Active and Passive Controls on Natural Convection of MHD Blasius and Sakiadis Flows with Variable Properties and Chemical Reaction

Consider an unsteady natural convection of Blasius and Sakiadis magnetohydrodynamic flows with variable properties (viscosity and thermal conductivity) and porous layers. To control thermal and concentration boundary layers we also considered the combined effect of Brownian motion and thermophoresis as well as viscous dissipation and thermal radiation with active and passive control of nanomaterial’s conditions. It coincides with the y=0 (Figure 1).

1.jpg

Figure 1. The physical model of the flow configuration

The Cartesian coordinate system has its origin located at the leading edge with the positive x-axis extending along the sheet in the upwards direction, while the y-axis is measured normal to the flow. We assume that for time t <0 the fluid. The unsteady fluid heat flows start at t = 0, the sheet is being stretched with the velocity $U_{w}(x, t)=\frac{a x}{1-c t}$ along the x-axis, keeping the origin fixed. The thermo-physical properties of the sheet or plate and the ambient fluid are assumed to be constant except density variations and the thermal conductivity which are assumed to vary linearly with temperature. The assumptions (with the Bossiness and boundary layer approximations) are, the governing equations for the convective flow and heat and mass transfer of the viscous fluid [20-24] are:

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

$\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}=\frac{1}{\rho} \frac{\partial}{\partial y}\left(\mu_{f} \frac{\partial u}{\partial y}\right)+$

$g\left(\beta_{c}\left(C-C_{\infty}\right)+\beta_{T}\left(T-T_{\infty}\right)\right)-\frac{\sigma B_{0}^{2}}{\rho} u-\frac{\vartheta}{K_{0}} u$    (2)

$\rho C_{p}\left(\frac{\partial T}{\partial t}+u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}\right)=\frac{\partial}{\partial y}\left(K(T) \frac{\partial T}{\partial y}\right)+$

$\mu_{f}\left(\frac{\partial u}{\partial y}\right)^{2}+\tau\left(\begin{array}{c}D_{B} \frac{\partial T}{\partial y} \frac{\partial C}{\partial y} \\ +\frac{D_{T}}{T_{\infty}}\left(\frac{\partial T}{\partial y}\right)^{2}\end{array}\right)+\frac{16 \sigma^{*} T_{\infty}^{3}}{3 k^{*}} \frac{\partial^{2} T}{\partial y^{2}}$    (3)

$\frac{\partial C}{\partial t}+u \frac{\partial C}{\partial x}+v \frac{\partial C}{\partial y}=D_{B} \frac{\partial^{2} C}{\partial y^{2}}-k_{0}\left(C-C_{\infty}\right)^{n}+$

$\frac{D_{T}}{T_{\infty}} \frac{\partial^{2} T}{\partial y^{2}}$    (4)

Subject to the boundary conditions are:

Passive control of nanomaterials:

1. Blasius problem:

$\left.\begin{array}{l}v=0, u=0, k_{\infty} \frac{\partial T}{\partial y}=-h_{f_{1}}\left(T-T_{\infty}\right) \\ D_{B} \frac{\partial C}{\partial y}+\frac{D_{T}}{T_{\infty}} \frac{\partial T}{\partial y}=0, a t y=0 \\ u=U_{w}, T=T_{\infty}, C=C_{\infty} \text { as } y \rightarrow \infty\end{array}\right\}$    (5)

2. Sakiadis problem:

$\left. \begin{align}  & v=0,\,u={{U}_{w}},{{k}_{\infty }}\frac{\partial T}{\partial y}=-{{h}_{{{f}_{1}}}}(T-{{T}_{\infty }}), \\  & {{D}_{B}}\frac{\partial C}{\partial y}+\frac{{{D}_{T}}}{{{T}_{\infty }}}\frac{\partial T}{\partial y}=0,\,at\,y=0, \\  & u=0,T={{T}_{\infty }},C={{C}_{\infty }}\,\,as\,\,y\to \infty  \\ \end{align} \right\}$    (6)

