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Lie Symmetry Analysis of Boundary Layer Stagnation-Point Flow and Heat Transfer of Non-Newtonian Power-Law Fluids Over a Nonlinearly Shrinking/Stretching Sheet with Thermal Radiation

  • G.C. Layek , Bidyut Mandal , Krishnendu Bhattacharyya EMAIL logo and Astick Banerjee
Published/Copyright: May 19, 2018
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 19 Issue 3-4

Abstract

A symmetry analysis of steady two-dimensional boundary layer stagnation-point flow and heat transfer of viscous incompressible non-Newtonian power-law fluids over a nonlinearly shrinking/stretching sheet with thermal radiation effect is presented. Lie group of continuous symmetry transformations is employed to the boundary layer flow and heat transfer equations, that gives scaling laws and self-similar equations for a special type of shrinking/stretching velocity (cx1/3) and free-stream straining velocity (ax1/3) along the axial direction to the sheet. The self-similar equations are solved numerically using very efficient shooting method. For the above nonlinear velocities, the unique self-similar solution is obtained for straining velocity being always less than the shrinking/stretching velocity for Newtonian and non-Newtonian power-law fluids. The thickness of velocity boundary layer becomes thinner with power-law index for shrinking as well as stretching sheet cases. Also, the thermal boundary layer thickness decreases with increasing values the Prandtl number and the radiation parameter.

Keywords: Lie group of symmetry transformations; boundary layer stagnation-point flow; heat transfer; non-Newtonian power-law fluids; nonlinearly shrinking/stretching sheet; thermal radiation

Acknowledgments

The authors would like to thank the editor and anonymous reviewers for their careful reading and constructive comments, which help a lot for improving the manuscript.

Appendix: Lie group theoretic method for differential equations

Lie group theoretic method is a very general tool for determining invariants or similarity solutions of differential equations. The key advantage of similarity solutions is that it reduces the number of independent variables for PDE and reduces the ODE to lower order. Fundamental theory of Lie symmetry analysis and some examples for problems such as heat equation can be found in Layek [27], Bluman and Kumei [29]. The book by Cantwell [28] is devoted to the application of Lie group analysis of some problems in fluid dynamics. Here we give a brief description of Lie group theory.

Consider the following PDE

[33]F(x,U,U(1),U(2),...,U(n))=0

where x=(x1,x2,...,xk) are the independent variables, u=(U1(x),U2(x),...,Ul(x)) the dependent variables,U(n)the nth order derivative of U. The main idea of Lie is to find transformations of variables that do not change the functional form of differential eq. [33]. A transformation X=(ϕ,ψ): D×R→D,D⊆Rn such that x∗=ϕx,U,ε and U∗=ψx,U,ε,ε∈S⊂R is a continuous parameter and forms a group, with x∗=x and U∗=U when ε=0 is called a symmetry group of transformation of eq. [33] if equivalence F(x,U,U(1),U(2),...,U(n))=0⇔F(x∗,U∗,U∗(1),U∗(2),...,U∗(n))=0 holds.

From Taylor series expansion about ε=0, the infinitesimal form of the transformations can be written as

[34]x∗=x+εξx,U+oε2andU∗=U+εηx,U+oε2,

where ξ=∂ϕ∂ε|ε=0 and η=∂ψ∂ε|ε=0 are called the infinitesimals of the transformations. The continuous form of the transformations can hence be determined by integrating the first-order system

∂x∗∂ε=ξ(x∗,U∗),∂y∗∂ε=η(x∗,U∗),

with the initial conditions x∗=x and y∗=y at ε=0. The corresponding infinitesimal generator is X=ξi∂/∂xi+ηj∂/∂Uj,i=1,...,k; j=1,2,...,l. The infinitesimals ξ,η can be determined using the defining equation XnF|F=0=0,Xn is the prolongation of X to all the derivatives of the dependent variables U up to order n of eq. [33]. The nth prolongation Xn is defined as

[35]X[n]=ξi∂∂xi+ηj∂∂Uj+ζi∂∂Uji+ζi1i2∂∂Uji1i2+...+ζi1i2...in∂∂Uji1i2...in

where Uji is the partial derivative of the variable Uj with respect to the variable xi, ζi1i2...ik=Di1...Dik(ηj−ξiUji)+ξiUjii1...ik,i,i1,...=1,2,..,k, and Dik denotes the total differential operator with respect to the variable xk, For the given problem, the coefficients ζi,ζij,ζijk,υi,ςi,ςij,ςijk can be written as

