Transport in Porous Media

, Volume 76, Issue 3, pp 309–325 | Cite as

Mixed Convection in Water Near the Density Extremum Along a Vertical Plate with Sinusoidal Surface Temperature Variation Embedded in a Porous Medium

  • Asterios PantokratorasEmail author


A steady laminar boundary layer flowing along a vertical plate immersed in a Darcy–Brinkman porous medium saturated with water at 4°C is studied. The plate temperature varies sinusoidally along the plate between 0 and 8°C where the density of water varies parabolically and is almost symmetrical at about 4°C. Except for the existence of the buoyancy force, it is assumed that either the plate moves upwards or the ambient water moves upwards (moving stream). The results are obtained with the direct numerical solution of the boundary layer equations taking into account the temperature dependence of water thermophysical properties (ρ, μ and c p). Results are presented for the wall temperature gradient and the wall shear stress along the plate for free convection and mixed convection. Temperature and velocity profiles are also presented.


Sinusoidal temperature Water Porous medium 

List of symbols




Specific heat under constant pressure


Half wavelength


Gravitational acceleration


Thermal conductivity


Mean bulk modulus of water


Porous medium permeability








Vertical velocity


Horizontal velocity


Specific volume of water


Vertical coordinate


Horizontal coordinate

Greek symbols


Dynamic viscosity


Kinematic viscosity




Shear stress








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  1. Blasius H.: Grenzschichten in Flussigkeiten mit kleiner Reibung. Z. Math. Phys. 56, 1–37 (1908)Google Scholar
  2. Fofonoff N.P.: Physical properties of seawater: a new salinity scale and equation of state for seawater. J. Geophys. Res. 90(C2), 3332–3342 (1985)CrossRefGoogle Scholar
  3. Holzbecher E.: Numerical studies on thermal convection in cold groundwater. Int. J. Heat Mass Transf. 40, 605–612 (1997)CrossRefGoogle Scholar
  4. Hooman K., Gurgenci H.: Effects of temperature-dependent viscosity on Benard convection in a porous medium using a non-Darcy model. Int. J. Heat Mass Transf. 51, 1139–1149 (2008a)CrossRefGoogle Scholar
  5. Hooman K., Gurgenci H.: Heatline visualization of natural convection in a porous cavity occupied by a fluid with temperature-dependent viscosity. J. Heat Transf. 130, 012501 (2008b)CrossRefGoogle Scholar
  6. Jian L., Ingham D.B., Pop I.: Natural convection from a vertical flat plate with a surface temperature oscillation. Int. J. Heat Mass Transf. 44, 2311–2322 (2001)CrossRefGoogle Scholar
  7. Kakac S., Yener Y.: Convective Heat Transfer, 2nd edn. CRC Press, Boca Raton (1995)Google Scholar
  8. Kandaswamy P., Eswaramurthi M.: Density maximum effect on buoyancy-driven convection of water in a porous cavity with variable side wall temperatures. Int. J. Heat Mass Transf. 51, 1955–1961 (2008)CrossRefGoogle Scholar
  9. Kao T.T.: Locally nonsimilar solution for laminar free convection adjacent to a vertical wall. ASME J. Heat Transf. 98, 321–322 (1976)Google Scholar
  10. Kaviany M.: Principles of Convective Heat Transfer, 2nd edn. Springer, New York (2001)Google Scholar
  11. Kukulka D.J., Gebhart B., Mollendorf J.C.: Thermodynamic and transport properties of pure and saline water. Adv. Heat Transf. 18, 325–363 (1987)Google Scholar
  12. Ling S.C., Nazar R., Pop I., Merkin J.H.: Steady mixed convection boundary-layer flow over a vertical flat surface in a porous medium filled with water at 4°C: variable surface heat flux. Transp. Porous Med. 70, 307–321 (2007)CrossRefGoogle Scholar
  13. Malkovsky V.I., Pek A.A.: Conditions for the onset of thermal convection of a homogeneous fluid in a vertical fault. Petrology 5, 381–387 (1997)Google Scholar
  14. Malkovsky V.I., Pek A.A.: Onset of thermal convection of a single-phase fluid in an open vertical fault. Izvestiya Phys. Solid Earth 40, 672–679 (2004)Google Scholar
  15. Na T.Y.: Numerical solution of natural convection flow past a non-isothermal vertical flat plate. Appl. Sci. Res. 33, 519–543 (1978)Google Scholar
  16. Nield, D.A., Bejan, A.: Convection in Porous Media, 3rd edn. Springer (2006)Google Scholar
  17. Ostrach, S.: An analysis of laminar free convection flow and heat transfer about a flat plate parallel to the direction of the generating body force. NASA Technical Report 1111 (1953)Google Scholar
  18. Pantokratoras A.: Laminar free-convection over a vertical plate with uniform blowing or suction in water with variable physical properties. Int. J. Heat Mass Transf. 45, 963–977 (2002)CrossRefGoogle Scholar
  19. Pantokratoras A.: Laminar natural convection in water near the density extremum along a vertical plate with sinusoidal surface temperature variation. Acta Mech. 172, 211–218 (2004a)CrossRefGoogle Scholar
  20. Pantokratoras A.: Laminar assisting mixed convection heat transfer from a vertical isothermal plate to water with variable physical properties. Heat Mass Transf. 40, 581–585 (2004b)CrossRefGoogle Scholar
  21. Patankar S.V., Spalding D.B.: Heat and Mass Transfer in Boundary Layers. Intertext, London (1970)Google Scholar
  22. Patankar S.V.: Numerical Heat Transfer and Fluid Flow. McGraw-Hill Book Company, New York (1980)Google Scholar
  23. Pop I., Gorla R.S.R., Rashidi M.: The effect of variable viscosity on flow and heat transfer to a continuous moving flat plate. Int. J. Eng. Sci. 30, 1–6 (1992)CrossRefGoogle Scholar
  24. Rees D.A.S.: The effect of streamwise surface temperature variations on vertical free convection. Int. J. Heat Mass Transfer 42, 2455–2464 (1999)CrossRefGoogle Scholar
  25. Saeid N.H.: Maximum density effects on natural convection in a porous cavity under thermal non-equilibrium condition. Acta Mech. 188, 55–68 (2007)CrossRefGoogle Scholar
  26. Sakiadis B.C.: Boundary layer behavior on continuous solid surfaces: the boundary layer on a continuous flat surface. AIChE J. 7, 221–225 (1961)CrossRefGoogle Scholar
  27. Schlichting H., Gersten K.: Boundary Layer Theory, 9th edn. Springer, Berlin (2003)Google Scholar
  28. Schmidt E., Beckmann W., Pohlhausen E.: Das Temperatur- und Geschwindigkeitsfeld von einer Warme abgebenden, senkrechten Platte bei naturlicher Konvektion. Forsch. Arb. Ing-Wes. 1, 391–404 (1930)Google Scholar
  29. Sparrow E.M., Gregg J.L.: Similar solutions for free convection from a nonisothermal vertical plate. Trans. Am. Soc. Mech. Eng. 80, 379–386 (1958)Google Scholar
  30. White F.: Viscous Fluid Flow, 2nd edn. McGraw-Hill, New York (1991)Google Scholar
  31. White F.: Viscous Fluid Flow, 3rd edn. McGraw-Hill, New York (2006)Google Scholar
  32. Yang J., Jeng D.R., DeWitt K.J.: Laminar free convection from a vertical plate with nonuniform surface conditions. Numer. Heat Transfer 5, 165–184 (1982)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.School of EngineeringDemocritus University of ThraceXanthiGreece

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