Abstract
This article investigates the natural convection flow of viscous incompressible fluid in a channel formed by two infinite vertical parallel plates. Fully developed laminar flow is considered in a vertical channel with steady-periodic temperature regime on the boundaries. The effect of internal heating by viscous dissipation is taken into consideration. Separating the velocity and temperature fields into steady and periodic parts, the resulting second order ordinary differential equations are solved to obtain the expressions for velocity, and temperature. The amplitudes and phases of temperature and velocity are also obtained as well as the rate of heat transfer and the skin friction on the plates. In presence of viscous dissipation, fluids of relatively small Prandtl number has higher temperature than the channel plates and as such, heat is being transferred from the fluid to the plate.
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E.M. Sparrow and J.L. Gregg, Nearly quasi-steady free convection heat transfer in gases, Trans. ASME. Ser. C. J. Heat Transfer, 1960, Vol. 82, No. 3, P. 258–260.
P.M. Chung and A.D. Anderson, Unsteady laminar free convection, Trans. ASME. Ser. C. J. Heat Transfer, 1961, Vol. 83, No. 4, P. 473–478.
R.J. Nanda and V.P. Sharma, Free convection laminar boundary layers in oscillatory flow, J. Fluid Mech., 1963, Vol. 15, Pt 3, P. 419–428.
M.D. Kelleher and K.T. Yang, Heat Transfer Response of Laminar Free convection boundary layers along a vertical heated plate to surface temperature oscillation, ZAMP, 1968, Vol. 19, No. 1, P. 31–44.
J.W. Yang, C. Scaccia, and J. Goodman, Laminar Natural Convection about vertical plates with oscillatory surface temperature, Trans. ASME. Ser. C. J. Heat Transfer, 1974, Vol. 96, No. 1, P. 9–14.
J.A. Schetz and R. Eichhorn, Unsteady natural convection in the vicinity of a doubly infinite vertical plate, Trans. ASME. Ser. C. J. Heat Transfer, 1962, Vol. 84, No. 3, P. 334–338.
P.K. Muhuri and A.S. Gupta, Free convection boundary layer on a flat plate due to small fluctuations in the surface temperature, ZAMM, 1979, Vol. 59, No. 2, P. 117–121.
N. Rott and M.L. Rosenzweig, On the response of the laminar boundary layer to small fluctuations of the free-stream velocity, J. Aerospace Sci., 1960, Vol. 27, No. 10, P. 741–748.
A. Bar-Cohen and W.M. Rohsenow, Thermally optimum spacing of vertical, natural convection cooled, parallel plates, Trans. ASME. Ser. C. J. Heat Transfer, 1984, Vol. 106, No. 1, P. 116–123.
C.Y. Wang, Free convection between vertical plates with periodic heat input, Trans. ASME. Ser. C. J. Heat Transfer, 1988, Vol. 110, No. 2, P. 508–511.
A. Barletta and E. Rossi di Schio, Mixed convection flow in a vertical circular duct with time-periodic boundary condition: steady-periodic regime, Int. J. Heat Mass Transfer, 2004, Vol. 47, No. 14–16, P. 3187–3195.
B.K. Jha and A.O. Ajibade, Free convective flow of heat generating /absorbing fluid between vertical porous plates with periodic heat input, Int. Comm. Heat Mass Transfer, 2009, Vol. 36, No. 6, P. 624–631.
A.Z. Sahin, The effect of variable viscosity on the entropy generation and pumping power in a laminar fluid flow through a duct subjected to constant heat flux, Int. Comm. Heat Mass Transfer, 1999, Vol. 35, No. 6, P. 499–506.
S.H. Tasnim and S. Mahmud, Entropy generation in a vertical concentric channel with temperature dependent viscosity, Int. Comm. Heat Mass Transfer, 2002, Vol. 29, No. 7, P. 907–918.
