On model for Darcy–Forchheimer 3D nanofluid flow subject to heat flux boundary condition


This paper provides a theoretical study of 3D nanofluid flow with zero nanoparticles mass and constant heat fluxes conditions. An incompressible Newtonian nanoliquid saturates the permeable media describing the Darcy–Forchheimer (DF) relation. A bidirectional stretchable sheet has been considered to produce the three-dimensional flow. Appropriate variables are considered to change the PDEs into the ODEs. The obtained nonlinear framework is computed by the optimal homotopic technique. Outcomes of numerous emerging flow factors on concentration and the temperature of nanoparticles are explored. Heat transport rate and skin frictions have been tabulated and analyzed. The presented data reveal that temperature distribution is upgraded for larger estimations of Forchheimer number. Furthermore, the heat transport rate reduces when thermophoresis number enhances.

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\(u, v, w\) :

Velocity components (m s−1)

\(C_{\infty }\) :

Ambient fluid concentration

\(k\) :

Thermal conductivity (W m−1 K−1)

\(\nu\) :

Kinematic viscosity (m2 s−1)

\(T\) :

Temperature (K)

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

Drag coefficient

\(\mu\) :

Dynamic viscosity (Pa s)

\(a, b\) :

Stretching constants (s−1)

\(K^{*}\) :

Permeability of porous medium

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

Heat capacity of liquid (J Kg−3 K−1)

\(f^{\prime}, g'\) :

Dimensionless velocities

\(\phi\) :

Dimensionless concentration

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

Local Nusselt number

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

Thermophoresis number

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

Brownian movement number

\(\lambda\) :

Porosity parameter

\(\alpha\) :

Ratio parameter

\({\text{Re}}_{\text{x}} ,\;{\text{Re}}_{\text{y}}\) :

Local Reynolds numbers

\(x, y, z\) :

Coordinate axes (m)

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

Thermal diffusivity (m2 s−1)

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

Base fluid density (kg m−3)

\(F\) :

Non-uniform inertia coefficient

C :


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

Brownian diffusion coefficient (m2 s−1)

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

Wall heat flux (W m−2)

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

Thermophoretic diffusion coefficient (m2 s−1)

\(T_{\infty }\) :

Ambient fluid temperature (K)

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

Nanoparticles effective heat capacity (J kg−3K−1)

\(\zeta\) :

Dimensionless variable

\(\theta\) :

Dimensionless temperature

\({\text{Sc}}\) :

Schmidt number

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

Local Sherwood number

\({ \Pr }\) :

Prandtl number

\(C_{\text{fx}} , C_{\text{fy}}\) :

Skin friction coefficients

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

Forchheimer parameter

\(N_{j}^{*}\) :

Arbitrary constants


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This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant Number (D-400-130-1441). The authors, therefore, gratefully acknowledge DSR technical and financial support.

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Correspondence to Taseer Muhammad.

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Ullah, M.Z., Muhammad, T. & Mallawi, F. On model for Darcy–Forchheimer 3D nanofluid flow subject to heat flux boundary condition. J Therm Anal Calorim (2020). https://doi.org/10.1007/s10973-020-09892-5

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  • 3D flow
  • Darcy–Forchheimer porous space
  • Nanoparticles
  • Flux conditions
  • OHAM