Asymptotic behavior of solutions to a BVP from fluid mechanics
Authors:
Susmita Sadhu and Joseph E. Paullet
Journal:
Quart. Appl. Math. 72 (2014), 703-718
MSC (2010):
Primary 34B15, 34B40, 34D05; Secondary 76S05, 76R10
DOI:
https://doi.org/10.1090/S0033-569X-2014-01359-X
Published electronically:
September 26, 2014
MathSciNet review:
3291823
Full-text PDF Free Access
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Additional Information
Abstract: In this paper we investigate a boundary value problem (BVP) derived from a model of boundary layer flow past a suddenly heated vertical surface in a saturated porous medium. The surface is heated at a rate proportional to $x^k$ where $x$ measures distance along the wall and $k>-1$ is constant. Previous results have established the existence of a continuum of solutions for $-1<k<-1/2$. Here we further analyze this continuum and determine that precisely one solution of this continuum approaches the boundary condition at infinity exponentially while all others approach algebraically. Previous results also showed that the solution to the BVP is unique for $-1/2 \leq k <0$. Here we extend the range of uniqueness to $0\leq k \leq 1$. Finally, the physical implications of the mathematical results are discussed and a comparison is made to the solutions for the related case of prescribed surface temperature on the surface.
References
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References
- J. V. Baxley, Some nonlinear third order boundary value problems on infinite intervals, Dynamic Systems and Applications, Vol. 6, Dynamic, Atlanta, GA, 2012, pp. 65–69. MR 3077398
- P. Cheng and W. J. Minkowycz, Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike, J. Geo. Res. 82 (1977), 2040-2044.
- D. M. Christopher, B. X. Wang, Similarity simulation for Marangoni convection around a vapor bubble during nucleation and growth, Int. J. Heat Mass Transfer 44 (2001), 799-810.
- D. M. Christopher and B. X. Wang, Prandtl number effects for Marangoni convection over a flat surface, Int. J. Therm. Sci. 40 (2001), 564-570.
- D. M. Christopher and B. X. Wang, Similarity solution for Marangoni convection over a flat surface, Trans. ASME J. Heat Transfer 124 (2002), 587-589.
- A. Davey, Boundary-layer flow at a saddle point of attachment, J. Fluid Mech. 10 (1961), 593–610. MR 0134089 (24 \#B142)
- Philip Hartman, On the asymptotic behavior of solutions of a differential equation in boundary layer theory, Z. Angew. Math. Mech. 44 (1964), 123–128 (English, with German and Russian summaries). MR 0175437 (30 \#5621)
- Philip Hartman, Ordinary differential equations, 2nd ed., Birkhäuser Boston, Mass., 1982. MR 658490 (83e:34002)
- D. R. Hartree, On an equation occurring in Falkner and Skan’s approximate treatment of the equations of the boundary layer, Proc. Camb. Phil. Soc. 33 (1937), 223-239.
- D. B. Ingham and S. N. Brown, Flow past a suddenly heated vertical plate in a porous medium, Proc. Roy. Soc. Lond. A403 (1986), 51-80.
- T. R. Mahapatra, S. K. Nandy, K. Vajravelu and R. A. Van Gorder, Stability analysis of fluid flow over a nonlinearly stretching sheet, Arch. Appl. Mech. 81 (2011), 1087-1091.
- J. H. Merkin and I. Pop, Natural convection boundary-layer flow in a porous medium with temperature-dependent boundary conditions, Transp. Porous Media 85 (2010), no. 2, 397–414. MR 2737544 (2011g:76155), DOI https://doi.org/10.1007/s11242-010-9569-9
- J. H. Merkin and G. Zhang, The boundary-layer flow past a suddenly heated vertical surface in a saturated porous medium, Heat and Mass Transfer 27 (1992), 299-304.
- Joseph E. Paullet, An uncountable number of solutions for a BVP governing Marangoni convection, Math. Comput. Modelling 52 (2010), no. 9-10, 1708–1715. MR 2719586 (2011g:76074), DOI https://doi.org/10.1016/j.mcm.2010.06.040
- Joseph E. Paullet and Joseph P. Previte, Comment on “Existence and uniqueness results for a nonlinear differential equation arising in viscous flow over a nonlinearly stretching sheet” [MR2735149], Appl. Math. Lett. 25 (2012), no. 8, 1114–1117. MR 2922522, DOI https://doi.org/10.1016/j.aml.2012.02.015
- I. Pop and D. B. Ingham, Convective Heat Transfer, Pergamon, 2001.
- Donald A. Nield and Adrian Bejan, Convection in porous media, 2nd ed., Springer-Verlag, New York, 1999. MR 1656781 (2000b:76107)
- William C. Troy, Edward A. Overman II, G. B. Ermentrout, and James P. Keener, Uniqueness of flow of a second-order fluid past a stretching sheet, Quart. Appl. Math. 44 (1987), no. 4, 753–755. MR 872826 (87m:76009)
- Lian Cun Zheng, Xiao Yan Sheng, and Xin Xin Zhang, Analytical approximate solutions for the Marangoni convection boundary layer equations, Acta Phys. Sinica 55 (2006), no. 10, 5298–5304 (Chinese, with English and Chinese summaries). MR 2287220
- Liancun Zheng, Xinxin Zhang, and Yingtao Gao, Analytical solution for Marangoni convection over a liquid-vapor surface due to an imposed temperature gradient, Math. Comput. Modelling 48 (2008), no. 11-12, 1787–1795. MR 2473408 (2010a:76047), DOI https://doi.org/10.1016/j.mcm.2008.04.003
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Additional Information
Susmita Sadhu
Affiliation:
Department of Mathematics, Georgia College & State University, Milledgeville, Georgia 31061
Email:
susmita.sadhu@gcsu.edu
Joseph E. Paullet
Affiliation:
School of Science, Pennsylvania State University at Erie, Erie, Pennsylvania 16563
Email:
jep7@psu.edu
Received by editor(s):
October 16, 2012
Received by editor(s) in revised form:
January 27, 2013
Published electronically:
September 26, 2014
Article copyright:
© Copyright 2014
Brown University