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Radiation, Velocity and Thermal Slips Effect Toward MHD Boundary Layer Flow Through Heat and Mass Transport of Williamson Nanofluid with Porous Medium

  • Research Article-Mechanical Engineering
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Abstract

The thermal radiation impact of MHD boundary layer flow of Williamson nanofluid along a stretching surface with porous medium taken into account of velocity and thermal slips is discussed numerically. This model aims to examine the phenomena of heat and mass transport caused by thermophoresis and Brownian motion. Through the help of similarity transformations, the governing system of PDEs is converted to a set of nonlinear ODE’s. Here, ordinary differential equations provide the mathematical formulation. The coupled system obtained has been analyzed using the Keller-Box technique; Newton's system dictates that the coefficients must be accurate and refined. To get a full understanding of the present situation, the effect of the flow regulating factors on relevant profiles is quantified and qualitatively assessed. The wall friction factor, heat, and mass transport coefficients are calculated graphically and tabulated. The findings reveal that when the slip and heat factor parameters improve, the boundary layer's thickness drops. Furthermore, the present findings indicate that raising the Williamson parameter enhances the concentration and temperature of the nanofluid. The validity of the outcomes is further shown by comparison to previously published data, which demonstrate good agreement.

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Abbreviations

\(\lambda = \sqrt {\frac{{2b^{3} }}{\nu }} \Gamma x\) :

Denotes the Williamson parameter

\(\Pr = \frac{\nu }{\alpha }\) :

Represents Prandtl number

\({\text{Nc}} = \frac{{\rho_{p} c_{p} }}{\rho c}\left( {C_{w} - C_{\infty } } \right)\) :

Denotes the heat capacities ratio parameter

\({\text{Nbt}} = \frac{{T_{\infty } D_{{\text{B}}} \left( {C_{w} - C_{\infty } } \right)}}{{D_{{\text{T}}} \left( {T_{w} - T_{\infty } } \right)}}\) :

Represents diffusivity ratio parameter

\({\text{Le}} = \frac{\alpha }{{D_{{\text{B}}} }}\) :

Denotes Lewis number

\({\text{Sc}} = \frac{\nu }{{D_{{\text{B}}} }}\) :

Is Schmidt number

\(M = \frac{{\sigma B_{0}^{2} }}{\rho b}\) :

Is magnetic field

\(Kp = \frac{\nu }{k^{\prime}b}\) :

Is the permeability parameter

\(R = \frac{{4\sigma^{*} T_{\infty }^{3} }}{{kk^{*} }}\) :

Indicates the radiation parameter

\(a,b\) :

Stretching rate constants

\(B^{*}\) :

Thermal slip factor

\(C\) :

Volume fraction of the nanoparticle

\(T\) :

Temperature of the fluid (K)

\(C_{p}\) :

Specific heat capacity of nanoparticle

\(C_{\infty }\) :

Ambient nanoparticle volume fraction (mol m−3)

\(D_{{\text{B}}}\) :

Brownian diffusion coefficient

\(D_{{\text{T}}}\) :

Thermophoresis diffusion coefficient

\(f\) :

Stream function in dimensionless form

\({\text{Re}}\) :

Reynolds number

\({\text{Sh}}\) :

Sherwood number

\(T_{w}\) :

Fluid temperature near sheet (K)

\(u\) :

Component of velocity along x-axis (m s−1)

\(v\) :

Component of velocity along y-axis (m s−1)

\(\alpha\) :

Thermal diffusivity of the nanofluid (m s−1)

\(\theta\) :

Temperature in dimensionless form (K)

\(C_{w}\) :

Nanoparticle volume fraction at the sheet (mol m−3)

\(\phi\) :

Nanoparticle volume fraction in dimensionless form (mol m−3)

\({\Gamma }\) :

Time constant

\(\delta\) :

Parameter of the velocity slip

\(\nu\) :

Kinematic viscosity (m2 s−1)

\(\mu\) :

Dynamic viscosity

\(\rho_{p}\) :

Nanoparticles density (kg/m3)

\(T_{\infty }\) :

Ambient fluid temperature (K)

\(\rho\) :

Nanofluid density (kg/m3)

\(\left( {\rho c} \right)_{f}\) :

Fluid heat capacity

\(\left( {\rho c} \right)_{p}\) :

Effective heat capacity of the nanoparticle

\(k^{\prime}\) :

Permeability of the porous medium

\(B_{0}\) :

Induced magnetic field (Tesla)

\(A^{*}\) :

Velocity slip factor

\(x,y -\) :

Coordinate axes (m)

\({\text{Nu}}\) :

Nusselt number

\(\sigma\) :

Electrical conductivity \((\Omega^{ - 1} \,{\text{m}}^{ - 1} )\)

\(\beta\) :

Thermal slip parameter

\(U_{w}\) :

Velocity along x-axis (m s−1)

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Acknowledgements

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work through Grant Code: (22UQU4240002DSR05).