H. Schlichting and K. Gersten, Boundary Layer Theory, 8th revised and enlarged edition, Springer-Verlag, New York, 2000.
S. Yasutomi, S. Bair, and W.O. Winer, An application of a free volume model to lubricant rheology. 1. Dependence of viscosity on temperature and pressure, J. Tribol. (Trans. ASME), 1984, Vol. 106, No. 2, P. 291–303.
B. Gebhart, Effects of viscous dissipation in natural convection, J. Fluid Mech., 1962, Vol. 14, Pt 2, P. 225–232.
O.D. Makinde, Thermal criticality in viscous reactive flows through channels with a sliding wall: An exploitation of the Hermite — Padé approximation method, Mathematical and Computer Modelling, 2008, Vol. 47, Nos. 5–6, P. 312–317.
M. Bister and K.A. Emanuel, Dissipative heating and hurricane intensity, Meteor. Atmos. Phys., 1998, Vol. 65, Nos. 3–4, P. 233–240.
E. Zanchini, Effect of viscous dissipation on mixed convection in a vertical channel with boundary condition of the third kind, Int. J. Heat Mass Transfer, 1998, Vol. 41, No. 23, P. 3949–3959.
A. Barletta, Laminar mixed convection with viscous dissipation in a vertical channel, Int. J. Heat Mass Transfer, 1998, Vol. 41, No. 22, P. 3501–3513.
A. Barletta and E. Rossi di Schio, Effect of viscous dissipation on mixed convection heat transfer in a vertical tube with uniform wall heat flux, Heat Mass Transfer, 2001, Vol. 38, No. 1–2, P. 129–140.
A. Barletta and E. Zanchini, Mixed convection with viscous dissipation in an inclined channel with prescribed wall temperatures, Int. J. Heat Mass Transfer, 2001, Vol. 44, No. 22, P. 4267–4275.
S. Businger and J.A. Businger, Viscous dissipation of turbulent kinetic energy in storms, J. Atmospheric Sci., 2001, Vol. 58, No. 28, P. 3793–3796.
E.M. Abo-Eldahab and M.A. El Aziz, Viscous dissipation and joule heating effects on MHD-free convection from a vertical plate with power-law variation in surface temperature in the presence of Hall and ion-slip currents, Appl. Math. Modelling, 2005, Vol. 29, No. 6, P. 579–595.
V.M. Soundalgekar, A.G. Uplekar, and B.S. Jaiswal, Transient free convection flow of dissipative fluid past an infinite vertical porous plate, Arch. Mech., 2004, Vol. 56, No. 1, P. 7–17.
T. Watanabe and I. Pop, Thermal boundary layers in magnetohydrodynamic flow over a flat plate in the presence of transverse magnetic field, Acta Mech., 1994, Vol. 105, No. 1–4, P. 233–238.
O.D. Makinde and R.L. Maserumule, Thermal criticality and entropy analysis for a variable viscosity Couette flow, Phys. Scr., 2008, Vol. 78, No. 1, P. 1–6.
O.D. Makinde, Irreversibility analysis of variable viscosity channel flow with convective cooling at the walls, Can. J. Phys., 2008, Vol. 86, No. 2, P. 383–389.
H. Hassan, N. Regnier, C. Pujos, and G. Defaye, The effect of viscous dissipation on the polymer temperature during injection molding, in: 5th Europ. Thermal Sci. Conf., The Netherlands, 2008.
J.H. Lienhard IV and J.H. Lienhard V, A Heat Transfer Textbook, 3rd edition (version 1.24), Phlogiston Press, Cambridge, MA, 2006, P. 707–718.
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Jha, B.K., Ajibade, A.O. Effect of viscous dissipation on natural convection flow between vertical parallel plates with time-periodic boundary conditions. Thermophys. Aeromech. 18, 561–571 (2011). https://doi.org/10.1134/S0869864311040056
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DOI: https://doi.org/10.1134/S0869864311040056