Non-symmetric flow over a stretching/shrinking surface with mass transfer
Abstract
The non-symmetric flow over a stretching/shrinking surface in an otherwise quiescent fluid is considered under the assumption that the surface can stretch or shrink in one direction and stretch in a direction perpendicular to this. The problem is reduced to similarity form, being described by two dimensionless parameters, γ the relative stretching/shrinking rate and S characterizing the fluid transfer through the boundary. Numerical solutions are obtained for representative values of γ and S, a feature of which are the existence of critical values 
of γ dependent on S, these being determined numerically. Asymptotic forms for large γ and S, for both fluid withdrawal, S > 0 and injection S < 0 are obtained and compared with the corresponding numerical results.
Keyword : non-symmetric flow, permeable surface, multiple solutions, asymptotic solutions
How to Cite
Lok, Y. Y., Merkin, J. H., & Pop, I. (2019). Non-symmetric flow over a stretching/shrinking surface with mass transfer. Mathematical Modelling and Analysis, 24(4), 617-634. https://doi.org/10.3846/mma.2019.037
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
NAG: Numerical Algorithm Group. Available from Internet: https://www.nag.co.uk
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P.D. Ariel. Generalized three-dimensional flow due to a stretching sheet. J. App. Math. Mech. (ZAMM), 83(12):844–852, 2003. https://doi.org/10.1002/zamm.200310052
W.H.H. Banks. Similarity solutions of the boundary-layer equations for a stretching wall. J. Theor. Appl. Mech., 2:375–392, 1983.
R.C. Bataller. Similarity solutions for flow and heat transfer of a quiescent fluid over a nonlinearly stretching surface. J. Mater. Processing Tech., 203(1–3):176– 183, 2008. https://doi.org/10.1016/j.jmatprotec.2007.09.055
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T. Fang. Boundary layer flow over a shrinking sheet with powerlaw velocity. Int J. Heat Mass Transfer, 51(25-26):5838–5843, 2008. https://doi.org/10.1016/j.ijheatmasstransfer.2008.04.067
T.G. Fang, J. Zhang and S.S. Yao. Viscous flow over an unsteady shrinking sheet with mass transfer. Chin. Phys. Lett., 26:014703, 2009. https://doi.org/10.1088/0256-307X/26/1/014703
E.G. Fisher. Extrusion of Plastics. Wiley, New York, 1976.
S. Goldstein. On backward boundary layers and flow in converging passages. J. Fluid Mech., 21(1):33–45, 1965. https://doi.org/10.1017/S0022112065000034
P.S. Gupta and A.S. Gupta. Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng., 55(6):744–746, 1977. https://doi.org/10.1002/cjce.5450550619
A. Ishak, R. Nazar and I. Pop. Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet. Heat Mass Transfer, 44:921–927, 2008. https://doi.org/10.1007/s00231-007-0322-z
S.J. Liao. A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat and Mass Transfer, 48(12):2529–2539, 2005. https://doi.org/10.1016/j.ijheatmasstransfer.2005.01.005
S.J. Liao and I. Pop. Explicit analytic solution for similarity boundary layer equations. Int. J. Heat Mass Transfer, 47(1):75–85, 2004. https://doi.org/10.1016/S0017-9310(03)00405-8
Y.Y. Lok, J.H. Merkin and I. Pop. Axisymmetric rotational stagnation point flow impinging on a permeable stretching/shrinking rotating disk. Eur. J. Mech. B/Fluids, 72(November-December):275–292, 2018. https://doi.org/10.1016/j.euromechflu.2018.05.013
E. Magyari and B. Keller. Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls. Eur. J. Mech. B/Fluids, 19(1):109–122, 2000. https://doi.org/10.1016/S0997-7546(00)00104-7
E. Magyari and P.Weidman. New solutions of the Navier-Stokes equations associated with flow above moving boundaries. Acta Mech., 228(10):3725–3733, 2017. https://doi.org/10.1007/s00707-017-1919-z
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J.H. Merkin. On dual solutions occurring in mixed convection in a porous medium. J. Engng Math., 20:171–179, 1986. https://doi.org/10.1007/BF00042775
J.H. Merkin and T. Mahmood. Mixed convection boundary layer similarity solutions: prescribed wall heat flux. J. Applied Math. Physics (ZAMP), 40:51– 68, 1989. https://doi.org/10.1007/BF00945309
J.H. Merkin and I. Pop. Natural convection boundary-layer flow in a porous medium with temperature-dependent boundary conditions. Transp. Porous Media, 85:397–414, 2010. https://doi.org/10.1007/s11242-010-9569-9
