MHD Flow Analysis of a Williamson Nanofluid due to Thomson and Troian Slip Condition

Abstract

In current challenge, the entropy generation approach is painted particularly as a powerful tool for the analysis of the brain function, in accordance with the theoretical and philosophical approach of Saint Thomas Aquinas. The present assessment is considered to look at the entropy minimization in MHD flow of a Williamson nanofluid flow past a stretchable surface under the effects of Thomson and Troian boundary condition. The flow is induced in the system because of the linear movement of stretched surface which is porous. Heat and mass transportations are the modern aspects in the flow, for this, theories of Cattaneo-Christov heat flux is used in this study. Moreover, viscous dissipation and thermal radiation impacts are taken place in the energy equation with linear expression. Additionally, the impact of heat generation or absorption with nonlinear expressions is also considered in the existing continuation which makes the study quite versatile. The finite difference strategy, for example bvp4c from Mat Lab is applied to solve the reduced ordinary differential equations. The proposed technique is more appropriate to solve the considered problem in the present analysis when compared to other previous works. From all these lines, observations are noted as due to the slip factor the velocity profile is declined and temperature distribution is improved. Because of utility of nanoparticles, Bejan number distribution is increased with the Weissenberg number and thermophoresis constant increments. From the analysis, present study is applicable in the fields of manufacturing processes and improvement in energy and heat resources.

Appendix

To calculate the Eq. (24), we need

$$ u = ax\,f^{\prime}(\eta ), $$

(41)

$$ \frac{\partial u}{{\partial x}} = a\,f^{\prime}(\eta ), $$

(42)

$$ \frac{\partial u}{{\partial y}} = ax\,f^{\prime \prime } (\eta )\sqrt {\frac{a}{\upsilon }} , $$

(43)

$$ \frac{{\partial^{2} u}}{{\partial y^{2} }} = \frac{{a^{2} x}}{\upsilon }\,f^{\prime \prime \prime } (\eta ), $$

(44)

$$ v = - \sqrt {a\upsilon } f(\eta ), $$

(45)

$$ \frac{\partial v}{{\partial x}} = - \sqrt {a\upsilon } f^{\prime}(\eta ) \times \frac{\partial \eta }{{\partial x}} = - \sqrt {a\upsilon } f^{\prime}(\eta ) \times 0 = 0, $$

(46)

$$ \frac{\partial v}{{\partial y}} = - \sqrt {a\upsilon } f^{\prime}(\eta ) \times \frac{\partial \eta }{{\partial y}} = - \sqrt {a\upsilon } f^{\prime}(\eta ) \times \sqrt {\frac{a}{\upsilon }} = - a\,f^{\prime}. $$

(47)

Now putting the above expression in Eq. (12), we get

$$ \begin{gathered} ax\,f^{\prime } (\eta ) \times a\,f^{\prime } (\eta ) - \sqrt {a\upsilon } f(\eta ) \times ax\,f^{\prime \prime } (\eta )\sqrt {\frac{a}{\upsilon }} = a^{2} x\,f^{\prime \prime \prime } (\eta ) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \sqrt 2 \upsilon \beta a^{3} x^{2} \sqrt {\frac{a}{\upsilon }} f^{\prime \prime } (\eta )f^{\prime \prime \prime } (\eta ) - \frac{{\sigma B_{0}^{2} }}{\rho }axf^{\prime } (\eta ) - \frac{\upsilon }{{k_{0} }}axf^{\prime } (\eta ). \hfill \\ \end{gathered} $$

(48)

Then further simplify we get

$$ a^{2} x\,\,(f^{\prime 2} - f\,f^{\prime \prime } ) = a^{2} x\left( {f^{\prime \prime \prime } + \sqrt {\frac{{2a^{3} }}{\upsilon }} \beta xf^{\prime \prime } \,f^{\prime \prime \prime } - \frac{{\sigma B_{0}^{2} }}{a\rho }f^{\prime } - \frac{\upsilon }{{ak_{0} }}f^{\prime } } \right). $$

