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Flow analysis by Cattaneo–Christov heat flux in the presence of Thomson and Troian slip condition

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Abstract

This communication considers entropy minimization in stagnation point flow of a hybrid nanofluid past a nonlinear permeable stretching sheet with Thomson and Troian boundary condition. Due to the porous medium, the Darcy–Forchheimer relation is added. The nonlinear thermal radiation, heat generation, and viscous dissipation by using Cattaneo–Christov heat flux model are explained. Further the influence of variable viscosity, activation energy, and variable mass diffusivity is taken into account. For first time, hybrid nanofluid consisting of carbon nanotubes with Thomson and Troian boundary conditions and induced MHD has been implemented and has not yet been studied. The finite difference method, i.e., bvp4c from Matlab, is utilized to solve the transformed ordinary differential equations (ODEs). This method has good certainty to solve this problem, compared to previous works. Comparison of simple nanofluid and hybrid nanofluid is graphically illustrated. It is noticed that the solid volume fraction decreases the velocity profile and enhances the temperature distribution. Further, compared to simple nanofluid, hybrid nanofluid has greater thermal conductivity and better heat transfer performance.

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Abbreviations

\(\hat{u}\) :

Along the x-axis velocity component

\(\hat{v}\) :

Along the y-axis velocity component

\(Q(x)\) :

Volumetric rate of heat source

\(K^{**}\) :

Permeability of porous medium

\(k^{*}\) :

Coefficient of mean absorption

\(F^{**}\) :

Non-uniform inertia coefficient

\(k_{r}\) :

Reaction rate constant

\(u_{e} (x)\) :

Free stream velocity of the fluid

\(E_{a}\) :

Activation energy

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

Mass diffusivity

\(\Pr\) :

Prandtl number

\(k({8}{\text{.61}} \times {10}^{ - 5} {\text{eV/K}})\) :

Boltzmann constant

\(L\) :

Diffusive constant parameter

\(E\) :

Non-dimensional activation energy

\(S_{c}\) :

Schmidt number

\(C_{f}\) :

Surface drag force

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

Nusselt number

\({\text{Br}}\) :

Brinkman number

\(F_{r}\) :

Inertia coefficient

\(E_{c}\) :

Eckert number

\(R_{d}\) :

Radiation parameter

\(R_{c}\) :

Dimensionless reaction rate

\(\hat{H}_{e}\) :

Magnetic field at free stream

\(M\) :

Magnetic parameter

\(m\) :

Power law parameter

\(\rho_{{{\text{hnf}}}}\) :

Hybrid nanofluid density

\(\sigma^{*}\) :

Stefan-Boltzmann constant

\(\mu_{{{\text{hnf}}}} (\hat{T})\) :

Hybrid nanofluid viscosity

\(\tau_{w}\) :

Shear stress

\(\alpha_{{{\text{hnf}}}}\) :

Hybrid nanofluid thermal diffusivity

\(\lambda\) :

Reciprocal magnetic Prandtl number

\((\rho C_{p} )_{{{\text{hnf}}}}\) :

Hybrid nanofluid heat capacity

\(\alpha_{1}\) :

Temperature difference

\(\mu_{\infty }\) :

Viscosity of fluid

\(\rho_{f}\) :

Density of fluid

\((\rho C_{p} )_{f}\) :

Heat capacity of fluid

\(D_{c}\) :

Dimensionless heat generation parameter

\(\mu_{0}\) :

Magnetic permeability

\(\alpha_{2}\) :

Concentration difference

\(\theta_{r}\) :

Variable viscosity parameter

\(\varepsilon_{1}\) :

Mass diffusivity parameter

\(\mu_{e}\) :

Magnetic diffusivity

\(\varepsilon\) :

Velocity ratio parameter

\(\gamma\) :

Parameter of thermal relaxation time

\(\xi\) :

Critical shear rate

\(\gamma_{1}\) :

Slip velocity parameter

\(\xi^{*}\) :

Critical shear rate (reciprocal)

\(\gamma^{*}\) :

Navier’s slip length

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Authors and Affiliations

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

    Shafiq Ahmad

  2. Mathematics and Its Applications in Life Sciences Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Sohail Nadeem

  3. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Sohail Nadeem

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  1. Shafiq Ahmad
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  2. Sohail Nadeem
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Ahmad, S., Nadeem, S. Flow analysis by Cattaneo–Christov heat flux in the presence of Thomson and Troian slip condition. Appl Nanosci 10, 4673–4687 (2020). https://doi.org/10.1007/s13204-020-01267-4

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