M. Miklavˇciˇc and C.Y. Wang. Viscous flow due to a shrinking sheet. Quart. Appl. Math., 64:283–290, 2006.
M. Sajid, N. Ali, Z. Abbas and T. Javed. Stretching flows with general slip boundary condition. Int. J. Mod. Phys. B., 24(30):5939–5947, 2010. https://doi.org/10.1142/S0217979210055512
L.F. Shampine, I. Gladwell and S. Thompson. Solving ODEs with Matlab. Cambridge University Press, Cambridge, 2003.
K. Vajravelu. Flow and heat transfer in a saturated porous medium over a stretching surface. J. App. Math. Mech. (ZAMM), 74(12):605–614, 1994. https://doi.org/10.1002/zamm.19940741209
C.Y. Wang. The three-dimensional flow due to a stretching flat surface. Phys. Fluids, 27(8):1915–1917, 1984. https://doi.org/10.1063/1.864868
C.Y. Wang. Review of similarity stretching exact solutions of the Navier-Stokes equations. Eur. J. Mech. B/Fluids, 30(5):475–479, 2011. https://doi.org/10.1016/j.euromechflu.2011.05.006
C.Y. Wang. Uniform flow over a bi-axial stretching surface. J. Fluids Engng., 137:084502, 2015. https://doi.org/10.1115/1.4029447
M.E. Ali. On thermal boundary layer on a power-law stretched surface with suction or injection. Int. J. Heat Fluid Flow, 16(4):280–290, 1995. https://doi.org/10.1016/0142-727X(95)00001-7
P.D. Ariel. Generalized three-dimensional flow due to a stretching sheet. J. App. Math. Mech. (ZAMM), 83(12):844–852, 2003. https://doi.org/10.1002/zamm.200310052
W.H.H. Banks. Similarity solutions of the boundary-layer equations for a stretching wall. J. Theor. Appl. Mech., 2:375–392, 1983.
R.C. Bataller. Similarity solutions for flow and heat transfer of a quiescent fluid over a nonlinearly stretching surface. J. Mater. Processing Tech., 203(1–3):176– 183, 2008. https://doi.org/10.1016/j.jmatprotec.2007.09.055
C.K. Chen and M.I. Char. Heat transfer of a continuous, stretching surface with suction or blowing. J. Math. Anal. Appl., 135(2):568–580, 1988. https://doi.org/10.1016/0022-247X(88)90172-2
R. Cortell. Viscous flow and heat transfer over a nonlinearly stretching surface. Appl. Math. Computation, 184(2):864–873, 2007. https://doi.org/10.1016/j.amc.2006.06.077
L.J. Crane. Flow past a stretching plate. J. Appl. Math. Phys. (ZAMP), 21(4):645–647, 1970. https://doi.org/10.1007/BF01587695
T. Fang. Boundary layer flow over a shrinking sheet with powerlaw velocity. Int J. Heat Mass Transfer, 51(25-26):5838–5843, 2008. https://doi.org/10.1016/j.ijheatmasstransfer.2008.04.067
T.G. Fang, J. Zhang and S.S. Yao. Viscous flow over an unsteady shrinking sheet with mass transfer. Chin. Phys. Lett., 26:014703, 2009. https://doi.org/10.1088/0256-307X/26/1/014703
E.G. Fisher. Extrusion of Plastics. Wiley, New York, 1976.
