}g\left(\frac{\partial T}{\partial x}\right)\\ +\frac{{\sigma }_{\text{nf}}{B}^2}{{\rho }_{\text{nf}}}\left(-\frac{\delta v}{\delta y}\sin \lambda \cos \lambda +\frac{\delta u}{\delta y}{\sin }^2\lambda +\frac{\delta u}{\delta x}\sin \lambda \cos \lambda -\frac{\delta v}{\delta x}{\cos }^2\lambda \right)\end{array}$

(6.65)

$\frac{\partial \psi }{\partial y}\frac{\partial T}{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial T}{\partial y}={\alpha }_{\text{nf}}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)$

(6.66)

$\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}=-\omega$

(6.67)

$X=\frac{x}{L},Y=\frac{y}{L},\Omega =\frac{\omega L^2}{{\alpha }_{\text{f}}},\Psi =\frac{\psi }{{\alpha }_{\text{f}}},\Theta =\frac{T-T_{\text{c}}}{\left(q^{″}L/k_{\text{f}}\right)},U=\frac{uL}{{\alpha }_{\text{f}}},V=\frac{vL}{{\alpha }_{\text{f}}}$

(6.68)

$\begin{array}{l}\frac{\partial \Psi }{\partial Y}\frac{\partial \Omega }{\partial X}-\frac{\partial \Psi }{\partial X}\frac{\partial \Omega }{\partial Y}=P{r}_{\text{f}}/\left({\left(1-\phi \right)}^{2.5}\left(\left(1-\phi \right)+\phi \frac{{\rho }_{\text{s}}}{{\rho }_{\text{f}}}\right)\right)\left(\frac{{\partial }^2\Omega }{\partial X^2}+\frac{{\partial }^2\Omega }{\partial Y^2}\right)\\ +R{a}_{\text{f}}P{r}_{\text{f}}\left[\left(1-\phi \right)+\phi \frac{{\beta }_{\text{s}}}{{\beta }_{\text{f}}}\right]\left(\frac{\partial \Theta }{\partial X}\right)\\ +H{a}^{2}P{r}_{\text{f}}\left(1+3\left({\sigma }_{\text{s}}/{\sigma }_{\text{f}}-1\right)\phi /\left(\left({\sigma }_{\text{s}}/{\sigma }_{\text{f}}+2\right)-\left({\sigma }_{\text{s}}/{\sigma }_{\text{f}}-1\right)\phi \right)\right)\\ \times 1/\left(\left(1-\phi \right)+\phi \frac{{\rho }_{\text{s}}}{{\rho }_{\text{f}}}\right)\left(-\frac{\delta V}{\delta Y}\tan \lambda +\right.\left.\frac{\delta U}{\delta Y}{\tan }^2\lambda +\frac{\delta U}{\delta X}\tan \lambda -\frac{\delta V}{\delta X}\right)\end{array}$

(6.69)

$\frac{\partial \Psi }{\partial Y}\frac{\partial \Theta }{\partial X}-\frac{\partial \Psi }{\partial X}\frac{\partial \Theta }{\partial Y}=\frac{k_{\text{nf}}}{k_{\text{f}}}/\left(\left(1-\phi \right)+\phi \frac{{\left(\rho Cp\right)}_{\text{s}}}{{\left(\rho Cp\right)}_{\text{f}}}\right)\left(\frac{{\partial }^2\Theta }{\partial X^2}+\frac{{\partial }^2\Theta }{\partial Y^2}\right)$

(6.70)

$\frac{{\partial }^2\Psi }{\partial X^2}+\frac{{\partial }^2\Psi }{\partial Y^2}=-\Omega$

(6.71)

where Ra_f = gβ_fL⁴q″/(k_f α_f v_f) is the Rayleigh number for the base fluid, $Ha=L{B}_x\sqrt{{\sigma }_f/{\mu }_f}$ is the Hartmann number, and Pr_f = υ_f /α_f is the Prandtl number for the base fluid. The boundary conditions as shown in Fig. 6.19 are as follows:

$\begin{array}{l}\frac{\partial \Theta }{\partial n}=-1\text{on}\text{the}\text{inner}\text{circular}\text{boundary}\\ \Theta =0.0\text{on}\text{the}\text{outer}\text{walls}\text{boundary}\\ \Psi =0.0\text{on}\text{all}\text{solid}\text{boundaries}\end{array}$

(6.72)

$N{u}_{\text{local}}=\left(\frac{k_{\text{nf}}}{k_{\text{f}}}\right){\left.\frac{1}{\Theta }\right|}_{\text{inner}\text{wall}}$

(6.73)

$N{u}_{\text{ave}}=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}N{u}_{\text{loc}}d\zeta$

(6.74)

$E=\frac{Nu\left(\phi =0\text{.04}\right)-Nu\left(\text{base fluid}\right)}{Nu\left(\text{base fluid}\right)}\times 100$

(6.75)

$\frac{\partial H}{\partial Y}=U\Theta -\frac{\partial \Theta }{\partial X},-\frac{\partial H}{\partial X}=V\Theta -\frac{\partial \Theta }{\partial Y}$

(6.76)

6.4.2. Effects of active parameters

The physical model along with the important geometrical parameters and the mesh of the enclosure used in the present CVFEM program are shown in Fig. 6.25 [10]. The width and height of the enclosure is H. The right and top wall of the enclosure are maintained at constant cold temperatures T_c, whereas the inner circular hot wall is maintained at constant hot temperature T_h and the two bottom and left walls with the length of H/2 are thermally isolated. Under all cases T_h > T_c condition is maintained. In this section r_in/r_out = 0.75. It is also assumed that the uniform magnetic field ($\overrightarrow{B}=B_x\overrightarrow{e_x}+B_y\overrightarrow{e_y}$) of constant magnitude $B=\sqrt{B_x^2+B_y^2}$ is applied, where $\overrightarrow{e_x}$ and $\overrightarrow{e_y}$ are unit vectors in the Cartesian coordinate system [11]. The governing equations are similar to those of exist in Section 6.4.1. The local Nusselt number of the nanofluid along the hot wall can be expressed as

$N{u}_{\text{local}}=\left(\frac{k_{\text{nf}}}{k_{\text{f}}}\right)\frac{\partial \Theta }{\partial r}$

(6.77)

$N{u}_{\text{ave}}=\frac{1}{\gamma }\underset{0}{\overset{\gamma }{\int }}N{u}_{\text{loc}}\left(\zeta \right)d\zeta$

(6.78)

6.5.2. Effects of active parameters

$\begin{array}{c}\hfill Nu=a_{13}+a_{23}{Y}_1+a_{33}{Y}_2+a_{43}{Y_1}^2+a_{53}{Y_2}^2+a_{63}{Y}_1{Y}_2\hfill \\ \hfill Y_1=a_{11}+a_{21}\zeta +a_{31}\phi +a_{41}{\zeta }^2+a_{51}{\phi }^2+a_{61}\zeta \phi \hfill \\ \hfill Y_2=a_{12}+a_{22}R{a}^{*}+a_{32}H{a}^{*}+a_{42}R{a}^{{*}^2}+a_{52}H{a}^{{*}^2}+a_{62}R{a}^{*}H{a}^{*}\hfill \end{array}$

(6.79)

$\begin{array}{ll}\hfill & E=b_{13}+b_{23}{Y}_1+b_{33}{Y}_2+b_{43}{Y_1}^2+b_{53}{Y_2}^2+b_{63}{Y}_1{Y}_2\hfill \\ \hfill & Y_1=b_{11}+b_{21}\zeta +b_{31}H{a}^{*}+b_{41}{\zeta }^2+b_{51}H{a}^{{*}^2}+b_{61}\zeta H{a}^{*}\hfill \\ \hfill & Y_2=b_{12}+b_{22}R{a}^{*}+b_{32}\zeta +b_{42}R{a}^{{*}^2}+b_{52}{\zeta }^2+b_{62}R{a}^{*}\zeta \hfill \end{array}$

(6.80)