Effect of Heat Source on MHD Flow Through Permeable Structure under Chemical Reaction and Oscillatory Suction

In this work, an unstable 2D flow of a laminar, viscid (Newtonian), electrically conducting fluid across a semi-limitless perpendicular permeable plate under motion in its plane (x-axis) embedded in a constant permeable structure was investigated. The medium is considered to be under a transverse magnetic field with concentrated buoyancy effects. Further, it is considered that no voltage is supplied which indicates that there is no electrical field. The fluid properties are considered to be uniform. The concentration of the imparting species is considered as C’w at the plate; the concentration of the specimens away from the wall, C’∞, is considered to be limitlessly less. Therefore, the Soret and the Dufour effects are ignored. The first-order chemical reaction is considered to be seen in the flow. Due to the semi-limitless plane surface considerations, the flow parameters are the functions of y’ and the time t’ only. The oscillatory suction velocity of the fluid at the plate normal to it is v’; initially, the plate relocates with the oscillatory velocity u’, in the direction of x that is in its plane. The pressure gradient is towards the x-axis. Further, there is a chemical reaction of the first order in the fluid. It is shown in Figure 1.

1.png

Figure 1. Model and coordinate system of the problem

The following governing equations considered in this work include the continuity equation (Eq. (1)), momentum equation (Eqns. (2)-(3)), temperature equation (Eq. (4)), and the equation of mass Eqns. (5)-(6).

$\frac{\partial v^{\prime}}{\partial y^{\prime}}=0$       (1)

$\frac{\partial u^{\prime}}{\partial t^{\prime}}+v^{\prime} \frac{\partial u^{\prime}}{\partial y^{\prime}}=-\frac{1}{\rho} \frac{\partial p^{\prime}}{\partial x^{\prime}}+\mathrm{v} \frac{\partial^{2} u^{\prime}}{\partial y^{\prime 2}}+\mathrm{g} \beta_{r}^{\prime}\left(T^{\prime}-T_{\infty}^{\prime}\right)+$ $\mathrm{g} \beta_{c}^{\prime}\left(C^{\prime}-C_{\infty}^{\prime}\right)-\frac{\sigma \beta_{0}^{2}}{\rho} \mathrm{u}^{\prime}-\frac{\mu}{\mathrm{k}} \mathrm{u}^{\prime}$                         (2)

where,

$-\frac{1}{\rho} \frac{\partial p^{\prime}}{\partial x^{\prime}}=-\frac{\partial U_{\infty}^{\prime}}{\partial t^{\prime}}+\left(\frac{\sigma \beta_{0}{ }^{2}}{\rho} u^{\prime}-\frac{\mu}{\mathrm{k}}\right) U_{\infty}{ }^{\prime}$                 (3)

$\frac{\partial \mathrm{T}^{\prime}}{\partial \mathrm{t}^{\prime}}=\frac{k}{\rho C_{p}} \frac{\partial^{2} T^{\prime}}{\partial y^{\prime 2}}+\frac{1}{\rho C_{p}} 4 \alpha_{1}^{2}\left(T^{1}-T_{0}^{\prime}\right)+\frac{Q^{\prime}\left(T^{\prime}-T_{0}^{\prime}\right)}{\rho C_{p}}-v^{\prime} \frac{\partial T^{\prime}}{\partial y^{\prime}}$                     (4)

$\frac{\partial \mathrm{C}^{\prime}}{\partial \mathrm{t}^{\prime}}+v^{\prime} \frac{\partial C^{\prime}}{\partial y^{\prime}}=D_{M} \frac{\partial^{2} C^{*}}{\partial y^{* 2}}-K_{T}\left(C^{\prime}-C_{\infty}^{\prime}\right)$                     (5)

$U^{\prime}\left(\mathrm{t}^{\prime}\right)=U_{\infty}{ }^{\prime}=U_{0}\left(1+\varepsilon e^{n^{\prime} t^{\prime}}\right), v^{\prime}=-V_{0}\left(1+\varepsilon A e^{n^{\prime} t^{\prime}}\right)$                 (6)

