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Research Papers: Conduction

Exact Multiple Solutions for the Slip Flow and Heat Transfer in a Converging Channel

[+] Author and Article Information
Mustafa Turkyilmazoglu

Department of Mathematics,
Hacettepe University,
Beytepe,
Ankara 06532, Turkey
e-mail: turkyilm@hacettepe.edu.tr

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received June 16, 2014; final manuscript received March 22, 2015; published online June 2, 2015. Assoc. Editor: Peter Vadasz.

J. Heat Transfer 137(10), 101301 (Oct 01, 2015) (8 pages) Paper No: HT-14-1406; doi: 10.1115/1.4030307 History: Received June 16, 2014; Revised March 22, 2015; Online June 02, 2015

A special case of Falkner–Skan flows past stretching boundaries is considered when the momentum and thermal slip boundary conditions are allowed at the boundary. Exact analytical solutions are found for the converging channel (wedge nozzle). The solutions are shown to be unique, double, or triple depending on the slip parameter and wall moving parameter. The provided closed-form analytical solutions are rare class of exact solutions for the Falkner–Skan flow equations. Thresholds of existence of multiple solutions are determined. For each flow solutions, the corresponding energy equation is also exactly solved when the internal heat generated by viscous dissipation can be neglected or numerically integrated when the viscous dissipation is significant. Analytic and numeric values of the rate of heat transfer affected by the presence of a surface temperature jump are also worked out. The possibility of realistic physical solution out of multiple solutions is finally discussed.

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References

Falkner, V. M., and Skan, S. W., 1931, “Some Approximate Solutions of the Boundary-Layer Equations,” Phiols. Mag., 12, pp. 865–896. [CrossRef]
Craven, A. H., and Peletier, L. A., 1972, “On the Uniqueness of Solutions of the Falkner–Skan Equation,” Mathematika, 19(1), pp. 135–138. [CrossRef]
Pantokratoras, A., 2006, “The Falkner–Skan Flow With Constant Wall Temperature and Variable Viscosity,” Int. J. Therm. Sci., 45(4), pp. 378–389. [CrossRef]
Riley, N., and Weidman, P. D., 1989, “Multiple Solutions of the Falkner–Skan Equation for Flow Past a Stretching Boundary,” SIAM J. Appl. Math., 49(5), pp. 1350–1358. [CrossRef]
Sparrow, E. M., and Abraham, J. P., 2005, “Universal Solutions for the Streamwise Variation of the Temperature of a Moving Sheet in the Presence of a Moving Fluid,” Int. J. Heat Mass Transfer, 48(15), pp. 3047–3056. [CrossRef]
Magyari, E., 2009, “Falkner–Skan Flows Past Moving Boundaries: An Exactly Solvable Case,” Acta Mech., 203(1–2), pp. 13–21. [CrossRef]
Turkyilmazoglu, M., 2011, “Multiple Solutions of Heat and Mass Transfer of MHD Slip Flow for the Viscoelastic Fluid Over a Stretching Sheet,” Int. J. Therm. Sci., 50(11), pp. 2264–2276. [CrossRef]
Turkyilmazoglu, M., 2012, “Exact Analytical Solutions for Heat and Mass Transfer of MHD Slip Flow in Nanofluids,” Chem. Eng. Sci., 84, pp. 182–187. [CrossRef]
Turkyilmazoglu, M., 2013, “Heat and Mass Transfer of MHD Second Order Slip Flow,” Comput. Fluids, 71, pp. 426–434. [CrossRef]
Turkyilmazoglu, M., 2015, “A Note on the Correspondence Between Certain Nanofluid Flows and Standard Fluid Flows,” ASME J. Heat Transfer, 137(2), p. 024501. [CrossRef]
Martin, M. J., and Boyd, I. D., 2010, “Falkner–Skan Flow Over a Wedge With Slip Boundary Conditions,” J. Thermophys. Heat Transfer, 24(2), pp. 263–270. [CrossRef]
Hutchins, D. K., Harper, M. H., and Felder, R. L., 1995, “Slip Correction Measurements for Solid Spherical Particles by Modulated Dynamic Light Scattering,” Aerosol Sci. Technol., 22(2), pp. 202–218. [CrossRef]
Harley, J. C., Huang, Y. F., Bau, H. H., and Zemel, J. N., 1995, “Gas-Flow in Microchannels,” J. Fluid Mech., 284, pp. 257–274. [CrossRef]
Mueller, T. J., and DeLaurier, J. D., 2003, “Aerodynamics of Small Vehicles,” Annu. Rev. Fluid Mech., 35, pp. 89–111. [CrossRef]
Turkyilmazoglu, M., 2015, “Slip Flow and Heat Transfer Over a Specific Wedge: An Exactly Solvable Falkner–Skan Equation,” J. Eng. Math. [CrossRef]
Pohlhausen, K., 1921, “Zur Näherungsweisen Integration der Differentialgleichung der Laminaren Grenzschicht,” J. Appl. Math. Mech. (ZAMM), 1, pp. 252–268. [CrossRef]
Magyari, E., 2007, “Backward Boundary Layer Heat Transfer in a Converging Channel,” Fluid Dyn. Res., 39(6), pp. 493–504. [CrossRef]
Turkyilmazoglu, M., 1998, “Linear Absolute and Convective Instabilities of Some Two- and Three Dimensional Flows,” Ph.D. thesis, University of Manchester, Manchester, UK.

Figures

Grahic Jump Location
Fig. 1

Sketch of the flow and coordinates

Grahic Jump Location
Fig. 2

(a) The existence domain of physical parameters L and F(0) for fixed Λ. (b) The critical parameters L and F(0) versus Λ.

Grahic Jump Location
Fig. 3

Dual velocity profiles for the specific values L = 2, Λ = −3, and Λ = −4

Grahic Jump Location
Fig. 4

(a) The existence domain of physical parameters L and F(0) for fixed Λ. (b) The critical parameters L and F(0) versus Λ.

Grahic Jump Location
Fig. 5

(a) Unique velocity profiles for the specific values L = 1, Λ = 2, and Λ = 4. (b) Triple velocity profiles for the specific values L = Λ = 2.

Grahic Jump Location
Fig. 6

Dual temperature profiles for the specific values L = 2 and Λ = −3 at two temperature jump conditions τ = 0 and τ = 2. (a) Pr = 1 and (b) Pr = 3.

Grahic Jump Location
Fig. 7

Unique temperature profile for the specific values L = 1 and Λ = 2 at two temperature jump conditions τ = 0 and τ = 2 with Pr = 1 and Pr = 3

Grahic Jump Location
Fig. 8

Triple temperature profiles for the specific values L = Λ = 2 at two temperature jump conditions τ = 0 and τ = 2. (a) Branch 1, (b) branch 2, and (c) branch 3.

Grahic Jump Location
Fig. 9

The rate of heat transfer -θ'(0) at two temperature jump conditions τ = 0 and τ = 2. (a) λ corresponding to Eq. (8) and (b) λ corresponding to Eq. (9).

Grahic Jump Location
Fig. 10

The effects on Eckert number Ec for L = 1 and Λ = 2 at two temperature jump conditions τ = 0 and τ = 2. (a) Pr = 1 and (b) Pr = 2.

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