BULLETIN OF THE TRANSILVANIA UNIVERSITY OF BRAŞOV
CRITERIONS AND VALIDITY CONDITIONS OF A
FLUID FLOW OR A HOT TRANSFER REGIME
M. LUPU* E. SCHEIBER*
Abstract: In this paper there are studied two kinds of problems about a fluid
flow or a hot transfer regime. The first problem refers to a parabolic type of a
laminar or viscous fluid flow or a hot transfer in a plane or cylindrical domain.
It is considered an inverse problem: knowing the data on the boundary of the
domain, a method to find the viscosity coefficient or the hot transfer coefficient is
given. The best numerical results may be obtained when the number of
measurements is increased. The second problem refers to the free boundary flow
in the presence of a curvilinear symmetrical obstacle. There are considered the
Helmholtz problem when the fluid is unlimited and the Retki – Jacob scheme
when a jet of fluid is forked by a symmetrical obstacle. The problems where
studied using singular integral equations and the shapes of the maximal drag
obstacle are obtained.
Keywords: Parabolic equations, the trapeze formula, viscosity coefficient,
inviscid jets, nonlinear operator, optimal design
1. Determining the coefficient from the parabolic type equation
The mathematical modeling of viscous fluid flows, the heating and cooling problems, metal
metal melting, ice melting, wood drying, dispersion or aggregated, mass and heat transfer
lead to partial derivate equations of a parabolic type. The viscosity coefficients appear in
these equations, humidity coefficients or dispersion – they can be both constant and variable.
We present a method to determine these coefficients when they are not stationary, in the
plane case or cylindrical, having input data regarding the speed or the tension on the
boundary, or the temperature and the thermal flow changed with the exterior. [8] [3]
A. We consider the parabolic equation:
(1)
w  2 w
k (t )
*
t

x 2
 F ( x, t ), w  w( x, t ), x  [0, l ], t  0
Faculty of Mathematics and Computer Sciences, Transilvania University of Brasov
Criterions and validity conditions of a fluid flow or a hot transfer regime
238
Representing the plane flow between two plates or the thermal process delimited by
the the x  0, x  h plates. For start we will consider the homogeneity case
E ( x, t )  0 and assume that we have the following information regarding w( x, t ) :
(2)
w( x, t  0)  f ( x)
(3)
w( x  0, t )  g (t ), w( x  l , t )  h(t )
(4)
w
w
 G(t ),
 H (t )
x x 0
x x l
The conditions (2)-(4) are actually Sturm – Liouville type conditions and have physical
meanings related to the according studied problems: so, viscous fluids the conditions (3) of
Couette type [3] show the mobility of the plates with speeds g (t ), h(t ) and the relation (4)
shows the tensions that emerge between the fluid and the plates. For the caloric problem w is
the temperature, (3) the walls heating, (4) the caloric flow between the inside and the outside
trough the walls.
Having these data we will present methods to find the coefficient k (t ) with the conditions
(2)-(4) or by weakening them. In order to perform calculations in these continous
environments we impose the non polarity conditions for data, meaning:
(5)
g (0)  f (0), h(0)  f (l ), G (0)  f '(0), H (0)  f '(l )
along with the continuity of the simple and partial derivates.
We integrate the equation (1) with respect to x
l
l
(6)
w
  w 
w
w
k (t ) 
0
t
dx  
0

 dx 
x  x 
x

x l
x
 H (t )  G (t )
x 0
We apply in the integral the trapeze formula and together with (3) we have:
l  w 
l
 w  
k (t )        k (t )  h '(t )  g '(t )  H (t )  G(t )
2  t  x l  t  x 0 
2
2  H (t )  G (t ) 
k (t ) 
l  h '(t )  g '(t ) 
(7)
(8)
l

If in (1) appears also F ( x, t ) then besides H-G we add U  F ( x, t )dx . We notice that
0
k (t ) is retrieved and we have not use (2) f(x) - although, this will definitely be used if
w( x, t ) is numerically determined in (8) we notice the validity condition H  G meaning
the plates are not moving parallel with Ox simultaneous with constant speed. The formula (8)
will be correct as much as in order to apply the trapeze formula we will consider on [0, l ]
w( xi , t )
l
multiple nodes having xi  i , i  0,1, , n with w( xi , t )  h( xi , t ), h '( xi , t ) 
.
n
t
In the end results, with these data, applying the trapeze formula,
Bulletin of the Transilvania University of Braşov • Vol. 13(48) - 2006
239
(9)
H (t )  G (t )
n 1
l  h '(t )  g '(t )