(49)

Now multiply the equation both sides by (1/a²x), thus

$$ f^{\prime \prime \prime } - f^{\prime 2} + f\,f^{\prime \prime } + \sqrt {\frac{{2a^{3} }}{\upsilon }} \beta xf^{\prime \prime } \,f^{\prime \prime \prime } - \frac{{\sigma B_{0}^{2} }}{a\rho }f^{\prime } - \frac{\upsilon }{{ak_{0} }}f^{\prime } = 0. $$

(50)

Using the thermophysical properties in Eq. (50), it becomes

$$ f^{\prime \prime \prime } - f^{\prime 2} + f\,f^{\prime \prime } + We\,f^{\prime \prime } \,f^{\prime \prime \prime } - M^{2} f^{\prime } - k_{p} f^{\prime } = 0. $$

(51)

Here, $We = \sqrt {\frac{{2a^{3} }}{\upsilon }} \beta x,\,\,M^{2} = \frac{{\sigma B_{0}^{2} }}{a\rho },\,k_{p} = \frac{\upsilon }{{ak_{0} }},$

Now to calculate the Eq. (25), we need

$$ \begin{gathered} T = \theta (\eta )\,\Delta T + T_{\infty } ,\,\,\Delta T = T_{w} - T_{\infty } , \hfill \\ C = \phi (\eta )\,C_{\infty } + C_{\infty } , \hfill \\ \end{gathered} $$

(52)

$$ \begin{gathered} \frac{\partial T}{{\partial x}} = \theta^{\prime } \left( \eta \right)\Delta T\frac{\partial \eta }{{\partial x}} = 0, \hfill \\ \frac{\partial T}{{\partial y}} = \theta^{\prime } \left( \eta \right)\Delta T\frac{\partial \eta }{{\partial y}} = \theta^{\prime } \left( \eta \right)\Delta T\sqrt {\frac{a}{\upsilon }} , \hfill \\ \frac{{\partial^{2} T}}{\partial x\partial y} = \theta^{\prime \prime } \left( \eta \right)\Delta T\sqrt {\frac{a}{\upsilon }} \frac{\partial \eta }{{\partial x}} = 0, \hfill \\ \frac{{\partial^{2} T}}{{\partial y^{2} }} = \theta^{\prime \prime } \left( \eta \right)\Delta T\sqrt {\frac{a}{\upsilon }} \frac{\partial \eta }{{\partial y}} = \theta^{\prime \prime } \left( \eta \right)\Delta T\frac{a}{\upsilon }, \hfill \\ \end{gathered} $$

(53)

$$ \frac{\partial C}{{\partial x}} = \phi^{\prime } \left( \eta \right)C_{\infty } \frac{\partial \eta }{{\partial x}} = 0, $$

$$ \frac{\partial C}{{\partial y}} = \phi^{\prime } \left( \eta \right)C_{\infty } \frac{\partial \eta }{{\partial y}} = \phi^{\prime } \left( \eta \right)C_{\infty } \sqrt {\frac{a}{\upsilon }} , $$

(54)

$$ \frac{{\partial^{2} C}}{{\partial y^{2} }} = \phi^{\prime \prime } \left( \eta \right)C_{\infty } \sqrt {\frac{a}{\upsilon }} \frac{\partial \eta }{{\partial y}} = \phi^{\prime \prime } \left( \eta \right)C_{\infty } \frac{a}{\upsilon }, $$