[36]ζ1=Dx(η1)−ψxDx(ξ1)−ψyDx(ξ2)ζ2=Dy(η1)−ψxDy(ξ1)−ψyDy(ξ2)ζ12=Dx(ζ2)−ψxyDx(ξ1)−ψyyDx(ξ2)ζ22=Dy(ζ2)−ψxyDy(ξ1)−ψyyDy(ξ2)ζ222=Dy(ζ22)−ψxyyDy(ξ1)−ψyyyDy(ξ2)υ1=Dx(η2)−UxDx(ξ1)−UyDx(ξ2)ς1=Dx(η3)−TxDx(ξ1)−TyDx(ξ2)ς2=Dy(η3)−TxDy(ξ1)−TyDy(ξ2)ς22=Dx(ς2)−TxyDy(ξ1)−TyyDy(ξ2)

The defining equation results in an over-determined set of linear homogeneous differential equations that is then solved for the infinitesimals ξ and η.

Eq. [33] invariant under the infinitesimal generator X is said to admit an invariant solution U=Ω(x) if and only if U=Ω(x) is a solution of eq. [33] and X(U−Ω(x))|U=Ω(x)=0, that is the solution is invariant under X. The invariant condition also known as invariant surface condition yields

[37]ξi∂Ωj(x)/∂xi=ηj

The method of characteristics then gives the characteristic equations

[38]dx1ξ1=...=dxiξi=dU1η1=...=dUjηj

Here Ω is replaced by U. Solving the above system, one would obtain invariants or self-similar solution of eq. [33].

References

[1] B.C. Sakiadis, Boundary layer behavior on continuous solid surfaces: I. Boundary layer equations for two-dimensional and axisymmetric flow, AIChE J. 7 (1961), 26–28.10.1002/aic.690070108Search in Google Scholar

[2] L.J. Crane, Flow past a stretching plate, Zamp. 21 (1970), 645–647.10.1007/BF01587695Search in Google Scholar

[3] C.Y. Wang, The three dimensional flow due to a stretching flat surface, Phys. Fluids. 27 (1984), 1915–1917.10.1063/1.864868Search in Google Scholar

[4] C.D. Surma Devi, H.S. Takhar and G. Nath, Unsteady, three-dimensional, boundary-layer flow due to a stretching surface, Int. J. Heat Mass. Transfer. 29 (12) (1986), 1996–1999.10.1016/0017-9310(86)90020-7Search in Google Scholar

[5] K.R. Rajagopal, T.Y. Na and A.S. Gupta, Flow of viscoelastic fluid over a stretching sheet, Rheol Acta. 23 (1984), 213–215.10.1007/BF01332078Search in Google Scholar

[6] W.C. Troy, E.A. Overman, H.G.B. Ermentrout and J.P. Keerner, Uniqueness of the flow of a second order fluid past a stretching sheet, Quart. Appl. Math. 44 (1987), 753–755.10.1090/qam/872826Search in Google Scholar

[7] B.S. Dandapat and A.S. Gupta, Flow and heat transfer in a viscoelastic fluid over a stretching sheet, Int. J. Non-Linear Mech. 24 (3) (1989), 215–219.10.1016/0020-7462(89)90040-1Search in Google Scholar

[8] K.R. Rajagopal, T.Y. Na and A.S. Gupta, A non-similar boundary layer on a stretching sheet in a non-Newtonian fluid with uniform free stream, J. Math. Phys. Sci. 21 (1987), 189–200.Search in Google Scholar

[9] R. Cortell, Viscous flow and heat transfer over a nonlinearly stretching sheet, Appl. Math. Comput. 184 (2007), 864–873.10.1016/j.amc.2006.06.077Search in Google Scholar

[10] H.I. Andersson and T.H. Toften, Numerical solution of a laminar boundary layer equations for power-law-fluid, J. non-Newtonian Fluid Mech. 32 (1989), 175–195.10.1016/0377-0257(89)85035-9Search in Google Scholar

[11] A.K. Sahu, M.N. Mathur, P. Chaturani and S.S. Bharatiya, Momentum and heat transfer from a continuous moving surface to a power law fluid, Acta Mech. 142 (2000), 119–131.10.1007/BF01190014Search in Google Scholar

