0
Research Papers: Fundamental Issues and Canonical Flows

An Unsteady Separated Stagnation-Point Flow Towards a Rigid Flat Plate

[+] Author and Article Information
S. Dholey

Department of Mathematics,
M.U.C. Women's College,
Burdwan 713 104, India
e-mail: sdholey@gmail.com

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received February 16, 2018; final manuscript received June 9, 2018; published online July 10, 2018. Assoc. Editor: Praveen Ramaprabhu.

J. Fluids Eng 141(2), 021202 (Jul 10, 2018) (11 pages) Paper No: FE-18-1104; doi: 10.1115/1.4040572 History: Received February 16, 2018; Revised June 09, 2018

This paper discusses an unsteady separated stagnation-point flow of a viscous fluid over a flat plate covering the complete range of the unsteadiness parameter β in combination with the flow strength parameter a (>0). Here, β varies from zero, Hiemenz's steady stagnation-point flow, to large β-limit, for which the governing boundary layer equation reduces to an approximate one in which the convective inertial effects are negligible. An important finding of this study is that the governing boundary layer equation conceives an analytic solution for the specific relation β = 2a. It is found that for a given value of β(0) the present flow problem always provides a unique attached flow solution (AFS), whereas for a negative value of β the self-similar boundary layer solution may or may not exist that depends completely on the values of a and β (<0). If the solution exists, it may either be unique or dual or multiple in nature. According to the characteristic features of these solutions, they have been categorized into two classes—one which is AFS and the other is reverse flow solution (RFS). Another interesting finding of this analysis is the asymptotic solution which is more practical than the numerical solutions for large values of β (>0) depending upon the values of a. A novel result which arises from the pressure distribution is that for a positive value of β the pressure is nonmonotonic along the stagnation-point streamline as there is a pressure minimum which moves toward the stagnation-point with an increasing value of β > 0.