LHS term of Eq. (22), we have

$$ \begin{aligned}& u\frac{\partial T}{{\partial x}} + v\frac{\partial T}{{\partial y}} + \lambda_{T} \left[ \begin{gathered} u\frac{\partial u}{{\partial x}}\frac{\partial T}{{\partial x}} + v\frac{\partial v}{{\partial y}}\frac{\partial T}{{\partial y}} + 2uv\frac{{\partial^{2} T}}{\partial x\partial y} \hfill \\ + u\frac{\partial v}{{\partial x}}\frac{\partial T}{{\partial y}} + v\frac{\partial u}{{\partial y}}\frac{\partial T}{{\partial x}} + u^{2} \frac{{\partial^{2} T}}{{\partial x^{2} }} + v^{2} \frac{{\partial^{2} T}}{{\partial y^{2} }} \hfill \\ \end{gathered} \right] \\ & = ax\,f^{\prime } \times 0 - \sqrt {a\upsilon } f \times \theta^{\prime } \Delta T\sqrt {\frac{a}{\upsilon }}\\ & + \lambda_{T} \left[ \begin{gathered} axf^{\prime } \times af^{\prime } \times 0 - \sqrt {a\upsilon } f \times - af^{\prime } \times \theta^{\prime } \Delta T\sqrt {\frac{a}{\upsilon }} - 2axf^{\prime } \times \sqrt {a\upsilon } f \times 0 \hfill \\ + axf^{\prime } \times 0 \times \theta^{\prime } \Delta T\sqrt {\frac{a}{\upsilon }} - \sqrt {a\upsilon } f \times axf^{\prime \prime } \sqrt {\frac{a}{\upsilon }} \times 0 + a\upsilon f^{2} \times \theta^{\prime \prime } \Delta T\frac{a}{\upsilon } \hfill \\ \end{gathered} \right] \hfill \\ & = - a(\Delta T)f\,\theta^{\prime } + \lambda_{T} \left[ {a^{2} f\,f^{\prime } \theta^{\prime } \Delta T + a^{2} f^{2} \theta^{\prime \prime } \Delta T} \right], \hfill \\ \end{aligned} $$

(55)

Multiply with $\frac{1}{a\Delta T}$

$$ - f\,\theta^{\prime } + \lambda_{T} a\left[ {f\,f^{\prime } \theta^{\prime } + f^{2} \theta^{\prime \prime } } \right] = - f\,\theta^{\prime } + \gamma \left[ {f\,f^{\prime } \theta^{\prime } + f^{2} \theta^{\prime \prime } } \right] $$

(56)

Here $\gamma = \lambda_{T} a$.

Right hand side term of Eq. (22), we have

$$ \begin{gathered} \frac{k}{{\rho c_{p} }}\frac{{\partial^{2} T}}{{\partial y^{2} }} - \frac{{16\sigma *T_{\infty }^{3} }}{{3\rho c_{p} k*}}\frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{ku_{w} }}{{x\rho c_{p} \upsilon }}\left[ {A(T_{w} - T_{\infty } )exp\left( { - \sqrt {\frac{a}{\upsilon }} y} \right) + (T - T_{\infty } )B} \right] \hfill \\ + \tau \left[ {D_{B} \frac{\partial C}{{\partial y}}\frac{\partial T}{{\partial y}} + \frac{{D_{T} }}{{T_{\infty } }}\left( {\frac{\partial T}{{\partial y}}} \right)^{2} } \right] + \frac{{\sigma B_{0}^{2} }}{{\rho c_{p} }}u^{2} + \frac{\mu }{{\rho c_{p} }}\left( {\frac{\partial u}{{\partial y}}} \right)^{2} + \frac{\mu }{{\rho c_{p} }}\beta \left( {\frac{\partial u}{{\partial y}}} \right)^{3} \, \hfill \\ = \frac{k}{{\rho c_{p} }}\left( {\theta^{\prime \prime } \Delta T\frac{a}{\upsilon }} \right) - \frac{{16\sigma *T_{\infty }^{3} }}{{3\rho c_{p} k*}}\left( {\theta^{\prime \prime } \Delta T\frac{a}{\upsilon }} \right) + \frac{kax}{{x\rho c_{p} \upsilon }}\left[ {A\,\Delta Texp\left( { - \eta } \right) + \theta \,\Delta T\,B} \right] \hfill \\ + \tau \left[ {D_{B} \phi^{\prime } \left( \eta \right)C_{\infty } \sqrt {\frac{a}{\upsilon }} \times \theta^{\prime } \left( \eta \right)\Delta T\sqrt {\frac{a}{\upsilon }} + \frac{{D_{T} }}{{T_{\infty } }}\theta^{\prime 2} \left( \eta \right)\Delta T^{2} \frac{a}{\upsilon }} \right] \hfill \\ + \frac{{a^{3} x^{2} }}{{c_{p} }}\left[ {\frac{{\sigma B_{0}^{2} }}{a\rho }f^{\prime 2} + \frac{\mu }{\rho \upsilon }f^{\prime \prime 2} + \frac{\mu ax}{{\rho \upsilon }}\beta \sqrt {\frac{a}{\upsilon }} f^{\prime \prime 3} } \right],\,\,\,\,\,\,\, \hfill \\ \end{gathered} $$

