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Research Papers: Forced Convection

# Numerical Simulation of the Effect of Transient Shear Stress and Rate of Heat Transfer Around Different Positions of Sphere in the Presence of Viscous Dissipation

[+] Author and Article Information

Department of Mathematics,
Faculty of Science,
University of Sargodha,
Sargodha 40100, Pakistan

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received May 22, 2016; final manuscript received November 23, 2017; published online February 28, 2018. Assoc. Editor: Antonio Barletta.

J. Heat Transfer 140(6), 061701 (Feb 28, 2018) (12 pages) Paper No: HT-16-1289; doi: 10.1115/1.4038841 History: Received May 22, 2016; Revised November 23, 2017

## Abstract

The present study is devoted to the problem of oscillatory convective flow in the presence of viscous dissipation around different positions of a sphere. The system of differential equations governing the flow phenomenon is transformed into dimensionless form by using suitable group of variables and then transformed into convenient form for integration by using primitive variable formulation. Numerical simulation based on finite difference method is carried out to analyze the mixed convection flow mechanism. Special focus is given on the transient shear stress and the rate of heat transfer characteristics and their dependency on various dimensionless parameters that is mixed convection parameter λ, Prandtl number Pr, dissipation parameter $N,$ and angular frequency parameter $ω$. The angles $X=30deg,90 deg$, and $360deg$ are the favorable positions around the sphere for different parameters, where the transient rate of shear stress and heat transfer is noted maximum. Later, the obtained results are presented graphically by using Tech Plot-360 and compared with the previous work given in the literature.

