Heat Transfer Exploration of MHD Flow Stream with Changing Viscosity and Thermal Conductivity due to Expandable Surface

Here steady two dimensional limit layer stream of an incompressible fluid streams past an extending sheet is thought of, equivalent and inverse powers acting along the x-axis and uniform attractive field B₀ is applied along y-axis normally. The variable thickness and variable warm conductivity are fluctuating and different boundaries are to be expected as constant. The governing equations are given by:

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

$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{1}{\rho }\frac{\partial }{\partial y}\left( \mu \frac{\partial u}{\partial y} \right)-\frac{\sigma {{B}_{0}}^{2}}{\rho }u$                (2)

$\rho {{c}_{p}}\left( u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y} \right)=\frac{\partial }{\partial y}\left( k\frac{\partial T}{\partial y} \right)$          (3)

The boundary restrictions are:

$\begin{align}  & u=Bx,\,\,\,\,\,v=0,\,\,\,\,\,T={{T}_{w}}\,\,\,\,\,at\,\,y=0 \\ & u\to 0,\,\,\,\,\,\,T\to {{T}_{\infty }}\,\,\,\,\,as\,\,\,\,\,y\to \infty  \\  \end{align}$           (4)

By introducing similarity transformation and dimensionless quantities as:

$\eta =\sqrt{\frac{B}{\upsilon }}y,\,\,\,u=BxF'(\eta ),\,\,\,\,\,\,v=-\sqrt{B\upsilon }F(\eta ), \\ \theta =\frac{T-{{T}_{\infty }}}{{{T}_{w}}-{{T}_{\infty }}}$         (5)

The temperature reliant on fluid viscosity is given by:

$\mu ={{\mu }^{*}}(a+b({{T}_{w}}-T))$        (6)

$\mu=\frac{1}{\left(b_{1}+b_{2} T\right)}$ is viscosity- temperature and can be expressed in the form:

$\mu=\frac{1}{b_{1}}\left(1+\frac{b_{2}}{b_{1}} T\right)^{-1}=\frac{1}{b_{1}}\left(1-\frac{b_{2}}{b_{1}} T+\frac{b_{2}^{2}}{b_{1}^{2}} T^{2}-\cdots \ldots . .\right) \cong \frac{1}{b_{1}}-\frac{b_{2}}{b_{1}} T$

$\mu=a-b T$

where, $a=\frac{1}{b_{1}}, b=\frac{b_{2}}{b_{1}}$.

The value of $b_{1}=53.41, b_{2}=2.43$ are to be taken and it gives $0^{0} \leq T \leq 23^{0}$. The viscosity-temperature relation satisfies the relation $\mu=e^{-a T}$. In this expression, the 2^nd and further order terms are ignored and the vary of temperature $\left(T_{w}-T_{\infty}\right)$ is considered i.e. $\left(0^{0}-23^{0}\right)$. We expect that thermal conductivity of the fluid is a linear function of the temperature and can be written as:

$k={{k}_{\infty }}\left( 1+\varepsilon \frac{T-{{T}_{\infty }}}{{{T}_{w}}-{{T}_{\infty }}} \right),\,\,\,\,\,\,k={{k}_{\infty }}\left( 1+\varepsilon \theta (\eta ) \right)$             (7)

The governing Eqns. (2)-(4) are converted by using Eq. (4)-(7) and can be transcribed as:

$\begin{align}  & \left[ a+A(1-\theta ) \right]F'''-AF''\theta ' \\ & +FF''-F{{'}^{2}}-MF'=0 \\  \end{align}$        (8)

$\left( 1+\varepsilon \theta  \right)\theta ''+\Pr F\theta '=0$        (9)

Here $A=b\left(T_{w}-T_{\infty}\right)$ is variable viscosity, $M=\frac{\sigma B_{0}^{2}}{\rho \beta}$ and $\operatorname{Pr}=\frac{\mu c_{p}}{\kappa_{\infty}}$.

