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.

## References

References
1.
,
L.
,
1963
,
Laminar Boundary Layer
,
Clarendon Press
,
Oxford, UK
.
2.
Smith
,
F. T.
,
1986
, “
,”
Ann. Rev. Fluid Mech.
,
18
(
1
), pp.
197
220
.
3.
Schlichting
,
H.
, and
Gersten
,
K.
,
2000
,
Boundary-Layer Theory
,
McGraw-Hill
,
New York
.
4.
White
,
F. M.
,
2006
,
Viscous Fluid Flow
,
McGraw-Hill
,
New York
.
5.
Stewartson
,
K.
,
1960
, “
The Theory of Unsteady Laminar Boundary Layers
,”
,
6
, pp.
1
37
.
6.
Riley
,
N.
,
1975
, “
,”
SIAM Rev.
,
17
(
2
), pp.
274
297
.
7.
Riley
,
N.
,
1990
, “
,”
Sci. Prog.
,
74
, pp.
361
377
.
8.
Telionis
,
D. P.
,
1981
,
,
Springer
,
New York
.
9.
Williams
,
J. C.
, and
Johnson
,
W. D.
,
1974
, “
Semisimilar Solutions to Unsteady Boundary Layer Flows Including Separation
,”
AIAA J.
,
12
(
10
), pp.
1388
1393
.
10.
Wang
,
C. Y.
,
1989
, “
Exact Solutions of the Unsteady Navier-Stokes Equations
,”
Appl. Mech. Rev
,
42
(
11S
), pp.
269
282
.
11.
Ingham
,
D. B.
,
1984
, “
,”
J. Comp. Phys.
,
53
(
1
), pp.
90
99
.
12.
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
.
13.
Blasius
,
H.
,
1908
, “
Grenzschichten in Flüssigkeiten Mit Kleiner Reibung
,”
Z. Math. Phys.
,
56
(
2
), pp.
1
37
.
14.
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
.
15.
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
.
16.
Williams
,
J. C.
, III
,
1968
, “
,”
AIAA J.
,
6
(
12
), pp.
2417
2419
.
17.
Rajappa
,
N. R.
,
1979
, “
Nonsteady Plane Stagnation-Point Flow With Hard Blowing
,”
ZAMP
,
59
(
9
), pp.
471
473
.
18.
Wang
,
C. Y.
,
1985
, “
,”
Phys. Fluids
,
28
(
7
), pp.
2046
2049
.
19.
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
.
20.
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
.
21.
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
.
22.
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
.
23.
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
.
24.
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
.
25.
Lok
,
Y. Y.
, and
Pop
,
I.
,
2014
, “
Stretching or Shrinking Sheet Problem for Unsteady Separated Stagnation-Point Flow
,”
Meccanica
,
49
(
6
), pp.
1479
1492
.
26.
Dholey
,
S.
,
2016
, “
Magnetohydrodynamic Unsteady Separated Stagnation-Point Flow of a Viscous Fluid Over of a Moving Plate
,”
ZAMM
,
96
(
6
), pp.
707
720
.
27.
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
.
28.
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
.
29.
Drazin
,
P. G.
, and
Riley
,
N.
,
2007
,
The Navier–Stokes Equations: A Classification of Flows and Exact Solutions
,
Cambridge University Press
,
Cambridge, UK
.
30.
Batchelor
,
G. K.
,
1997
,
An Introduction to Fluid Mechanics
,
Cambridge University Press
,
Cambridge, UK
.