The schematic diagram and the mesh of the semiannulus enclosure used in the present CVFEM program are shown in Fig. 7.16 [7]. The inner wall is maintained at constant temperatures T_h and the other walls are maintained at constant temperature T_c (T_h > T_c). The boundary conditions for concentration are similar to those of temperature. For the expression of the magnetic field strength, it can be considered that the magnetic source represents a magnetic wire placed vertically to the x–y plane at the point $\left(\overline{a},\overline{b}\right)$. The components of the magnetic field intensity $\left(\overline{H_x},\overline{H_y}\right)$ and the magnetic field strength $(\overline{H})$ can be considered as [3]:

$\overline{H_x}=\frac{{\gamma }^{\prime }}{2\pi }\frac{1}{{\left(x-\overline{a}\right)}^2+{\left(y-\overline{b}\right)}^2}\left(y-\overline{b}\right)$

(7.40)

$\overline{H_y}=-\frac{{\gamma }^{\prime }}{2\pi }\frac{1}{{\left(x-\overline{a}\right)}^2+{\left(y-\overline{b}\right)}^2}\left(x-\overline{a}\right)$

(7.41)

$\overline{H}=\sqrt{{\overline{H}}_x^2+{\overline{H}}_y^2}=\frac{{\gamma }^{\prime }}{2\pi }\frac{1}{\sqrt{{\left(x-\overline{a}\right)}^2+{\left(y-\overline{b}\right)}^2}}$

(7.42)

where γ′ is the magnetic field strength at the source (of the wire) and $\left(\overline{a},\overline{b}\right)$ is the position where the source is located. The contours of the magnetic field strength are shown in Fig. 7.17. In this chapter, magnetic source is located at (−0.01 cols, 0.5 rows). The upper wall is Lid-driven with velocity of U_Lid. In this section, γ is equal to 45°.

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

(7.43)

${\rho }_{\text{f}}\left\{u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right\}=-\frac{\partial P}{\partial x}+\mu \left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)-\sigma B_y^{2}u+\sigma B_x{B}_{y}v$

(7.44)

${\rho }_{\text{f}}\left\{u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right\}=-\frac{\partial P}{\partial y}+\mu \left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)-\sigma B_x^{2}v+\sigma B_x{B}_{y}u$

(7.45)

$\begin{array}{l}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)+\frac{{\left(\rho c\right)}_{\text{p}}}{{\left(\rho c\right)}_{\text{f}}}\left[\begin{array}{l}{D}_{\text{B}}\left\{\frac{\partial \phi }{\partial x}.\frac{\partial T}{\partial x}+\frac{\partial \phi }{\partial y}.\frac{\partial T}{\partial y}\right\}\\ +(D_{\text{T}}/T_{\text{c}})\left\{{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2\right\}\end{array}\right]\\ +\sigma {\left(u{B}_y-v{B}_x\right)}^2\\ +\mu \left\{2{\left(\frac{\partial u}{\partial x}\right)}^2+2{\left(\frac{\partial v}{\partial y}\right)}^2+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2\right\}\end{array}$

(7.46)

$u\frac{\partial \phi }{\partial x}+v\frac{\partial \phi }{\partial y}=D_{\text{B}}\left\{\frac{{\partial }^2\phi }{\partial x^2}+\frac{{\partial }^2\phi }{\partial y^2}\right\}+\left(\frac{D_{\text{T}}}{T_{\text{c}}}\right)\left\{\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right\}$

(7.47)

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

(7.48)

$\begin{array}{l}X=\frac{x}{L},\hspace{1em}Y=\frac{y}{L},\hspace{1em}\Omega =\frac{\omega L}{U_{Lid}},\hspace{1em}\Psi =\frac{\psi }{U_{Lid}L},\hspace{1em}\Theta =\frac{T-T_c}{T_{\text{h}}-T_{\text{c}}},\hspace{1em}\Phi =\frac{\phi -{\phi }_{\text{c}}}{{\phi }_{\text{h}}-{\phi }_{\text{c}}},\\ U=\frac{u}{U_{Lid}},\hspace{1em}V=\frac{v}{U_{Lid}},\hspace{1em}H=\frac{\overline{H}}{\overline{H_0}},\hspace{1em}{H}_x=\frac{\overline{H_x}}{\overline{H_0}},\hspace{1em}{H}_y=\frac{\overline{H_y}}{\overline{H_0}}\end{array}$