In Eqns. (1)-(6), k refers to the permeability of the porous medium. ρ, μ, v, β_c, and σ refer to the density, dynamic viscosity, kinematic viscosity, coefficient of volume expansion due to concentration, and the coefficient of electrical conductivity of the fluid respectively. D_M refers to the mass diffusion coefficient. K_r refers to the first-order chemical reaction coefficient. V₀ refers to the mean suction velocity and ε refers to a minimal quantity less than unity, and A refers to a positive constant such that εA<1. The negative sign shows that the suction velocity is directed towards the plate. The boundary conditions are considered as shown in Eq. (7). 

$\mathrm{t}^{\prime} \geq 0: \mathrm{u}^{\prime}=U_{0}$ at, $y^{\prime}=0 ; C^{\prime}=C_{w}^{\prime}+\varepsilon\left(C_{w}^{\prime}-C_{\infty}^{\prime}\right) e^{n^{\prime} t^{\prime}}$ at

$y^{\prime}=0, \mathrm{u}^{\prime} \rightarrow U^{\prime}\left(\mathrm{t}^{\prime}\right), C^{\prime} \rightarrow C_{\infty}^{\prime}$ as $y^{\prime} \rightarrow \infty$                       (7)

In Eq. (7), $U^{\prime}\left(\mathrm{t}^{\prime}\right)=U_{\infty}^{\prime}=U_{0}\left(1+\varepsilon e^{n^{\prime} t^{\prime}}\right)$.

The dimensionless quantities are considered as shown in Eq. (8) and the dimensionless parameters are given in Eqns. (9)-(12). 

$\mathrm{y}=\frac{V_{0}}{v} y^{\prime}, \mathrm{u}=\frac{u^{\prime}}{V_{0}}, \mathrm{t}=\frac{V_{0}{ }^{2}}{v} t^{\prime}, n=\frac{v n^{\prime}}{V_{0}{ }^{2}}, v^{\prime}=V_{0}\left(1+\varepsilon A e^{n t}\right)$                (8)

$K=\frac{K^{\prime} V_{0}{ }^{2}}{v^{2}}, M=\frac{\sigma \beta_{0}{ }^{2} v}{\rho V_{0}{ }^{2}}, C=\frac{C^{\prime}-C_{\infty}{ }^{\prime}}{C_{w}{ }^{\prime}-C_{\infty}{ }^{\prime}}, S c=\frac{v}{D_{M}}, G m=\frac{g \beta_{c} v\left(C_{w}{ }^{\prime}-c_{\infty}{ }^{\prime}\right)}{U_{0} V_{0}{ }^{2}}, K r=\frac{K_{r}{ }^{\prime}}{V_{0}{ }^{2}}$                           (9)

$\frac{\partial u}{\partial t}-\left(1+\varepsilon A e^{n t}\right) \frac{\partial u}{\partial y}=\frac{\partial^{2} u}{\partial y^{2}}+G m C+G r T-\left(M+\frac{1}{k}\right)(u-v)$                   (10)

$\frac{\partial T}{\partial t}-\left(1+\varepsilon A e^{n t}\right) \frac{\partial T}{\partial y}=\frac{1}{P_{r}} \frac{\partial^{2} T}{\partial y^{2}}+N^{2} T-\mathrm{S} T$        (11)

$\frac{\partial C}{\partial t}-\left(1+\varepsilon A e^{n t}\right) \frac{\partial C}{\partial y}=\frac{1}{S_{c}} \frac{\partial^{2} C}{\partial y^{2}}-K_{r} C$          (12)

Further, the other boundary conditions that are used are shown in Eqns. (13)-(14).