The boundary conditions (4) can be reduced:

$F^{\prime}=1, F=0, \theta=1$ at $\eta=0$

$F^{\prime} \rightarrow 0, \theta \rightarrow 0$ as $\eta \rightarrow \infty$               (10)

The physical quantities of friction coefficent $C_{f}$ and the local Nusselt number $N u_{x}$ which are defined as:

${{C}_{f}}=\frac{{{\tau }_{w}}}{\rho u_{e}^{2}(x)},\,\,\,\,\,\,\,N{{u}_{x}}=\frac{x{{q}_{w}}}{k({{T}_{w}}-{{T}_{\infty }})}$            (11)

Here $\tau_{w}$ and $q_{w}$ are sheared stress and heat flux at the wall is given by:

${{\tau }_{w}}=\mu {{\left( \frac{\partial u}{\partial y} \right)}_{y=0}},\,{{q}_{w}}=-k{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}$           (12)

Substituting (5) into (11) and (12):

$\operatorname{Re}_{x}^{\frac{1}{2}}{{C}_{f}}=F''(0),\,\,\,\,\,\,\operatorname{Re}_{x}^{-\frac{1}{2}}N{{u}_{x}}=\theta '(0)\,\,$             (13)

Here $R e_{x}=\frac{B}{v} x^{2}$ is local Reynolds number.

The current outcomes are awesome concurrence with existing outcomes. Table 2 exhibits the estimations of friction coefficient -F''(0) and the wall temperature gradient -θ'(0) for various standards of viscosity factor A and thermal conductivity parameter ε with a=1 and Pr=0.71, M=0.5. In this table in the event that expanding of thickness boundary, at that point both friction coefficient and wall temperature, inclination are improvements and expanding of thermal conductivity boundary, the wall temperature gradient -θ'(0) diminishes.

Table 3 presents the estimations of friction factor -F''(0) and the wall temperature, inclination -θ'(0) for various estimations of Prandtl group Pr, Magnetic factor M and thermal conductivity factor ε with a=1 and A=1. In this table expanding of Prandtl number, then both friction coefficient and wall temperature gradient are improvements. Expanding of Magnetic boundary, the skin contact coefficient increments and wall temperature inclination diminish. Both friction coefficient and the wall temperature, angles are diminished by expanding of thermal conductivity.

Table 2. Friction coefficient -F''(0), wall temperature gradient -θ'(0) standards with a=1 and Pr=0.71, M=0.5

Table 3. Friction coefficient -F''(0) and the wall temperature gradient -θ'(0) standards with a=1 and A=1

Expanding of thermal conductivity, then Figure 1 outlines the impacts of temperature reliant on variable viscosity factor A in the velocity profile. The graph illustrates that the velocity sketch increments with expanding movable viscosity parameter. The impacts of a magnetic factor if there should be an occurrence both unchanging viscosity and adjustable viscosity, the velocity profile diminishes with expanding of magnetic parameter, it appears in Figure 2 and Figure 3.

The impact of Prandtl number Pr in the event of adjustable viscosity on velocity study is appeared in Figure 4.

The impacts are improved when the Prandtl number is increments. Figure 5 displays that the influence of variable viscosity A on temperature profile impacts is reduced by extending adjustable viscosity. Figure 6 exhibits that the temperature profile improves with extending thermal conductivity limit ε if there ought to emerge an event of adjustable viscosity.

The impacts of magnetic boundary M in both uniform viscosity and variable viscosity, the temperature profile increments with expanding attractive boundary and it tends to be appeared in Figure 7 and Figure 8. Figure 9 and Figure 10 display that the temperature profile diminishes if there should arise an occurrence of both uniform viscosity and variable viscosity when Prandtl number Pr increments.

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Figure 1. Velocity profile for distinct estimations of variable viscosity A when a=1, Pr=0.71, M=0.5 & ε=0.5

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Figure 2. Velocity profile for various estimations of Magnetic Factor M in the case of uniform viscosity when a=1, Pr=0.71, ε=0.5 & A=0

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Figure 3. Velocity profile for distinct estimations of Magnetic factor M In the case of variable viscosity when a=1, Pr=0.71, ε=0.5 & A=1

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Figure 4. Velocity description for different estimations of Magnetic factor M in case of variable viscosity when a=1, M=0.5, ε=0.5 & A=0

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Figure 5. Temperature profile for distinct standards of variable viscosity when a=1, Pr=0.71, M=0.5 & ε=0.5

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Figure 6. Temperature sketches for various estimations of movable thermal Conductivity ε in the case of variable viscosity when a=1, Pr=0.71, M=0.5 & A=1

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Figure 7. Temperature descriptions for many estimations of a magnetic factor M In the case of uniform viscosity when a=1, Pr=0.71, ε=0.5 & A=0

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Figure 8. Temperature sketches for distinct estimations of a magnetic factor M In the case of variable viscosity when a=1, Pr=0.71, ε=0.5 & A=1

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Figure 9. Temperature sketches for many estimations of a magnetic factor M In the case of uniform viscosity when a=1, M=0.5, ε=0.5 & A=0

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Figure 10. Temperature sketches for many estimations of a magnetic factor M In the case of variable viscosity when a=1, M=0.5, ε=0.5 & A=0