Homotopy solutions for a generalized second-grade fluid past a porous plate. (English) Zbl 1094.76005

Summary: We study the flow of a second-grade fluid past a porous plate subject to either suction or blowing at the plate. A modified model of second-grade fluid that has shear-dependent viscosity and can predict the normal stress difference is used. The differential equations governing the flow are solved using homotopy analysis method. Expressions for the velocity have been constructed and discussed with the help of graphs. Analysis of the obtained results shows that the flow is appreciably influenced by the material and normal stress coefficients. Several results are deduced as particular cases of the presented analysis.


76A05 Non-Newtonian fluids
76S05 Flows in porous media; filtration; seepage
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
Full Text: DOI


[1] Bird, R. B., Armstrong, R. C., and Hassager, J., Dynamics of Polymeric Liquids., Vol. 1, Wiley, New York, 1977.
[2] Slattery, J. C., Advanced Transport Phenomena., Cambridge University Press, New York, 1999. · Zbl 0963.76003
[3] Dunn, J. E. and Fosdick, R. L., ’Thermodynamics stability, and boundedness of fluids of complexity 2 and fluids of second grade,’ Arch. Rat. Mech. Anal.. 56., 1974, 191–252. · Zbl 0324.76001
[4] Dunn, J. E. and Rajagopal, K. R., ’Fluids of differential type: Critical review and thermodynamic analysis,’ Int. J. Eng. Sci.. 33., 1995, 689–729. · Zbl 0899.76062
[5] Fosdick, R. L. and Rajagopal, K. R., ’Anomalous features in the model of second order fluids,’ Arch. Rat. Mech. Anal.. 70., 1979, 145–152. · Zbl 0427.76006
[6] Rajagopal, K. R., ’A note on unsteady unidirectional flows of a non-Newtonian fluid,’ Int. J. Non-Linear Mech.. 17., 1982, 369–373. · Zbl 0527.76003
[7] Rajagopal, K. R., ’On the creeping flow of the second grade fluid,’ J. Non-Newtonian Fluid Mech.. 15., 1984, 239–246. · Zbl 0568.76015
[8] Rajagopal, K. R., ’Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid,’ Acta Mech.. 49., 1983, 282–285. · Zbl 0539.76005
[9] Hayat, T., Asghar, S., and Siddiqui, A. M., ’On the moment of a plane disk in a non-Newtonian fluid,’ Acta Mech.. 136., 1999, 125–131. · Zbl 0934.76003
[10] Hayat, T., Asghar, S., and Siddiqui, A. M., ’Periodic unsteady flows of a non-Newtonian fluid,’ Acta Mech.. 131., 1998, 169–175. · Zbl 0939.76002
[11] Hayat, T., Asghar, S., and Siddiqui, A. M., ’Some unsteady unidirectional flows of a non-Newtonian fluid,’ Int. J. Eng. Sci.. 38., 2000, 337–346. · Zbl 1210.76015
[12] Hayat, T., Khan, M., Siddiqui, A. M., and Asghar, S., ’Transient flows of a second grade fluid,’ Int. J. Non-Linear Mech.. 39., 2004, 1621–1633. · Zbl 1348.76017
[13] Benharbit, A. M. and Siddiqui, A. M., ’Certain solutions of the equations of the planar motion of a second grade fluid for steady and unsteady cases,’ Acta Mech.. 94., 1992, 85–96. · Zbl 0749.76002
[14] Rajagopal, K. R. and Gupta, A. S., ’An exact solution for the flow of a non-Newtonian fluid past an infinite plate,’ Meccanica. 19., 1984, 158–160. · Zbl 0552.76008
[15] Bandelli, R. and Rajagopal, K. R., ’On the falling of objects in non-Newtonian fluids,’ Ann. Ferrara. 39., 1996, 1–18. · Zbl 0858.76004
[16] Bandelli, R., ’Unsteady unidirectional flows of second grade fluids in domains with heated boundaries,’ Int. J. Non-Linear Mech.. 30., 1995, 263–269. · Zbl 0837.76004
[17] Fetecau, C. and Zierep, J., ’On a class of exact solutions of the equations of motion of a second grade fluid,’ Acta Mech.. 150., 2001, 135–138. · Zbl 0992.76006