(7.49)

$\begin{array}{l}\left[\frac{\partial \Psi }{\partial Y}\frac{\partial \Omega }{\partial X}-\frac{\partial \Psi }{\partial X}\frac{\partial \Omega }{\partial Y}\right]=\frac{1}{\mathrm{Re}}\left(\frac{{\partial }^2\Omega }{\partial X^2}+\frac{{\partial }^2\Omega }{\partial Y^2}\right)\\ -\frac{H{a}^2}{\mathrm{Re}}\times \left\{\frac{\partial V}{\partial X}{H}_x^2+V\left(2H_x\frac{\partial H_x}{\partial X}\right)-\frac{\partial U}{\partial X}{H}_x{H}_y-U\frac{\partial H_x}{\partial X}{H}_y-U\right.\frac{\partial H_y}{\partial X}{H}_x\\ \left.-\frac{\partial U}{\partial Y}{H}_y^2-U\left(2H_y\frac{\partial H_y}{\partial Y}\right)+\frac{\partial V}{\partial Y}{H}_x{H}_y+V\frac{\partial H_x}{\partial Y}{H}_y+V\frac{\partial H_y}{\partial Y}{H}_x\right\}\end{array}$

(7.50)

$\begin{array}{l}\frac{\partial \Psi }{\partial Y}\frac{\partial \Theta }{\partial X}-\frac{\partial \Psi }{\partial X}\frac{\partial \Theta }{\partial Y}=\frac{1}{\mathrm{Re}\mathrm{Pr}}\left(\frac{{\partial }^2\Theta }{\partial X^2}+\frac{{\partial }^2\Theta }{\partial Y^2}\right)+\frac{Nb}{\mathrm{Re}}\left(\frac{\partial \Phi }{\partial X}\frac{\partial \Theta }{\partial X}+\frac{\partial \Phi }{\partial Y}\frac{\partial \Theta }{\partial Y}\right)+\frac{Nt}{\mathrm{Re}}\left({\left(\frac{\partial \Theta }{\partial X}\right)}^2+{\left(\frac{\partial \Theta }{\partial Y}\right)}^2\right)\\ +H{a}^{2}Ec{\left\{U{H}_y-V{H}_x\right\}}^2\\ +Ec\left\{2{\left(\frac{\partial U}{\partial X}\right)}^2+2{\left(\frac{\partial V}{\partial Y}\right)}^2+{\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)}^2\right\}\end{array}$

(7.51)

$\frac{\partial \Psi }{\partial Y}\frac{\partial \Phi }{\partial X}-\frac{\partial \Psi }{\partial X}\frac{\partial \Phi }{\partial Y}=\frac{1}{Le\mathrm{Re}}\left(\frac{{\partial }^2\Phi }{\partial X^2}+\frac{{\partial }^2\Phi }{\partial Y^2}\right)+\frac{Nt}{NbLe\mathrm{Re}}\left(\frac{{\partial }^2\Theta }{\partial X^2}+\frac{{\partial }^2\Theta }{\partial Y^2}\right)$

(7.52)

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

(7.53)

where Prandtl number, the Brownian motion parameter, the thermophoretic parameter, Lewis number, Hartmann number, Eckert number, and Reynolds number are defined as: Pr = μ/ρ_fα, $Nb={(\rho c)}_{\text{p}}{D}_{\text{B}}({\phi }_{\text{h}}-{\phi }_{\text{c}})\text{/}({(\rho c)}_{\text{f}}\alpha ),Nt={(\rho c)}_{\text{p}}{D}_{\text{T}}(T_{\text{h}}-T_{\text{c}})\text{/}[{(\rho c)}_{\text{f}}\alpha T_{\text{c}}],$ $Le=\alpha \text{/}{D}_{\text{B}},Ha=L{H}_0{\mu }_0\sqrt{\sigma \text{/}\mu },$ $\mathrm{Ec}=\left(U_{Lid}^2\right)/\left[\left(C_{\text{P}}\right)∆T\right]$, and $\mathrm{Re}=\frac{{\rho }_{\text{f}}L{U}_{Lid}}{\mu }$, respectively.

