This investigation deals with some exact analytical solutions of the incompressible second grade fluid by using the method based on the separation of variables. In many cases, this method can derive exact analytical solutions easier than other methods. A family of solutions is derived in this paper, which governs scientific and engineering experimentations. The derived solutions represent the flows having streamlines as a family of ellipses, parabolas, concentric circles, and rectangular hyperbolas. From practical point of view, these flows have applications in many manufacturing processes in industry. Some physical features of the derived solutions are also illustrated by their contour plots.
Issue Section:
Fundamental Issues and Canonical Flows
References
1.
Dunn
, J. E.
, and Fosdick
, R. L.
, 1974
, “Thermodynamics, Stability and Boundedness of Fluids of Complexity 2 and Fluids of Second Grade
,” Arch. Ration. Mech. Anal.
, 56
(3
), pp. 191
–252
.10.1007/BF002809702.
Kaloni
, P. N.
, and Siddiqui
, A. M.
, 1983
, “The Flow of a Second Grade Fluid
,” Int. J. Eng. Sci.
, 21
(10
), pp. 1157
–1169
.10.1016/0020-7225(83)90080-03.
Berker
, R.
, 1963
, “Intégration des Équations du Mouvement d'un Fluide Visqueux Incompressible
,” Handbuch der Physik
, Vol. VIII/2
, S.
Flugge
, ed., Springer
, Berlin
.4.
Berker
, R.
, 1979
, “A New Solution of the Navier—Stokes Equation for the Motion of a Fluid Contained Between Two Parallel Plates Rotating About the Same Axis
,” Arch. Mech. Stosow.
, 31
(2
), pp. 265
–280
.5.
Berker
, R.
, 1982
, “An Exact Solution of the Navier—Stokes Equation: The Vortex With Curvilinear Axis
,” Int. J. Eng. Sci.
, 20
(2
), pp. 217
–230
.10.1016/0020-7225(82)90017-96.
Wang
, C. Y.
, 1989
, “Exact Solutions of the Unsteady Navier—Stokes Equations
,” ASME Appl. Mech. Rev.
, 42
(11
), pp. 268
–282
.10.1115/1.31524007.
Wang
, C. Y.
, 1990
, “Exact Solutions of the Navier—Stokes Equations—The Generalized Beltrami Flows: Review and Extension
,” Acta Mech.
, 81
(1–2
), pp. 69
–74
.10.1007/BF011745568.
Wang
, C. Y.
, 1991
, “Exact Solutions of the Steady-State Navier—Stokes Equations
,” Annu. Rev. Fluid Mech.
, 23
(1
), pp. 159
–177
.10.1146/annurev.fl.23.010191.0011119.
Drazin
, P.
, and Riley
, N.
, 2006
, The Navier—Stokes Equations: A Classification of Flows and Exact Solutions
, Cambridge University
, Cambridge, UK
.10.1017/CBO978051152645910.
Rajagopal
, K. R.
, and Gupta
, A. S.
, 1984
, “An Exact Solution for the Flow of a Non-Newtonian Fluid Past an Infinite Porous Plate
,” Meccanica
, 19
(2
), pp. 158
–160
.10.1007/BF0156046411.
Labropulu
, F.
, 2000
, “Exact Solutions of Non-Newtonian Fluid Flows With Prescribed Vorticity
,” Acta Mech.
, 141
(1–2
), pp. 11
–20
.10.1007/BF0117680412.
Majda
, A. J.
, and Bertozzi
, A.
, 2001
, Vorticity and Incompressible Flow
, Cambridge University
, Cambridge, UK
.10.1017/CBO978051161320313.
Le-Cao
, K.
, Mai-Duy
, N.
, and Tran-Cong
, T.
, 2009
, “An Effective Integrated-RBFN Cartesian-Grid Discretization for the Stream Function-Vorticity-Temperature Formulation in Nonrectangular Domains
,” Numer. Heat Transfer, Part B
, 55
(6
), pp. 480
–502
.10.1080/1040779090282747014.
Wright
, G. B.
, and Fornberg
, B.
, 2006
, “Scattered Node Compact Finite Difference-Type Formulas Generated From Radial Basis Functions
,” J. Comput. Phys.
, 212
(1
), pp. 99
–123
.10.1016/j.jcp.2005.05.03015.
Pu
, J. H.
, Shao
, S.
, Huang
, Y.
, and Hussain
, K.
, 2013
, “Evaluations of SWEs and SPH Numerical Modeling Techniques for Dam Break Flows
,” Eng. Appl. Comput. Fluid Mech.
, 7
(4
), pp. 544
–563
.10.1080/19942060.2013.1101549216.
