4.1.1. Problem definition

The numerical model consists in a two-dimensional square cavity with side equal to L which represents the characteristic dimension of the problem (Fig. 4.1A) [1]. The heat source is centrally located on the bottom surface and its length L/3. The cooling is achieved by the two vertical walls. The heat source has constant heat flux q″, while the cooling walls have a constant temperature T_c; all the other surfaces are adiabatic. 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 $\overrightarrow{e_y}$ 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 = cot⁻¹ (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.

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

(4.1)

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

(4.2)

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

(4.3)

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

(4.4)

where the radiation heat flux q_r is considered according to Rosseland approximation such that $q_{\text{r}}=-\frac{4{\sigma }_{\text{e}}}{3{\beta }_{\text{R}}}\frac{\partial T^4}{\partial y}$ where σ₀ and β_R are the Stefan–Boltzmann constant and the mean absorption coefficient, respectively. Following Raptis [2], the fluid-phase temperature differences within the flow are assumed to be sufficiently small, so that T⁴ may be expressed as a linear function of temperature. This is done by expanding T⁴ in a Taylor series about the temperature T_c and neglecting higher order terms to yield, $T^4\cong 4T_{\text{c}}^{3}T-3T_{\text{c}}^4$.

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

(4.5)

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

(4.6)

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

(4.7)

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

(4.8)

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

(4.9)

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

(4.10)

$\frac{\partial \psi }{\partial y}\frac{\partial T}{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial T}{\partial y}={\alpha }_{\text{nf}}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)-\frac{1}{{\left(\rho C_{\text{p}}\right)}_{\text{nf}}}\frac{\partial q_{\text{r}}}{\partial y}+\frac{{\mu }_{\text{nf}}}{{\left(\rho C_{\text{p}}\right)}_{\text{nf}}}\left(u^2+v^2\right)$

(4.11)

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

(4.12)

$\begin{array}{l}X=\frac{x}{L},\hspace{1em}Y=\frac{y}{L},\hspace{1em}\Omega =\frac{\omega L^2}{{\alpha }_{\text{f}}},\hspace{1em}\Psi =\frac{\psi }{{\alpha }_{\text{f}}},\hspace{1em}\\ \Theta =\frac{T-T_{\text{c}}}{(q^{″}L\text{/}{k}_{\text{f}})},\hspace{1em}U=\frac{uL}{{\alpha }_{\text{f}}},\hspace{1em}V=\frac{vL}{{\alpha }_{\text{f}}}\end{array}$

(4.13)

$\begin{array}{l}\frac{\partial \Psi }{\partial Y}\frac{\partial \Omega }{\partial X}-\frac{\partial \Psi }{\partial X}\frac{\partial \Omega }{\partial Y}=P{r}_{\text{f}}\left(\frac{{\mu }_{\text{nf}}}{{\mu }_{\text{f}}}\frac{{\rho }_f}{{\rho }_{\text{nf}}}\frac{k_{\text{f}}}{k_{\text{nf}}}\frac{{\left(\rho C_{\text{p}}\right)}_{\text{nf}}}{{\left(\rho C_{\text{p}}\right)}_{\text{f}}}\right)\left(\frac{{\partial }^2\Omega }{\partial X^2}+\frac{{\partial }^2\Omega }{\partial Y^2}\right)\\ +R{a}_{\text{f}}P{r}_{\text{f}}\frac{{\beta }_{\text{nf}}}{{\beta }_{\text{f}}}\left(\frac{\partial \Theta }{\partial X}\right)\\ +H{a}^{2}P{r}_{\text{f}}\frac{{\sigma }_{\text{nf}}}{{\sigma }_{\text{f}}}\frac{{\rho }_{\text{f}}}{{\rho }_{\text{nf}}}\left(-\frac{\delta V}{\delta Y}\tan {\theta }_{\text{M}}+\right.\left.\frac{\delta U}{\delta Y}{\tan }^2{\theta }_{\text{M}}+\frac{\delta U}{\delta X}\tan {\theta }_{\text{M}}-\frac{\delta V}{\delta X}\right)\end{array}$

(4.14)