(57)

Multiply with $\frac{1}{a\Delta T}$

$$ \begin{gathered} \frac{k}{{\upsilon \rho c_{p} }}\left( {\theta^{\prime \prime } } \right) - \frac{k}{{\upsilon \rho c_{p} }}\frac{{16\sigma *T_{\infty }^{3} }}{3kk*}\left( {\theta^{\prime \prime } } \right) + \frac{k}{{\rho c_{p} \upsilon }}\left[ {A\,exp\left( { - \eta } \right) + \theta \,\,B} \right] \hfill \\ + \frac{{\tau D_{B} }}{\upsilon }C_{\infty } \theta^{\prime } \phi^{\prime } + \frac{{\tau D_{T} }}{{T_{\infty } \upsilon }}\Delta T\theta^{\prime 2} + \frac{{u_{w}^{2} }}{{c_{p} \Delta T}}\left[ {\frac{{\sigma B_{0}^{2} }}{a\rho }f^{\prime 2} + f^{\prime \prime 2} + \frac{1}{\sqrt 2 }\beta x\sqrt {\frac{{2a^{3} }}{\upsilon }} f^{\prime \prime 3} } \right],\,\,\,\,\, \hfill \\ \end{gathered} $$

(58)

Simply this we get

$$ \begin{gathered} \frac{1}{\Pr }\theta^{\prime \prime } - \frac{1}{\Pr }\frac{4}{3}Rd\theta^{\prime \prime } + Nb\,\theta^{\prime } \phi^{\prime } + Nt\,\theta^{\prime 2} \hfill \\ + \frac{1}{\Pr }\left[ {A\,exp\left( { - \eta } \right) + \theta \,\,B} \right] + Ec\left[ {M\,f^{\prime 2} + f^{\prime \prime 2} + \frac{we}{{\sqrt 2 }}f^{\prime \prime 3} } \right],\,\,\,\,\, \hfill \\ \end{gathered} $$

(59)

where $\Pr = \frac{{\mu c_{p} }}{k},Rd = \frac{{4\sigma *T_{\infty }^{3} }}{kk*},Nb = \frac{{\tau D_{B} }}{\upsilon }C_{\infty } ,Nt = \frac{{\tau D_{T} }}{{T_{\infty } \upsilon }}\Delta T,Ec = \frac{{u_{w}^{2} }}{{c_{p} \Delta T}},$

Finally, to calculate Eq. (26), we have

$$ \begin{gathered} u\frac{\partial C}{{\partial x}} + v\frac{\partial C}{{\partial y}} = D_{B} \frac{{\partial^{2} C}}{{\partial y^{2} }} + \frac{{D_{T} }}{{T_{\infty } }}\frac{{\partial^{2} T}}{{\partial y^{2} }}, \hfill \\ ax\,f^{\prime } \times 0 - \sqrt {a\upsilon } f \times \phi^{\prime } C_{\infty } \sqrt {\frac{a}{\upsilon }} = D_{B} \phi^{\prime \prime } C_{\infty } \frac{a}{\upsilon } + \frac{{D_{T} }}{{T_{\infty } }}\theta^{\prime \prime } \Delta T\frac{a}{\upsilon }, \hfill \\ - aC_{\infty } f\phi^{\prime } = D_{B} \frac{{aC_{\infty } }}{\upsilon }\phi^{\prime \prime } + \frac{{D_{T} }}{{T_{\infty } }}\frac{a\Delta T}{\upsilon }\theta^{\prime \prime } , \hfill \\ \end{gathered} $$