$N{u}_{\text{loc}}=\frac{\partial \Theta }{\partial r}$

(7.54)

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

(7.55)

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

(7.56)

7.3.2. Effects of active parameters

$r=r_{\text{in}}+A\cos \left(N\left(\zeta \right)\right)$

(7.57)

$\begin{array}{l}\rho =\phi {\rho }_{\text{p}}+(1-\phi ){\rho }_{\text{f}}\\ \cong \phi {\rho }_{\text{p}}+(1-\phi )\left\{{\rho }_{{\text{f}}_0}\left(1-\beta (T-T_{\text{c}}\right)\right\}\end{array}$

(7.58)

where ρ_f is the base fluid’s density, T_c is a reference temperature, ${\rho }_{{\text{f}}_0}$ is the base fluid’s density at the reference temperature, and β is the volumetric coefficient of expansion. Taking the density of base fluid as that of the nanofluid, the density becomes

$\rho \cong \phi {\rho }_{\text{p}}+(1-\phi )\left\{{\rho }_0\left(1-\beta (T-T_{\text{c}}\right)\right\}$

(7.59)

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

(7.60)

${\rho }_{\text{f}}\left\{u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right\}=-\frac{\partial P}{\partial x}+\mu \left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)+{\mu }_{0}M\frac{\partial \overline{H}}{\partial x}-\sigma B_y^{2}u+\sigma B_x{B}_{y}v$

(7.61)

$\begin{array}{l}{\rho }_{\text{f}}\left\{u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right\}=-\frac{\partial P}{\partial y}+\mu \left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)-\left(\phi -{\phi }_{\text{c}}\right)\left({\rho }_{\text{p}}-{\rho }_{{\text{f}}_0}\right)g+(1-{\phi }_{\text{c}}){\rho }_{{\text{f}}_0}\left(T-T_{\text{c}}\right)g\\ +{\mu }_{0}M\frac{\partial \overline{H}}{\partial y}-\sigma B_x^{2}v+\sigma B_x{B}_{y}u\end{array}$

(7.62)

$\begin{array}{l}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)+\frac{{\left(\rho c\right)}_{\text{p}}}{{\left(\rho c\right)}_{\text{f}}}\left[\begin{array}{l}{D}_{\text{B}}\left\{\frac{\partial \phi }{\partial x}.\frac{\partial T}{\partial x}+\frac{\partial \phi }{\partial y}.\frac{\partial T}{\partial y}\right\}\\ +(D_{\text{T}}/T_{\text{c}})\left\{{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2\right\}\end{array}\right]\\ +{\left(u{B}_y-v{B}_x\right)}^2\\ -{\mu }_{0}T\frac{\partial M}{\partial T}\left(u\frac{\partial \overline{H}}{\partial x}+v\frac{\partial \overline{H}}{\partial y}\right)+\mu \left\{2{\left(\frac{\partial u}{\partial x}\right)}^2+2{\left(\frac{\partial v}{\partial y}\right)}^2+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2\right\}\end{array}$

(7.63)

$u\frac{\partial \phi }{\partial x}+v\frac{\partial \phi }{\partial y}=D_{\text{B}}\left\{\frac{{\partial }^2\phi }{\partial x^2}+\frac{{\partial }^2\phi }{\partial y^2}\right\}+\left(\frac{D_{\text{T}}}{T_{\text{c}}}\right)\left\{\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right\}$

(7.64)

For the variation of the magnetization M, with the magnetic field intensity $\overline{H}$ and temperature T, the following relation derived experimentally in [3] is considered:

$M=K^{\prime }\overline{H}\left({T^{\prime }}_c-T\right)$

(7.65)

where K′ is a constant and ${T^{\prime }}_{\text{c}}$ is the Curie temperature.