Author information

Authors and Affiliations

  1. Department of Mathematics, Anurag University, Venkatapur, Hyderabad, 500 088, India

    Y. Dharmendar Reddy

  2. Departement of Physics, Faculty of Sciences, University of 20 Août 1955-Skikda, Skikda, Algeria

    Fateh Mebarek-Oudina

  3. Department of Mathematics, JNTUH College of Engineering, Hyderabad, Telangana State, 500085, India

    B. Shankar Goud

  4. Mechanical Engineering Department, College of Engineering and Islamic Architecture, Umm Al-Qura University, Makkah, P. O. Box 5555, Mecca, Saudi Arabia

    A. I. Ismail

  5. Mathematics Department, Faculty of Science, Tanta University, P.O. Box 31527, Tanta, Egypt

    A. I. Ismail

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  1. Y. Dharmendar Reddy
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  2. Fateh Mebarek-Oudina
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  3. B. Shankar Goud
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  4. A. I. Ismail
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Correspondence to Fateh Mebarek-Oudina.

Appendix

Appendix

After using the similarity transformations

$$\begin{aligned} & u = bxf^{\prime}\left( \eta \right),\quad v = - \left( {b\nu } \right)^{\frac{1}{2}} f\left( \eta \right),\quad \eta = \sqrt {\frac{b}{\nu }} y \\ & \theta \left( \eta \right) = \frac{{T - T_{\infty } }}{{T_{w} - T_{\infty } }},\quad \phi \left( \eta \right) = \frac{{C - C_{\infty } }}{{C_{w} - C_{\infty } }} \\ \end{aligned}$$

The following Partial differential Eqs. (1)–(4) are transformed into Ordinary differential equations:

First we need to find the following differentials of \(\eta , u,v,T\;{\text{and}}\;C\) with respect to \(x \& y.\)

$$\begin{aligned} & \frac{\partial \eta }{{\partial x}} = 0,\quad \frac{\partial \eta }{{\partial y}} = \sqrt {\frac{b}{\nu }} , \\ & \quad \frac{\partial u}{{\partial x}} = bf^{\prime}\left( \eta \right),\quad \frac{\partial u}{{\partial y}} = bxf^{\prime\prime}\left( \eta \right)\sqrt {\frac{b}{\nu }} \\ & \frac{\partial v}{{\partial x}} = 0, \quad \frac{\partial v}{{\partial y}} = - bf^{\prime}\left( \eta \right),\quad \frac{{\partial^{2} u}}{{\partial y^{2} }} = \frac{{xb^{2} }}{\nu }f^{\prime\prime\prime}\left( \eta \right), \\ & \quad \frac{\partial T}{{\partial x}} = 2ax\theta^{\prime}\left( \eta \right), \quad \frac{\partial T}{{\partial y}} = \sqrt {\frac{b}{\nu }} ax^{2} \theta ^{\prime}\left( \eta \right) \\ & \frac{{\partial^{2} T}}{{\partial y^{2} }} = \frac{b}{\nu }ax^{2} \theta^{\prime\prime}\left( \eta \right),\quad \frac{\partial C}{{\partial x}} = 2ax\phi \left( \eta \right) , \\ & \quad \frac{\partial C}{{\partial y}} = \sqrt {\frac{b}{\nu }} ax^{2} \phi^{^{\prime}\left( \eta \right)} ,\quad \frac{{\partial^{2} C}}{{\partial y^{2} }} = \frac{b}{\nu }ax^{2} \phi^{\prime\prime}\left( \eta \right) \\ \end{aligned}$$

Then Eq. (1) itself satisfies

$$\frac{\partial u }{{\partial x}} + \frac{\partial v }{{\partial y}} = bf^{\prime}\left( \eta \right) + ( - bf^{\prime}\left( \eta \right) = 0$$

Equation (2) \(u\frac{\partial u}{{\partial x}} + v\frac{\partial u}{{\partial y}} = \nu \frac{{\partial^{2} u}}{{\partial y^{2} }} + \sqrt 2 \nu {\Gamma }\frac{\partial u}{{\partial y}}\frac{{\partial^{2} u}}{{\partial y^{2} }} - \frac{{\sigma B^{2}_{0} }}{\rho }u - \frac{v}{{k^{\prime}}}u\) transformed to the following form.

