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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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Figures

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Fig. 1

Sketch of the flow and coordinates

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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.

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Fig. 3

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

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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.

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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.

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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.

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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

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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.

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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).

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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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