Chau
, K. W.
, and Jiang
, Y. W.
, 2004
, “A Three-Dimensional Pollutant Transport Model in Orthogonal Curvilinear and Sigma Coordinate System for Pearl River Estuary
,” Int. J. Environ. Pollut.
, 21
(2
), pp. 188
–198
.10.1504/IJEP.2004.00418517.
Liu
, T.
, and Yang
, J.
, 2014
, “Three-Dimensional Computations of Water-Air Flow in a Bottom Spillway During Gate Opening
,” Eng. Appl. Comput. Fluid Mech.
, 8
(1
), pp. 104
–115
.10.1080/19942060.2014.1101550118.
Wu
, C. L.
, and Chau
, K. W.
, 2006
, “Mathematical Model of Water Quality Rehabilitation With Rainwater Utilization—A Case Study at Haigang
,” Int. J. Environ. Pollut.
, 28
(3–4
), pp. 534
–545
.10.1504/IJEP.2006.01122719.
Lai
, W.
, and Khan
, A. A.
, 2012
, “Discontinuous Galerkin Method for 1D Shallow Water Flows in Natural Rivers
,” Eng. Appl. Comput. Fluid Mech.
, 6
(1
), pp. 74
–86
.10.1080/19942060.2012.1101540420.
Chau
, K. W.
, and Jiang
, Y. W.
, 2001
, “3D Numerical Model for Pearl River Estuary
,” ASCE J. Hydraul. Eng.
, 127
(1
), pp. 72
–82
.10.1061/(ASCE)0733-9429(2001)127:1(72)21.
Cai
, R.
, and Zhang
, N.
, 2002
, “Explicit Analytical Solutions of Incompressible Unsteady 2–D Laminar Flow With Heat Transfer
,” Int. J. Heat Mass Transfer
, 45
(12
), pp. 2623
–2627
.10.1016/S0017-9310(01)00340-422.
Cai
, R.
, and Gou
, C.
, 2006
, “Algebraically Explicit Analytical Solutions for Unsteady Non-Newtonian Swirling Flow in an Annular Pipe
,” Sci. China, Ser. G
, 49
(4
), pp. 396
–400
.10.1007/s11433-006-2001-323.
Cai
, R.
, and Liu
, Q.
, 2008
, “A New Method for Deriving Analytical Solutions of Partial Differential Equations—Algebraically Explicit Analytical Solutions of Two-Buoyancy Natural Convection in Porous Media
,” Sci. China, Ser. G
, 51
(11
), pp. 1733
–1744
.10.1007/s11433-008-0174-724.
Le-Cao
, K.
, Mai-Duy
, N.
, Tran
, C. D.
, and Tran-Cong
, T.
, 2011
, “Numerical Study of Stream-Function Formulation Governing Flows in Multiply-Connected Domains by Integrated RBFs and Cartesian Grids
,” Comput. Fluids
, 44
(1
), pp. 32
–42
.10.1016/j.compfluid.2010.11.02725.
Saccomandi
, G.
, 1994
, “Some Unsteady Exact Pseudo-Plane Solutions for the Navier—Stokes Equations
,” Meccanica
, 29
(3
), pp. 261
–269
.10.1007/BF0146143926.
Saccomandi
, G.
, 1994
, “Some Exact Pseudo-Plane Solutions of the First Kind for the Navier—Stokes Equations
,” ZAMM-Z. Angew. Math. Mech.
, 45
(6
), pp. 978
–985
.10.1007/BF0095208827.
Rivlin
, R. S.
, and Ericksen
, J. L.
, 1955
, “Stress-Deformation Relations for Isotropic Materials
,” J. Ration. Mech. Anal.
, 4
(2
), pp. 323
–425
.10.1512/iumj.1955.4.5401128.
Dunn
, J. E.
, and Rajagopal
, K. R.
, 1995
, “Fluids of Differential Type: Critical Review and Thermodynamic Analysis
,” Int. J. Eng. Sci.
, 33
(5
), pp. 689
–729
.10.1016/0020-7225(94)00078-X29.
Destrade
, M.
, and Saccomandi
, G.
, 2006
, “Solitary and Compact-Like Shear Waves in the Bulk of Solids
,” Phys. Rev. E
, 73
, p. 065604
.10.1103/PhysRevE.73.06560430.
Miller
, W.
, Jr., 1989
, “Mechanism for Variable Separation in Partial Differential Equations and Their Relationship to Group Theory
,” Symmetries and Nonlinear Phenomena
, D.
Levi
and P.
Winternitz
, eds., World Scientific
, London
.Copyright © 2015 by ASME
You do not currently have access to this content.
