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Research Papers: Jets, Wakes, and Impingment Cooling

# Axisymmetric Stagnation Flow and Heat Transfer of a Compressible Fluid Impinging on a Cylinder Moving Axially

[+] Author and Article Information
Asghar B. Rahimi

Professor
Faculty of Engineering,
Ferdowsi University of Mashhad,
P. O. Box No. 91775-1111,
Mashhad 1111, Iran
e-mail: rahimiab@yahoo.com

Hamid Mohammadiun

Assistant Professor
Department of Mechanical Engineering,
Shahrood Branch,
Islamic Azad University,
Shahrood 3619633619, Iran.

Mohammad Mohammadiun

Assistant Professor
Department of Mechanical Engineering,
Shahrood Branch,
Islamic Azad University,
Shahrood 3619633619, Iran

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received March 2, 2014; final manuscript received July 15, 2015; published online October 6, 2015. Assoc. Editor: William P. Klinzing.

J. Heat Transfer 138(2), 022201 (Oct 06, 2015) (9 pages) Paper No: HT-14-1106; doi: 10.1115/1.4031130 History: Received March 02, 2014; Revised July 15, 2015

## Abstract

The steady-state viscous flow and also heat transfer in the vicinity of an axisymmetric stagnation point on a cylinder moving axially with a constant velocity are investigated. Here, fluid with temperature-dependent density is considered. The impinging freestream is steady and with a constant strain rate (strength) $k¯$. An exact solution of the Navier–Stokes equations and energy equation is derived in this problem. A reduction of these equations is obtained by use of appropriate transformations. The general self-similar solution is obtained when the wall temperature of the cylinder or its wall heat flux is constant. All the solutions above are presented for Reynolds numbers, $Re=k¯a2/2υ$, ranging from 0.1 to 1000, low Mach number, selected values of compressibility factor, and different values of Prandtl numbers where $a$ is cylinder radius and $υ$ is kinematic viscosity of the fluid. Shear stress is presented as well. Axial movement of the cylinder does not have any effect on heat transfer but its increase increases the axial component of fluid velocity field and the shear stress.

