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Applied Mathematics E-Notes, 14(2014), 123-126 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
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Nonexistence For The “Missing”Similarity
Boundary-Layer Flow
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Joseph Edward Paullety
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Received 12 July 2014
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Abstract
This note considers the boundary value problem
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00
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( )+
0
( ) + ( )2 = 0;
0;
> 0;
subject to
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(0) = 1 and (1) = 0;
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which arises in certain situations of boundary layerp‡ow. Previous work on the
problem established the existence of a min 2 [1; 2= 3] such that solutions exist
for
< min
min . It has been conjectured that for
p no solution exists. We
partially resolve this conjecture by proving that for
2=3 :8165 no solution
to the boundary value problem exists.
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1
Introduction
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In [1] and [2], Magyari et al. consider the boundary value problem (BVP):
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0
( ) + ( )2 = 0 for
0;
(1)
subject to
(0) = 1 and (1) = 0:
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( )+
(2)
This BVP arises in two distinct physical situations. One is in steady boundary-layer
‡ow due to a moving permeable ‡at surface in a quiescent viscous ‡uid [1]. The other
is in free convection boundary-layer ‡ow of a Darcy-Boussinesq ‡uid from a heated
vertical permeable plate [2]. In both of these situations, the usual similarity variable
transformation produces a valid reduced model most of the time. However, in the
…rst situation, when the surface is stretching with inverse-linear velocity, Magyari et
al. [1] show that a logarithmic term in the wall coordinate must be added to the
usual expression for the stream function in order to obtain a correct reduction; that
given by (1-2). A similar term must be included in the second situation when the
wall temperature distribution is inverse-linear [2]; again resulting in the BVP (1-2).
Mathematics Subject Classi…cations: 34B15
of Science, Pennsylvania State University at Erie, The Behrend College, Erie, Pennsylvania
16563-0203, USA.
y School
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Nonexistence for the "Missing" Boundary-Layer Flow
Magyari et al. call these two special cases “missing” boundary-layer ‡ows because the
usual similarity variable reduction misses valid and physically relevant results.
For the BVP (1-2), Magyari et al. show that no solution exists for
0. Numerically, they …nd a value min 1:079131 such that a unique solution exists for = min
and multiple solutionsp exist for all > min . Recently, Zhang [3] proved that there
exists a min 2 [1; 2= 3] such that for
min a solution to the BVP (1-2) exists.
Existence or nonexistence of solutions for 0 < < min remains
p an open question. We
partially resolve this question by proving that for 0 <
2=3 0:8165 no solution
to the BVP (1-2) exists.
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The following Theorem is our main result.
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Nonexistence Result
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( ) + ( )2 = 0 for
0;
(3)
subject to
(0) = 1 and
0
(0) = ;
(4)
where is a free parameter. By standard existence and uniqueness theory, the IVP
(3-4) will have a unique local solution for any value of . We will show that there is
no value of such that the solution of the IVP (3-4) will exist for all
0 and satisfy
the desired boundary condition at in…nity, (1) = 0.
We begin by listing some properties that such a solution must satisfy. First note
that the ODE (3) implies that ( ) cannot have a minimum. Thus, if
0 gives a
solution to the BVP, then ( ) is monotonically decreasing for all > 0 and tends to
zero as ! 1. If > 0 gives a solution, then ( ) must attain a positive maximum
and then monotonically decrease to zero as goes to in…nity.
A di¤erentiation of (3) yields
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0
( )+
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2=3 no solution to the boundary value problem (1-2)
PROOF. Consider the initial value problem (IVP) given by
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p
THEOREM. For 0 <
exists.
( )+
00
( ) + 2 ( ) 0 ( ) = 0:
Note that after is ultimately decreasing, 0 cannot have a maximum. Thus for a
solution, 0 is ultimately monotonically increasing and bounded above by zero. Thus
0
(1) 0 exists, and since (1) also exists, we must then have 0 (1) = 0.
Next we derive several integral relationships that any solution must satisfy. An
integration of the ODE (3) from 0 to gives
0
( )
+
( )
+
Z
0
(t)2 dt = 0:
J. E Paullet
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Letting
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tend to in…nity and solving for the integral results in
Z
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1
(t)2 dt =
+ :
(5)
0
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0
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0
Multiplying the ODE (3) by
Again letting
( )2
2
and integrating from 0 to
Z
2
+
0
0
Z
1
0
Finally, multiplying the ODE (3) by
0 to we obtain
Z
( ) 0( )
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0
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Letting
( )2
2
(t)2 dt +
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Z
1
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(t)3 dt =
+
+
Z
(t)3 dt = 0:
0
+
2
+
Z
1
0
(t)2 dt:
(7)
0
2
+
2+3
6
2
<
+ ;
(8)
p
2
2
or, after
2=3. Thus for
p rearranging terms, 2 + 3 < 3 ; which cannot hold if
2=3 no solution to the BVP (1-2) exists for which
0.
Next consider the possibility that > 0 gives a solution. As noted earlier, such
a solution must attain a positive maximum, necessarily above one, and then decrease
monotonically toward zero. Thus there exists a point 0 > 0 at which ( ) decreases
through one. In the above expressions, we can integrate from 0 to > 0 and in (5),
(6) and (7) replace the lower limit of integration with 0 and replace with 0 ( 0 ).
Thus, for > 0 we again have 0 < ( ) < 1 and the exact same argument now implies
that
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Now, if
0 gives a solution to the BVP (1-2), then the solution is monotonically
decreasing and 0 < ( ) < 1 for all > 0. Thus ( )3 < ( )2 for all > 0. Using
this fact along with (5), (6) and (7) we obtain
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(6)
tend to in…nity results in
0
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:
and integrating, by parts where necessary, from
0
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= 0:
2
2+3
6
(t)2 dt =
0
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1
tend to in…nity and solving for the integral gives
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( )3
3
(t)2 dt +
we obtain
which is again contradicted if
2 + 3 0 ( 0 )2 < 3 2 ;
p
2=3, proving the theorem.
(9)
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Nonexistence for the "Missing" Boundary-Layer Flow
References
[1] E. Magyari, I. Pop and B. Keller, The “missing”similarity boundary-layer ‡ow over
a moving plane surface, Z. Angew. Math. Mech., 53(2002), 782–793.
[2] E. Magyari, I. Pop and B. Keller, The "missing” self-similar free convection
boundary-layer ‡ow over a vertical permeable surface in a porous medium,Transp.
Porous Media, 46(2002), 91–102.
[3] Z. Zhang, A two-point boundary value problem arising in boundary layer theory,
J. Math. Anal. Appl., 417(2014), 361–375.