$\begin{array}{l}\frac{\partial \Psi }{\partial Y}\frac{\partial \Theta }{\partial X}-\frac{\partial \Psi }{\partial X}\frac{\partial \Theta }{\partial Y}=\frac{k_{\text{nf}}}{k_{\text{f}}}\frac{{\left(\rho C_{\text{p}}\right)}_{\text{f}}}{{\left(\rho C_{\text{p}}\right)}_{\text{nf}}}\left(\frac{{\partial }^2\Theta }{\partial X^2}\right)\\ +\left(\frac{k_{\text{nf}}}{k_{\text{f}}}\frac{{\left(\rho C_{\text{p}}\right)}_{\text{f}}}{{\left(\rho C_{\text{p}}\right)}_{\text{nf}}}+\frac{4}{3}Rd{\left(\frac{k_{\text{nf}}}{k_{\text{f}}}\right)}^{-1}\right)\frac{{\partial }^2\Theta }{\partial Y^2}\\ +\varepsilon \left(\frac{k_{\text{nf}}}{k_{\text{f}}}\right)\frac{{\left(\rho C_{\text{p}}\right)}_{\text{f}}}{{\left(\rho C_{\text{p}}\right)}_{\text{nf}}}\frac{{\mu }_{\text{nf}}}{{\mu }_{\text{f}}}\left(U^2+V^2\right)\end{array}$

(4.15)

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

(4.16)

where Ra_f = gβ_fL⁴q″/(k_f α_f, υ_f) is the Rayleigh number for the base fluid, $Ha=H{B}_x\sqrt{{\sigma }_{\text{f}}/{\mu }_{\text{f}}}$ is the Hartmann number, and Pr_f = υ_f/α_f is the Prandtl number for the base fluid. Also, $Rd=4{\sigma }_{\text{e}}{T}_{\text{c}}^3/\left({\beta }_{\text{R}}k\right)$ and $M=K^{\prime }\overline{H}\left({T^{\prime }}_{\text{c}}-T\right)$ are radiation parameter and viscous dissipation parameter, respectively. The boundary conditions as shown in Fig. 4.1 are as follows:

$\begin{array}{ll}\partial \Theta /\partial n=-1.0\hfill & \text{on}\text{the}\text{heat}\text{source}\hfill \\ \Theta =0.0\hfill & \text{on}\text{the}\text{left}\text{and}\text{right}\hfill \\ \partial \Theta /\partial n=0.0\hfill & \text{on}\text{all}\text{the}\text{other}\text{adiabatic}\text{surfaces}\hfill \\ \Psi =0.0\hfill & \text{on}\text{all}\text{solid}\text{boundaries}\hfill \end{array}$

(4.17)

$N{u}_{\text{local}}=\left(\frac{k_{\text{nf}}}{k_{\text{f}}}\right){\left.\left(1+\frac{4}{3}Rd{\left(\frac{k_{\text{nf}}}{k_{\text{f}}}\right)}^{-1}\right)\frac{1}{\theta }\right|}_{L/3<X<2L/3,Y=0}$

(4.18)

$N{u}_{\text{ave}}=\frac{1}{L/3}\underset{L/3}{\overset{2L/3}{\int }}N{u}_{\text{loc}}\left(X\right)dX$

(4.19)

To estimate the enhancement of heat transfer between the case of $\phi =0.04$ and the pure fluid (base fluid) case, the enhancement is defined as:

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

(4.20)

4.1.2. Effects of active parameters

Heat and mass transfer analysis in the unsteady two-dimensional squeezing nanofluid flow between the infinite parallel plates is considered (Fig. 4.7) [4]. The two plates are placed at $\ell {(1-\gamma t)}^{1/2}=h(t)$. When γ > 0 the two plates are squeezed until they touch t = 1/γ, and for γ < 0 the two plates are separated. The viscous dissipation effect, the generation of heat due to friction caused by shear in the flow, is retained.