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## References

Hiemenz, K. , 1911, “ Die Grenzchicht an Einem in den Gleichformingen Flussigkeitsstrom Eingetauchten Geraden KreisZylinder,” Dinglers Polytechnic J., 326, pp. 321–410.
Homann, F. Z. , 1936, “ Der Einfluss Grosser Zahighkeit bei der Strmung um den Zylinder und um die Kugel,” Z. Angew. Math. Mech., 16(3), pp. 153–164.
Howarth, L. , 1951, “ The Boundary Layer in Three-Dimensional Flow. Part II—The Flow Near Stagnation Point,” Phios. Mag., 42(335), pp. 1433–1440.
Davey, A. , 1961, “ Boundary Layer Flow at a Saddle Point of Attachment,” J. Fluid Mech., 10(4), pp. 593–610.
Wang, C. Y. , 1974, “ Axisymmetric Stagnation Flow on a Cylinder,” Q. Appl. Math., 32, pp. 207–213.
Wang, C. Y. , 1973, “ Axisymmetric Stagnation Flow Towards a Moving Plate,” Am. Inst. Chem. Eng. J., 19(5), pp. 1080–1082.
Gorla, R. S. R. , 1976, “ Heat Transfer in an Axisymmetric Stagnation Flow on a Cylinder,” Appl. Sci. Res., 32(5), pp. 541–553.
Gorla, R. S. R. , 1977, “ Unsteady Laminar Axisymmetric Stagnation Flow Over a Circular Cylinder,” Development Mech., 9, pp. 286–288.
Gorla, R. S. R. , 1978, “ Nonsimilar Axisymmetric Stagnation Flow on a Moving Cylinder,” Int. J. Sci., 16(6), pp. 397–400.
Gorla, R. S. R. , 1978, “ Transient Response Behavior of an Axisymmetric Stagnation Flow on a Circular Cylinder Due to Time-Dependent Free Stream Velocity,” Lett. Appl. Eng. Sci., 16(7), pp. 493–502.
Gorla, R. S. R. , 1979, “ Unsteady Viscous Flow in the Vicinity of an Axisymmetric Stagnation-Point on a Cylinder,” Int. Sci., 17(1), pp. 87–93.
Cunning, G. M. , Davis, A. M. J. , and Weidman, P. D. , 1998, “ Radial Stagnation Flow on a Rotating Cylinder With Uniform Transpiration,” J. Eng. Math., 33(2), pp. 113–128.
Grosch, C. E. , and Salwen, H. , 1982, “ Oscillating Stagnation-Point Flow,” Proc. R. Soc. London, A384(1786), pp. 175–190.
Takhar, H. S. , Chamkha, A. J. , and Nath, J. , 1999, “ Unsteady Axisymmetric Stagnation-Point Flow of a Viscous Fluid on a Cylinder,” Int. J. Eng. Sci., 37(15), pp. 1943–1957.
Saleh, R. , and Rahimi, A. B. , 2004, “ Axisymmetric Stagnation-Point Flow and Heat Transfer of a Viscous Fluid on a Moving Cylinder With Time-Dependent Axial Velocity and Uniform Transpiration,” ASME J. Fluids Eng., 126(6), pp. 997–1005.
Rahimi, A. B. , and Saleh, R. , 2007, “ Axisymmetric Stagnation-Point Flow and Heat Transfer of a Viscous Fluid on a Rotating Cylinder With Time-Dependent Angular Velocity and Uniform Transpiration,” ASME J. Fluids Eng., 129(1), pp. 106–115.
Rahimi, A. B. , and Saleh, R. , 2008, “ Similarity Solution of Unaxisymmetric Heat Transfer in Stagnation-Point Flow on a Cylinder With Simultaneous Axial and Rotational Movements,” ASME J. Heat Transfer, 130(5), p. 054502.
Shokrgozar Abbasi, A. , and Rahimi, A. B. , 2009, “ Non-Axisymmetric Three-Dimensional Stagnation-Point Flow and Heat Transfer on a Flat Plate,” ASME J. Fluids Eng., 131(7), p. 074501.
Shokrgozar Abbasi, A. , and Rahimi, A. B. , 2009, “ Three-Dimensional Stagnation-Point Flow and Heat Transfer on a Flat Plate With Transpiration,” J. Thermophys. Heat Transfer, 23(3), pp. 513–521.
Shokrgozar Abbasi, A. , Rahimi, A. B. , and Niazman, H. , 2011, “ Exact Solution of Three-Dimensional Unsteady Stagnation Flow on a Heated Plate,” J. Thermodyn. Heat Transfer, 25(1), pp. 55–58.
Shokrgozar Abbasi, A. , and Rahimi, A. B. , 2012, “ Investigation of Two-Dimensional Stagnation-Point Flow and Heat Transfer Impinging on an Accelerated Flat Plate,” ASME J. Heat Transfer, 134(6), p. 064501.
Shokrgozar Abbasi, A. , and Rahimi, A. B. , 2013, “ Solidification of Two-Dimensional Viscous, Incompressible Stagnation Flow,” ASME J. Heat Transfer, 135(7), p. 072301.
Karwe, M. V. , and Jaluria, Y. , 1988, “ Fluid Flow and Miaxed Convection Transport From a Plate in Rolling and Extrusion Process,” ASME J. Heat Transfer, 110(3), pp. 655–661.
Kang, B. H. , Yoo, J. , and Jaluria, Y. , 1994, “ Experimental Study of the Convective Cooling of a Heated Continuously Moving Material,” ASME J. Heat Transfer, 116(1), pp. 199–208.
Choudhury, S. R. , and Jaluria, Y. , 1994, “ Forced Corrective Heat Transfer From a Continuously Loving Heated Cylindrical Rod in Materials Processing,” ASME J. Heat Transfer, 116(3), pp. 724–734.
Karwe, M. V. , and Jaluria, Y. , 1991, “ Numerical Simulation of Thermal Transport Associated With a Continuously Moving Flat Sheet in Material Processing,” ASME J. Heat Transfer, 113(3), pp. 612–619.
Wang, C. Y. , and Chiu-On, Ng. , 2013, “ Stagnation Flow on a Heated Vertical Plate With Surface Slip,” ASME J. Heat Transfer, 135(7), p. 074505.
Hong, L. , and Wang, C. Y. , 2009, “ Annular Axisymmetric Stagnation Flow on a Moving Cylinder,” Int. J. Eng. Sci., 47(1), pp. 141–152.
Memon, N. , and Jaluria, Y. , 2011, “ Flow Structure and Heat Transfer in a Stagnation Flow CVD Reactor,” ASME J. Heat Transfer, 133(8), p. 082501.
Hayat, T. , Anwar, M. S. , Farooq, M. , and Alsaedi, A. , 2014, “ MHD Stagnation Point Flow of Second Grade Fluid Over a Stretching Cylinder With Heat and Mass Transfer,” Int. J. Nonlinear Sci. Numer. Simul, 15(6), pp. 365–376.
Bono, G. , and Awruch, A. M. , 2008, “ An Adaptive Mesh Strategy for High Compressible Flows Based on Nodal Re-Allocation,” J. Braz. Soc. Mech. Sci. Eng., 30(3), pp. 189–195.
Subhashini, S. V. , and Nath, G. , 1999, “ Unsteady Compressible Flow in the Stagnation Region of Two-Dimensional and Axisymmetric Bodies,” Acta Mech., 134(3), pp. 135–145.
Kumari, M. , and Nath, G. , 1980, “ Unsteady Compressible 3-Dimensional Boundary Layer Flow Near an Axisymmetric Stagnation Point With Mass Transfer,” Int. J. Eng. Sci., 18(11), pp. 1285–1300.
Kumari, M. , and Nath, G. , 1981, “ Self-Similar Solution of Unsteady Compressible Three-Dimensional Stagnation-Point Boundary Layers,” J. Appl. Math. Phys., 32(3), pp. 267–276.
Katz, A. , 1972, “ Transformations of the Compressible Boundary Layer Equations,” SIAM J. Allied Math., 22(4), pp. 604–611.
Afzal, N. , and Ahmad, S. , 1975, “ Effect of Suction and Injection on Self-Similar Solutions of Second-Order Boundary Layer Equations,” Int. J. Heat Mass Transfer, 18(5), pp. 607–614.
Libby, P. A. , 1967, “ Heat and Mass Transfer at a General Three-Dimensional Stagnation Point,” AIAA J., 5(3), pp. 507–517.
Gersten, K. , Papenfuss, H. D. , and Gross, J. F. , 1978, “ Influence of the Prandtl Number on Second-Order Heat Transfer Due to Surface Curvature at a Three-Dimensional Stagnation Point,” Int. J. Heat Mass Transfer, 21(3), pp. 275–284.
Mozayyeni, H. R. , and Rahimi, A. B. , 2013, “ Three-Dimensional Stagnation Flow and Heat Transfer of a Viscous, Compressible Fluid on a Flat Plate,” ASME J. Heat Transfer, 135(10), p. 101702.
Mozayyeni, H. R. , and Rahimi, A. B. , 2014, “ Unsteady Two-Dimensional Stagnation-Point Flow and Heat Transfer of a Viscous, Compressible Fluid on an Accelerated Flat Plate,” ASME J. Heat Transfer, 136(4), p. 041701.
Mohammadiun, H. , and Rahimi, A. B. , 2012, “ Axisymmetric Stagnation-Point Flow and Heat Transfer of a Viscous, Compressible Fluid on a Cylinder,” J. Thermophys. Heat Transfer, 26(3), pp. 494–502.
Press, W. H. , Flannery, B. P. , Teukolsky, S. A. , and Vetterling, W. T. , 1997, Numerical Recipes, the Art of Scientific Computing, Cambridge University Press, Cambridge, UK.

