zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Series solutions of nano boundary layer flows by means of the homotopy analysis method. (English) Zbl 1135.76016
Summary: We present a `similar’ solution for nano boundary layers with nonlinear Navier boundary condition. Three types of flows are considered: (i) the flow past a wedge; (ii) the flow in a convergent channel; (iii) the flow driven by an exponentially-varying outer flows. The resulting differential equations are solved by homotopy analysis method. Different from the perturbation methods, the present method is independent of small physical parameters so that it is applicable to not only weak but also strong nonlinear flow phenomena. Numerical results are compared with available exact results to demonstrate the validity of the present solution. The effects of the slip length $\ell $, the index parameters $n$ and $m$ on the velocity profile and tangential stress are discussed.

MSC:
76D10Boundary-layer theory, separation and reattachment, etc. (incompressible viscous fluids)
76M55Dimensional analysis and similarity (fluid mechanics)
WorldCat.org
Full Text: DOI
References:
[1] Navier, C. L. M.H.: Mémoire sur LES lois du mouvement des fluids, Mém. acad. Roy. sci. Inst. France 6, 389-440 (1823)
[2] Shikhmurzaev, Y. D.: The moving contact line on a smooth solid surface, Int. J. Multiphase flow 19, 589-610 (1993) · Zbl 1144.76452 · doi:10.1016/0301-9322(93)90090-H
[3] C.H. Choi, J.A. Westin, K.S. Breuer, To slip or not to slipwater flows in hydrophilic and hydrophobic microchannels, in: Proceedings of IMECE 2002, New Orleans, LA, Paper No. 2002-33707
[4] Matthews, M. T.; Hill, J. M.: Nano boundary layer equation with nonlinear Navier boundary condition, J. math. Anal. appl. 333, 381-400 (2007) · Zbl 1207.76050 · doi:10.1016/j.jmaa.2006.08.047
[5] Schlichting, H.: Boundary layer theory, (1979) · Zbl 0434.76027
[6] M.T. Matthews, J.M. Hill, A note on the boundary layer equations with linear slip boundary layer condition, Appl. Math. Lett. (2007), doi:10.1016/j.aml.2007.09.002, in press · Zbl 05228006
[7] S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University, 1992
[8] Liao, S. J.: Beyond perturbation: introduction to the homotopy analysis method, (2003)
[9] Liao, S. J.: An explicit, totally analytic approximation of Blasius viscous flow problems, Internat. J. Non-linear mech. 34, No. 4, 759-778 (1999) · Zbl 05137896
[10] Liao, S. J.: On the homotopy analysis method for nonlinear problems, Appl. math. Comput. 147, 499-513 (2004) · Zbl 1086.35005 · doi:10.1016/S0096-3003(02)00790-7
[11] Liao, S. J.; Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations, Stud. appl. Math. 119, 297-355 (2007)
[12] Liao, S. J.: Beyond perturbation: review on the basic ideas of the homotopy analysis method and its applications, Adv. mech. 38, No. 1, 1-34 (2008)
[13] Hayat, T.; Javed, T.; Sajid, M.: Analytic solution for rotating flow and heat transfer analysis of a third-grade fluid, Acta mech. 191, 219-229 (2007) · Zbl 1117.76069 · doi:10.1007/s00707-007-0451-y
[14] Hayat, T.; Khan, M.; Sajid, M.; Asghar, S.: Rotating flow of a third grade fluid in a porous space with Hall current, Nonlinear dynam. 49, 83-91 (2007) · Zbl 1181.76149 · doi:10.1007/s11071-006-9105-1
[15] Hayat, T.; Sajid, M.: On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder, Phys. lett. A 361, 316-322 (2007) · Zbl 1170.76307 · doi:10.1016/j.physleta.2006.09.060
[16] Hayat, T.; Sajid, M.: Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet, Int. J. Heat mass transfer 50, 75-84 (2007) · Zbl 1104.80006 · doi:10.1016/j.ijheatmasstransfer.2006.06.045
[17] Hayat, T.; Abbas, Z.; Sajid, M.; Asghar, S.: The influence of thermal radiation on MHD flow of a second grade fluid, Int. J. Heat mass transfer 50, 931-941 (2007) · Zbl 1124.80325 · doi:10.1016/j.ijheatmasstransfer.2006.08.014
[18] Hayat, T.; Sajid, M.: Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid, Internat. J. Engrg. sci. 45, 393-401 (2007) · Zbl 1213.76137 · doi:10.1016/j.ijengsci.2007.04.009
[19] Hayat, T.; Ahmed, N.; Sajid, M.; Asghar, S.: On the MHD flow of a second grade fluid in a porous channel, Comput. math. Appl. 54, 407-414 (2007) · Zbl 1123.76072 · doi:10.1016/j.camwa.2006.12.036
[20] Hayat, T.; Khan, M.; Ayub, M.: The effect of the slip condition on flows of an Oldroyd 6-constant fluid, J. comput. Appl. 202, 402-413 (2007) · Zbl 1147.76550 · doi:10.1016/j.cam.2005.10.042
[21] Sajid, M.; Siddiqui, A. M.; Hayat, T.: Wire coating analysis using MHD Oldroyd 8-constant fluid, Int. J. Engrg. sci. 45, 381-392 (2007)
[22] Sajid, M.; Hayat, T.; Asghar, S.: Non-similar analytic solution for MHD flow and heat transfer in a third-order fluid over a stretching sheet, Int. J. Heat mass transfer 50, 1723-1736 (2007) · Zbl 1140.76042 · doi:10.1016/j.ijheatmasstransfer.2006.10.011
