- Email: [email protected]
On solutions of the integral form of the second-order boundary-layer equations
Int. J. Engn~
Sci. Vol. 8. pp. 15 1-l 56.
Pergamon
Press
1970.
Printed
in Great
Britain
ON SOLUTIONS OF THE INTEGRAL FORM OF THE SECOND-ORDER BOUNDARY-LAYER EQUATIONS MICHAEL Virginia
Polytechnic
Institute,
J. WERLEt Engineering
and
R. THOMAS
Mechanics
Department,
DAVIS Blacksburg,
Va. 24061,
U.S.A
Abstract- Approximate solutions to the second-order boundary-layer equations for incompressible flow are considered. A simplified form of the integral momentum equation for two-dimensional and axisymmetric flow is presented. Solutions to the problem of shear flow over a flat plate are obtained for five different polynomial approximations to the second-order boundary-layer velocity profile. It is found that the results obtained depend rather erratically on the polynomial degree employed.
INTRODUCTION THE
PURPOSE of this note is to bring to light results of an investigation concerned with the approximate solution of the integral form of the second-order boundary-layer equations for laminar incompressible flow. In particular, attention will be centered on the sensitivity of the method to the order of the polynomial used to represent the firstand second-order boundary-layer velocity profiles. The insensitivity of the first-order solutions to the polynomial order (see Schlichting, [ 11) certainly gives impetus to the present approach because it indicates that it is more important to satisfy the integral momentum equation than to be concerned with the details of the velocity profile employed. It is not immediately apparent that this latter fact is directly applicable to the second-order solutions and it is felt that an investigation of this point offers one means of testing the validity of the polynomial approximations.
GOVERNING
EQUATIONS
The most general form of the integrated boundary-layer equations for two-dimensional or axisymmetric incompressible how was presented in reference [2]. Subsequent study (see reference [3]) has shown that considerable simplification is achieved through a simple redefinition of terms involved. For the sake of completeness, the general problem is briefly restated here. Thus, the first-order (or classical) integrated boundarylayer equation may be stated as+:
*Former address: Applied Aerodynamics Division, U.S. Naval Ordnance Laboratory, Silver Spring. Md. 20910, U.S.A. Sunless, otherwise noted, the nomenclature used here is identical to that of reference [2] with one added simplification, i.e. the inviscid flow properties evaluated at the surface will be written without their arguments. Thus, for example, U, is in reality, C/, (s, 0). 151
and. as was shown
in reference
[3]. the second-order
equation
can be written
as
In equation (2). the second-order contributions due to vorticity interaction can be identified by the coefficient B’(O), the displacement speed effect by iJz, the longitudinal curvature by K, and the transverse curvature by u& The origins of the new boundarylayer thicknesses, & and 6,. introduced in equation (2) are best understood in terms of the classical concepts of the displacement and momentum thicknesses. Simply stated. the displacement thickness is that amount by which a body must be thickened in order to correctly calculate the inviscid flow field about that shape. The appropriate asymptotic expansion for large Reynolds numbers is then given as; 6-
c6, +E’&+.
. ..
(3)
where 1 E-S
(4)
P
and R, is the flow’s characteristic Reynolds number. The related to the second-order displacement thickness i& through
new thickness the relation
8, is then
(5) These
two thicknesses
are defined
by the integrals
(6)
(7)
In an analogous
manner,
one can define a momentum
o- Eo,+&&+. where 0, is related
to H, through
thickness, ...
0 such that: (8)
the relation
(9) and
(10)
Second-order
boundary-layer
equations
153
with
TEST
PROBLEM
The test problem considered here is that of a semi-infinite flat plate placed in a stream which has a uniform (but symmetric) shear. The exact solution for such was first given by Murray [4] and was later discussed by Van Dyke [S]. Devan and Oberai [6] found that they could match these exact solutions within a few percent through use of fourth-order polynomial approximations to both the first- and second-order velocity profiles in conjunction with the appropriate integral momentum equations. In the present work the integral equations will be solved for six different polynomial approximations to the second-order velocity profile in order to test the validity of the concept being applied. For a free-stream velocity distribution of the form U(--3c,_v) = I $_oJ\>I
(12)
where o is constant. it has been shown by Van Dyke[5] that the solution of the firstorder boundary-layer equations is identical to that forw = 0, and that there is no secondorder correction to the inviscid flow. The vorticity interaction produces the only second-order viscous effect and the integral equation reduces to simply de, -72,.-w, ds where 6, is given by (see equation 8x-J;:
(13)
( I I ))
(1~,--wN)(I-2rr,)dN-t2~J;;
(I-u,)NdN,
( 14)
and
For the first-order profile, the Karman so that withr) = N/l, = N/C,.s”‘.