The active control of nanomaterial

3. Blasius problem:

$\left.\begin{array}{l}v=0, u=0, k_{\infty} \frac{\partial T}{\partial y}=-h_{f_{1}}\left(T-T_{\infty}\right), \\ C=C_{w} \text { at } y=0, \\ u=U_{w}, T=T_{\infty}, C=C_{\infty} \text { as } y \rightarrow \infty\end{array}\right\}$    (7)

4. Sakiadis problem:

$\left. \begin{align}  & v=0,\,u={{U}_{w}},{{k}_{\infty }}\frac{\partial T}{\partial y}=-{{h}_{{{f}_{1}}}}(T-{{T}_{\infty }}), \\  & C={{C}_{w}}\,at\,y=0, \\  & u=0,T={{T}_{\infty }},C={{C}_{\infty }}\,\,as\,\,y\to \infty  \\ \end{align} \right\}$    (8)

Here $u$ and $v$ are the velocity components in the $x$ and $y$ directions, respectively, $v$ is the kinematic viscosity, $g$ is the acceleration due to gravity, $\beta_{T}, \beta_{C}$ is the coefficient of thermal and concentration expansions, $T$ is the fluid temperature, $T_{\infty}, C_{\infty}$ is the ambient temperature and concentration, $\rho$ is the density, $c_{p}$ is the specific heat at constant pressure, $K(T)$ is the variable thermal conductivity, $k_{n}$ is the chemical reaction rate, $\sigma^{*}$ is the Boltzmann constant, $k^{*}$ is Stefan constant, $h_{f_{1}}$ are the heat transfer coefficients due to thermal. In this paper the thermal conductivity and viscosity is assumed to vary linearly with temperature as:

$K(T)=K_{\infty}\left(1+\frac{\varepsilon}{\Delta T}\left(T-T_{\infty}\right)\right)$    (9)

$\mu_{f}(T)=\frac{\mu_{\infty}}{\left(1+\omega\left(T-T_{\infty}\right)\right)}$    (10)

Here $\Delta T=(T w-T \infty), \quad T_{w}$ is the surface temperature, $\varepsilon$ and $\omega$ are small parameter known as the conductivity and viscosity variation parameters, $k_{\infty}, \mu_{\infty}$ are the thermal conductivity and viscosity of the fluid far away from the surface. These particular forms of a $U_{w}(x, t)$ and $T_{w}(x, t)$ have been chosen in order to obtain a new similarity transformation, which transforms the governing partial differential Eqns. $(2)-(4)$ into a set of coupled ordinary differential equations. Defining the following dimensionless functions $\mathrm{f}$ and $\theta,$ and the similarity variable $\zeta$ as:

$\zeta=\left(\frac{a}{v(1-c t)}\right)^{\frac{1}{2}} y, \quad \psi=\left(\frac{v a}{(1-c t)}\right)^{\frac{1}{2}} x f(\zeta)$

$\theta(\zeta)=\frac{\left(T-T_{\infty}\right)}{\left(T_{w}-T_{\infty}\right)}$    (11)

where, $\quad \psi(x, y, t)$ is a stream function defined as $(u, v)=(\partial \psi / \partial y,-\partial \psi / \partial x) .$ The Substituting Eq. (11) into Eqns. $(2)-(4)$ and making use of Eqns. (7),(8) and (9) we obtain

$\frac{1}{(1+E \theta)} f^{\prime \prime \prime}-\frac{E}{(1+E \theta)^{2}} \theta^{\prime} f^{\prime \prime}+f f^{\prime \prime}-f^{\prime 2}-$

$A\left(f^{\prime}+\frac{1}{2} \zeta f^{\prime \prime}\right)+\lambda_{T} \theta+\lambda_{c} \phi-(M+K) f^{\prime}=0$    (12)

$\frac{A}{2} \zeta \theta^{\prime}-f \theta^{\prime}-\frac{1}{\operatorname{Pr}}\left(\theta^{\prime \prime}+\frac{\varepsilon}{1+\varepsilon \theta} \theta^{\prime 2}\right)-N b \theta^{\prime} \phi^{\prime}$

$-N t \theta^{\prime 2}-E c f^{\prime \prime 2}-\frac{1}{\operatorname{Pr}} \operatorname{Nr} \theta^{\prime \prime}=0$    (13)