$\mathrm{t} \geq 0: u=1, T=\left(1+\varepsilon e^{n t}\right), C=\left(1+\varepsilon e^{n t}\right)$ at $y=0$;                       (13)

$u \rightarrow\left(1+\varepsilon e^{n t}\right), T \rightarrow 0, C \rightarrow 0$ as $y \rightarrow \infty$                           (14)

The model is developed using the perturbation method as follows.

Initially, the Eqns. (15)-(17) are considered.

$u(y, t)=u_{0}(\mathrm{y})+\varepsilon e^{n t} u_{1}(y)+O\left(\varepsilon^{2}\right)$;                 (15)

$T(y, t)=T_{0}(\mathrm{y})+\varepsilon e^{n t} T_{1}(y)+O\left(\varepsilon^{2}\right)$;              (16)

$c(y, t)=C_{0}(\mathrm{y})+\varepsilon e^{n t} C_{1}(y)+O\left(\varepsilon^{2}\right)$;             (17)

Thereafter, the Eq. (15) and Eq. (16) are substituted in Eq. (10) and Eq. (12). Then, by comparing the coefficients of ε^0 and ε^1, Eqns. (18)-(23) are obtained. 

$u_{0}^{\prime \prime}+u_{0}^{\prime}-\left(M+\frac{1}{K}\right) u_{0}=-\left(M+\frac{1}{K}\right)-G m C_{0}-G r T_{0}$               (18)

$u_{1}^{\prime \prime}+u_{1}^{\prime}-\left(M+\frac{1}{K}+n\right) u_{1}=-\left(M+\frac{1}{K}+n\right)-A u_{0}{ }^{\prime}-G m C_{1}-G r T_{1}$              (19)

$T_{0}{ }^{\prime \prime}+P r T_{0}{ }^{\prime}+\left(N^{2}-\mathrm{S}\right) \operatorname{Pr} T_{0}=0$                (20)

$T_{1}{ }^{\prime \prime}-\left(n-N^{2}+S\right) \operatorname{Pr} T_{1}=-\operatorname{Pr} A T_{0}{ }^{\prime}$                   (21)

$C_{0}^{\prime \prime}+S c C_{0}{ }^{\prime }-\operatorname{KrSc} C_{0}=0$                  (22)

$C_{1}^{\prime \prime}+S c C_{1}^{\prime}-(K r+n) S c C_{1}=-A S c C_{0}^{\prime}$                 (23)

Then, the boundary conditions as shown in Eq. (13) and Eq. (14) become as shown in Eq. (24).

$\mathrm{t} \geq 0: u_{0}=1, u_{1}=0 ; T_{0}=1, T_{1}=1 ; C_{0}=1, C_{1}=1$ at $y=0$;

$u_{0} \rightarrow 1, u_{1} \rightarrow 1 ; T_{0} \rightarrow 0, T_{1} \rightarrow 0 ; C_{0} \rightarrow 0, C_{1} \rightarrow 0$, as $y \rightarrow \infty$                      (24)

Further, by solving Eqns. (18)-(23) using the boundary conditions as shown in Eq. (24), Eqns. (25)-(30) are obtained.

$C_{0}=e^{m_{2} y}$      (25)

$C_{1}=B_{4} e^{m_{4} y}+\left(1-B_{4}\right) e^{m_{2} y}$           (26)

$T_{0}=e^{m_{6} y}$       (27)

$T_{1}=B_{8} e^{m_{8} y}+\left(1-B_{8}\right) e^{m_{6} y}$            (28)

$u_{0}=L_{1}\left(e^{m_{6} y}-e^{m_{10} y}\right)+L_{2}\left(e^{m_{2} y}-e^{m_{10} y}\right)+1$                    (29)

$u_{1}=L_{3}\left(e^{m_{10} y}-e^{m_{12} y}\right)+L_{4}\left(e^{m_{4} y}-e^{m_{12} y}\right)+L_{5}\left(e^{m_{8} y}-e^{m_{12} y}\right)+L_{6}\left(e^{m_{6} y}-e^{m_{12} y}\right)+L_{7}\left(e^{m_{2} y}-e^{m_{12} y}\right)$                   (30)