In the aforementioned equations, μ₀ is the magnetic permeability of vacuum (4π × 10⁻⁷ Tm/A), $\overline{H}$ is the magnetic field strength, $\overline{B}$ is the magnetic induction $\left(\overline{B}={\mu }_0\overline{H}\right)$, and the bar above the quantities denotes that they are dimensional.

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

(7.66)

$\begin{array}{l}X=\frac{x}{L},Y=\frac{y}{L},\Omega =\frac{\omega L^2}{\alpha },\Psi =\frac{\psi }{\alpha },\Theta =\frac{T-T_{\text{c}}}{T_{\text{h}}-T_{\text{c}}},\Phi =\frac{\phi -{\phi }_{\text{c}}}{{\phi }_{\text{h}}-{\phi }_{\text{c}}},U=\frac{uL}{\alpha },V=\frac{vL}{\alpha }\\ H=\frac{\overline{H}}{\overline{H_0}},H_x=\frac{\overline{H_x}}{\overline{H_0}},H_y=\frac{\overline{H_y}}{\overline{H_0}}\end{array}$

(7.67)

$\begin{array}{l}\left[\frac{\partial \Psi }{\partial Y}\frac{\partial \Omega }{\partial X}-\frac{\partial \Psi }{\partial X}\frac{\partial \Omega }{\partial Y}\right]=\mathrm{Pr}\left(\frac{{\partial }^2\Omega }{\partial X^2}+\frac{{\partial }^2\Omega }{\partial Y^2}\right)+\mathrm{Pr}Ra\left(\frac{\partial \Theta }{\partial X}-Nr\frac{\partial \Phi }{\partial X}\right)\\ +M{n}_{\text{F}}\mathrm{Pr}\left\{\frac{\partial H}{\partial X}\frac{\partial \Theta }{\partial Y}-\frac{\partial H}{\partial Y}\frac{\partial \Theta }{\partial X}\right\}H\\ -H{a}^2\mathrm{Pr}\times \left\{\frac{\partial V}{\partial X}{H}_x^2+V\left(2H_x\frac{\partial H_x}{\partial X}\right)-\frac{\partial U}{\partial X}{H}_x{H}_y-U\frac{\partial H_x}{\partial X}{H}_y-U\right.\frac{\partial H_y}{\partial X}{H}_x\\ \left.-\frac{\partial U}{\partial Y}{H}_y^2-U\left(2H_y\frac{\partial H_y}{\partial Y}\right)+\frac{\partial V}{\partial Y}{H}_x{H}_y+V\frac{\partial H_x}{\partial Y}{H}_y+V\frac{\partial H_y}{\partial Y}{H}_x\right\}\end{array}$

(7.68)

$\begin{array}{l}\frac{\partial \Psi }{\partial Y}\frac{\partial \Theta }{\partial X}-\frac{\partial \Psi }{\partial X}\frac{\partial \Theta }{\partial Y}=\left(\frac{{\partial }^2\Theta }{\partial X^2}+\frac{{\partial }^2\Theta }{\partial Y^2}\right)+Nb\left(\frac{\partial \Phi }{\partial X}\frac{\partial \Theta }{\partial X}+\frac{\partial \Phi }{\partial Y}\frac{\partial \Theta }{\partial Y}\right)+Nt\left({\left(\frac{\partial \Theta }{\partial X}\right)}^2+{\left(\frac{\partial \Theta }{\partial Y}\right)}^2\right)\\ +H{a}^{2}Ec{\left\{U{H}_y-V{H}_x\right\}}^2+M{n}_{\text{F}}Ec\left\{U\frac{\partial H}{\partial X}+V\frac{\partial H}{\partial Y}\right\}H\left({\varepsilon }_1+\Theta \right)\\ +Ec\left\{2{\left(\frac{\partial U}{\partial X}\right)}^2+2{\left(\frac{\partial V}{\partial Y}\right)}^2+{\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)}^2\right\}\end{array}$

(7.69)

