We present a purely analytic solution to the steady three-dimensional viscous stagnation point flow of second grade fluid over a heated flat plate moving with some constant speed. The analytic solution is obtained by a newly developed analytic technique, namely, homotopy analysis method. By giving a comparison with the existing results, it is shown that the obtained analytic solutions are highly accurate and are in good agreement with the results already present in literature. Also, the present analytic solution is uniformly valid for all values of the dimensionless second grade parameter . The effects of and the Prandtl number Pr on velocity and temperature profiles are discussed through graphs.
1.
Blasius
, H.
, 1908, “Grenzschichten in Flussigkeiten mit Kleiner Reibung
,” Z. Math. Phys.
, 56
, pp. 1
–17
.2.
Heimenz
, K.
, 1911, “Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder
,” Ph.D. thesis, Gottingen.3.
Glauert
, M. B.
, 1956, “The Laminar Boundary Layer on Oscillating Plates and Cylinders
,” J. Fluid Mech.
0022-1120, 1
, pp. 97
–110
.4.
Stuart
, J. T.
, 1959, “The Viscous Flow Near a Stagnation Point When the External Flow Has Uniform Vorticity
,” J. Aerosp. Sci.
0095-9820, 26
, pp. 124
–125
.5.
Davey
, A.
,1961, “Boundary Layer Flow at a Saddle Point of Attachment
,” J. Fluid Mech.
0022-1120, 10
, pp. 593
–610
.6.
Tamada
, K. J.
, 1979, “Two-Dimensional Stagnation Point Flow Impinging Obliquely on a Plane Wall
,” J. Phys. Soc. Jpn.
0031-9015, 46
, pp. 310
–311
.7.
Dorepaal
, J. M.
, 1986, “An Exact Solution of the Navier-Stokes Equation Which Describes Non-Orthogonal Stagnation-Point Flow in two Dimensions
,” J. Fluid Mech.
0022-1120, 163
, pp. 141
–147
.8.
Zumbrunnen
, D. A.
, 1991, “Convective Heat and Mass Transfer in The Stagnation Region of a Laminar Planar jet Impinging on a Moving Surface
,” ASME J. Heat Transfer
0022-1481, 113
, pp. 563
–570
.9.
Wang
, C. Y.
, 1985, “The Unsteady Oblique Stagnation Point Flow
,” Phys. Fluids
0031-9171, 28
, pp. 2046
–2049
.10.
Ariel
, P. D.
, 1994, “Three-Dimensional Stagnation Point Flow of a Viscoelastic Fluid
,” Mech. Res. Commun.
0093-6413, 21
, pp. 389
–396
.11.
Ariel
, P. D.
, 1994, “Stagnation Point Flow With Suction: An Approximate Solution
,” ASME J. Appl. Mech.
0021-8936, 61
, pp. 976
–978
.12.
Ariel
, P. D.
, 2001, “Axisymmetric Flow of a Second Grade Fluid Past a Stretching Sheet
,” Int. J. Eng. Sci.
0020-7225, 39
, pp. 529
–553
.13.
Bian
, X.
, and Rangel
, R. H.
, 1996, “The Viscous Stagnation Flow Solidification Problem
,” Int. J. Heat Mass Transfer
0017-9310, 39
, pp. 3581
–3594
.14.
Mahapatra
, T. R.
, and Gupta
, A. S.
, 2002, “Heat Transfer in Stagnation Point Flow Towards a Stretching Sheet
,” Heat Mass Transfer
0947-7411, 38
, pp. 517
–521
.15.
Baris
, S.
, 2003, “Steady Three-Dimensional Flow of a Second Grade Fluid Towards a Stagnation Point at a Moving Flat Plate
,” Turk. J. Eng. Environ. Sci.
1300-0160, 27
, pp. 21
–29
.16.
Fosdick
, R. L.
, and Rajagopal
, K. R.
, 1978, “Uniqueness and Drag for Fluids of Second Grade in Steady Motion
,” Int. J. Non-Linear Mech.
0020-7462, 13
, pp. 131
–137
.17.
Dunn
, J. E.
, and Fosdick
, R. L.
, 1974, “Thermodynamics, Stability and Boundedness of Fluids of Complexity and Fluids of Second Grade
,” Arch. Ration. Mech. Anal.
0003-9527, 56
, pp. 191
–252
).18.
Dunn
, J. E.
, and Rajagopal
, K. R.
, 1995, “Fluids of Different Type—Critical Review and Thermodynamic Analysis
,” Int. J. Eng. Sci.
0020-7225, 33
, pp. 689
–729
.19.
Nayfeh
, A. H.
, 2000, Perturbation Methods
, Wiley
, New York
.20.
Liao
, S. J.
, 2003, Beyond Perturbation: Introduction to Homotopy Analysis Method
, Chapman and Hall∕CRC
, Boca Raton, FL
.21.