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## References

Gebhart, B. , 1962, “ Effects of Viscous Dissipation in Natural Convection,” J. Fluid Mech., 14(2), pp. 225–232.
Van Dorn, W. G. , 1966, “ Boundary Dissipation of Oscillatory Waves,” J. Fluid Mech., 24(4), pp. 769–779.
Gebhart, B. , and Mollendorf, J. , 1969, “ Viscous Dissipation in External Natural Convection Flows,” J. Fluid Mech., 38(1), pp. 97–107.
Hieber, C. A. , and Gebhart, B. , 1969, “ Mixed Convection From a Sphere at Small Reynolds and Grashof Numbers,” J. Fluid Mech., 38(1), pp. 137–159.
Ishigaki, H. , 1970, “ Periodic Boundary Layer Near Two Dimensional Stagnation Points,” J. Fluid Mech., 43(3), pp. 477–486.
Iqbal, M. , Aggarwala, B. D. , and Rokerya, M. S. , 1970, “ Viscous Dissipation Effects on Combined Free and Forced Convection Through Vertical Circular Tubes,” ASME J. Appl. Mech., 37(4), pp. 931–935.
Ishigaki, H. , 1971, “ An Exact Periodic Solution of the Energy Equation,” J. Fluid Mech., 50(4), pp. 657–668.
Ishigaki, H. , 1972, “ Heat Transfer in a Periodic Boundary Layer Near Two Dimension Stagnation Points,” J. Fluid Mech., 56(4), pp. 619–627.
Graham, W. , 1974, “ A Separated Flow in Mixed Convection,” J. Fluid Mech., 62(2), pp. 359–369.
Turcotte, D. L. , Hsui, A. T. , Torrance, K. E. , and Schubert, G. , 1974, “ Influence of Viscous Dissipation on Benard Convection,” J. Fluid Mech., 64(2), pp. 369–374.
Van der Borght, R. , 1975, “ The Effect of Viscous Dissipation on Non-Linear Convection at High Rayleigh Number,” J. Aust. Math. Soc., Ser. B. Appl. Math., 19(2), pp. 165–172.
Sparrow, E. M. , and Lee, L. , 1976, “ Analysis of Mixed Convection About Horizontal Cylinder,” Int. J. Heat Mass Transfer, 19(2), pp. 229–232.
Dowden, J. , 1981, “ The Stability of a Periodically Heated Layer of Fluid,” J. Fluid Mech., 110, pp. 149–159.
Menendez, A. N. , and Ramaprian, B. R. , 1984, “ Prediction of Periodic Boundary Layers,” Int. J. Numer. Method Fluids, 4(8), pp. 781–80.
Philip, H. , 1984, “ On the Stability of the Unsteady Boundary Layer on a Cylinder Oscillating Transversely in a Viscous Fluid,” Int. J. Fluid Mech., 146, pp. 347–367.
Golra, R. S. R. , Jankoswki, F. , and Textort, D. , 1988, “ Periodic Boundary Layer Near an Axisymmetric Stagnation Point on a Circular Cylinder,” Int. J. Heat Fluid Flow, 9(4), pp. 421–426.
Golra, R. S. R. , Jankoswki, F. , and Textort, D. , 1988, “ Thermal Response of Periodic Boundary Layer Near an Axisymmetric Stagnation Point on a Circular Cylinder,” Int. J. Heat Fluid Flow, 9(4), pp. 427–430.
Borisevich, V. D. , and Potanin, E. P. , 1988, “ Effects of Viscous Dissipation and Joule Heat on Heat Transfer Near a Rotating Disk in the Presence of Intensive Suction,” J. Eng. Phy., 55(5), pp. 1220–1223.
Duck, P. W. , 1989, “ A Numerical Method for Treating Time Periodic Boundary Layers,” J. Fluid Mech., 204, pp. 549–561.
Cheng, C.-H. , Kou, H.-S. , and Hiang, W.-H. , 1990, “ Flow Reversal and Heat Transfer of Fully Developed Mixed Convection in Vertical Channel,” J. Thermophys. Heat Transfer, 4(3), pp. 375–383.
Nguyen, H. D. , Paik, S. , and Chung, J. N. , 1993, “ Unsteady Mixed Convection Heat Transfer From a Solid Sphere: The Conjugate Problem,” Int. J. Heat Mass Transfer, 36(18), pp. 4443–4453.
Zanchini, E. , 1998, “ Effect of Viscous Dissipation on Mixed Convection in a Vertical Channel With Boundary Conditions of the Third Kind,” Int. J. Heat Mass Transfer, 41(23), pp. 3949–3959.
Barletta, A. , 1998, “ Laminar Mixed Convection With Viscous Dissipation in a Vertical Channel,” Int. J. Heat Mass Transfer, 41(22), pp. 3501–3513.
Barletta, A. , 1999, “ Combined Forced and Free Convection With Viscous Dissipation in a Vertical Circular Duct,” Int. J. Heat Mass Transfer, 42(12), pp. 2243–2253.