## Figures

Fig. 1

A schematic mechanism of the radially impinging flow production

Fig. 2

Schematic diagram of an axially moving cylinder

Fig. 3

Schematic diagram of inviscid flow on cylinder

Fig. 4

Variation of dimensionless axial movement function (H/V) in terms of η for Tw = 500 K, T∞ = 300 K, β = 0.0033, Re = 1, and selected values of Prandtl number

Fig. 5

Variations of dimensionless axial movement function (H/V) in terms of η for selected values of Pr for Re = 10, β = 0.0033, and γ = 100

Fig. 6

Variation of θ in terms of η at, Tw = 500 K, T∞ = 300 K, Re = 10.0, β = 0.0033, and for different values of Prandtl numbers

Fig. 7

Variations of dimensionless axial movement function (H/V) in terms of η for selected values of β for Re = 100, Pr = 1, and Tw = 500 K

Fig. 8

Variation of shear stress versus Reynolds number at Pr = 0.7, γ = 50, V  =  5 m/s for selected values of compressibility factor

Fig. 9

Variations of θ in terms of η at Pr = 1.0, Tw = 500 K, T∞ = 500 K, Re = 1.0, and for different values of compressibility factor

Fig. 10

Variations of θ in terms of η at γ = 10.0 and Pr = 0.7, Re = 10 for different values of compressibility factor

Fig. 11

Variation of shear stress against wall temperature at V  =  5 m/s and V  =  10 m/s, Pr = 0.7, β  =  0.0033, and for selected values of Reynolds numbers

Fig. 12

Variation of shear stress against γ for selected values of Reynolds number at V = 5 m/s and V = 10 m/s, Pr = 0.7, β = 0.0033, and for selected values of Reynolds number

Fig. 13

Variation of pressure function in terms of η at, Pr = 0.7, Tw = 500 K, T∞ = 300 K, β = 0.0033, and for different values of Reynolds numbers

Fig. 14

Variation of f in terms of η at Tw = 300 K, β = 0.0033, and Pr = 0.7 for different values of Reynolds numbers

Fig. 15

Variation of shear stress against Reynolds number at V = 5 m/s, Pr = 0.7, β = 0.0033, and for selected values of wall temperature

Fig. 16

Variations of dimensionless axial movement function (H/V) in terms of η for β = 0 and for selected values of Reynolds number

Fig. 17

The normalized stream function ψ̂ = ψ/0.5k¯a3 = 2f(η)(z/a) with, Re = 1, α = 0. Fluid is injected from the outer cylinder at η = 2 toward the inner cylinder.

Fig. 18

The normalized stream function ψ̂ = (ψ/0.5k¯a3) = 2f(η)(z/a)−α∫1ηH/Vdη with Re = 1. Fluid is injected from the outer cylinder at η = 2 toward the inner cylinder.

## Errata

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