×

zbMATH — the first resource for mathematics

On the analytic solutions of the nonhomogeneous Blasius problem. (English) Zbl 1071.65108
Summary: A totally analytic solution of the nonhomogeneous Blasius problem is obtained using the homotopy analysis method. This solution converges for \(0\leqslant \eta < \infty\). Existence and nonuniqueness of solution is also discussed. An implicit relation between the velocity at the wall \(\lambda\) and the shear stress \(\alpha=f''(0)\) is obtained. The results presented here indicate that two solutions exist in the range \(0 < \lambda < \lambda_c\), for some critical value \(\lambda_c\) one solution exists for \(\lambda=\lambda_c\), and no solution exists for \(\lambda > \lambda_c\). An analytical value of the critical value of \(\lambda_c\) is also obtained for the first time.

MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adomian, G., A review of the decomposition method in applied mathematics, J. math. anal. appl., 135, 501-544, (1988) · Zbl 0671.34053
[2] Adomian, G., A review of the decomposition method and some recent results for nonlinear equations, Comput. math. appl., 21, 5, 101-127, (1991) · Zbl 0732.35003
[3] Adomian, G., Solving frontier problems of physicsthe decomposition method, (1994), Kluwer Academic Publishers Boston
[4] Adomian, G., Solution of physical problems by decomposition, Comput. math. appl., 27, 9/10, 145-154, (1994) · Zbl 0803.35020
[5] Adomian, G., Solution of the thomas – fermi equation, Appl. math. lett., 11, 3, 131-133, (1998) · Zbl 0947.34501
[6] Allan, F.M., On the similarity solutions of the boundary layer problem over a moving surface, Appl. math. lett., 10, 2, 81-85, (1997) · Zbl 0895.76025
[7] F.M. Allan, R.M. Abu-Saris, On the existence and non-uniqueness of non-homogeneous Blasius problem, Proceedings of the second Pal. International Conference, Gorden and Breach, 1999.
[8] Ayub, M.; Rasheed, A.; Hayat, T., Exact flow of a third grade fluid past a porous plate using homotopy analysis method, Internat. J. engrg. sci., 41, 2091-2103, (2003) · Zbl 1211.76076
[9] Coppel, W.A., On a differential equation of boundary layer theory, Phil. trans. roy. soc. London, ser. A, 253, 101-136, (1960) · Zbl 0093.19105
[10] Hayat, T.; Khan, M.; Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant fluid, Internat. J. engrg. sci., 42, 123-135, (2004) · Zbl 1211.76009
[11] Hayat, T.; Khan, M.; Asghar, S., Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta mech., 168, 213-232, (2004) · Zbl 1063.76108
[12] Hussaini, M.Y.; Lakin, W.D.; Nachman, On similarity solution of a boundary layer problem with upstream moving wall, A., SIAM J. appl. math., 7, 4, 699-709, (1987) · Zbl 0634.76034
[13] Liao, S.J., An explicit totally analytic approximate solution for Blasius viscous flow problems, Internat. J. non-linear mech., 34, 759-778, (1999) · Zbl 1342.74180
[14] Liao, S.J., A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate, J. fluid mech., 385, 101-128, (1999) · Zbl 0931.76017
[15] Liao, S.J., Beyond perturbationintroduction to the homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton
[16] Liao, S.J., On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J. fluid mech., 488, 189-212, (2003) · Zbl 1063.76671
[17] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl. math. comput., 147/2, 499-513, (2004) · Zbl 1086.35005
[18] Liao, S.J.; Campo, A., Analytic solutions of the temperature distribution in Blasius viscous flow problems, J. fluid mech., 453, 411-425, (2002) · Zbl 1007.76014
[19] Liao, S.J.; Cheung, K.F., Homotopy analysis of nonlinear progressive waves in deep water, J. engrg. math., 45, 2, 105-116, (2003) · Zbl 1112.76316
[20] Liao, S.J.; Pop, I., Explicit analytic solution for similarity boundary layer equations, Internat. J. heat mass transfer, 47/1, 75-85, (2004) · Zbl 1045.76008
[21] Schlichting, H., Boundary layer theory, (1979), McGraw-Hill New York
[22] M. Syam, A domain decomposition method for approximating the solution of the Korteweg – deVries equation, Appl. Math. Comput. (2004), in press.
[23] Weyl, H., On the differential equation of the simplest boundary-layer problems, Ann. math., 43, 381-407, (1942) · Zbl 0061.18002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.