[23] Sajid, M.; Hayat, T.; Asghar, S.: Non-similar solution for the axisymmetric flow of a third-grade fluid over radially stretching sheet, Acta mech. 189, 193-205 (2007) · Zbl 1117.76006 · doi:10.1007/s00707-006-0430-8
[24] Abbasbandy, S.: Soliton solutions for the 5th-order KdV equation with the homotopy analysis method, Nonlinear dynam. 51, 83-87 (2008) · Zbl 1170.76317 · doi:10.1007/s11071-006-9193-y
[25] Abbasbandy, S.: The application of the homotopy analysis method to solve a generalized Hirota -- satsuma coupled KdV equation, Phys. lett. A 361, 478-483 (2007) · Zbl 1273.65156
[26] Y.P. Liu, Z.B. Li, The homotopy analysis method for approximating the solution of the modified Korteweg -- de Vries equation, Chaos Solitons Fractals, in press · Zbl 1197.65166 · doi:10.1016/j.chaos.2007.01.148
[27] Zou, L.; Zong, Z.; Wang, Z.; He, L.: Solving the discrete KdV equation with homotopy analysis method, Phys. lett. A 370, 287-294 (2007) · Zbl 1209.65122 · doi:10.1016/j.physleta.2007.05.068
[28] Song, L.; Zhang, H. Q.: Application of homotopy analysis method to fractional KdV -- Burgers -- Kuramoto equation, Phys. lett. A 367, 88-94 (2007) · Zbl 1209.65115 · doi:10.1016/j.physleta.2007.02.083
[29] Abbasbandy, S.: The application of the homotopy analysis method to nonlinear equations arising in heat transfer, Phys. lett. A 360, 109-113 (2006) · Zbl 1236.80010
[30] Abbasbandy, S.: Homotopy analysis method for heat radiation equations, Int. commun. Heat mass transfer 34, 380-387 (2007)
[31] Sajid, M.; Hayat, T.; Asghar, S.: Comparison between the HAM and HPM solutions of tin film flows of non-Newtonian fluids on a moving belt, Nonlinear dynam. 50, 27-35 (2007) · Zbl 1181.76031 · doi:10.1007/s11071-006-9140-y
[32] M. Sajid, T. Hayat, Comparison of HAM and HPM methods for nonlinear heat conduction and convection equations, Nonlinear Anal. Real World Appl., in press · Zbl 1156.76436 · doi:10.1016/j.nonrwa.2007.08.007
[33] Zhu, S. P.: An exact and explicit solution for the valuation of American put options, Quant. finance 6, 229-242 (2006) · Zbl 1136.91468 · doi:10.1080/14697680600699811
[34] Zhu, S. P.: A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield, Anziam j. 47, 477-494 (2006) · Zbl 1147.91336 · doi:10.1017/S1446181100010087 · http://www.austms.org.au/Publ/ANZIAM/V47P4/2378.html
[35] Y. Wu, K.F. Cheung, Explicit solution to the exact Riemann problems and application in nonlinear shallow water equations, Int. J. Numer. Methods Fluids, in press · Zbl 1210.76033 · doi:10.1002/fld.1696
[36] Yamashita, M.; Yabushita, K.; Tsuboi, K.: An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method, J. phys. A 40, 8403-8416 (2007) · Zbl 05178236
[37] Bouremel, Y.: Explicit series solution for the glauert-jet problem by means of the homotopy analysis method, Commun. nonlinear sci. Numer. simul. 12, No. 5, 714-724 (2007) · Zbl 1115.76065 · doi:10.1016/j.cnsns.2005.07.001
[38] Tao, L.; Song, H.; Chakrabarti, S.: Nonlinear progressive waves in water of finite depth --- an analytic approximation, Clastal engrg. 54, 825-834 (2007)
[39] Song, H.; Tao, L.: Homotopy analysis of 1D unsteady, nonlinear groundwater flow through porous media, J. coastal res. 50, 292-295 (2007)
[40] A. Molabahrami, F. Khani, The homotopy analysis method to solve the Burgers -- Huxley equation, Nonlinear Anal. B Real World Appl. (2007), doi:10.1016/j.nonrwa.2007.10.014, in press · Zbl 1167.35483
[41] Bataineh, A. S.; Noorani, M. S. M.; Hashim, I.: Solutions of time-dependent Emden -- Fowler type equations by homotopy analysis method, Phys. lett. A 371, 72-82 (2007) · Zbl 1209.65104 · doi:10.1016/j.physleta.2007.05.094
[42] Wang, Z.; Zou, L.; Zhang, H.: Applying homotopy analysis method for solving differential -- difference equation, Phys. lett. A 369, 77-84 (2007) · Zbl 1209.65119 · doi:10.1016/j.physleta.2007.04.070
[43] Inc., Mustafa: On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method, Phys. lett. A 365, 412-415 (2007) · Zbl 1203.65275 · doi:10.1016/j.physleta.2007.01.069
[44] W.H. Cai, Nonlinear dynamics of thermal-hydraulic networks, PhD thesis, University of Notre Dame, 2006
[45] Song, Y.; Zheng, L. C.; Zhang, X. X.: On the homotopy analysis method for solving the boundary layer flow problem over a stretching surface with suction and injection, J. univ. Sci. technol. Beijing 28, 782-784 (2006)
[46] Liao, S. J.; Magyari, E.: Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones, Z. angew. Math. phys. 57, No. 5, 777-792 (2006) · Zbl 1101.76056 · doi:10.1007/s00033-006-0061-x
[47] Liao, S. J.: A new branch of solutions of boundary-layer flows over a permeable stretching plate, Internat. J. Non-linear mech. 42, 819-830 (2007) · Zbl 1200.76046 · doi:10.1016/j.ijnonlinmec.2007.03.007