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Research Papers: Forced Convection

Fractional Boundary Layer Flow and Heat Transfer Over a Stretching Sheet With Variable Thickness

[+] Author and Article Information
Lin Liu

School of Mathematics and Physics,
University of Science and Technology Beijing,
Beijing 100083, China;
Beijing Key Laboratory for
Magneto-Photoelectrical Composite
and Interface Science,
Beijing 100083, China

Liancun Zheng

School of Mathematics and Physics,
University of Science and Technology Beijing,
Beijing 100083, China;
Beijing Key Laboratory for
Magneto-Photoelectrical Composite
and Interface Science,
Beijing 100083, China

Yanping Chen

School of Mathematics and Physics,
University of Science and Technology Beijing,
Beijing 100083, China

Fawang Liu

School of Mathematical Sciences,
Queensland University of Technology,
GPO Box 2434,
Brisbane 4001, Queensland, Australia

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 26, 2017; final manuscript received January 11, 2018; published online May 7, 2018. Assoc. Editor: Guihua Tang.

J. Heat Transfer 140(9), 091701 (May 07, 2018) (9 pages) Paper No: HT-17-1504; doi: 10.1115/1.4039765 History: Received August 26, 2017; Revised January 11, 2018

The paper gives a comprehensive study on the space fractional boundary layer flow and heat transfer over a stretching sheet with variable thickness, and the variable magnetic field is applied. Novel governing equations with left and right Riemann–Liouville fractional derivatives subject to irregular region are formulated. By introducing new variables, the boundary conditions change as the traditional ones. Solutions of the governing equations are obtained numerically where the shifted Grünwald formulae are applied. Good agreement is obtained between the numerical solutions and exact solutions which are constructed by introducing new source items. Dynamic characteristics with the effects of involved parameters on the velocity and temperature distributions are shown and discussed by graphical illustrations. Results show that the velocity boundary layer is thicker for a larger fractional parameter or a smaller magnetic parameter, while the temperature boundary layer is thicker for a larger fractional parameter, a smaller exponent parameter, or a larger magnetic parameter. Moreover, it is thicker at a smaller y and thinner at a larger y for the velocity boundary layer with a larger exponent parameter while for the velocity and temperature boundary layers with a smaller weight coefficient.

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Figures

Grahic Jump Location
Fig. 1

Schematic of a stretching sheet with variable thickness

Grahic Jump Location
Fig. 2

Comparisons of the exact solutions and numerical solutions

Grahic Jump Location
Fig. 3

Velocity distributions versus y with different α when n=2, x=1, u0=1, r1=0.5, and M=1

Grahic Jump Location
Fig. 4

Temperature distributions versus y with different β when M=1, x=1, u0=1, n=2, α=0.9, r1=0.5, and r2=0.5

Grahic Jump Location
Fig. 5

Velocity distributions versus y with different M when n=2, x=1, u0=1, r1=0.5, and α=0.9

Grahic Jump Location
Fig. 6

Temperature distributions versus y with different M when β=0.9, x=1, u0=1, n=2, α=0.9, r1=0.5, and r2=0.5

Grahic Jump Location
Fig. 7

Velocity distributions versus y with different n when M=0.5, x=1, u0=1, r1=0.5, and α=0.9

Grahic Jump Location
Fig. 8

Temperature distributions versus y with different n when β=0.9, x=1, u0=1, M=0.5, α=0.9, r1=0.5, and r2=0.5

Grahic Jump Location
Fig. 9

Velocity distributions versus y with different γ1 when M=1, x=1, u0=1, n=2, and α=0.9

Grahic Jump Location
Fig. 10

Temperature distributions versus y with different γ2 when β=0.9, x=1, u0=1, M=1, α=0.9, n=2, and r1=0.5

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