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Research Papers: Natural and Mixed Convection

Effects of Second-Order Slip and Magnetic Field on Mixed Convection Stagnation-Point Flow of a Maxwellian Fluid: Multiple Solutions

[+] Author and Article Information
M. M. Rahman

Department of Mathematics and Statistics,
College of Science,
Sultan Qaboos University,
P.O. Box 36,
P.C. 123 Al-Khod,
Muscat 123, Sultanate of Oman
e-mail: mansurdu@yahoo.com

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 3, 2016; final manuscript received July 6, 2016; published online August 16, 2016. Assoc. Editor: Guihua Tang.

J. Heat Transfer 138(12), 122503 (Aug 16, 2016) (10 pages) Paper No: HT-16-1002; doi: 10.1115/1.4034161 History: Received January 03, 2016; Revised July 06, 2016

In this paper, we investigate the effects of second-order slip and magnetic field on the nonlinear mixed convection stagnation-point flow toward a vertical permeable stretching/shrinking sheet in an upper convected Maxwell (UCM) fluid with variable surface temperature. Numerical results are obtained using the bvp4c function from matlab for the reduced skin-friction coefficient, the rate of heat transfer, the velocity, and the temperature profiles. The results indicate that multiple (dual) solutions exist for a buoyancy opposing flow for certain values of the parameter space irrespective to the types of surfaces whether it is stretched or shrinked. It is found that an applied magnetic field compensates the suction velocity for the existence of the dual solutions. Depending on the parametric conditions; elastic parameter, magnetic field parameter, first- and second-order slip parameters significantly controls the flow and heat transfer characteristics. The illustrated streamlines show that for upper branch solutions, the effects of stretching and suction are direct and obvious as the flow near the surface is seen to suck through the permeable sheet and drag away from the origin of the sheet. However, aligned but reverse flow occurs for the case of lower branch solutions when the mixed convection effect is less significant.

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Figures

Grahic Jump Location
Fig. 1

Variations of Rex1/2Cf for different values of s and λ when M=0.5, K=0.1, γ=0.5, a=0.5, and b=0

Grahic Jump Location
Fig. 2

Variations of Rex−1/2Nux for different values of s and λ when M=0.5, K=0.1, γ=0.5, a=0.5, and b=0

Grahic Jump Location
Fig. 3

Variations of Rex1/2Cf for different values of s and λ when M=0.5, K=0.1, γ=0.5, a=0.5, and b=−0.05

Grahic Jump Location
Fig. 4

Variations of Rex−1/2Nux for different values of s and λ when M=0.5, K=0.1, γ=0.5, a=0.5, and b=−0.05

Grahic Jump Location
Fig. 5

Variations of Rex1/2Cf for different values of s and λ when M=0.5, K=0.1, γ=0.5, a=0, and b=−0.05

Grahic Jump Location
Fig. 6

Variations of Rex−1/2Nux for different values of s and λ when M=0.5, K=0.1, γ=0.5, a=0, and b=−0.05

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Fig. 7

Variations of Rex1/2Cf for different values of s and M at λ=−5.315 when K=0.1, γ=0.5, a=0.5, and b=0

Grahic Jump Location
Fig. 8

Variations of Rex−1/2Nux for different values of s and M at λ=−5.315 when K=0.1, γ=0.5, a=0.5, and b=0

Grahic Jump Location
Fig. 9

Variations of Rex1/2Cf for different values of K and λ when M=0.5, s=0.8, γ=0.5, a=0.5, and b=−0.05

Grahic Jump Location
Fig. 10

Variations of Rex−1/2Nux for different values of K and λ when M=0.5, s=0.8, γ=0.5, a=0.5, and b=−0.05

Grahic Jump Location
Fig. 11

Variations of Rex1/2Cf for different values of λ and γ when M=0.5, K=0.1, s=0.8, a=0, and b=−0.05

Grahic Jump Location
Fig. 12

Variations of Rex−1/2Nux for different values of λ and γ, when M=0.5, K=0.1, s=0.8, a=0, and b=−0.05

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Fig. 13

Velocity profiles for different values of b at λ=−5.315, when M=0.5,K=0.1,γ=0.5, s=0.8, and a=0.5

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Fig. 14

Temperature profiles for different values of b at λ=−5.315, when M=0.5,K=0.1,γ=0.5, s=0.8, and a=0.5

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Fig. 15

Velocity profiles for different values of M at λ=−5.315, when K=0.1,γ=0.5, s=0.8, a=0.5, and b=−0.05

Grahic Jump Location
Fig. 16

Temperature profiles for different values of M at λ=−5.315, when K=0.1,γ=0.5, s=0.8, a=0.5, and b=−0.05

Grahic Jump Location
Fig. 17

Velocity profiles for different values of K at λ=−5.315, when M=0.5,γ=0.5, s=0.8, a=0.5, and b=−0.05

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Fig. 18

Temperature profiles for different values of K at λ=−5.315, when M=0.5,γ=0.5, s=0.8, a=0.5, and b=−0.05

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Fig. 19

Velocity profiles for different values of s at λ=−5.315, when M=0.5,γ=0.5, K=0.1, a=0.5, and b=−0.05

Grahic Jump Location
Fig. 20

Temperature profiles for different values of s at λ=−5.315, when M=0.5,γ=0.5, K=0.1, a=0.5, and b=−0.05

Grahic Jump Location
Fig. 21

Streamlines for upper branch solutions when λ=−5.315,K=0.1,M=0.5,s=0.8, γ=0.5, a=0.5, and b=−0.05

Grahic Jump Location
Fig. 22

Streamlines for lower branch solutions, when λ=−5.315,K=0.1,M=0.5,s=0.8, γ=0.5, a=0.5, and b=−0.05

Grahic Jump Location
Fig. 23

Streamlines for upper branch solutions, when λ=−1,K=0.1,M=0.5,s=0.8, γ=0.5, a=0.5, and b=−0.05

Grahic Jump Location
Fig. 24

Streamlines for lower branch solutions, when λ=−1,K=0.1,M=0.5,s=0.8, γ=0.5, a=0.5, and b=−0.05

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