# Mixed convection heat transfer of water about a vertical surface of variable heat flux with density inversion

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## Abstract

Laminar mixed convection boundary layer flow of water about a vertical plate is studied including density inversion effects. The plate surface is subjected to an arbitrary heat flux that is a function of vertical distance from the leading edge. Both aiding and opposing mixed convection situations are considered in the study. The dimensionless forms of stream function and temperature are expanded in terms of perturbation elements and universal functions. The differentials of the heat flux, which are functions of vertical distance, are used as perturbation elements to obtain universal functions. The results for universal functions required to find temperature and velocity profiles are obtained. The obtained universal functions are valid for any arbitrary wall heat flux variation. The universal applicability of results is demonstrated for power-law variation of wall heat flux. The velocity and temperature variation, boundary layer thickness, Nusselt numbers and skin friction coefficient are presented for various values of mixed convection parameter, wall flux power index, for both aiding and opposing mixed convection. For a given combination of Grashof and Reynolds numbers, the heat transfer rates and skin friction coefficient are found to increase almost linearly with wall flux power index, in the parameter range of the study, for both the cases of aiding and opposing mixed convection. The present results of special cases are found to match well with the results available in the literature.

## Keywords

Mixed convection variable wall heat flux density inversion of water heat transfer rates## References

- 1.Wilks G 1974 The flow of a uniform stream over a semi-infinite vertical flat plate with uniform surface heat flux.
*Int. J. Heat Mass Transfer*17: 743–753CrossRefGoogle Scholar - 2.Carey V P and Gebhart B 1982 Transport at large down-stream distances in mixed convection flow adjacent to a vertical uniform heat flux surface.
*Int. J. Heat Mass Transfer*25(2): 255–266CrossRefGoogle Scholar - 3.Moulic S G and Yao L S 2009 Mixed convection along a semi-infinite vertical flat plate with uniform surface heat flux.
*J. Heat Transfer—Trans. ASME*131: 225021–225028Google Scholar - 4.Merkin J H and Mahmood T 1989 Mixed convection boundary layer similarity solutions: prescribed wall heat flux.
*J. Appl. Math. Phys.*40: 51–68MathSciNetzbMATHGoogle Scholar - 5.Merkin J H, Pop I and Mahmood T 1991 Mixed convection on a vertical surface with a prescribed heat flux: the solution for small and large Prandtl numbers.
*J. Eng. Math.*25: 165–190MathSciNetCrossRefGoogle Scholar - 6.Wickern G 1991 Mixed convection from an arbitrarily inclined semi-infinite flat plate-I, the influence of the inclination angle.
*Int. J. Heat Mass Transfer*34(8): 1935–1945CrossRefGoogle Scholar - 7.Wickern G 1991 Mixed convection from an arbitrarily inclined semi-infinite flat plate-II. The influence of the Prandtl number.
*Int. J. Heat Mass Transfer*34(8): 1947–1957CrossRefGoogle Scholar - 8.Yeh H M, Tsai S W and Yang C C 1987 Heat and mass transfer in mixed convection over a horizontal plane.
*Numer. Heat Transfer*12(2): 229–242CrossRefGoogle Scholar - 9.Armaly B F, Chen T S and Ramachandran N 1987 Correlations for laminar mixed convection on vertical, inclined and horizontal flat plates with uniform surface heat flux.
*Int. J. Heat Mass Transfer*30(2): 405–405CrossRefGoogle Scholar - 10.Trimbitas R, Grosan T and Pop I 2015 Mixed convection boundary layer flow past vertical flat plate in nano fluid: case of prescribed wall heat flux.
*Appl. Math. Mech.*36(8): 1091–1104MathSciNetCrossRefGoogle Scholar - 11.Ranganathan P and Viskanta R 1984 Mixed convection boundary-layer flow along a vertical surface in a porous medium.
*Numer. Heat Transfer*7(3): 305–317CrossRefGoogle Scholar - 12.Ahmad S and Pop I 2010 Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nano fluids.
*Int. Commun. Heat Mass Transfer*37: 987–991CrossRefGoogle Scholar - 13.Mahmoud M A A 2010 Variable fluid properties and thermal radiation effects on mixed convection flow over a horizontal surface.
*Int. J. Comput. Methods Eng. Sci. Mech.*11: 299–303CrossRefGoogle Scholar - 14.Sawant S M and Rao C G 2010 Combined conduction–mixed convection–surface radiation from a uniformly heated vertical plate.
*Chem. Eng. Commun.*197: 881–899CrossRefGoogle Scholar - 15.Pal D 2015 Unsteady convective boundary layer flow and heat transfer over a stretching surface with non-uniform heat source/sink and thermal radiation.
*Int. J. Comput. Methods Eng. Sci. Mech.*16(3): 170–181Google Scholar - 16.Vighnesam N V and Soundalgekar V M 1997 Combined free and forced convection flow of water at 4\(^\circ \)C from a vertical plate with variable temperature.
*Indian J. Eng. Mater. Sci.*5: 124–126Google Scholar - 17.Lin D S and Gebhart B 1986 Buoyancy-induced flow adjacent to a horizontal surface submerged in porous medium saturated with cold water.
*Int. J. Heat Mass Transfer*29(4): 611–623CrossRefGoogle Scholar - 18.Kumaran V and Pop I 2006 Steady free convection boundary layer over a vertical flat plate embedded in a porous medium filled with water at 4\(^\circ \)C.
*Int. J. Heat Mass Transfer*49: 3240–3252CrossRefGoogle Scholar - 19.Hussain T and Afzal N 1988 Mixed convection boundary layer flow on a horizontal plate in a uniform stream.
*Int. J. Heat Mass Transfer*31(12): 2505–2516CrossRefGoogle Scholar - 20.Kumari M 2001 Variable viscosity effects on free and mixed convection boundary-layer flow from a horizontal surface in a saturated porous medium – variable heat flux.
*Mech. Res. Commun.*28(3): 339–348CrossRefGoogle Scholar - 21.Gavara M R, Dutta P and Seetharamu K N 2012 Mixed convection adjacent to non-isothermal vertical surfaces.
*Int. J. Heat Mass Transfer*55(17–18): 4580–4587CrossRefGoogle Scholar - 22.Seetharamu K N and Dutta P 1988 Mixed convection about a non-isothermal vertical surface in a porous medium.
*Int. J. Numer. Methods Fluids*8(6): 723–735CrossRefGoogle Scholar - 23.Tong W and Koster J N 1993 Natural convection of water in a rectangular cavity including density inversion.
*Int. J. Heat Fluid Flow*14(4): 366–375CrossRefGoogle Scholar - 24.Schlichting H and Gersten K 2001
*Boundary layer theory*, 8th ed. Berlin, Heidelberg: Springer-VerlagzbMATHGoogle Scholar - 25.Van Dyke M D 1975
*Perturbation methods in fluid mechanics*. New York: Parabolic PresszbMATHGoogle Scholar - 26.Sparrow E M and Lin S H 1965 Boundary layers with prescribed heat flux—application to simultaneous convection and radiation.
*Int. J. Heat Mass Transfer*8(3): 437–448CrossRefGoogle Scholar