[12] L.C. Zhang and X.X. Zhang, Skin friction and heat transfer in a power law fluid in a laminar boundary layer along a moving surface, Int. J. Heat Mass. Transfer. 45 (2002), 2667–2672.10.1016/S0017-9310(01)00363-5Search in Google Scholar

[13] J. Nagler, Generalized similarity transformation model for Power-law laminar boundary layer fluids with non-linear dynamic viscosity, WSEAS Trans. Fluid Mech. 9 (2014), 168–177.Search in Google Scholar

[14] B.S. Reddy, N. Kishan and M.N. Rajasekhar, MHD boundary layer flow of a non-Newtonian power-law fluid on a moving at plate, Adv. Appl. Sci. Res. 3 (2012), 1472–1481.Search in Google Scholar

[15] H.I. Andersson, K.H. Bech and B.S. Dandapat, Magnetohydrodynamic flow of a power-law fluid over a stretching sheet, Int. J. Non-Linear Mech. 27 (6) (1992), 929–936.10.1016/0020-7462(92)90045-9Search in Google Scholar

[16] R. Cortell, A note on magnetohydrodynamic flow of a power-law fluid over a stretching sheet, Appl. Math. Comput. 168 (2005), 557–566.10.1016/j.amc.2004.09.046Search in Google Scholar

[17] K.V. Prasad, S.R. Santhi and P.S. Datti, Non-Newtonian power-law fluid flow and heat transfer over a non-linearly stretching surface, Appl. Math. 3 (2012), 425–435.10.4236/am.2012.35065Search in Google Scholar

[18] M. Miklavčič and C.Y. Wang, Viscous flow due a shrinking sheet, Q. Appl. Math. 64 (2006), 283–290.10.1090/S0033-569X-06-01002-5Search in Google Scholar

[19] C.Y. Wang, Stagnation flow towards a shrinking sheet, Int. J. Nonlinear Mech. 43 (2008), 377–382.10.1016/j.ijnonlinmec.2007.12.021Search in Google Scholar

[20] A. Ishak, Y.Y. Lok and I. Pop, Stagnation-point flow over a shrinking sheet in a micropolar fluid, Chem. Eng. Commun. 197 (2010), 1417–1427.10.1080/00986441003626169Search in Google Scholar

[21] K. Bhattacharyya and G.C. Layek, Effects of suction/blowing on steady boundary layer stagnation-point flow and heat transfer towards a shrinking sheet with thermal radiation, Int. J. Heat Mass. Transfer. 54 (2011), 302–307.10.1016/j.ijheatmasstransfer.2010.09.043Search in Google Scholar

[22] K. Bhattacharyya, S. Mukhopadhyay and G.C. Layek, Slip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet, Int. J. Heat Mass. Transfer. 54 (2011), 308–313.10.1016/j.ijheatmasstransfer.2010.09.041Search in Google Scholar

[23] Y.Y. Lok, A. Ishak and I. Pop, MHD stagnation-point flow towards a shrinking sheet, Int. J. Numer. Meth. Heat Fluid Flow. 21 (2011), 61–72.10.1108/09615531111095076Search in Google Scholar

[24] N.A. Yacob and A. Ishak, Flow and heat transfer of a power-law fluid over a permeable shrinking sheet, Sains Malaysiana. 43 (3) (2014), 491–496.Search in Google Scholar

[25] T.R. Mahapatra, S.K. Nandy, K. Vajravelu and R.A. Van Gorder, Dual solutions for the magnetohydrodynamic stagnation-point flow of a power-law fluid over a shrinking sheet, ASME J. Appl. Mech. 79 (2012), 024503.10.1115/1.4005584Search in Google Scholar

[26] N. Bachok and A. Ishak, Similarity solutions for the stagnation-point flow and heat transfer over a nonlinearly stretching/shrinking sheet, Sains Malaysiana. 40 (11) (2011), 1297–1300.Search in Google Scholar

[27] G.C. Layek, An introduction to dynamical system and chaos, India: Springer, 2015.10.1007/978-81-322-2556-0Search in Google Scholar

[28] B.J. Cantwell, Introduction to symmetry analysis, Cambridge, New York: Cambridge University Press, 2002.Search in Google Scholar

[29] G.W. Bluman and S. Kumei, Symmetries and differential equations, Springer, New York, 1989.10.1007/978-1-4757-4307-4Search in Google Scholar