FIGURES IN THIS ARTICLE
<>
Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Rosenhead, L. , 1963, Laminar Boundary Layer, Clarendon Press, Oxford, UK.
Smith, F. T. , 1986, “ Steady and Unsteady Boundary Layer Separation,” Ann. Rev. Fluid Mech., 18(1), pp. 197–220. [CrossRef]
Schlichting, H. , and Gersten, K. , 2000, Boundary-Layer Theory, McGraw-Hill, New York. [CrossRef] [PubMed] [PubMed]
White, F. M. , 2006, Viscous Fluid Flow, McGraw-Hill, New York.
Stewartson, K. , 1960, “ The Theory of Unsteady Laminar Boundary Layers,” Adv. Appl. Mech., 6, pp. 1–37. [CrossRef]
Riley, N. , 1975, “ Unsteady Laminar Boundary Layers,” SIAM Rev., 17(2), pp. 274–297. [CrossRef]
Riley, N. , 1990, “ Unsteady Viscous Flows,” Sci. Prog., 74, pp. 361–377.
Telionis, D. P. , 1981, Unsteady Viscous Flow, Springer, New York. [CrossRef]
Williams, J. C. , and Johnson, W. D. , 1974, “ Semisimilar Solutions to Unsteady Boundary Layer Flows Including Separation,” AIAA J., 12(10), pp. 1388–1393. [CrossRef]
Wang, C. Y. , 1989, “ Exact Solutions of the Unsteady Navier-Stokes Equations,” Appl. Mech. Rev, 42(11S), pp. 269–282. [CrossRef]
Ingham, D. B. , 1984, “ Unsteady Separation,” J. Comp. Phys., 53(1), pp. 90–99. [CrossRef]
Ludlow, D. K. , Clarkson, P. A. , and Bassom, A. P. , 2000, “ New Similarity Solutions of the Unsteady Incompressible Boundary Layer Equations,” Q. J. Mech. Appl. Math., 53(2), pp. 175–206. [CrossRef]
Blasius, H. , 1908, “ Grenzschichten in Flüssigkeiten Mit Kleiner Reibung,” Z. Math. Phys., 56(2), pp. 1–37.
Burde, G. I. , 1995, “ Nonsteady Stagnation-Point Flows Over Permeable Surfaces: Explicit Solutions of the Navier-Stokes Equations,” ASME J. Fluids Eng., 117(1), pp. 189–191. [CrossRef]
Dholey, S. , and Gupta, A. S. , 2013, “ Unsteady Separated Stagnation-Point Flow of an Incompressible Viscous Fluid on the Surface of a Moving Porous Plate,” Phys. Fluids, 25(2), pp. 1–18. [CrossRef]
Williams, J. C., III , 1968, “ Nonsteady Stagnation-Point Flow,” AIAA J., 6(12), pp. 2417–2419. [CrossRef]
Rajappa, N. R. , 1979, “ Nonsteady Plane Stagnation-Point Flow With Hard Blowing,” ZAMP, 59(9), pp. 471–473.
Wang, C. Y. , 1985, “ The Unsteady Oblique Stagnation-Point Flow,” Phys. Fluids, 28(7), pp. 2046–2049. [CrossRef]
Fang, T. , Lee, C. F. , and Zhang, J. , 2011, “ The Boundary Layers of an Unsteady Incompressible Stagnation-Point Flow With Mass Transfer,” Int. J. Nonlinear Mech., 46(7), pp. 942–948. [CrossRef]
Nazar, R. , Amin, N. , Filip, D. , and Pop, I. , 2004, “ Unsteady Boundary Layer Flow in the Region of the Stagnation Point on a Stretching Sheet,” Int. J. Eng. Sci., 42(11–12), pp. 1241–1253. [CrossRef]
Zhong, Y. , and Fang, T. , 2011, “ Unsteady Stagnation Point Flow Over a Plate Moving Along the Direction of Flow Impingement,” Int. J. Heat Mass Transfer, 54, pp. 3103–3108. [CrossRef]
Magyari, E. , and Weidman, P. D. , 2012, “ Comment on ‘Unsteady Stagnation Point Flow Over a Plate Moving Along the Direction of Flow Impingement’ by Y. Zhong and T. Fang, Int. J. Heat and Mass Trans., 54 (2011) 3103–3108,” Int. J. Heat Mass Transfer, 55(4), pp. 1423–1424. [CrossRef]
Zhong, Y. , and Fang, T. , 2012, “ Reply to Comment on ‘Unsteady Stagnation Point Flow Over a Plate Moving Along the Direction of Flow Impingement’ by Y. Zhong and T. Fang, Int. J. Heat and Mass Trans., 54 (2011) 3103–3108,” Int. J. Heat Mass Transfer, 55(4), pp. 1425–1426. [CrossRef]
Ma, K. H. P. , and Hui, W. H. , 1990, “ Similarity Solutions of the Two-Dimensional Unsteady Boundary-Layer Equations,” J. Fluid Mech., 66(1), pp. 537–559. [CrossRef]
Lok, Y. Y. , and Pop, I. , 2014, “ Stretching or Shrinking Sheet Problem for Unsteady Separated Stagnation-Point Flow,” Meccanica, 49(6), pp. 1479–1492. [CrossRef]
Dholey, S. , 2016, “ Magnetohydrodynamic Unsteady Separated Stagnation-Point Flow of a Viscous Fluid Over of a Moving Plate,” ZAMM, 96(6), pp. 707–720. [CrossRef]
Dholey, S. , 2015, “ The Boundary Layers of an Unsteady Separated Stagnation-Point Flow of a Viscous Incompressible Fluid Over of a Moving Plate,” Fluid Dyn. Res., 47(3), pp. 1–22. [CrossRef]
Dholey, S. , 2018, “ Unsteady Separated Stagnation-Point Flow and Heat Transfer of a Viscous Fluid Over of a Moving Flat Surface,” Phys. Fluids, 30(4), pp. 1–11. [CrossRef]
Drazin, P. G. , and Riley, N. , 2007, The Navier–Stokes Equations: A Classification of Flows and Exact Solutions, Cambridge University Press, Cambridge, UK.
Batchelor, G. K. , 1997, An Introduction to Fluid Mechanics, Cambridge University Press, Cambridge, UK.

Figures

Grahic Jump Location
Fig. 1

Solution domain for the transformed boundary layer Eq.(16) subjected to the boundary conditions (17). The self-similar boundary layer solution is terminated at the value of γc = −4.50660. Separation occurs at the values of γ0 = −3.16240 and γ1 = −0.54778 since F¨(0) becomes zero, even though the boundary layer solutions still exists there. The values of F¨(0)=(0.92321, −0.98514) and 1.23259 for the values of γ = −1 and 0 are in excellent agreement with the corresponding valuesof Dholey and Gupta [15] and Drazin and Riley [29], respectively.