$$\begin{aligned} & \Rightarrow bxf^{\prime}\left( \eta \right).bf^{\prime}\left( \eta \right) + \left[ { - \left( {b\nu }\right)^{\frac{1}{2}} f\left( \eta \right)} \right].\left[ {bx\sqrt {\frac{b}{\nu }} f^{\prime\prime}\left( \eta \right)} \right]\\&\quad = \nu \left[ {\frac{{b^{2} x}}{\nu }f^{\prime\prime\prime}\left( \eta \right)} \right ] \\ & \qquad +\sqrt 2 \nu {\Gamma} {bx\sqrt {\frac{b}{\nu }} }f^{\prime\prime}( \eta ) {\frac{{b^{2} x}}{\nu }}f^{\prime\prime\prime}\left( \eta \right) - \frac{{\sigma B_{0}^{2} }}{\rho }bxf^{\prime}\left( \eta \right)\\ &\qquad -\frac{\nu }{{k^{\prime}}}bxf^{\prime}\left( \eta \right) \\ &\Rightarrow b^{2} xf^{{\prime}{2}} \left( \eta \right) - b^{2}xf\left( \eta \right)f^{\prime\prime}\left( \eta \right) = b^{2}xf^{\prime\prime\prime}\left( \eta \right) \\ & \quad + \sqrt {\frac{{2b^{3} }}{\nu }} b^{2} x{\Gamma }f^{^{\prime\prime}} \left(\eta \right)f^{\prime\prime\prime}\left( \eta \right) - \frac{{b^{2}x\sigma B_{0}^{2} }}{\rho b}f^{\prime}\left( \eta \right) -\frac{{\nu b^{2} x}}{{k^{\prime}b}}f^{\prime}\left( \eta \right) \\& \Rightarrow f^{{\prime}{2}} \left( \eta \right) - f\left( \eta \right)f^{\prime\prime}\left( \eta \right) =f^{\prime\prime\prime}\left( \eta \right) + \sqrt {\frac{{2b^{3}}}{\nu }} {\Gamma }f^{\prime\prime}\left( \eta \right)f^{\prime\prime\prime}\left( \eta \right)\\ &\quad -\frac{{\sigma B_{0}^{2} }}{\rho b}f^{\prime}\left( \eta \right) -\frac{\nu }{{k^{\prime}b}}f^{\prime}\left( \eta \right) \\ &\Rightarrow \user2{f^{\prime\prime\prime}}\left( {\varvec{\eta}}\right) + \user2{\lambda f^{\prime\prime}}\left( {\varvec{\eta}}\right)\user2{f^{\prime\prime\prime}}\left( {\varvec{\eta}} \right)\\ &\quad - \user2{f^{\prime}}^{2} \left( {\varvec{\eta}} \right) +{\varvec{f}}\left( {\varvec{\eta}}\right)\user2{f^{\prime\prime}}\left( {\varvec{\eta}} \right) -\user2{Mf^{\prime}}\left( {\varvec{\eta}} \right) -\user2{Kpf^{\prime}}\left( {\varvec{\eta}} \right) = 0 \\\end{aligned}$$

The Energy Eq. (3)

\(u\frac{\partial T}{{\partial x}} + v\frac{\partial T}{{\partial y}} = \alpha \frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{\left( {\rho c} \right)_{p} }}{{\left( {\rho c} \right)_{f} }}\left[ {D_{{\text{B}}} \frac{\partial C}{{\partial y}}\frac{\partial T}{{\partial y}} + \frac{{D_{T} }}{{D_{\infty } }}\left( {\frac{\partial T}{{\partial y}}} \right)^{2} } \right] + \frac{{16\sigma^{*} T_{\infty }^{3} }}{{3k^{*} \left( {\rho c} \right)_{f} }}\frac{{\partial^{2} T}}{{\partial y^{2} }}\) is transformed to the following form:

$$\begin{aligned} & \Rightarrow bxf^{\prime}\left( \eta \right). 2ax\theta^{\prime}\left( \eta \right) + \left[ { - \left( {b\nu } \right)^{\frac{1}{2}} f\left( \eta \right)} \right].\left[ {ax^{2} \sqrt {\frac{b}{\nu }} \theta^{^{\prime}\left( \eta \right)} } \right]\\ & \quad = \alpha \left[ {\frac{{abx^{2} }}{\nu }\theta^{^{\prime\prime}\left( \eta \right)} } \right] \\ & \qquad + \frac{{\left( {\rho C} \right)_{p} }}{{\left( {\rho C} \right)_{f} }}\left[ {D_{{\text{B}}} ax^{2} \sqrt {\frac{b}{\nu }} \theta^{\prime}\left( \eta \right).ax^{2} \sqrt {\frac{b}{\nu }} \phi^{\prime}\left( \eta \right) + \frac{{D_{{\text{T}}} }}{{T_{\infty } }}\left( {ax^{2} \sqrt {\frac{b}{\nu }} \theta^{\prime}\left( \eta \right)} \right)^{2} } \right]\\ & \qquad + \frac{{16\sigma^{*} T_{\infty }^{3} }}{{\left( {\rho C} \right)_{f} 3k^{*} \nu }}ax^{2} b \theta ^{\prime\prime}\left( \eta \right). \\ & \Rightarrow 2abx^{2} f^{\prime}\theta^{\prime} - abx^{2} f\theta^{\prime} = \alpha \frac{{abx^{2} }}{\nu }\theta^{\prime\prime} + \frac{{\left( {\rho C} \right)_{p} }}{{\left( {\rho C} \right)_{f} }}D_{{\text{B}}} a^{2} x^{4} \frac{b}{\nu }\theta^{\prime}\phi^{\prime} \\ & \qquad + \frac{{\left( {\rho C} \right)_{p} }}{{\left( {\rho C} \right)_{f} }}a^{2} x^{4} \frac{b}{\nu }\frac{{D_{{\text{T}}} }}{{T_{\infty } }}\theta^{{^{{\prime}{2}} }} + \frac{{16\sigma^{*} T_{\infty }^{3} }}{{\left( {\rho C} \right)_{f} 3k^{*} \nu }}ax^{2} b \theta ^{\prime\prime} \\ & \Rightarrow 2f^{\prime}\theta - f\theta^{\prime} = \frac{\alpha }{\nu }\theta^{\prime\prime} + \frac{{\left( {\rho C} \right)_{p} }}{{\left( {\rho C} \right)_{f} }}D_{{\text{B}}} \frac{{ax^{2} }}{\nu }\theta^{\prime}\phi^{\prime} \\ &\qquad + \frac{{\left( {\rho C} \right)_{p} }}{{\left( {\rho C} \right)_{f} }}\frac{{ax^{2} }}{\nu }\frac{{D_{{\text{T}}} }}{{T_{\infty } }}\theta^{{^{{\prime}{2}} }} + \frac{{16\sigma^{*} T_{\infty }^{3} }}{{\left( {\rho C} \right)_{f} 3k^{*} \nu }} \theta ^{\prime\prime} \\ & \Rightarrow 2f^{\prime}\theta - f\theta^{\prime} = \frac{1}{\Pr }\theta^{\prime\prime} + \frac{{{\text{Nc}}}}{{{\text{Le}}.\Pr }}\theta^{\prime}\phi^{\prime} + \frac{{{\text{Nc}}}}{{\Pr .{\text{Le}}.{\text{Nbt}}}}\theta^{{^{{\prime}{2}} }} + \frac{4}{3\Pr }R \theta ^{\prime\prime} \\ & \Rightarrow 2f^{\prime}\theta - f\theta^{\prime} = \frac{1}{\Pr }\theta^{\prime\prime} + \frac{4}{3\Pr }R \theta ^{\prime\prime} + \frac{{{\text{Nc}}}}{{\Pr .{\text{Le}}}}\theta^{\prime}\phi^{\prime} + \frac{{{\text{Nc}}}}{{\Pr .{\text{Le}}.{\text{Nbt}}}}\theta^{{^{{\prime}{2}} }} \\ & \Rightarrow 2f^{\prime}\theta - f\theta^{\prime} = \frac{1}{\Pr }\left( {1 + \frac{4}{3}R} \right) \theta ^{\prime\prime} + \frac{{{\text{Nc}}}}{{\Pr .{\text{Le}}}}\theta^{\prime}\phi^{\prime} + \frac{{{\text{Nc}}}}{{\Pr .{\text{Le}}.{\text{Nbt}}}}\theta^{{^{{\prime}{2}} }} \\ & \Rightarrow \left( {\frac{{3 + 4{\varvec{R}}}}{3}} \right)\user2{\theta^{\prime\prime}} + {\varvec{Pr}}.\user2{f\theta^{\prime}} - 2{\varvec{Pr}}.\user2{f^{\prime}\theta } + \left( {\frac{{{\varvec{Nc}}}}{{{\varvec{Le}}}}} \right)\user2{\theta^{\prime}}\phi^{\prime}\\ &\qquad + \frac{{{\varvec{Nc}}}}{{\left( {{\varvec{Le}}} \right)\left( {{\varvec{Nbt}}} \right)}}\left( {\user2{\theta^{\prime}}} \right)^{2} = 0\user2{ } \\ \end{aligned}$$

Similarly, we get

Species concentration equation from (4) in the following form

$$\phi^{\prime\prime} + {\varvec{Sc}}.{\varvec{f}}\phi^{\prime} + \left( {\frac{1}{{{\varvec{Nbt}}}}} \right)\user2{\theta^{\prime\prime}} = 0$$

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Reddy, Y.D., Mebarek-Oudina, F., Goud, B.S. et al. Radiation, Velocity and Thermal Slips Effect Toward MHD Boundary Layer Flow Through Heat and Mass Transport of Williamson Nanofluid with Porous Medium. Arab J Sci Eng 47, 16355–16369 (2022). https://doi.org/10.1007/s13369-022-06825-2

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