  ht' ( xi , t ) 

n
2
i 1

Lets’ obtain k (t ) if one information, either H (t ) or G (t ) is missing- the case when G (t ) is
k (t ) 
missing from (3), (4); we apply the same operations:
w
( s, t )ds

x
0
h(t )  g (t )  w(l , t )  w(0, t )  
(10)
2w
w
w
 x2 dx   W (s, t )ds  x (l, t )  x ( , t )
(11)
w
2w
w
( , t )  H (t )   W ( s, t )ds, 2  W ( x, t )  k (t )
x
x
t

(12)
l
l
l
l
We replace (12) in (10) and we will apply the Dirichlet formula for the iterate integral
b

b
a
a
a
b
b

a
 d  W (s, t )ds   ds  W (s, t )d   (b   )W (s, t )ds
l
l2
h(t )  g (t )  lH (t )   (l  s )W ( s, t )ds  lH (t )  W (0, t )
2
0
h(t )  g (t )  lH (t )  k (t )
k (t ) 
l 2 w
l 2 g '(t )
 lH (t )  k (t )
2 t x 0
2
2 lH (t )  h(t )  g (t )
l 2 g '(t )
 kH (t )
(13)
(14)
(15)
In (13) we applied the trapeze formula, formulas (1) and (3), obtaining (15) comparable with
(8); the trapeze formula can be applied on more nodes obtaining a formula comparable with
(9). If G (t ) is present and H (t ) missing the reasoning in identical obtaining:
(16)
2  h(t )  g (t )  lG(t )
k (t ) 
l 2 g '(t )
 kG (t )
It can be noticed that the formulas (8) (15) (16) are not identical; for a greater precision we
must use the (9) type formulas; because of the data structure and presentation we can build an
optimal control problem to exactly determine k (t ) [1]: Determine the k control that
minimizes the functional I (k )
2
2
T
(17)
 w

w

 

I ( k )   
(0, t )  G (t )   
(l , t )  H (t )  dt 
 min
  x
 
 x
0 
With the restrictions (1) (2) (3) if minimum is zero then w is the solution satisfying (1) (2)
(3).
Criterions and validity conditions of a fluid flow or a hot transfer regime
240
B. The case of cylindrical coordinates (coaxial cylinders with radius r1  r2 ) or the circular
crown domain; it is studied considering the equation and conditions [3] [8] for Couette type
movements or with gravity and pressure difference trough F ( r , t )
(18)
w 1   w 

r
  F (r , t ), w  w(r , t ), r1  r  r2
t r r  r 
w(r , t  0)  f (r )
w(r1 , t )  g1 (t ), w(r2 , t )  g 2 (t )
k (t )
(19)
(20)
w
w
 G1 (t ),
 G2 (t )
r r  r1
r r r2
(21)
Using the same rationale we get k (t ) : we apply the trapeze formula after we integrate with
respect to r (18).
r2
(22)
w   w 
w
rk (t )
t

1 1
r
 ; k (t )   (  , t )d  r2G2  rG
r  r 
t
r1
r2  r1  w
w

r1
(r1 , t )  r2
(r2 , t )   r2G2 (t )  r1G1 (t )

2  t
t

2 r2G2 (t )  r1G1 (t )
k (t ) 
r2  r1 r1 g1 '(t )  r2 g 2 '(t )
k (t )
meaning
(23)
For a greater precision we split the interval [r1 , r2 ] in n parts with ri  i
r2  r1
and the
n
integral from (22) we apply the trapeze formula, obtaining a type (9) formula
k (t ) 
2n
r2  r1
r2G2 (t )  r1G1 (t )
(23’)
n 1
r1 g1 '(t )  r2 g 2 '(t )  2 ri g i '(t )
i 1
If in (18) appears F ( r , t ) then in formulas (23) (23’) (25) U is added the the upper side of
the
fraction
r2G2  r1G1  U (r, t )
r2
where
U (r , t )   rF (r , t )dr
and
r1
R
G2 (t ) 
1
rF (r , t )dr . It can be noticed that in (23) (23’) we have the validity
R 0
condition for the experimental case so that r2G2 (t )  rG
1 1 (t ) .
If we have the circular cylinder of radius R , r1  0, r2  R and G1 (t ) is missing, we
integrate (18) and apply the trapeze formula on the interval [0, R ] obtaining k (t ) :
r R
w
w
k (t )  r (r , t )dr  r
 RG2 (t )
t
t r 0
0
R
Bulletin of the Transilvania University of Braşov • Vol. 13(48) - 2006
k (t ) 
241
(24)
2 G2 (t )
R g1 (t )
Since the approximation is wide, we split [0, R ] in n parts with ri  i
R
obtaining with
n
data w(ri , t )  gi (t )
G2 (t )
k (t ) 
n 1
R 