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

(4.21)

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

(4.22)

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

(4.23)

$\begin{array}{l}\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)+\frac{\mu }{{\left(\rho c_{\text{p}}\right)}_{\text{f}}}\left(4{\left(\frac{\partial u}{\partial x}\right)}^2\right)\\ +\frac{{\left(\rho c_{\text{P}}\right)}_{\text{p}}}{{\left(\rho c_{\text{P}}\right)}_{\text{f}}}\left[\begin{array}{l}{D}_{\text{B}}\left\{\frac{\partial C}{\partial x}.\frac{\partial T}{\partial x}+\frac{\partial C}{\partial y}.\frac{\partial T}{\partial y}\right\}\\ +(D_{\text{T}}/T_{\text{c}})\left\{{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2\right\}\end{array}\right]-\frac{1}{{\left(\rho C_{\text{p}}\right)}_{\text{f}}}\frac{\partial q_{\text{r}}}{\partial y}\end{array}$

(4.24)

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

(4.25)

where the radiation heat flux q_r is considered according to Rosseland approximation such that $q_{\text{r}}=-\frac{4{\sigma }_{\text{e}}}{3{\beta }_{\text{R}}}\frac{\partial T^4}{\partial y}$ where σ_e and β_R are the Stefan–Boltzmann constant and the mean absorption coefficient, respectively. The fluid-phase temperature differences within the flow are assumed to be sufficiently small, so that T⁴ may be expressed as a linear function of temperature. This is done by expanding T⁴ in a Taylor series about the temperature T_c and neglecting higher order terms to yield, $T^4\cong 4T_{\text{c}}^{3}T-3T_{\text{c}}^4$.

Here u and $v$ are the velocities in the x- and y-directions, respectively. T is the temperature, C is the concentration, P is the pressure, ρ_f is the base fluid’s density, μ is the dynamic viscosity, k is the thermal conductivity, c_P is the specific heat of nanofluid, D_B is the diffusion coefficient of the diffusing species. The relevant boundary conditions are as follows:

$\begin{array}{l}C=0,v=v_{\text{w}}=dh/dt,T=T_{\text{H}},C=C_{\text{H}}\text{at}y=h(t),\\ v=\partial u/\partial y=\partial T/\partial y=\partial C/\partial y=0\text{at}y=0.\end{array}$

(4.26)

$\begin{array}{l}\eta =\frac{y}{[\ell {(1-\gamma t)}^{1/2}]},u=\frac{\gamma x}{[2(1-\gamma t)]}{f}^{\prime }(\eta ),\\ v=-\frac{\gamma \ell }{[2{(1-\gamma t)}^{1/2}]}f(\eta ),\theta =\frac{T}{T_{\text{H}}},\\ \phi =\frac{C}{C_{\text{h}}}.\end{array}$

(4.27)

$f^{iv}-S\left(\eta f^{‴}+3f^{″}+f^{\prime }{f}^{″}-f{f}^{‴}\right)=0\text{,}$

(4.28)

$\left(1+\frac{4}{3}Rd\right){\theta }^{″}+\mathrm{Pr}S\left(f{\theta }^{\prime }-\eta {\theta }^{\prime }\right)+\mathrm{Pr}Ec\left({f^{″}}^2\right)+Nb{\phi }^{\prime }{\theta }^{\prime }+Nt{{\theta }^{\prime }}^2=0\text{,}$

(4.29)

${\phi }^{″}+S.Sc\left(f{\phi }^{\prime }-\eta {\phi }^{\prime }\right)+\frac{Nt}{Nb}{\theta }^{″}=0\text{,}$

(4.30)

$\begin{array}{l}f\left(0\right)=0,f^{″}\left(0\right)=0,{\theta }^{\prime }\left(0\right)=0,\\ {\phi }^{\prime }\left(0\right)=0,f\left(1\right)=1,f^{\prime }\left(1\right)=0,\\ \theta \left(1\right)=\phi \left(1\right)=1,\end{array}$

(4.31)

where $S$ is the squeeze number, Pr is the Prandtl number, Ec is the Eckert number, Sc is the Schmidt number, Rd is radiation parameter, Nb is the Brownian motion parameter, and Nt is the thermophoretic parameter, which are defined as:

$\begin{array}{l}S=\frac{\beta l^2}{2\mu }{\rho }_{\text{f}},\mathrm{Pr}=\frac{\mu }{{\rho }_{\text{f}}\alpha },Ec=\frac{1}{c_{\text{p}}}{\left(\frac{\beta x}{2\left(1-\beta t\right)}\right)}^2\hfill \\ Sc=\frac{\mu }{{\rho }_{\text{f}}D},Rd=4{\sigma }_{\text{e}}{T}_{\text{c}}^3/\left({\beta }_{\text{R}}k\right),\hfill \\ Nb={(\rho c)}_{\text{p}}{D}_{\text{B}}(C_{\text{h}})/\left({(\rho c)}_{\text{f}}\alpha \right),\hfill \\ Nt={(\rho c)}_{\text{p}}{D}_{\text{T}}(T_{\text{H}})/[{(\rho c)}_{\text{f}}\alpha T_{\text{c}}].\hfill \end{array}$

(4.32)

$Nu=\frac{-\ell k_{\text{nf}}{\left(\frac{\partial T}{\partial y}\right)}_{y=h\left(t\right)}}{k_{\text{f}}{T}_{\text{H}}}\left(1+\frac{4}{3}Rd\right)$

(4.33)

$N{u}^{*}=-\left(1+\frac{4}{3}Rd\right){\theta }^{\prime }\left(1\right)$

(4.34)

4.2.2. Effects of active parameters

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

(4.35)

${\rho }_{\text{f}}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+2\Omega w\right)=-\frac{\partial p^{*}}{\partial x}+\mu \left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)-\sigma B_0^{2}u\text{,}$

(4.36)

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

(4.37)

${\rho }_{\text{f}}\left(u\frac{\partial w}{\partial x}+\nu \frac{\partial w}{\partial y}-2\Omega u\right)=\mu \left(\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2}\right)-\sigma B_0^{2}w\text{,}$

(4.38)

$\begin{array}{l}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}=\alpha \left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\partial }^{2}T}{\partial z^2}\right)+\\ +\frac{{\left(\rho c_{\text{P}}\right)}_{\text{p}}}{{\left(\rho c_{\text{P}}\right)}_{\text{f}}}\left[\begin{array}{l}{D}_B\left\{\frac{\partial C}{\partial x}.\frac{\partial T}{\partial x}+\frac{\partial C}{\partial y}.\frac{\partial T}{\partial y}+\frac{\partial C}{\partial z}.\frac{\partial T}{\partial z}\right\}\\ +(D_{\text{T}}/T_{\text{c}})\left\{{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2+{\left(\frac{\partial T}{\partial z}\right)}^2\right\}\end{array}\right]-\frac{1}{{\left(\rho C_{\text{p}}\right)}_{\text{f}}}\frac{\partial q_{\text{r}}}{\partial y}\text{,}\end{array}$

(4.39)

$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+w\frac{\partial C}{\partial z}=D_{\text{B}}\left(\frac{{\partial }^{2}C}{\partial x^2}+\frac{{\partial }^{2}C}{\partial y^2}+\frac{{\partial }^{2}C}{\partial z^2}\right)+\left(\frac{D_{\text{T}}}{T_0}\right)\left\{\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\partial }^{2}T}{\partial z^2}\right\}$

(4.40)

where the radiation heat flux q_r is considered according to Rosseland approximation such that $q_{\text{r}}=-\frac{4{\sigma }_{\text{e}}}{3{\beta }_{\text{R}}}\frac{\partial T^4}{\partial y}$ where σ_e and β_R are the Stefan–Boltzmann constant and the mean absorption coefficient, respectively. The fluid-phase temperature differences within the flow are assumed to be sufficiently small, so that T⁴ may be expressed as a linear function of temperature [6]. This is done by expanding T⁴ in a Taylor series about the temperature T_c and neglecting higher order terms to yield, $T^4\cong 4T_{\text{c}}^{3}T-3T_{\text{c}}^4$.