$\frac{\partial \Psi }{\partial Y}\frac{\partial \Phi }{\partial X}-\frac{\partial \Psi }{\partial X}\frac{\partial \Phi }{\partial Y}=\frac{1}{Le}\left(\frac{{\partial }^2\Phi }{\partial X^2}+\frac{{\partial }^2\Phi }{\partial Y^2}\right)+\frac{Nt}{NbLe}\left(\frac{{\partial }^2\Theta }{\partial X^2}+\frac{{\partial }^2\Theta }{\partial Y^2}\right)$

(7.70)

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

(7.71)

where thermal Rayleigh number, the buoyancy ratio number, Prandtl number, the Brownian motion parameter, the thermophoretic parameter, Lewis number, Hartmann number, Eckert number, and Magnetic number arising from FHD of nanofluid are defined as: $Ra=\left(1-{\phi }_{\text{c}}\right){\rho }_{f_0}g\beta L^3\left(T_{\text{h}}-T_{\text{c}}\right)/\left(\mu \alpha \right)$, $Nr=\left({\rho }_{\text{p}}-{\rho }_0\right)\left({\phi }_{\text{h}}-{\phi }_{\text{c}}\right)/[\left(1-{\phi }_{\text{c}}\right){\rho }_{{\text{f}}_0}\beta L\left(T_{\text{h}}-T_{\text{c}}\right)]$, $\mathrm{Pr}=\mu /{\rho }_{\text{f}}\alpha$, $Nb={(\rho c)}_{\text{p}}{D}_{\text{B}}({\phi }_{\text{h}}-{\phi }_{\text{c}})/\left({(\rho c)}_{\text{f}}\alpha \right)$, $Nt={(\rho c)}_{\text{p}}{D}_{\text{T}}(T_{\text{h}}-T_{\text{c}})/[{(\rho c)}_{\text{f}}\alpha T_{\text{c}}]$, $Le=\alpha /D_{\text{B}},Ha=L{H}_0{\mu }_0\sqrt{\sigma /\mu },\text{Ec}=\left(\alpha \mu \right)/\left[\left(\rho C_{\text{P}}\right)∆T{L}^2\right]$, and $M{n}_{\text{F}}={\mu }_0{H}_0^2{K}^{\prime }\left(T_{\text{h}}-T_{\text{c}}\right)L^2/\left(\mu \alpha \right)$, respectively.

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

(7.72)

$N{u}_{\text{ave}}=\frac{1}{0.5\pi }\underset{0}{\overset{0.5\pi }{\int }}N{u}_{\text{loc}}d\gamma$

(7.73)

7.4.2. Effects of active parameters

The schematic diagram and the mesh of the semiannulus enclosure used in the present CVFEM program are shown in Fig. 7.31 [9]. The inner wall is maintained at constant temperatures T_h and the other walls are maintained at constant temperature T_c (T_h > T_c). For the expression of the magnetic field strength, it can be considered that the magnetic source represents a magnetic wire placed vertically to the x–y plane at the point $\left(\overline{\text{a}},\overline{b}\right)$. The contours of the magnetic field strength are shown in Fig. 7.32. In this chapter, magnetic source is located at (−0.01 cols, 0.5 rows). The upper wall is Lid-driven with velocity of U_Lid.

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

(7.74)

${\rho }_{\text{nf}}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=-\frac{\partial P}{\partial x}+\eta \left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)+{\mu }_{0}M\frac{\partial \overline{H}}{\partial x}-{\sigma }_{\text{nf}}{B}_y^{2}u+{\sigma }_{\text{nf}}{B}_x{B}_{y}v$

(7.75)

$\begin{array}{l}{\rho }_{\text{nf}}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{\partial P}{\partial y}+\eta \left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)+{\mu }_{0}M\frac{\partial \overline{H}}{\partial y}-{\sigma }_{\text{nf}}{B}_x^{2}v+{\sigma }_{\text{nf}}{B}_x{B}_{y}u\\ +{\rho }_{\text{nf}}{\beta }_{\text{nf}}g\left(T-T_{\text{c}}\right)\end{array}$

(7.76)