$\frac{A}{2} \phi^{\prime} \zeta-f \phi^{\prime}=\frac{1}{L e}\left(\phi^{\prime \prime}+\frac{N t}{N b} \theta^{\prime \prime}\right)-K r \phi^{n}$    (14)

The boundary conditions are

1. Blasius problem:

$f(0)=0, f^{\prime}(0)=0, N b \phi^{\prime}(0)+N t \theta^{\prime}(0)=0$

$\theta^{\prime}(0)=-\left(1-B i_{1} \theta(0)\right)$

$\phi(0)=1, f^{\prime}(\infty)=1, \theta(\infty)=0, \phi(\infty)=0$    (15)

2. Sakiadis problem:

$f(0)=0, f^{\prime}(0)=1, N b \phi^{\prime}(0)+N t \theta^{\prime}(0)=0$

$\theta^{\prime}(0)=-\left(1-B i_{1} \theta(0)\right)$

$\phi(0)=1, f^{\prime}(\infty)=0, \theta(\infty)=0, \phi(\infty)=0$    (16)

where, a prime denotes differentiation with respect to $\zeta A$ $=c / a$ is that unsteady parameter, $E=\frac{\omega}{\left(T_{w}-T_{\infty}\right)}$ is viscosity variation parameter, $M=\frac{\sigma B_{0}^{2}(1-c t)}{\rho a}$ is the magnetic field parameter, $K=\frac{v(1-c t)}{k_{0} a}$ is the porosity parameter, $E c=$ $\frac{U_{w}^{2}}{c_{p}\left(T_{\infty}-T_{r}\right)},$ is Eckert number, $U_{w}^{2}=\left(\frac{a x}{1-c t}\right)^{2}, \quad N r=\frac{16 \sigma^{*} T_{\infty}^{3}}{3 k^{*} K}$ is thermal radiation, $\mathrm{Pr}=v / \alpha_{\infty}$ is the Prandtl number, $B i_{1}=$ $\frac{h_{f_{1}}}{k_{\infty}} \sqrt{\frac{v(1-c t)}{a c}}$ is thermal Biot number, $\lambda_{T}=\frac{g \beta_{T}\left(T_{w}-T_{\infty}\right)(1-c t)^{2}}{a^{2} x}$ is a thermal buoyancy, $\lambda_{C}=\frac{g \beta_{c}\left(c_{w}-c_{\infty}\right)(1-c t)^{2}}{a^{2} x},$ is concentration buoyancy, $N b=\frac{\tau D_{B}}{v \rho C_{p}}\left(C_{w}-C_{\infty}\right), N t=\frac{\tau D_{T}}{T_{\infty} v \rho C_{p}}\left(T_{w}-T_{\infty}\right)$

$L e=\frac{v}{D_{B}}$

From the engineering point of view, the important characteristics of the flow are the skin-friction coefficient, the local Nusselt and Sherwood numbers, respectively defined as:

$C f=\frac{\tau_{w}}{\rho U_{w}^{2} / 2}, N u_{x}=\frac{x q_{w}}{k_{\infty}\left(T_{w}-T_{\infty}\right)}$

$S h_{x}=\frac{x j_{w}}{D_{B}\left(C_{w}-C_{\infty}\right)}$    (17)

where, the skin friction $\tau_{w}$ and the heat and mass transfer $q_{w}, j_{w}$ from the surface are given by

$\tau_{w}=\mu\left(\frac{\partial u}{\partial y}\right)_{y=0} ; q_{w}=-k\left(\frac{\partial T}{\partial y}\right)_{y=0}$

$j_{w}=-D_{B}\left(\frac{\partial C}{\partial y}\right)_{y=0}$    (18)

Substituting Eq. (18) into (17), we obtain

$\mathrm{Re}_{x}^{1 / 2} C f=\left(1+\frac{1}{(1+E \theta)}\right) f^{\prime \prime}(0)$,

$\mathrm{Re}_{x}^{-1 / 2} N u=-\left(1+\frac{4}{3} R+\varepsilon \theta^{\prime}(0)^{2}+\varepsilon\right) \theta^{\prime}(0)$,

$\mathrm{Re}_{x}^{-1 / 2} S h=-\phi^{\prime}(0)$    (19)

where, $\operatorname{Re}_{x}=U_{w} x / v_{f}$ is the local Reynolds number.