(60)

Multiply with $\frac{1}{{aC_{\infty } }}$

$$ - f\phi^{\prime } = \frac{{D_{B} }}{\upsilon }\left( {\phi^{\prime \prime } + \frac{{D_{T} }}{{T_{\infty } }}\frac{\Delta T}{{D_{B} C_{\infty } }}\theta^{\prime \prime } } \right), $$

(61)

$$ - {\text{PrLe}} f\phi^{\prime } = \phi^{\prime \prime } + \frac{Nt}{{Nb}}\theta^{\prime \prime } , $$

(62)

where $Le = \frac{\alpha }{{D_{B} }}$.

Boundary conditions

$$\begin{aligned} & u = u_{w} + u_{l} = ax + \gamma *\left( {1 - \xi *\frac{\partial u}{{\partial y}}} \right)^{1/2} \frac{\partial u}{{\partial y}},\,\,v = 0,\\ & T = T_{w} ,\,\,D_{B} \frac{\partial C}{{\partial y}} + \frac{{D_{T} }}{{T_{\infty } }}\frac{\partial T}{{\partial y}} = 0,\quad {\text{at y}} = 0 \\ \end{aligned}$$

(63)

$$ \begin{aligned} ax\,f^{\prime } &= ax + \gamma *\left( {1 - \xi *ax\,f^{{\prime \prime }} \sqrt {\frac{a}{\upsilon }} } \right)^{{1/2}} ax\,f^{{\prime \prime }} \sqrt {\frac{a}{\upsilon }} , \\ &\quad - \sqrt {a\upsilon } f = 0,\,\,\theta \,(T_{w} - T_{\infty } ) + T_{\infty } = T_{w} ,\,\,D_{B} \phi ^{\prime } C_{\infty } \sqrt {\frac{a}{\upsilon }} + \frac{{D_{T} }}{{T_{\infty } }}\theta ^{\prime } \Delta T\sqrt {\frac{a}{\upsilon }} = 0, \\&\Rightarrow f^{\prime } = 1 + \gamma *\sqrt {\frac{a}{\upsilon }} \left( {1 - \xi *u_{w} \,\sqrt {\frac{a}{\upsilon }} f^{{\prime \prime }} } \right)^{{1/2}} \,f^{{\prime \prime }} ,\,\,f = 0,\,\,\theta (T_{w} - T_{\infty } ) \\&= T_{w} - T_{\infty } ,\,\,\,D_{B} C_{\infty } \phi ^{\prime } + \frac{{D_{T} }}{{T_{\infty } }}\theta ^{\prime } \Delta T = 0, \\ \\& \Rightarrow f^{\prime } = 1 + \gamma _{1} \left( {1 - L_{1} f^{{\prime \prime }} } \right)^{{1/2}} \,f^{{\prime \prime }} ,\,\,f = 0,\,\,\theta = 1,\,\,\,\frac{{\tau D_{B} C_{\infty } }}{\upsilon }\phi ^{\prime } + \frac{{\tau D_{T} }}{{T_{\infty } \upsilon }}\Delta T\theta ^{\prime } = 0, \end{aligned} $$

(64)

at y = 0

$$ f^{\prime } = f^{\prime } = 1 + \gamma_{1} \left( {1 - L_{1} f^{\prime \prime } } \right)^{1/2} \,f^{\prime \prime } ,\,\,f = 0,\,\,\theta = 1,\,\,\,Nb\,\phi^{\prime } + Nt\theta^{\prime } = 0, $$

(65)

where $\gamma_{1} = \gamma *\sqrt {\frac{a}{\upsilon }} ,\,\,L_{1} = \xi *u_{w} \sqrt {\frac{a}{\upsilon }} .$