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Analysis of chemically reactive species with mixed convection and Darcy–Forchheimer flow under activation energy: a novel application for geothermal reservoirs

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Abstract

In the present article, an analysis has been performed to discuss the impact of steady mixed convection with Darcy–Forchheimer flow towards linear surface. Investigation has been achieved in the presence of Arrhenius activation energy and radiative heat flux which are associated with the heat and mass transport analysis which has not been performed so far. Porous media features are elaborated by utilizing Darcy–Forchheimer relation. Boundary-layer idea is employed for the simplification of governing expressions. The resulting set of mathematical expression is now solved with the help of bvp4c MATLAB package which applies a three-stage Lobatto IIIa finite-difference collocation scheme. Diagrams are drawn against pertinent parameters such as buoyancy forces ratio parameter, mixed convection parameter, porosity parameter, local inertia coefficient, activation energy, chemical reaction rate constant, Schmidt number, temperature difference ratio, exponentially fitted constant, magnetic parameter, radiation parameter, first-order and second-order slip parameter, suction or injection parameter and Prandtl number. It is observed that both mixed convection and activation energy parameters have an opposite impact on species profile. Also the present results are compared with those available in the literature for some cases, and an excellent agreement is found between them.

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Abbreviations

\(\bar{A}\), \(\bar{B}\) :

Constant

\(B(x)\) :

Magnetic field

\(C_{\text{w}}\) :

Wall concentration

\(C_{\text{o}}\) :

Reference concentration

\(U_{\text{w}}\) :

Wall velocity

\(U_{0}\) :

Reference velocity

\(T\) :

Temperature (K)

\(T_{\text{w}}\) :

Wall temperature

\(T_{\text{o}}\) :

Reference temperature

\(T_{\infty }\) :

Free-stream temperature

C p :

Specific heat \(({\text{J}}\,{\text{kg}}^{ - 1} \,{\text{K}}^{ - 1} )\)

C f :

Skin friction

f :

Dimensionless stream function

\(F_{\text{r}}\) :

Local inertial coefficient

k :

Thermal conductivity \(({\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1} )\)

K n :

Knudsen number

\(K_{\text{p}}\) :

Porosity parameter

L :

Characteristic length

N :

Ratio of thermal and solutal expansions

\(N_{\text{ux}}\) :

Local Nusselt number

Pr:

Prandtl number

R :

Radiation parameter

M :

Magnetic field \(({\text{A/m}})\)

Rex :

Local Reynolds number

\(V_{\text{o}}\) :

Initial strength of suction

E :

Activation energy

Sc:

Schmidt number

(u, v):

Components of velocity \(({\text{m}}\,{\text{s}}^{ - 1} )\)

(x, y):

Coordinate axes normal to sheet (m)

\(\mu\) :

Dynamic viscosity \(({\text{N}}\,{\text{s}}\,{\text{m}}^{ - 1} )\)

\(\theta\) :

Dimensionless temperature

S :

Suction/injection parameter

\(\gamma\) :

First-order slip

\(\delta\) :

Second-order slip

\(\lambda_{1}\) :

Mean free molecular path

\(\alpha\) :

Momentum accommodation

\(\varGamma\) :

Temperature difference ratio

\(\rho\) :

Density \(({\text{k}}\,{\text{g}}\,{\text{m}}^{ - 3} )\)

\(k^{ * }\) :

Mean absorption coefficient

\(\sigma\) :

Chemical reaction rate constant

\(\sigma_{1}\) :

Stefan–Boltzmann constant

\(\psi\) :

Stream function \(({\text{m}}^{ 2} \,{\text{s}}^{ - 1} )\)

\(\lambda\) :

Mixed convection parameter

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Acknowledgements

The corresponding author is profoundly grateful to the Higher Education Commission (HEC) for their financial support under Start-Up Research Grant Program (SRGP) with Project No. 2495.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Bacha Khan University, Charsadda, 24420, KPK, Pakistan

    Aaqib Majeed

  2. Department of Mathematics and Statistics, FBAS, IIUI, H-10, Islamabad, 44000, Pakistan

    Ahmad Zeeshan

  3. Department of Informatics, Faculty of Mathematics and Natural Sciences, University of Oslo, Oslo, Norway

    Farzan Majeed Noori

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Majeed, A., Zeeshan, A. & Noori, F.M. Analysis of chemically reactive species with mixed convection and Darcy–Forchheimer flow under activation energy: a novel application for geothermal reservoirs. J Therm Anal Calorim 140, 2357–2367 (2020). https://doi.org/10.1007/s10973-019-08978-z

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