Adomian
, G.
, 1975, “Nonlinear Stochastic Differential Equations
,” J. Math. Anal. Appl.
0022-247X, 55
, pp. 414
–452
.22.
Karmishin
, A. V.
, Zhukov
, A. T.
, and Kolosov
, V. G.
, 1990, Methods of Dynamics Calculation and Testing for Thin-Walled Structures
, Mashiniostronie
, Moscow
, in Russian.23.
Lyapunov
, A. M.
, 1992, General Problems on Stability of Motions
, Taylor and Francis
, London
, English translation.24.
Liao
, S. J.
, 2006, “An Analytic Solution of Unsteady Boundary-Layer Flows Caused by an Impulsively Stretching Plate
,” Commun. Nonlinear Sci. Numer. Simul.
1007-5704, 11
, pp. 326
–339
.25.
Xu
, H.
, and Liao
, S. J.
, 2005, “Series Solutions of Unsteady Magnetohydrodynamic Flows of Non-Newtonian Fluids Caused by an Impulsively Stretching Plate
,” J. Non-Newtonian Fluid Mech.
0377-0257, 129
, pp. 46
–55
.26.
Ali
, A.
, and Mehmood
, A.
, 2008, “Homotopy Analysis of Unsteady Boundary-Layer Flow Adjacent to Permeable Stretching Surface in a Porous Medium
,” Commun. Nonlinear Sci. Numer. Simul.
1007-5704, 13
(2
), pp. 340
–349
.27.
Mehmood
, A.
, and Ali
, A.
, 2006, “Analytic Solution of Generalized Three-dimensional Flow and Heat Transfer Over a Stretching Plane Wall
,” Int. Commun. Heat Mass Transfer
0735-1933, 33
, pp. 1243
–1252
.28.
Liao
, S. J.
, 1999, “A Uniformly Valid Analytical Solution of 2D Viscous Flow Past a Semi Infinite Flat Plate
,” J. Fluid Mech.
0022-1120, 385
, pp. 101
–128
.29.
Liao
, S. J.
, 1999, “An Explicit, Totally Analytic Approximate Solution for Blasius Viscous Flow Problems
,” Int. J. Non-Linear Mech.
0020-7462, 34
, pp. 759
–778
.30.
Liao
, S. J.
, and Pop
, I.
, 2004, “Explicit Analytic Solution for Similarity Boundary-Layer Equations
,” Int. J. Heat Mass Transfer
0017-9310, 47
(1
), pp. 75
–85
.31.
Liao
, S. J.
, and Campo
, A.
, 2002, “Analytic Solutions of the Temperature Distribution in Blasius Viscous Flow Problems
,” J. Fluid Mech.
0022-1120, 453
, pp. 411
–425
.32.
Liao
, S. J.
, and Cheung
, K. F.
, 2003, “Homotopy Analysis of Nonlinear Progressive Waves in Deep Water
,” J. Eng. Math.
0022-0833, 45
(2
), pp. 105
–116
.33.
Xu
, H.
, 2004, “An Explicit Analytic Solution for Free Convection About a Vertical Flat Plate Embedded in a Porous Media by Means of Homotopy Analysis Method
,” Appl. Math. Comput.
0096-3003, 158
, pp. 433
–443
.34.
Liao
, S. J.
, 2004, “On the Homotopy Analysis Method for Nonlinear Problems
,” Appl. Math. Comput.
0096-3003, 147
, pp. 499
–513
.35.
Liao
, S. J.
, 2003, “On the Analytic Solution of Magnetohydrodynamic Flows of Non-Newtonian Fluid Over a Stretching Sheet
,” J. Fluid Mech.
0022-1120, 488
, pp. 189
–212
.36.
Liao
, S. J.
, 2005, “A New Branch of Solutions of Boundary-layer Flows Over an Impermeable Stretched Plate
,” Int. J. Heat Mass Transfer
0017-9310, 48
(12
), pp. 2529
–2539
.37.
Yang
, C.
, and Liao
, S. J.
, 2006, “On the Explicit, Purely Analytic Solution of Von Kármán Swirling Viscous Flow
,” Commun. Nonlinear Sci. Numer. Simul.
1007-5704, 11
, pp. 83
–93
.38.
Wang
, C.
, Zhu
, J. M.
, Liao
, S. J.
, and Pop
, I.
, 2003, “On the Explicit Analytic Solutions of Cheng-Chang Equations
,” Int. J. Heat Mass Transfer
0017-9310, 46
(10
), pp. 1855
–1860
.39.
Allan
, F. M.
, and Syam
, M. I.
, 2005, “On Analytic Solution of the Non-Homogeneous Blasius Problem
,” J. Comput. Appl. Math.
0377-0427, 182
, pp. 355
–365
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