2 ri gi '(t )  Rg R '(t ) 

2n  i 1

(25)
In an analog manner we can formulate the optimal control problem for the determination of
k (t ) : Determine the control k (t ) which minimize the functional I [k ] [1]:
2
2
 w
  w
 
I ( k )   
(r1 , t )  G1 (t )   
(r2 , t )  G2 (t )  dt 
 min
  r
 
 r
0 
T
with the restrictions (18) (19) (20). This method can be applied numerical and experimental
in biology, thermal processes, drying, melting, diffusion etc. with practical data observable
on measure machines, determining this way the coefficient k (t ) .
2. The study of potential plane flows with stream lines for incompressible fluids in the
presence of symmetrical curvilinear profiles.
We consider the potential plane flows, with free lines, for incompressible fluids in the
presence of symmetrical profiles in two cases: first case – curvilinear obstacle in free
unlimited stream (based on Helmholtz schema) [9] and the second case – fluid streams,
upstream delimited by the height h, forked by a symmetrical obstacle, the stream lines at
infinity having the asymptotic angle  (the Rethi – Jacob model) [2]
The obstacles have the same length L and in each situation we’ll have three sub-cases: 1st
case – curvilinear obstacle convex toward downstream, 2nd case - straight perpendicular plate
and 3rd case curvilinear obstacle convex toward upstream (fig. I1, I2, I3) [6] and for the
second situation we have the same obstacles in the presence of the limit stream with  ,
asymptotic angle (fig. II1, II2, II3).
The studies we made [4], [5] considered the as helping canonical domain – the half plane
V0
 i , V  V 0 where
V
df
V 0 is the fluid speed on the free lines, V is the speed modulus w  V , w  u  iv 
,
dz
  arg w, w  V 0e . We’ll consider the physical domain D z , D f the potential domain,
    i ,   0 and the Jukovski function  ( )  t  i  ln
D the hodographic domain and D the canonical domain, where:
Criterions and validity conditions of a fluid flow or a hot transfer regime
242
f ( z )   ( x, y )  i ( x, y ),
w
(26)
df
 Ve i
dz
the complex potentials were determined f  f ( ) for the two situations f I ( )  A ,
q
iq
ln(   a) 
where A, a are the parameters a  1 and q is the stream debit
2
2
of width h . We determine    ( ) and so W  W ( ); in this manner, from
df
df d
we get
and the correspondent D z , D f , D , Dw cu D - conformal

z  z ( )
dz d dz
transformations, and trough transitivity f  f ( z ), w  w( z ) .
f II ( ) 
  1 1 t ( s) ds
 1
i 1 s  1 s  
The singular integral equations resulted from determining  ( ) which links dual  ( )
with t ( ) ,   (1,1) are, for the two situations the followings [4] [5] [6] (obtained with
 ( ) 
1
1
 ( s)
ds
,
1 s s 
 ( ) 
Sohotski – Plemelj formula)
t ( ) 
1

1

1
 (s)
ds
,
1 s s 
  1 1 t ( s) ds
 ( )  
 1 s  1 s  
(27)
For the 1st situation the upstream asymptotic angle is   0 and for the 2nd situation:
 (a) 
a 1

1

1
t ( s)
ds
;
s 1 a  s
L 1 V 0 d
1
 

h  1 V ( ) a   
1
k
e t ( ) d
 a 
1
1
(28)
Where k is the connection report from which we find the values for a parameter, knowing
k and the speed distribution on the profile V  V ( ) or t  t ( ),   (1,1) .
The formulas (27) allow solving reverse problems, with t ( ) sau  ( ) given on the profile,
the profile shape is determined, for cases (26), (28), V  V ( ) will be given and
 ( ) determined, and for (27) of the plate    2 is determined V  V ( ), t  t ( ) .
The aero dynamical and geometrical parameters for the presented 6 cases are represented in
the tables below [4] [5] [6].
Cases I1, I2, I3 based on Helmholtz schema[6]
Bulletin of the Transilvania University of Braşov • Vol. 13(48) - 2006
Curvilinear Profile
V
L
0
V
O
The impermeable
parachute
Helmholtz Schema
B
0
L
V
O
x
243
0
L
O
x
x
A0
Fig. I3
Fig. I2
Fig. I1
Parameters in f  f ( ) will be
LV 0
f ( )  A , A 
4
f ( )  A , A 
2 LV 0
f ( )  A , A 
 4
LV 0
4e
The speed distribution on the profile
V1  V
0
 2  1
V2  V 0 
 2  1