Hossain, M. A. , and Rees, D. A. S. , 1999, “ Combined Heat and Mass Transfer in Natural Convection Flow From a Vertical Wavy Surface,” Acta Mech., 136(3–4), pp. 133–141.
Barletta, A. , and Zanchini, E. , 2001, “ Mixed Convection With Viscous Dissipation in an Inclined Channel With Prescribed Wall Temperatures,” Int. J. Heat Mass Transfer, 44(22), pp. 4267–4275.
Hossain, M. A. , Kabir, S. , and Rees, D. A. S. , 2002, “ Natural Convection of Fluid With Variable Viscosity From a Heated Vertical Wavy Surface,” J. Z. Angew. Math. Phys., 53(1), pp. 48–57.
Antar, M. A. , and El-Shaarawi, M. A. I. , 2002, “ Mixed Convection Around a Liquid Sphere in an Air Stream,” Int. J. Heat Mass Transfer, 38(4–5), pp. 419–424.
Joseph, D. D. , and Wong, J. , 2004, “ The Dissipation Approximation and Viscous Potential Flow,” J. Fluid Mech., 505, pp. 365–377.
Pantokratoras, A. , 2005, “ Effect of Viscous Dissipation in Natural Convection along a Heated Vertical Plate,” J. Appl. Math. Model., 29(6), pp. 553–564.
Rieutord, M. , and Valdettaro, L. , 2010, “ Viscous Dissipation by Tidally Forced Inertial Modes in a Rotating Spherical Shell,” Int. J. Fluid Mech., 643, pp. 363–394.
Elgazery, N. S. , and Abd Elazem, Y. , 2011, “ Effect of Viscous Dissipation and Joule Heating on Natural Convection Flow of a Viscous Fluid From Heated Vertical Wavy Surface,” Z. Naturforch, 66a, pp. 427–440.
Jha, B. K. , and Ajibade, A. O. , 2012, “ Effect of Viscous Dissipation on Natural Convection Flow Between Vertical Parallel Plates With Time-Periodic Boundary Conditions,” J. Commun. Nonlinear Sci. Numer. Simul., 17(4), pp. 1576–1587.
Praveen, N. , Nath, S. , and Abdul Alim, M. , 2014, “ Viscous Dissipation Effects on a Natural Convection Flow Along a Vertical Wavy Surface,” Proceedia Eng., 90, pp. 294–300.
Juncu, G. , 2015, “ The Influence of Viscous Dissipation on the Unsteady Conjugate Forced Convection Heat Transfer From a Fluid Sphere,” Int. J. Heat Mass Transfer, 90, pp. 542–555.
Hossain, M. Z. , and Floryan, J. M. , 2015, “ Mixed Convection in a Periodically Heated Channel,” J. Fluid Mech., 768, pp. 51–90.
Miklavic, M. , 2015, “ Stability Analysis of Some Fully Developed Mixed Convection Flow in a Vertical Channel,” Z. Angew. Math. Mech., 95(9), pp. 982–986.
Reger, K. , and Gorder, R. A. V. , 2013, “ Lane-Emden Equations of Second Kind Modeling Thermal Expansion in Infinite Cylinder and Sphere,” Appl. Math. Mech., 34(12), pp. 1439–1452.
Ravindran, R. , and Ganapathirao, M. , 2013, “ Non-Uniform Slot Suction/Injection Into Mixed Convection Boundary Layer Flow Over Vertical Cone,” Appl. Math. Mech., 34(11), pp. 1327–1338.
Aman, F. , Ishaq, A. , and Pop, I. , 2011, “ Mixed Convection Boundary Layer Flow Near Stagnation Point on Vertical Surface With Slip,” Appl. Math. Mech., 32(12), pp. 1599–1606.
Ali, F. M. , Nazar, R. , Arifin, N. M. , and Pop, I. , 2014, “ Mixed Convection Stagnation Point Flow on Vertical Stretching Sheet With External Magnetic Field,” Appl. Math. Mech., 35(2), pp. 155–166.
Wei, L. , Lu, Y. , and Wei, J. , 2014, “ Numerical Study on Laminar Free Convection Heat Transfer Between Sphere Particle and High Pressure Water in Pseudo-Critical Zone,” Therm. Sci., 18(4), pp. 1293–1303.
Ashraf, M. , Ahmad, M. , and Anwar Hossain, M. , 2015, “ Numerical Simulation of Radiative Natural Convection Flow Around a Magnetized Sphere,” J. Thermophys. Heat Transfer, 29(3), pp. 602–609.
Ashraf, M. , Fatima, A. , and Gorla, R. S. R. , 2017, “ Periodic Momentum and Thermal Boundary Layer Mixed Convection Flow Around Surface of the Sphere in the Presence of Viscous Dissipation,” Can. J. Phys., 95(10), pp. 976–986.
Sphaier, L. A. , Barletta, A. , and Celli, M. , 2015, “ Unstable Mixed Convection in Heated Inclined Porous Channel,” J. Fluid Mech., 778, pp. 428–450.