Grahic Jump Location
Fig. 2

Any pair of values (a,β) lying on and above the solid line correspond to at least one solution of Eq. (9) that satisfies the conditions (8). Any line implies the faster convergence of thesolution to the free boundary condition of Eq. (8). Below the solid line no self-similar boundary layer solution will be found. For a = 1, the values of βc, β0, and β1 are same as the corresponding values of γc, γ0, and γ1 as mentioned in the caption of Fig. 1.

Grahic Jump Location
Fig. 4

Variation of f′(η) with η for various values of β (≫1) when a = 5. For a given value of β (≥55), the numerical solution of the governing boundary layer Eq. (9) subjected to Eq. (8) does not capture the correct asymptotic behavior owing to the heavy pressure effect in the boundary layer region. However, the perturbed solution obtained from Eq. (9) secures the smooth matching of this solution with the outer boundary condition of Eq. (8) for large values of β depending upon the values of a (see Fig. 9).

Grahic Jump Location
Fig. 3

Variation of f′(η) with η for various values of β (≪−1) when a = 25 in cases of both AFS and RFS. For the fixed value of a = 25, AFS does not exist one below the value of β (=β0) ≈−79.05945. For both types of flows, the boundary layer thickness decreases rapidly with an increasing value of |β| when β < 0.

Grahic Jump Location
Fig. 5

Dimensionless displacement thickness δ as a function of β for several values of a in the case of AFS flow only. The decreasing rate of δ with β is very prominent for a small value of a, while for a large value of a it is insignificant owing to the saturation of the boundary layer flow. When a = 1, the values of δ = 0.69389 and 0.64790 for the values of β = −1 and 0 exhibit anoutstanding agreement with the corresponding values of Dholey and Gupta [15] and Drazin and Riley [29], respectively. Moreover, the values of δ = 0.81650 and 0.57735 for the pair of values (a = 0.5, β = 1) and (a = 1, β = 2) coincide with our earlier result δ=1/3a which is obtained from the analytic solution (19).

Grahic Jump Location
Fig. 7

(a)–(d) Streamlines of the unsteady separated stagnation-point flow for various values of β (≪−1) with a = 25 and t = 5 in the case of RFS flow only. From left to right, the numerical values of the streamlines are ψ = −1.0, −0.8, −0.6, −0.4, −0.2, −0.1, 0.0, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0. The nonzero stagnation-point approaches the plate surface along with the detachment of the other streamlines from the stagnation-point streamline (ψ = 0) with an increasing value of |β| when β < 0.

Grahic Jump Location
Fig. 8

(a)–(d) Streamlines of the unsteady separated stagnation-point flow for several values of time t with a = 25 and β = −25 in the case of RFS flow only. From left to right the numerical values of the streamlines are ψ = −0.8, −0.6, −0.4, −0.2, −0.1, 0.0, 0.1, 0.2, 0.4, 0.6, 0.8. The location of the nonzero stagnation-point remains the same for all values of time t, although, the other streamlines get detached from the stagnation-point streamline (ψ = 0) with an increasing value of time t.

Grahic Jump Location
Fig. 6

Centerline pressure distribution for three dissimilar (negative, zero, and positive) values of β when a = 1. For a positive value of β, the centerline pressure in nonmonotonic in nature, there is a pressure minimum which approaches the origin along the stagnation-point streamline with an increasing value of β.

Grahic Jump Location
Fig. 9

Asymptotic profiles of f′(η) which are obtained from Eq.(31) at four different values of β (= 50, 80, 150, and 500) when a = 5. Each profile matches correctly with the outer boundary condition of (8). This means that each solution follows the correct asymptotic behavior for the values of β≥50. Thus, we see that the asymptotic solution (31) must be followed to ensure full compliance with the numerical method which is unable to produce the correct asymptotic solution of this flow problem for large values of β (>55) when a = 5. Hence, for a fixed value of a, there is a definite value of β beyond which Eq.(31) provides the correct asymptotic solutions.

Grahic Jump Location
Fig. 10

Asymptotic profiles of f(η) which are obtained from Eq.(31) for several positive values of β when a = 5. As expected, velocity at a given location increases with the increase of β.

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In