1 
2

V0
2




1
2

1 
2
V 0 1 
V3 
e
2
The curvilinear obstacle’s tangent angle
3 
1 
 1   
T

2
2

 2

2 

ln
2


1
2 
2  1 
2  1

 1   
T

2

 2
Pressures resultant
V L

     i

-1 1
P1 
A0
O
02
B
C
P2 
V02 L 2
2  4
 1 t ( s)



ds
 1 s 

J t    1 1
t (s)
 e ds
2
V 0 L
2
P3 
2
1
The distribution V3 was obtained by extending J [t ] with the Jensen inequality for
J t 
Criterions and validity conditions of a fluid flow or a hot transfer regime
244
2
[5]
1 
t 3  1  ln
The resistance coefficient
C 1x 
2

 0,637
C x2 
2
 0,87980
 4
C x3 
8
 0,936797
e
2P
C 1x  C x2  C x3 (maximal)
0
V L
Cases II1, II2, II3 limited stream of width h , forked by obstacle
Cx 
V0
V0
D 
A
h
A
A

D
E
h
h
d
O
M
1

B
E
V
D
O
V0
1
A
0
L
V
M
O
0
Fig. II3
Fig. II1
Fig. II2
 2  1
V2  V 
 2  1

1 
2
V1  V0




0
1
2

V0
2
1 
V 0 1 
V

2 3
e
2
V1  V2  V3
The value of angle  on the profile
h 
 1   
h   T 

2
 2  2

T [u ] 
h 
1


1  u du
,
u
0
2

1

ln
2  1 
2  1
 1   
T

 2  2
2
ln
  1 u

u  [0,1],
T
0
u
The asymptotic angle  (h) downstream


2 

 a 1
 1h (a )  T 
 2 h (a)  arctg
2
a 1
2 

 a 1
 3h (a )  T 

1
ln
2
a 1  2
a 1  2

Bulletin of the Transilvania University of Braşov • Vol. 13(48) - 2006
245
 h (a) is descendant with respect to h and ascendant for  ( ) [6] [7]
 1h   2h   3h   max


0  3 
0  1 
0 2 
4
2

    i
a
-1
1
CD
A0
O
B

A
With lim  (h(a ))  0 we get case I from II
h  , a 
L
are given u * is found and then a * :
h
1
u 1
1
1 u
e
1 u
k1(u )  u ln
k 2(u )  1  1  u 2  u ln
k 3(u )  u ln

u 1

1 u

1 u
L
2
k   k1  k 2  k 3 with u* 
 1 . Given k  u  u*  a  a *
h
a 1
C x (3h) 
2[1  cos  1 (h(a*))]
2[1  cos  2 (h(a*))]
C x (1h) 
C x (2h) 
2[1  cos  3 (h(a*))]
k1

k2
k3
If the reports k 
C 1x (h)  C x2 (h)  C x3 (h)
For
We
k
fixed we get:  1 (h)   2 (h)   3 (h)
and
L
can
notice
that
C x (h) is ascendant with respect to
 (h),
C x (h)
 0,
 (h)
C x (h)
 0 then lim C x (h)  C x
h 
 (h)
2
8
C x2 (h) h

C x3 (h) h



 4
e
meaning C x (h) is ascendant in h and, if h   ,
C 1x (h) h
 C 1x 

2

As a result we obtain a way to go from the 2nd case, trough flow’s width enlargement
h   ,to the 1st situation – obstacle in unlimited fluid (Helmholtz version) [7] [6]
We represent C x (h) with the help of the asimptotic angle

