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Modeling and analysis of Maxwell nanofluid considering mixed convection and Darcy–Forchheimer relation

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Abstract

This research describes mixed convective Maxwell nanoliquid stretching flow subject to Newtonian heating. Porous medium characteristics are elaborated utilizing non-Darcian relation. Modeling is based on Brownian movement, heat generation, thermophoresis and chemical reaction aspects. Boundary-layer idea is employed for simplification of governing expressions. Optimal homotopy algorithm is implemented for computations of nonlinear problems. Besides, the Sherwood and Nusselt numbers along with temperature, velocity and nanoparticle concentration are analyzed. Our outcomes reveal opposite characteristics for porosity factor and Deborah number against velocity field.

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Abbreviations

\(u,\,\;v\) :

Velocity components

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

Brownian movement coefficient

\(x,\,\;y\) :

Space coordinates

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

Thermophoresis diffusion coefficient

\(\nu\) :

Kinematic viscosity

\({k_1}\) :

Reaction rate

\({\rho _f}\) :

Base liquid density

\({u_w}(x)\) :

Stretching velocity

\(\mu\) :

Dynamic viscosity

\({h_c}\) :

Heat transfer coefficient

\(\alpha\) :

Thermal diffusivity

\({h_t}\) :

Mass transfer coefficient

\(k\) :

Thermal conductivity

\(c\) :

Stretching rate

\(\tau\) :

Heat capacity ratio

\(\beta\) :

Deborah number

\({\left( {\rho c} \right)_f}\) :

Liquid heat capacity

\(\delta\) :

Mixed convection variable

\({\left( {\rho c} \right)_p}\) :

Nanoparticle effective heat capacity

\(G{r_x}\) :

Thermal buoyancy variable

\(K\) :

Porous medium permeability

\(N\) :

Ratio of solutal to thermal buoyancy

\({\lambda _1}\) :

Relaxation time

\(Gr_{x}^{ * }\) :

Concentration buoyancy variable

\(g\) :

Gravitational acceleration

\({F_r}\) :

Local inertia coefficient

\({\Lambda _1}\) :

Thermal expansion coefficient

\(C_{b}^{ * }\) :

Drag coefficient per unit length

\({\Lambda _2}\) :

Solutal expansion coefficient

\(Nt\) :

Thermophoretic variable

\({C_b}\) :

Drag coefficient

\(\Pr\) :

Prandtl number

\({Q_0}\) :

Coefficient of heat (absorption, generation)

\(Nb\) :

Brownian motion variable

\(T\) :

Temperature

\(S\) :

Heat generation variable

\(C\) :

Concentration

\(\gamma\) :

Chemical reaction variable

\({\gamma _1}\) :

Thermal conjugate number

\({\gamma _2}\) :

Solutal conjugate number

\(Sc\) :

Schmidt number

\({\text{N}}{{\text{u}}_x}Re_{x}^{{ - \tfrac{1}{2}}}\) :

Dimensionless Nusselt number

\({\text{S}}{{\text{h}}_x}Re_{x}^{{ - \tfrac{1}{2}}}\) :

Dimensionless Sherwood number

\(R{e_x}\) :

Local Reynolds number

\(f\) :

Dimensionless velocity

\(\theta\) :

Dimensionless temperature

\(\phi\) :

Dimensionless concentration

\({\hbar _f},\;{\hbar _\theta },\;{\hbar _\phi }\) :

Auxiliary variables

\(\eta\) :

Dimensionless variable

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Acknowledgements

The authors wish to express their thanks for the financial supports received from King Fahd University of Petroleum and Minerals under grant IN171009.

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Authors and Affiliations

  1. Department of Mathematics, DCC-KFUPM, Box 5084, 31261, Dhahran, Saudi Arabia

    M. Adil Sadiq

  2. Department of Mathematics and Statistics, The University of Lahore Gujrat Campus, Gujrat, 50700, Pakistan

    M. Waqas

  3. Department of Mathematics, Quaid-I-Azam University 45320, Islamabad, 44000, Pakistan

    M. Waqas & T. Hayat

  4. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    T. Hayat & A. Alsaedi

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  1. M. Adil Sadiq
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Correspondence to M. Waqas.

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Sadiq, M.A., Waqas, M., Hayat, T. et al. Modeling and analysis of Maxwell nanofluid considering mixed convection and Darcy–Forchheimer relation. Appl Nanosci 9, 1155–1162 (2019). https://doi.org/10.1007/s13204-019-00968-9

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