## Figures

Fig. 1

Coordinate system and flow configuration

Fig. 2

Geometrical interpretation of (a) transient shear stress and (b) transient rate of heat transfer against τ for different values of mixed convection parameter λ = 0.3, 0.5, and 0.7 when Pr = 7.0, angular frequency ω = 0.05, and viscous dissipation parameter N = 0.015 at position of the sphere X = 0 deg

Fig. 3

Geometrical interpretation of (a) transient shear stress and (b) transient rate of heat transfer against τ for different values of mixed convection parameter λ = 0.3, 0.5 and 0.7 when Pr = 7.0, angular frequency ω = 0.05, and viscous dissipation parameter N = 0.015 at position of the sphere X = 30 deg

Fig. 4

Geometrical interpretation of (a) transient shear stress and (b) transient rate of heat transfer against τ for different values of mixed convection parameter λ = 0.3, 0.5, and 0.7 when Pr = 7.0, angular frequency ω = 0.05, and viscous dissipation parameter N = 0.015 at position of the sphere X = 90 deg

Fig. 5

Geometrical interpretation of (a) transient shear stress and (b) transient rate of heat transfer against τ for different values of viscous dissipation parameter N = 0.01, 0.015, and 0.1 when Pr = 7.0, angular frequency ω = 0.05 and mixed convection parameter λ = 0.5 at position of the sphere X = 0 deg

Fig. 6

Geometrical interpretation of (a) transient shear stress and (b) transient rate of heat transfer against τ for different values of viscous dissipation parameter N = 0.01, 0.015 and 0.1 when Pr = 7.0, angular frequency ω = 0.05, and mixed convection parameter λ = 0.5 at position of the sphere X = 90 deg

Fig. 7

Geometrical interpretation of (a) transient shear stress and (b) transient rate of heat transfer against τ for different values of viscous dissipation parameter N = 0.01, 0.015 and 0.1 when Pr = 7.0, angular frequency ω = 0.05, and mixed convection parameter λ = 0.5 at position of the sphere X = 360 deg

Fig. 8

Geometrical interpretation of (a) transient shear stress and (b) transient rate of heat transfer against τ for different values of frequency parameter ω = 0.05, 0.07, 0.1 when N = 0.015, Prandtl number Pr = 7.0, and mixed convection parameter λ = 0.5 at position of the sphere X = 0 deg

Fig. 9

Geometrical interpretation of (a) transient shear stress and (b) transient rate of heat transfer against τ for different values of Prandtl number Pr = 5.0, 7.0, 10.0 when N = 0.015, angular frequency ω = 0.05, and mixed convection parameter λ = 0.8 at position of the sphere X = 0 deg

Fig. 10

Geometrical interpretation of (a) transient shear stress and (b) transient rate of heat transfer against τ for different values of Prandtl number Pr = 5.0, 7.0, 10.0 when N = 0.015, angular frequency ω = 0.05, and mixed convection parameter λ = 0.8 at position of the sphere X = 90 deg

Fig. 11

Geometrical interpretation of (a) transient shear stress and (b) transient rate of heat transfer against τ for different values of Prandtl number Pr = 5.0, 7.0, 10.0 when N = 0.015, angular frequency ω = 0.05, and mixed convection parameter λ = 0.8 at position of the sphere X = 360 deg

Fig. 12

Geometrical interpretation of (a) transient shear stress and (b) transient rate of heat transfer against τ for different values of frequency parameter ω = 0.05, 0.07, 0.1 when N = 0.015, Prandtl number Pr = 7.0, and mixed convection parameter λ = 0.5 at position of the sphere X = 90 deg

Fig. 13

Geometrical interpretation of (a) transient shear stress and (b) transient rate of heat transfer against τ for different values of frequency parameter ω = 0.05, 0.07, 0.1 when N = 0.015, Prandtl number Pr = 7.0, and mixed convection parameter λ = 0.5 at position of the sphere X = 360 deg

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