$\begin{array}{l}{\left(\rho C_{\text{p}}\right)}_{\text{nf}}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=k_{\text{nf}}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)+{\sigma }_{\text{nf}}{\left(u{B}_y-v{B}_x\right)}^2\\ -{\mu }_{0}T\frac{\partial M}{\partial T}\left(u\frac{\partial \overline{H}}{\partial x}+v\frac{\partial \overline{H}}{\partial y}\right)+\eta \left\{2{\left(\frac{\partial u}{\partial x}\right)}^2+2{\left(\frac{\partial v}{\partial y}\right)}^2+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2\right\}\end{array}$

(7.77)

where $\eta =(1+\overrightarrow{\delta }.\overrightarrow{B}){\mu }_{\text{nf}}$, the variation of magnetic field-dependent (MFD) viscosity (δ) has been taken to be isotropic, δ₁ = δ₂ = δ₃= δ. For the variation of the magnetization M, with the magnetic field intensity $\overline{H}$ and temperature T, the following relation, derived experimentally in [3] and also used in [4], is considered:

$M=K^{\prime }\overline{H}\left({T^{\prime }}_{\text{c}}-T\right)$

(7.78)

where K′ is a constant and ${T^{\prime }}_{\text{c}}$ is the Curie temperature.

In the aforementioned equations, μ₀ is the magnetic permeability of vacuum (4π × 10⁻⁷ Tm/A), $\overline{H}$ is the magnetic field strength, $\overline{B}$ is the magnetic induction $\left(\overline{B}={\mu }_0\overline{H}\right)$, and the bar above the quantities denotes that they are dimensional.

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

(7.79)

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

(7.80)

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

(7.81)

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

(7.82)

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

(7.83)

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

(7.84)

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

(7.85)

$\begin{array}{l}X=\frac{x}{L}, Y=\frac{y}{L}, \Theta =\frac{T-T_{\text{c}}}{T_{\text{h}}-T_{\text{c}}},U=\frac{u}{U_{Lid}},\\ V=\frac{v}{U_{Lid}},H=\frac{\overline{H}}{{\overline{H}}_0},H_x=\frac{{\overline{H}}_x}{{\overline{H}}_0},H_y=\frac{{\overline{H}}_y}{{\overline{H}}_0}\end{array}$

(7.86)

where in Eq. (7.86) $\overline{H_0}=\overline{H}\left(\overline{a},0\right)=\frac{\gamma }{2\pi \left|b\right|}$ and L = r_out − r_in = r_in.

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

(7.87)

$\begin{array}{l}U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{1}{\mathrm{Re}}\left[\frac{{\mu }_{\text{nf}}/{\mu }_{\text{f}}}{{\rho }_{\text{nf}}/{\rho }_{\text{f}}}\right]\left(1+{\delta }^{*}\left(H_x+H_y\right)\right)\left(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right)\\ +M{n}_{\text{F}}\left(\frac{{\rho }_{\text{f}}}{{\rho }_{\text{nf}}}\right)\left({\varepsilon }_2-{\varepsilon }_1-\Theta \right)H\frac{\partial H}{\partial X}-\frac{H{a}^2}{\mathrm{Re}}\left[\frac{{\sigma }_{\text{nf}}/{\sigma }_{\text{f}}}{{\rho }_{\text{nf}}/{\rho }_{\text{f}}}\right]\left(H_y^{2}U-H_x{H}_{y}V\right)\end{array}$

(7.88)

$\begin{array}{l}U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{1}{\mathrm{Re}}\left[\frac{{\mu }_{\text{nf}}\text{/}{\mu }_{\text{f}}}{{\rho }_{\text{nf}}\text{/}{\rho }_{\text{f}}}\right](1+{\delta }^{*}(H_x+H_y))\left(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right)\\ +M{n}_{\text{F}}\left(\frac{{\rho }_{\text{f}}}{{\rho }_{n\text{f}}}\right)({\epsilon }_2-{\epsilon }_1-\Theta )H\frac{\partial H}{\partial Y}-\frac{H{a}^2}{\mathrm{Re}}\left[\frac{{\sigma }_{\text{nf}}\text{/}{\sigma }_{\text{f}}}{{\rho }_{\text{nf}}\text{/}{\rho }_{\text{f}}}\right](H_x^{2}V-H_x{H}_{y}U)\\ +\frac{Gr}{R{e}^2}\left[\frac{{\beta }_{\text{nf}}}{{\beta }_{\text{f}}}\right]\Theta \end{array}$