Here u, $v$, and $z$ are the velocities in the x-, y-, and z-directions, respectively; T is the temperature; C is the concentration; ρ_f is the base fluid’s density; μ is the dynamic viscosity; k is the thermal conductivity; c_P is the specific heat of nanofluid; and D_B is the diffusion coefficient of the diffusing species. Also, p* is the modified fluid pressure. The absence of $\frac{\partial p*}{\partial z}$ in Eq. (4.38) implies that there is a net cross-flow along the z-axis. The relevant boundary conditions are as follows:

$\begin{array}{cc}\hfill u=ax,v=0,w=0,T=T_{\text{h}},C=C_h\hfill & \hfill \text{at}y=0\hfill \\ \hfill u=0,v=0,w=0,T=T_0,C=C_0\hfill & \hfill \text{at}y=+h\hfill \end{array}$

(4.41)

$\begin{array}{l}\eta =\frac{y}{h},u=axf\text{'}(\eta ),\nu =-ahf(\eta ),w=axg(\eta )\\ \theta (\eta )=\frac{T-T_{\text{h}}}{T_0-T_{\text{h}}},\phi (\eta )=\frac{C-C_{\text{h}}}{C_0-C_{\text{h}}}\end{array}$

(4.42)

$-\frac{1}{\rho h}\frac{\partial p^{*}}{\partial \eta }=a^{2}x\left[f^{\prime }-f{f}^{″}-\frac{f^{‴}}{R}+\frac{M}{R}+\frac{2K_{\text{r}}}{R}g\right]\text{,}$

(4.43)

$-\frac{1}{\rho h}\frac{\partial p^{*}}{\partial \eta }=a^{2}h\left[f{f}^{\prime }+\frac{1}{R}{f}^{″}\right]\text{,}$

(4.44)

$g^{″}-R(f^{\prime }g-f{g}^{\prime })+2K_{\text{r}}{f}^{\prime }-Mg=0$

(4.45)

$R=\frac{a{h}^2}{\upsilon },M=\frac{\sigma B_0^2{h}^2}{\rho \upsilon },K_{\text{r}}=\frac{\Omega h^2}{\upsilon }$

(4.46)

$f^{‴}-R[{f^{\prime }}^2-f{f}^{″}]-2K_{\text{r}}^{2}g-M^2{f}^{\prime }=A$

(4.47)

Differentiation of Eq. (4.47) with respect to $\eta$ gives:

$f^{iv}-R(f^{\prime }{f}^{″}-f{f}^{″})-2K_{\text{r}}{g}^{\prime }-M{f}^{″}=0$

(4.48)

$f^{iv}-R(f^{\prime }{f}^{″}-f{f}^{″})-2K_{\text{r}}{g}^{\prime }-M{f}^{″}=0$

(4.49)

$g^{″}-R(f^{\prime }g-f{g}^{\prime })+2K_{\text{r}}{f}^{\prime }-Mg=0$

(4.50)

$\left(1+\frac{4}{3}Rd\right){\theta }^{″}+\mathrm{Pr}Rf{\theta }^{\prime }+Nb{\phi }^{\prime }{\theta }^{\prime }+Nt{{\theta }^{\prime }}^2=0\text{,}$

(4.51)

${\phi }^{″}+R.Scf{\phi }^{\prime }+\frac{Nt}{Nb}{\theta }^{″}=0\text{,}$

(4.52)

$\begin{array}{l}f=0,f\text{'}=1,g=0,\theta =1,\phi =1\text{at}\eta =0\\ f=0,f\text{'}=0,g=0,\theta =0,\phi =0\text{at}\eta =1\end{array}$

(4.53)

$\begin{array}{l}\mathrm{Pr}=\frac{\mu }{{\rho }_{\text{f}}\alpha },Sc=\frac{\mu }{{\rho }_{\text{f}}D},Rd=4{\sigma }_{\text{e}}{T}_{\text{c}}^3/\left({\beta }_{\text{R}}k\right),\hfill \\ Nb={(\rho c)}_{\text{p}}{D}_{\text{B}}(C_{\text{h}})/\left({(\rho c)}_{\text{f}}\alpha \right),\hfill \\ Nt={(\rho c)}_{\text{p}}{D}_{\text{T}}(T_{\text{H}})/[{(\rho c)}_{\text{f}}\alpha T_{\text{c}}].\hfill \end{array}$

(4.54)

${\tilde{C}}_{\text{f}}=(Rx/h)C_{\text{f}}=f^{″}=0,Nu=-\left(1+\frac{4}{3}Rd\right)q^{\prime }(0)$

(4.55)

4.3.2. Effects of active parameters