S. Goldstein. On backward boundary layers and flow in converging passages. J. Fluid Mech., 21(1):33–45, 1965. https://doi.org/10.1017/S0022112065000034
P.S. Gupta and A.S. Gupta. Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng., 55(6):744–746, 1977. https://doi.org/10.1002/cjce.5450550619
A. Ishak, R. Nazar and I. Pop. Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet. Heat Mass Transfer, 44:921–927, 2008. https://doi.org/10.1007/s00231-007-0322-z
S.J. Liao. A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat and Mass Transfer, 48(12):2529–2539, 2005. https://doi.org/10.1016/j.ijheatmasstransfer.2005.01.005
S.J. Liao and I. Pop. Explicit analytic solution for similarity boundary layer equations. Int. J. Heat Mass Transfer, 47(1):75–85, 2004. https://doi.org/10.1016/S0017-9310(03)00405-8
Y.Y. Lok, J.H. Merkin and I. Pop. Axisymmetric rotational stagnation point flow impinging on a permeable stretching/shrinking rotating disk. Eur. J. Mech. B/Fluids, 72(November-December):275–292, 2018. https://doi.org/10.1016/j.euromechflu.2018.05.013
E. Magyari and B. Keller. Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls. Eur. J. Mech. B/Fluids, 19(1):109–122, 2000. https://doi.org/10.1016/S0997-7546(00)00104-7
E. Magyari and P.Weidman. New solutions of the Navier-Stokes equations associated with flow above moving boundaries. Acta Mech., 228(10):3725–3733, 2017. https://doi.org/10.1007/s00707-017-1919-z
J.B. McLeod and K.R. Rajagopal. On the uniqueness of flow of a Navier-Stokes fluid due to a stretching boundary. Arch. Ration. Mech. Anal., 98:385–393, 1987. https://doi.org/10.1007/BF00276915
J.H. Merkin. On dual solutions occurring in mixed convection in a porous medium. J. Engng Math., 20:171–179, 1986. https://doi.org/10.1007/BF00042775
J.H. Merkin and T. Mahmood. Mixed convection boundary layer similarity solutions: prescribed wall heat flux. J. Applied Math. Physics (ZAMP), 40:51– 68, 1989. https://doi.org/10.1007/BF00945309
J.H. Merkin and I. Pop. Natural convection boundary-layer flow in a porous medium with temperature-dependent boundary conditions. Transp. Porous Media, 85:397–414, 2010. https://doi.org/10.1007/s11242-010-9569-9
M. Miklavˇciˇc and C.Y. Wang. Viscous flow due to a shrinking sheet. Quart. Appl. Math., 64:283–290, 2006.
M. Sajid, N. Ali, Z. Abbas and T. Javed. Stretching flows with general slip boundary condition. Int. J. Mod. Phys. B., 24(30):5939–5947, 2010. https://doi.org/10.1142/S0217979210055512
L.F. Shampine, I. Gladwell and S. Thompson. Solving ODEs with Matlab. Cambridge University Press, Cambridge, 2003.
K. Vajravelu. Flow and heat transfer in a saturated porous medium over a stretching surface. J. App. Math. Mech. (ZAMM), 74(12):605–614, 1994. https://doi.org/10.1002/zamm.19940741209
C.Y. Wang. The three-dimensional flow due to a stretching flat surface. Phys. Fluids, 27(8):1915–1917, 1984. https://doi.org/10.1063/1.864868
C.Y. Wang. Review of similarity stretching exact solutions of the Navier-Stokes equations. Eur. J. Mech. B/Fluids, 30(5):475–479, 2011. https://doi.org/10.1016/j.euromechflu.2011.05.006
C.Y. Wang. Uniform flow over a bi-axial stretching surface. J. Fluids Engng., 137:084502, 2015. https://doi.org/10.1115/1.4029447