Pohlhausen
fourth-order
profile
is employed
11, = 27) - 2$ - n4. Solution of equation ( I) using equations momentum equation now becomes
Before
d
m
-[I ds
0
(16)
(6) and ( 16) gives C, = 5.84. The integral
(l-++4$+2r)4)(~~X-~N)dN
considering
solutions
of equation
(I 7) via polynomial
approximations
to
153
hl. J. WERLE
Us. it is fruitful to first consider be employed here are
and R. I’. DAVIS
an appropriate
Np(S,
set of boundary
conditions.
The ones to
0)
=
0,
0)
=
-+y
12)
=
cfJLJ.
(20)
&Js,
1,)
=
w.
(21)
L4,,v,,(s,
12)
=
0,
(22)
u,,v,v(s,
u,(s,
( IX)
( 19)
and
where I, represents the value of N at which the second-order quantities take on theii limit values (note C, is a constant resulting from the first-order solutions, C, = 1.752). With respect to these conditions. equation ( 18) represents the no-slip condition, equation (19) is required to satisfy the boundary-layer equation at the wall, equation (20) represents the outer matching condition and equations (21) and (22) are required to satisfy the boundary-layer equations at N = I,?. While admittedly, one could generate an infinite number of boundary conditions by successive differentiation of the secondorder boundary-layer equation: those presented above represent a natural set. That second-order profile which satisfies all of the conditions of equations (I 8-22), plus the integral momentum equation, will be the second-order counterpart of the first-order Karman Pohlhausen solutions. Since both 14, and uq appear within the integral of equation (I 7). it will be necessary to investigate the relationship between I, and i2. The possibility that 1, > I, has already been eliminated and it remains to consider the two distinct cases of I, = I, or I, < I,. This investigation begins by considering I, = I2 and employs polynomial profiles in 5 = N/l, = 7, taking the general second-order velocity profile to be of the form
(23) With equation
(23), the integral
momentum
equation
(equation
(I 7)) reduces
to
(24) where I = I,~f’(~)[l-4~+4~3-2~“] The second-order
d&
(25)
wall shear is given by
tThese conditions are based on the assumption that the first-order velocity goes to its limit value at or before the second-order velocity does so. This assumption appears a logical one since the first-order quantities in fact appear as coefficients in the second-order equations and therefore serve to drive the second-order velocity to its limit value.
Second-order
boundary-layer
Equation (24) has been solved for five separate in equation (25) and the results are given in Table conditions which have been used.
Boundary
Shear stress - ~~,,,/w
(IX). (20) (181, (20). (21)
-i)
l-5)’ Ic,GY+t)+t515(1-5)’
(18-21) (18-22)
[C,(31+l)+f515(1-i)” ioriginally
The exact solution
conditions
(18)
Cl5 c‘,i(l C,C(
polynomial representations off(c) 1, along with a list of the boundary
I
Table Profile-J‘(c)
due to Devan
and Oberia,
was given by Murray, T
1.55
equations
1.337 2.209 3.510 3.022t 3.734
(6).
141, as
pUz= 3.126 w,
(27)
which was predicted quite accurately using the fourth-order profile above. The rather erratic dependence of the solutions on the profile order certainly does little to build one’s confidence in this approach. As a final test of the polynomial approach a solution has also been obtained taking I, > I,, and a velocity distribution of the form (note that 77no longer equals 5)
Fitting a fourth-order (18-22)) yields
polynomial
f(5)
in 5 to all of the boundary
=85(1-L)“.
conditions
(equations
(29)
and the shear stress at the wall as rq(. --
2.414~.
(30)
Certainly this is a poor approximation to the exact result (equation (27)). While the above results serve as a test only of the vorticity interaction solution using the polynomial approximations, the conclusions reached above can be extended to some extent with little difficulty. As discussed in reference [7], the linearity of the second-order equations produces a breakdown of the general problem into four separate contributions: the displacement speed effect. the vorticity interaction effect, the effects due to longitudinal curvature, and those effects due to the presence of transverse curvature. As was shown in reference [3], the latter three of these problems all have velocity profiles of the same general form. Since the polynomial approach failed for the vorticity problem, and both the longitudinal and transverse curvature profiles are of the same form, (but with different slopes as N + m) it is reasonable to assume equally poor results will be obtained for the latter as well.
I”/,
kl..l.
and
wL-Kl.t~
I<. I.
