Self-Similar Solution of Radial Stagnation Point Flow and Heat Transfer of a Viscous, Compressible Fluid Impinging on a Rotating Cylinder

  • Asghar B. Rahimi
  • Hamid MohammadiunEmail author
  • Mohammad Mohammadiun
Research Paper


In this study, the radial stagnation point flow of strain rate \(\bar{k}\) impinging on a cylinder rotating at constant angular velocity ω and its heat transfer are investigated. Reduction in the Navier–Stokes equations and energy equation to primary nonlinear ordinary differential equation systems is obtained by use of appropriate transformations when the angular velocity and wall temperature or wall heat flux all are constant. The impinging free stream is steady and normal to the surface from all sides, and the range of Reynolds number variation (\(Re = \bar{k}a^{2} /2\upsilon\)) is 0.1–1000 in which a and υ are cylinder radius and kinematic viscosity, respectively. Flow results are presented for selected values of compressibility factor and different values of Prandtl numbers along with shear stress and Nusselt number. For all values of Reynolds numbers and surface temperature or surface heat flux, as compressibility factor increases the radial velocity field, the heat transfer coefficient and the wall shear stress increase, whereas the angular velocity field decreases. The rotating movement of the cylinder does not have any effect on the radial component of the velocity, but its increase increases the angular component of the fluid velocity field and the surface shear stress.


Stagnation point flow Constant angular velocity Heat transfer Compressibility factor Constant wall temperature and heat flux 

List of symbols


Cylinder radius

\(c(\eta )\)

Density ratio

\(f(\eta )\)

Function of \(\eta\)

\(G(\eta )\)

Function of \(\eta\)


Thermal conductivity


Free stream strain rate


Fluid pressure


Non-dimensional pressure


Prandtl number


Heat flux at the wall

\(r,\phi ,z\)

Cylindrical coordinates

\(Re = \frac{{\bar{k}a^{2} }}{2\upsilon }\)

Reynolds number




Wall temperature

\(T_{\infty }\)

Free stream temperature


Radial component of the velocity


Angular component of the velocity


Axial component of the velocity


Nusselt number

Greek symbols


Compressibility factor

\(\varGamma (\eta )\)

Function related to density


Similarity variable

\(\theta (\eta )\)

Non-dimensional temperature




Kinematic viscosity


Angular velocity of the cylinder

\(\rho (\eta )\)

Fluid density

\(\rho_{\infty }\)

Free stream density


Shear stress


Stream function

\(\hat{\psi } = \frac{\psi }{{0.5\bar{k}a^{3} }}\)