[30] M. Oberlack, A unified approach for symmetries in plane parallel turbulent shear flows, J. Fluid Mech. 427 (2001), 299–328.10.1017/S0022112000002408Search in Google Scholar

[31] Z.-S. She, X. Chen and F. Hussain, Lie-group derivation of a multi-layer mixing length formula for turbulent channel and pipe flows, arXiv, 1112.6312v3 [physics.flu-dyn].Search in Google Scholar

[32] Z.-S. She, X. Chen and F. Hussain, Quantifying wall turbulence via a symmetry approach: A Lie group theory, J. Fluid Mech. 827 (2017), 322–356.10.1017/jfm.2017.464Search in Google Scholar

[33] X. Chen and F. Hussain, Similarity transformation for equilibrium boundary layers, including effects of blowing and suction, Phys. Rev. Fluids. 2 (2017), art. id 034605.10.1103/PhysRevFluids.2.034605Search in Google Scholar

[34] M. Jalil and S. Asghar, Flow and heat transfer of Powell-Eyring fluid over a stretching surface: A Lie group analysis, J. Fluids Eng. 135 (2013), art. id 121201.10.1115/1.4025097Search in Google Scholar

[35] H.S. Hassan, S.A. Mahrous, A. Sharara and A. Hassan, A study for MHD boundary layer flow of variable viscosity over a heated stretching sheet via Lie-group method, Appl. Math. Inf. Sci. 9 (2015), 1327–1338.10.12785/amis/091L06Search in Google Scholar

[36] M.Q. Brewster, Thermal radiative transfer properties, Chichester: John Wiley and Sons, 1972.Search in Google Scholar

[37] K. Bhattacharyya and G.C. Layek, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Phys. Res. Int. 2014 (2014), art id 592536.10.1155/2014/592536Search in Google Scholar

Received: 2017-09-27
Accepted: 2018-01-20
Published Online: 2018-05-19
Published in Print: 2018-06-26

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijnsns-2017-0211
Keywords for this article
Lie group of symmetry transformations; boundary layer stagnation-point flow; heat transfer; non-Newtonian power-law fluids; nonlinearly shrinking/stretching sheet; thermal radiation

Articles in the same Issue

  1. Frontmatter
  3. A Robust Algorithm for Nonlinear Variable-Order Fractional Control Systems with Delay
  4. Numerical Methods for the Derivative Nonlinear Schrödinger Equation
  5. Lax Integrability and Exact Solutions of a Variable-Coefficient and Nonisospectral AKNS Hierarchy
  6. Modeling of Supersonic/Hypersonic Boundary Layer Transition Using a Single-Point Approach
  7. A Novel Macromodel based on Krylov Subspace Projection Method for Micromixers with Serpentine Channels
  8. Approaches to the Numerical Estimates of Grid Convergence of NSE in the Presence of Singularities
  9. Numerical Solutions of Stochastic Volterra–Fredholm Integral Equations by Hybrid Legendre Block-Pulse Functions
  10. Positivity and Stability of Standard and Fractional Descriptor Continuous-Time Linear and Nonlinear Systems
  11. Dynamics of Almost Periodic Solution for a Delayed Facultative Mutualism Model Involving Negative Feedback Terms
  12. Controllability of Fractional Evolution Inclusions with Noninstantaneous Impulses
  13. Analysis of a Delayed Predator–Prey System with Harvesting
  14. Nonlinear Bending of Rectangular Magnetoelectroelastic Thin Plates with Linearly Varying Thickness
  15. Numerical Method for a Class of Nonlinear Singularly Perturbed Delay Differential Equations Using Parametric Cubic Spline
  16. Numerical Simulation for Shale Gas Flow in Complex Fracture System of Fractured Horizontal Well
  17. Real-Time Control of a Rotary Inverted Pendulum using Robust LQR-based ANFIS Controller
  18. A Study of an Extended Generalized (2+1)-dimensional Jaulent–Miodek Equation
  19. RBFPUM with QR Factorization for Solving Water Flow Problem in Multilayered Soil
  20. Classical Magnetism and an Integral Formula Involving Modified Bessel Functions
  21. Lie Symmetry Analysis of Boundary Layer Stagnation-Point Flow and Heat Transfer of Non-Newtonian Power-Law Fluids Over a Nonlinearly Shrinking/Stretching Sheet with Thermal Radiation
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