(7.89)

$\begin{array}{l}U\frac{\partial \Theta }{\partial X}+V\frac{\partial \Theta }{\partial Y}=\frac{1}{\mathrm{Pr}\mathrm{Re}}\left[\frac{\frac{k_{\text{nf}}}{k_{\text{f}}}}{\frac{{\left(\rho C_{\text{P}}\right)}_{\text{nf}}}{{\left(\rho C_{\text{P}}\right)}_{\text{f}}}}\right]\left(\frac{{\partial }^2\Theta }{\partial X^2}+\frac{{\partial }^2\Theta }{\partial Y^2}\right)\\ +H{a}^2\frac{Ec}{\mathrm{Re}}\left[\frac{\frac{{\sigma }_{\text{nf}}}{{\sigma }_{\text{f}}}}{\frac{{\left(\rho C_{\text{P}}\right)}_{\text{nf}}}{{\left(\rho C_{\text{P}}\right)}_{\text{f}}}}\right]{\left\{U{H}_y-V{H}_x\right\}}^2+M{n}_{\text{F}}Ec\frac{{\left(\rho C_{\text{P}}\right)}_{\text{f}}}{{\left(\rho C_{\text{P}}\right)}_{\text{nf}}}\left\{U\frac{\partial H}{\partial X}+V\frac{\partial H}{\partial Y}\right\}H\left({\varepsilon }_1+\Theta \right)\\ +\frac{Ec}{\mathrm{Re}}\left[\frac{\frac{{\mu }_{\text{nf}}}{{\mu }_{\text{f}}}}{\frac{{\left(\rho C_{\text{P}}\right)}_{\text{nf}}}{{\left(\rho C_{\text{P}}\right)}_{\text{f}}}}\right]\left(1+{\delta }^{*}\left(H_x+H_y\right)\right)\left\{2{\left(\frac{\partial U}{\partial X}\right)}^2+2{\left(\frac{\partial V}{\partial Y}\right)}^2+{\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)}^2\right\}\end{array}$

(7.90)

where $\mathrm{Re}=\frac{{\rho }_{\text{f}}L{U}_{Lid}}{{\mu }_{\text{f}}},Gr=g\beta ∆T{L}^3/{\upsilon }^2,Ha=L{H}_0{\mu }_0\sqrt{{\sigma }_f/{\mu }_f},{\varepsilon }_1=T_1/∆T$, $\mathrm{Ec}=\left({\rho }_{\text{f}}{U}_{Lid}^2\right)/\left[{\left(\rho C_{\text{P}}\right)}_{\text{f}}∆T\right]$, $Ri=\raisebox{1ex}{$Gr$}\!\left/ \!\raisebox{-1ex}{$R{e}^2$}\right.,{\delta }^{*}=\delta {\mu }_0{H}_0$, and $M{n}_{\text{F}}={\mu }_0{H}_0^2{K}^{\prime }\left(T_{\text{h}}-T_{\text{c}}\right)/\left({\rho }_{\text{f}}{U}_{Lid}^2\right)$ are the Reynolds number, Grashof number, Hartmann number arising from MHD, temperature number, Eckert number, Richardson number, viscosity parameter, and Magnetic number arising from FHD the for the base fluid, respectively. The stream function and vorticity are defined as:

$\begin{array}{l}u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x},\omega =\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\\ \Omega =\frac{\omega L}{U_{Lid}},\Psi =\frac{\psi }{L{U}_{Lid}}\end{array}$

(7.91)

$\begin{array}{ll}\Theta =1.0\hfill & \text{on}\text{the}\text{inner}\text{circular}\text{boundary}\hfill \\ \Theta =0.0\hfill & \text{on}\text{the}\text{other}\text{walls}\hfill \end{array}$

(7.92)