Normalized stream function


  1. Abbasi AS, Rahimi AB (2009a) Non-axisymmetric three-dimensional stagnation-point flow and heat transfer on a flat plate. J Fluids Eng 131(7):074501.1–074501.5Google Scholar
  2. Abbasi AS, Rahimi AB (2009b) Three-dimensional stagnation-point flow and heat transfer on a flat plate with transpiration. J Thermophys Heat Transfer 23(3):513–521Google Scholar
  3. Abbasi AS, Rahimi AB (2012) Investigation of two-dimensional stagnation-point flow and heat transfer impinging on a flat plate. J Heat Transf Tech Brief 134(6):064501Google Scholar
  4. Abbasi AS, Rahimi AB, Niazman H (2011) Exact solution of three-dimensional unsteady stagnation flow on a heated plate. J Thermodyn Heat Transf 25(1):55–58Google Scholar
  5. Afzal N, Ahmad S (1975) Effect of suction and injection on self-similar solutions of second-order boundary layer equations. Int J Heat Mass Transf 18:607–614zbMATHGoogle Scholar
  6. Chamkha AJ (2000) Flow of two-immiscible fluids in porous and nonporous channels. J Fluids Eng 122(1):117–124Google Scholar
  7. Chamkha AJ, Ahmed SE (2011) Similarity solution for unsteady MHD flow near a stagnation point of a three-dimensional porous body with heat and mass transfer, heat generation/absorption and chemical reaction. J Appl Fluid Mech 4(1):87–94zbMATHGoogle Scholar
  8. Chamkha AJ, Khaled AA (2000) Similarity solutions for hydromagnetic mixed convection heat and mass transfer for Hiemenz flow through porous media. Int J Numer Meth Heat Fluid Flow 10(1):94–115zbMATHGoogle Scholar
  9. Cunning GM, Davis AMJ, Weidman PD (1998) Radial stagnation flow on a rotating cylinder with uniform transpiration. J Eng Math 33:113–128MathSciNetzbMATHGoogle Scholar
  10. Davey A (1951) Boundary layer flow at a Saddle point of attachment. J Fluid Mech 10:593–610MathSciNetzbMATHGoogle Scholar
  11. Gersten K, Papenfuss HD, Gross JF (1978) Influence of the Prandtl number on second-order heat transfer due to surface curvature at a three-dimensional stagnation point. Int J Heat Mass Transf 21:275–284Google Scholar
  12. Gorla RSR (1976) Heat transfer in an axisymmetric stagnation flow on a cylinder. Appl Sci Res 32:541–553Google Scholar
  13. Gorla RSR (1977) Unsteady laminar axisymmetric stagnation flow over a circular cylinder. Dev Mech 9:286–288Google Scholar
  14. Gorla RSR (1978a) Nonsimilar axisymmetric stagnation flow on a moving cylinder. Int J Sci 16:392–400zbMATHGoogle Scholar
  15. Gorla RSR (1978b) Transient response behavior of an axisymmetric stagnation flow on a circular cylinder due to time-dependent free stream velocity. Lett Appl Eng Sci 16:493–502zbMATHGoogle Scholar
  16. Gorla RSR (1979) Unsteady viscous flow in the vicinity of an axisymmetric stagnation-point on a cylinder. Int. Sci. 17:87–93Google Scholar
  17. Grosch CE, Salwen H (1982) Oscillating stagnation-point flow. Proc R Soc Lond A384:175–190MathSciNetzbMATHGoogle Scholar
  18. Hiemenz K (1911) Die grenzchicht an einem in den gleichformingen Flussigkeitsstrom eingetauchten geraden KreisZylinder. Dinglers Polytech J 326:321–410Google Scholar
  19. Homann FZ (1936) Der EINFLUSS GROSSER Zahighkeit bei der Strmung um den Zylinder und um die Kugel. Z Angew Math Mech 16:153–164zbMATHGoogle Scholar
  20. Hong L, Wang CY (2009) Annular axisymmetric stagnation flow on a moving cylinder. Int J Eng Sci 47:141–152MathSciNetzbMATHGoogle Scholar
  21. Howarth L (1951) The boundary layer in three-dimensional flow. Part II. The flow near stagnation point. Philos Mag 42:1433–1440MathSciNetzbMATHGoogle Scholar
  22. Katz A (1972) Transformations of the compressible boundary layer equations. SIAM J Allied Math 22(4):604–611zbMATHGoogle Scholar
  23. Kumari M, Nath G (1980) Unsteady compressible 3-dimensional boundary layer flow near an axisymmetric stagnation point with mass transfer. Int J Eng Sci 18:1285–1300zbMATHGoogle Scholar
  24. Kumari M, Nath G (1981) Self-similar solution of unsteady compressible three-dimensional stagnation-point boundary layers. J Appl Math Phys 32:267–276Google Scholar
  25. Libby PA (1967) Heat and mass transfer at a general three-dimensional stagnation point. AIAA J 5(3):507–517zbMATHGoogle Scholar
  26. Magyari E, Chamkha AJ (2008) Exact analytical results for the thermosolutal MHD Marangoni boundary layers. Int J Therm Sci 47(7):848–857Google Scholar
  27. Mohammadiun H, Rahimi AB (2012) Stagnation-point flow and heat transfer of a viscous, compressible fluid on a cylinder. J Thermophys Heat Transfer 26(3):494–502Google Scholar
  28. Mohammadiun H, Rahimi AB, Kianifar A (2013) Axisymmetric stagnation-point flow and heat transfer of a viscous, compressible fluid on a cylinder with constant heat flux. Sci Iran B 20(1):185–194Google Scholar
  29. Mudhaf AA, Chamkha AJ (2005) Similarity solutions for MHD thermosolutal Marangoni convection over a flat surface in the presence of heat generation or absorption effects. Heat Mass Transf 42:112–121Google Scholar
  30. Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1997) Numerical recipes, the art of scientific computing. Cambridge University Press, CambridgezbMATHGoogle Scholar
  31. Rahimi AB, Saleh R (2007) Axisymmetric stagnation-point flow and heat transfer of a viscous fluid on a rotating cylinder with time-dependent angular velocity and uniform transpiration. J Fluids Eng 129:107–115Google Scholar
  32. Rahimi AB, Saleh R (2008) Similarity solution of unaxisymmetric heat transfer in stagnation-point flow on a cylinder with simultaneous axial and rotational movements. J Heat Transf Tech Brief 130:054502-1–054502-5Google Scholar
  33. Rahimi AB, Mohammadiun H, Mohammadiun M (2016) Axisymmetric stagnation flow and heat transfer of a compressible fluid impinging on a cylinder moving axially. ASME J Heat Transf 138(2):022201Google Scholar
  34. Saleh R, Rahimi AB (2004) Axisymmetric stagnation-point flow and heat transfer of a viscous fluid on a moving cylinder with time-dependent axial velocity and uniform transpiration. J Fluids Eng 126:997–1005Google Scholar
  35. Subhashini SV, Nath G (1999) Unsteady compressible flow in the stagnation region of two-dimensional and axisymmetric bodies. Acta Mech 134:135–145zbMATHGoogle Scholar
  36. Takhar HS, Chamkha AJ, Nath J (1999) Unsteady axisymmetric stagnation-point flow of a viscous fluid on a cylinder. Int J Eng Sci 37:1943–1957zbMATHGoogle Scholar
  37. Takhar HS, Chamkha AJ, Nath G (2000) Combined heat and mass transfer along a vertical moving cylinder with a free stream. Heat Mass Transf 36:237–246Google Scholar
  38. Takhar HS, Chamkha AJ, Nath G (2001) Unsteady three-dimensional MHD-boundary-layer flow due to the impulsive motion of a stretching surface. Acta Mech 146(1):59–71zbMATHGoogle Scholar
  39. Takhar HS, Chamkha AJ, Nath G (2002) Natural convection on a vertical cylinder embedded in a thermally stratified high-porosity medium. Int J Therm Sci 41(1):83–93Google Scholar
  40. Umavathi JC, Chamkha AJ (2005) Unsteady two-fluid flow and heat transfer in a horizontal channel. Heat Mass Transf 42:81–90Google Scholar
  41. Wang CY (1973) Axisymmetric stagnation flow towards a moving plate. Am Inst Chem Eng J 19(5):1080–1082Google Scholar
  42. Wang CY (1974) Axisymmetric stagnation flow on a cylinder. Q Appl Math 32:207–213zbMATHGoogle Scholar

Copyright information

© Shiraz University 2018

Authors and Affiliations

  • Asghar B. Rahimi
    • 1
  • Hamid Mohammadiun
    • 2
    Email author
  • Mohammad Mohammadiun
    • 2
  1. 1.Faculty of EngineeringFerdowsi University of MashhadMashhadIran
  2. 2.Department of Mechanical EngineeringShahrood Branch, Islamic Azad UniversityShahroodIran

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