[18] Fetecau, C., Fetecau, C., and Zierep, J., ’Decay of a potential vortex and propagation of a heat wave in second grade fluid,’ Int. J. Non-Linear Mech.. 37., 2002, 1051–1056. · Zbl 1346.76033
[19] Man, C. S., ’Nonsteady channel flow of ice as a modified second grade fluid with power law viscosity,’ Arch. Rat. Mech. Anal.. 119., 1992, 35–57. · Zbl 0757.76001
[20] Massoudi, M. and Phuoc, T. X., ’Fully developed flow of a modified second grade fluid with temperature dependent viscosity,’ Acta Mech.. 150., 2001, 23–37. · Zbl 0993.76005
[21] Straughan, B., ’Energy stability in the Benard problem for a fluid of second grade,’ J. Appl. Math. Phys. (ZAMP). 34., 1983, 502–508. · Zbl 0557.76011
[22] Straughan, B., The Energy Method, Stability and Nonlinear Convection., Springer, New York, 1992. · Zbl 0743.76006
[23] Gupta, G. and Massoudi, M., ’Flow of a generalized second grade fluid between heated plates,’ Acta Mech.. 99., 1993, 21–33. · Zbl 0774.76005
[24] Franchi, H. and Straughan, B., ’Stability and nonexistence results in the generalized theory of a fluid of second grade,’ J. Math. Anal. Appl.. 180., 1993, 122–137. · Zbl 0797.76002
[25] Liao, S. J., ’The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,’ Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
[26] Liao, S. J. and Campo, A., ’Analytic solutions of the temperature distribution in Blasius viscous flow problems,’ J. Fluid Mech.. 453., 2002, 411–425. · Zbl 1007.76014
[27] Liao, S. J., ’On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet,’ J. Fluid Mech.. 488., 2003, 189–212. · Zbl 1063.76671
[28] Liao, S. J., ’An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying amplitude,’ Int. J. Non-Linear Mech.. 38., 2003, 1173–1183. · Zbl 1348.74225
[29] Liao, S. J. and Pop, I., ’Explicit analytic solution for similarity boundary layer equations,’ Int. J. Heat Mass Trans.. 47., 2004, 75–85. · Zbl 1045.76008
[30] Ayub, M., Rasheed, A., and Hayat, T., ’Exact flow of a third grade fluid past a porous plate using homotopy analysis method,’ Int. J. Eng. Sci.. 41., 2003, 2091–2103. · Zbl 1211.76076
[31] Hayat, T., Khan, M., and Ayub, M., ’On the explicit analytic solutions of an Oldroyd 6-constant fluid,’ Int. J. Eng. Sci.. 42., 2004, 123–135. · Zbl 1211.76009
[32] Liao, S. J., Beyond Perturbation: Introduction to Homotopy Analysis Method., Chapman & Hall/CRC Press, Boca Raton, FL, 2003.
[33] Liao, S. J., ’On the homotopy analysis method for nonlinear problems,’ Appl. Math. Comput.. 147., 2004, 499–513. · Zbl 1086.35005
[34] Liao, S. J. and Cheung, K. F., ’Homotopy analysis of nonlinear progressive waves in deep water,’ J. Eng. Math.. 45., 2003, 105–116. · Zbl 1112.76316
[35] Hayat, T., Khan, M., and Asghar, S., ’Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid,’ Acta Mech.. 168., 2004, 213–232. · Zbl 1063.76108
[36] Rivilin, R. S. and Ericksen, J. L., ’Stress deformation relations for isotropic materials,’ J. Rat. Mech. Anal.. 4., 1955, 323–425.
[37] Man, C. S., Shields, D. H., Kjartanson, B., and Sun, Q., ’Creeping of ice as a fluid of complexity 2: The pressuremeter problem,’ in Proceedings 10th Canadian Congress on Applied Mechanics., Vol. 1, H. Rasmussen (ed.), Univ. W. Ontario, London, Ontario, 1985.
[38] Siddiqui, A. M., Hayat, T., and Asghar, S., ’Periodic flows of a non-Newtonian fluid between two parallel plates,’ Int. J. Non-Linear Mech.. 34., 1999, 895–899. · Zbl 1006.76004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.