Numerical study of nonlinear mixed convection inside stagnation-point flow over surface-reactive cylinder embedded in porous media

  • Kun Hong
  • Rasool Alizadeh
  • Mostafa Valizadeh Ardalan
  • Amireh Nourbakhsh
  • Nader Karimi
  • Yang Yang
  • Qingang XiongEmail author


Nonlinear mixed convection of heat and mass in a stagnation-point flow of an impinging jet over a solid cylinder embedded in a porous medium is investigated by applying a similarity technique. The problem involves a heterogenous chemical reaction on the surface of the cylinder and nonlinear heat generation in the porous solid. The conducted analysis considers combined heat and mass transfer through inclusions of Soret and Dufour effects and predicts the velocity, temperature and concentration fields as well as the average Nusselt and Sherwood number. It is found that intensification of the nonlinear convection results in development of higher axial velocities over the cylinder and reduces the thickness of thermal and concentration boundary layers. Hence, consideration of nonlinear convection can lead to prediction of higher Nusselt and Sherwood numbers. Further, the investigation reveals that the porous system deviates from local thermal equilibrium at higher Reynolds numbers and mixed convection parameter.


Nonlinear mixed convection Stagnation-point flow Local thermal non-equilibrium Nonlinear heat generation Soret effect Dufour effect 

List of symbols


Cylinder radius (m)


Interfacial surface area per unit volume of the porous medium (m−1)


Biot number \({\text{Bi}} = \frac{{h_{\text{sf}} a_{\text{sf}} \cdot a}}{{4k_{\text{f}} }}\)


Fluid concentration (kg m−3)


Specific heat at constant pressure (J K−1 kg−1)


Concentration (kg m−3)


Molecular diffusion coefficient (m2 s−1)


Dufour number \({\text{Df}} = \frac{{D \cdot k_{\text{f}} }}{{C_{\text{s}} \cdot C_{\text{p}} }}\frac{{C_{\infty } }}{{\left( {T_{\text{w}} - T_{\infty } } \right)\upsilon }}\)

\(f(\eta )\)

Function related to u-component of velocity

\(f^{\prime } (\eta )\)

Normalised velocity related to w component


Heat transfer coefficient (W K−1 m−2)


Interstitial heat transfer coefficient (W K−1 m−2)


Thermal conductivity (W K−1 m−2)


Freestream strain rate (s−1)


Permeability of the porous medium (m2)


Mass transfer coefficient (m s−1)


Kinetic constant (kg m−2 s−1)


Thermal diffusion ratio


Ratio of concentration to thermal buoyancy forces \(N^{*} = \frac{{\beta_{3} \cdot C_{\infty } }}{{\beta_{1} \left( {T_{\text{w}} - T_{\infty } } \right)}}\)


Nusselt number


Nusselt number averaged over the surface of the cylinder


Pressure (Pa)


Dimensionless fluid pressure


The initial pressure (Pa)

\({ \Pr }\)

Prandtl number


Heat source parameter \(Q_{\text{h}} = \frac{{Q_{1} a^{2} }}{{4k_{\text{f}} }}\)


Mass flow at the wall (kg m−2 s−1)


Heat flow at the wall (W m−2)


Radial coordinate


Freestream Reynolds number \({\text{Re}} = \frac{{\bar{k} \cdot a^{2} }}{2\upsilon }\)


Schmidt number \({\text{Sc}} = \frac{\upsilon }{D}\)


Soret number \({\text{Sr}} = \frac{{D \cdot k_{\text{f}} }}{{T_{\infty } }}\frac{{\left( {T_{\text{w}} - T_{\infty } } \right)}}{{C_{\infty } \cdot \alpha }}\)


Sherwood number


Average Sherwood number


Mean fluid temperature (K)

\(u, w\)

Velocity components along (\(r - \varphi - z\))-axis (m s−1)


Axial coordinate

Greek symbols


Thermal diffusivity (m2 s−1)


Nonlinear mixed convection parameter for concentration \(\beta_{\text{C}} = \frac{{\beta_{4} \cdot C_{\infty } }}{{\beta_{3} }}\)


Nonlinear mixed convection parameter for temperature \(\beta_{\text{t}} = \frac{{\beta_{2} \left( {T_{\text{w}} - T_{\infty } } \right)}}{{\beta_{1} }}\)


Modified conductivity ratio \(\gamma = \frac{{k_{\text{f}} }}{{k_{\text{s}} }}\)


Damköhler number \(\gamma^{*} = \frac{{k_{\text{R}} \cdot a}}{2D}\frac{1}{{C_{\infty } }}\)


Similarity variable, \(\eta = \left( {\frac{r}{a}} \right)^{2}\)

\(\theta (\eta )\)

Non-dimensional temperature


Permeability parameter, \(\lambda = \frac{{a^{2} }}{{4k_{1} }}\)




Dimensionless temperature difference \(\varLambda = \frac{{\left( {T_{\text{w}} - T_{\infty } } \right)}}{{T_{\infty } }}\)


Dynamic viscosity of fluid (N s m−2)


Kinematic viscosity of the fluid (m2 s−1)


Density of fluid (kg m−3)


Dimensionless concentration


Angular (circumferential) coordinate



Condition on the surface of the cylinder


Far field







This work was supported by the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (17KJA530001) and Foundation of Huaian Municipal Science and Technology Bureau (HAA201734). Dr. Hong thanks the support of Six Talent Peaks Project of Jiangsu Province (2018-XNY-004).


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2020

Authors and Affiliations

  • Kun Hong
    • 1
  • Rasool Alizadeh
    • 2
  • Mostafa Valizadeh Ardalan
    • 3
  • Amireh Nourbakhsh
    • 4
  • Nader Karimi
    • 5
  • Yang Yang
    • 6
  • Qingang Xiong
    • 1
    • 7
    Email author
  1. 1.National and Local Joint Engineering Research Center for Mineral Salt Deep Utilization, Key Laboratory for Palygorskite Science and Applied Technology of Jiangsu ProvinceHuaiyin Institute of TechnologyHuai’anChina
  2. 2.Department of Mechanical Engineering, Quchan BranchIslamic Azad UniversityQuchanIran
  3. 3.Department of Mechanical EngineeringShahrood University of TechnologyShahroodIran
  4. 4.Department of Mechanical EngineeringBu-Ali Sina UniversityHamedanIran
  5. 5.School of EngineeringUniversity of GlasgowGlasgowUK
  6. 6.Stanley Black & Decker, Inc.TownsonUSA
  7. 7.IT Innovation CenterGeneral MotorsWarrenUSA

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