In Eqns. (25)-(30),

$m_{2}=\frac{-S c-\sqrt{S c^{2}+4 K r S c}}{2}, m_{4}=\frac{-S c-\sqrt{S c^{2}+4 S c(K r+n)}}{2}$,

$m_{6}=\frac{-P r-\sqrt{P r^{2}-4 \operatorname{Pr}\left(N^{2}-S\right)}}{2}, m_{8}=-\sqrt{\operatorname{Pr(n-N^{2}+S)},}$

$m_{10}=\frac{-1-\sqrt{1-4\left(M+\frac{1}{K}\right)}}{2}\,\,\, , m_{12}=\frac{-1-\sqrt{1+4\left(M+\frac{1}{K}+n\right)}}{2}$

$B_{4}=1-\frac{-A S c m_{2}}{m_{2}{ }^{2}+S c m_{2}-E_{2}} \,\,\, , B_{8}=1-\frac{-P r A m_{6}}{m_{6}{ }^{2}-P r\left(n-N^{2}+S\right)}$  ,

$L_{1}=\frac{-G r}{m_{6}{ }^{2}+m_{6}-\left(M+\frac{1}{K}\right)}, L_{2}=\frac{-G m}{m_{2}{ }^{2}+m_{2}-\left(M+\frac{1}{K}\right)}$,

$L_{3}=\frac{A m_{10}\left(L_{1}+L_{2}\right)}{m_{10}{ }^{2}+m_{10}-\left(M+\frac{1}{K}+n\right)}\,\,\, , L_{4}=\frac{-G m B_{4}}{m_{4}{ }^{2}+m_{4}-\left(M+\frac{1}{K}+n\right)}$

$L_{5}=\frac{-G r B_{8}}{m_{8}{ }^{2}+m_{8}-\left(M+\frac{1}{K}+n\right)}\,\,\, , L_{6}=\frac{-G r\left(1-B_{8}\right)-A L_{1} m_{6}}{m_{6}{ }^{2}+m_{6}-\left(M+\frac{1}{K}+n\right)}$    ,

$L_{7}=\frac{-G m\left(1-B_{4}\right)-A L_{2} m_{2}}{m_{2}{ }^{2}+m_{2}}-\left(M+\frac{1}{K}+n\right)$

Thereafter, the skin friction, rate of heat transfer, and rate of mass transfer are obtained as shown in Eq. (31), Eq. (32), and Eq. (33) respectively.

$\left(\frac{\partial u}{\partial y}\right)_{y=0}$          (31)

$\left(\frac{\partial T}{\partial y}\right)_{y=0}$         (32)

$\left(\frac{\partial C}{\partial y}\right)_{y=0}$        (33)

In this work, a free convective flow of a viscid compact, electrically conductive fluid was analyzed during its flow through a plate in permeable condition with oscillatory suction with first-order temperature and chemical reaction, and the transverse magnetic field. The influence of the flow variables on the velocity field, temperature field, and concentration dispersion was analyzed and the results were depicted graphically. From the results, it was observed that with the increase in the value of the magnetic parameter, Schmidt number, and the chemical reaction parameter Kr, the velocity of the flow field showed a decrease at all points. Therefore, the thickness of the velocity boundary layer decreased. Further, it was found that with the increase in the value of the porosity parameter K and the Grashoff number for mass transfer Gm, the velocity of the flow field showed an increase at all points. Thus, the thickness of the velocity boundary layer was found to increase. Moreover, it is identified that as the value of Schmidt number increased, the chemical reaction parameter Kr and the concentration of the flow field decreased at all points and hence the thickness of the concentration boundary layer decreased. Furthermore, it is found that as the porosity parameter K and the Grashoff number for the mass transfer Gm, Schmidt number Sc, and the Oscillatory Suction parameter ‘n’ increased, the concentration of the flow field was found to be unaltered at all points and therefore the thickness of the concentration boundary layer also remained unaltered.