:    (u)  u  f
1
( )
Criterions and validity conditions of a fluid flow or a hot transfer regime
246
 1 (h)  T (u)  f1 (u)
 2 (h)  arctg
u
1 u
 f 2 (u )
1 u
 f 3 (u )
1 u
u  f 31 ( 3 )
u  f 21 ( 2 )
21  cosf 2 (u )
C x2 (h) 
k 2 (u )
u  f11 ( 1 )
21  cosf1 (u)
C 1x (h) 
k1 (u)
1
2
 3 (h)  T (u )  ln
C x3 (h) 
21  cosf 3 (u )
k 3 (u )
C x2 (h)  C x2 ( 2 ) 
C 1x (h)  C 1x ( 1 )
2

1
2

ctg
C x3 (h)  C x3 ( 3 )

  
ln tg   
2
2 4
We’ve got the general case C x (h)  C x ( ) including the C. Iacob formula for
C x2 (h)  C x (h)  C x2 ( ) .
Analyzing the values of C x (h), C x (k ) for the plate, obtained by C. Iacob [2], S. Popp [9],
and the tables of C x (h) for the curvilinear obstacle C x h( ) [4], in the last situation II3 there
is a case where L  1 when    ; in this case  (a*)    a*  1,005; k *  6,412 and
h*  1 k*  0,156. We get the following conclusions regarding the movement behaviour
and validity conditions for C x (h) with respect to C x (unlimited stream), the optimal
deflector for maximal resistance [4].
If:
a) 0  h  k *  C xh  2h(1  cos  3 )
b) k  k *  6,412  C xh  C xh*  4h*  0,623;   
c) h  h * and h    C x (h)  C x3 
k

a*
Cx
8
 0,936 in this case h  0,156 L
e
2.245
3
4
5
6.142 k 0
1.571
1.494
0.891
1.1871
1.219
0.864
2.251
1.074
0.814
2.623
1.024
0.747
3.142
1.005
0.624
Table 1
7
3.359
1.002
0.565
Table 2
Bulletin of the Transilvania University of Braşov • Vol. 13(48) - 2006

6

4

3

2
2
3
3
4
5
6
11
12

0.287
11.747
0.933
0.623
5.217
0.928
1.088
2.974
0.919
2.245
1.494
0.891
3.584
1.116
0.837
4.282
1.054
0.797
4.989
1.024
0.748
5.700
1.011
0.690
6.412
1.005
0.624
k

a*
Cx
247
References
1. Barbu, V.: Metode matematice in optimizarea sistemelor diferentiale, Bucuresti, Editura
Academiei, 1989
2. Iacob, C.: Introduction mathematique a la mechanique des fluides,Paris, Editura Gautier
Villars, 1959
3. Ionescu, D.: Introducere in mecanica fluidelor, Bucuresti, Editura Tehnica, 2004
4. Lupu, M., Scheiber, E.: Analytical method for airfoils optimization in the case of
nonlinear problems in jet aerodynamics In: Mathematical Reports, Romanian Academy,
nr1, 2001, p. 33-43
5. Lupu, M., Scheiber, E.: Exact solution for an optimal impermeable parachute problem,
In: Journal of Computational and Applied Mathematics, Elsevier, vol 147, 2002, p. 277286
6. Lupu, M., Isaia, F.: Utilizarea problemelor la limita inverse la optimizarea operatorilor
neliniari in hidroaerodinamica, In: Bulletin of the Transilvania University of Braşov,
2003, p. 133-142
7. Maklakov, D. V.: Nonlinear problems in hydro aerodynamics, Moscow, Editura Yamus
K., 1997
8. Oroveanu, T.: Mecanica fluidelor vascoase, Bucuresti, Editura Academiei, 1967
9. Popp, S.: Mathematical models in cavity flow theory, Bucuresti, Editura Tehnica, 1985
Criterii si conditii de validitate pentru regimuri de curgeri fluide sau
transfer termic
Rezumat: Lucrarea trateaza doua tipuri de probleme privind regimul de curgere sau
transfer termic. Prima problema se refera la ecuatiile de tip parabolic ce modeleaza
curgeri laminare ale fluidelor vascoase, nestationare sau transfer termic; domeniile
sunt plane sau cilindrice. A doua problema se refera la curgeri cu suprafete libere
plane in prezenta unor obstacole curbilinii simetrice in doua variante: varianta I
obstacool in fluid nelimitat (problema Helmholtz) si varianta II jetul de latime h
bifurcat de obstacol simetric (schema Rathi – Jacob). Sunt folosite ecuatii integrale
singulare pentru obtinerea coeficientului de rezistenta maximala.
Cuvinte cheie: ecuatii parabolice, formula trapezului, coeficient de vascozitate,
operatori neliniari