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

(7.93)

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

(7.94)

7.5.2. Effects of active parameters

$S(z)=-\frac{d_i}{2}-\delta \sin \left[\frac{2\pi (z-L_{\text{S}})}{L_{\text{w}}}\right],L_{\text{S}}\le z\le L_{\text{e}}$

(7.95)

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

(7.96)

${\rho }_{\text{m}}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=-\frac{\partial p}{\partial x}+{\mu }_{\text{m}}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}\right)+F_K(x)$

(7.97)

${\rho }_{\text{m}}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)=-\frac{\partial p}{\partial x}+{\mu }_{\text{m}}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}+\frac{{\partial }^{2}v}{\partial z^2}\right)+F_K(y)$

(7.98)

${\rho }_{\text{m}}\left(u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)=-\frac{\partial p}{\partial x}+{\mu }_{\text{m}}\left(\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2}+\frac{{\partial }^{2}w}{\partial z^2}\right)$

(7.99)

${\left({\rho }_{\text{m}}{C}_{\text{p}}\right)}_{\text{m}}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}\right)=k_{\text{m}}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\partial }^{2}T}{\partial z^2}\right)$

(7.100)

The terms F_K(x) and F_K(y) are related to FHD due to the existence of the magnetic gradient and are called the Kelvin force. ${\mu }_{0}M\frac{\partial H}{\partial x}$ and ${\mu }_{0}M\frac{\partial H}{\partial y}$ are the components of Kelvin force in the x- and y-directions, respectively. They are resulted from the electric current flowing through the wire. Therefore, it is needed to define the magnetic field of electric current. The components of the magnetic field H_x, H_y in the x-and y-directions are calculated as follows [3]:

$H_x(x,y)=\frac{I}{2\pi }\frac{(x-a)}{{(x-a)}^2+{(y-b)}^2}$

(7.101)

$H_y(x,y)=\frac{I}{2\pi }\frac{(y-b)}{{(x-a)}^2+{(y-b)}^2}$

(7.102)

$H_x(x,y,z)=\frac{I}{2\pi }\frac{1}{\sqrt{{(x-a)}^2+{(y-b)}^2}}$

(7.103)

It is needed to be mentioned that the term ${\mu }_{0}M\frac{\partial H}{\partial z}$ should be added to the momentum equation in the z-direction when axial nonuniform magnetic gradient is existed in the domain. M is the magnetization and is defined as [11]

$M=\frac{6m_{\text{p}}}{\pi d_{\text{p}}^3}\left[\text{cot}h(\xi )-\frac{1}{\xi }\right]$

(7.104)

$m_{\text{p}}=\frac{4{\mu }_{\text{B}}\pi d_{\text{p}}^3}{6\times 91.25\times {10}^{-30}}$

(7.105)

$\xi =\frac{{\mu }_0{m}_{\text{p}}H}{k_{\text{B}}T}$

(7.106)

$Mn=\frac{{\mu }_0\chi H_{\text{r}}^2{h}^2}{{\rho }_{\text{m}}{\alpha }_{\text{m}}^2}$

(7.107)

where χ is the magnetic susceptibility of ferrofluid. As mentioned earlier, the magnetic susceptibility of a ferrofluid containing 4 vol.% with a mean diameter of 10 nm is in the order of χ = 0.348586 [12]. H_r is the characteristic of magnetic field strength and calculated by $H_{\text{r}}=H(a,0)=\frac{1}{2\pi b}$. The mixture and physical properties in the aforementioned equations are calculated as follows.

$R{e}_{\text{m}}=\frac{{\rho }_{\text{m}}{v}_{\text{m}}{d}_i}{{\mu }_{\text{m}}}$

(7.108)

$N{u}_{\text{hot}}=\frac{q^{″}{D}_{{\text{h}}_i}}{k_{\text{m}}(T_{\text{w}}-T_{\text{b}})}$

(7.109)

where T_b is bulk temperature and $D_{{\text{h}}_i}$ is expressed by following equation:

$D_{{\text{h}}_i}=d_i+2\delta$

(7.110)

7.6.2. Effects of active parameters