I).\\
IS
It hxwld appear then that the polynomial approach needs little further investigation. the rather accurate results given by Devan and Oberai [6] must. unfortunately. he viewed as being rather t’ortuitou% in light of the present study. ;tnd
K El’ EK t-:N(‘t:S
[ I ] 11.S<‘HLICHTING. 121 M. J. WER1.E 131M. J. WERLE,
Borrrrc/tr,?-Ltrv~,~ 7‘lr~or~. p. 205. Pergamon Pre\\ and R. T. DAVIS. /nt../. E/Is/?~.SC,~. 4. 423 (I 9hh). U.S. Ntr~:n/Ord. Lab.R~,J. No. TR 6%‘; I 196X).
[4] J. D. b1URRAY.J. 151 M. VAN DYKE,J. [6] [7]
(I’)551
FlrritlM~~~h. II. 309( I‘961). FluidMcvl~. 14.481 ( IYhZ).
I.. DEVAN and M. M. ORERAI.A/AA.//r/2. 1X38( IYh4). hl.V.AN I)YKF../.FI/A;~M~‘~./~.I~. Ihl (IYh2).
H&um6Les auteurs Ctudient des solutions approchkea des iquations du second ordre de la couche limite d’un Ccoulement de fluide incompressible et proposent une forme simplifiee de I’inttgrale de la quantitC de mouvement applicable aux Ccoulements :I deux dimensions ou de rCvolution. Ils obtiennent des solutions du problkme de la couche de choc sur une plaque plane pour cinq approximations diffkrentes de la rCpartition des vitesses dans la couche limite effectuees g I’aide d’un polynbme et trouvent que les rCsultats obtenuh dtpendent assez irrCgul&ement du degrC du polynBme employ6.
Zusammenfassung-
Ungefahre Liiungen fiir die Grenzschichtgleichungen zweiter Ordnung fiir inkompressible StrGmung werden untersucht. Eine vereinfachte Form der integralen Momentgleichung fiir zweidimensionale und achsensymmetrische Striimung wird vorgelegt. Liisungen fiir das Problem der ScherstrC mung iiber eine flache Platte werden fiir fiinf verschiedene polynomische Anngherungen an das Grenzschichtgeschwindigkeitsprofil zweiter Ordnung erhalten. Es wird gefunden, dass die erhaltenen Resultate errdtisch vom polynomischen Grad, der verwendet wird, abhgngen. Summarioconsiderano le aoluzioni approssimate delle equazioni di strata limite di seconda grandezra per un flusso incomprimibile. Si presenta una forma semplificata dell’equazione di moment0 integrato per un flusso bidimensionale ed assisimmetrico. Si ottengono soluzioni al problema di flusso su un piatto liscio per cinque approssimazioni polinomiali differenti al profilo di velocith dello strato limite di second’ordine. Si scoprc the i risultati ottenuti dipendono in manieraalquanto erratica dal grado polinomiale impiegato. A6CTpPKT-PPaCCMaTpuBaroTCrr IlOp!$LIKa n0
KOflWECTBJ
‘GlRaVH .IJfR
IIJISI HPA2KHMat?MOTO
n0
CKOPOCTA
OIlIM60~HOfi
MnororneHa.
i-PaHH’IHOrO
MHHMYto
PelIIeHMs
npenCTaBiWHa
LlBkGKKeHFiHLIJIR ABj’Xpa3MepHOrO
TtYEHHEO 06TeKafOLWMy
KOHTYPa
CKOpee
npu6nEixeHHble TeSCHBII.
ypaBHeHH8
yIlpOIIZHH%l
11 OCeCAMMeTPH’lHOrO
IIJlOCKyIO IIJIrlTy rIpEi IllTA CJlOIl
3aBNCF,MOCTb
“0
BTOpOrO
IIOpJlAKa.
nOJ,yYeHHblX
TplHHSHOMY
+OpMa
PaSJIA’IHbIX
Te’ieHPiB.
OT
YTO
CTeIleHA
BTOpOrO
ypaBH’ZHEiSl
nOJIj”faIoTCR
MHO,-O’WeHHblX
YCTaHaBJlHBaeTCSl, pe3yJIbTaTOB
CJIOH)
AHTerpaJIbHOrO
pWleHrtH
npFf6mmemnx CJIeAyeT
C’IATaTb
yrIOTpe6neHHOrO
Sci. Vol. 8. pp. 15 1-l 56.
Pergamon
Press
1970.
Printed
in Great
Britain
ON SOLUTIONS OF THE INTEGRAL FORM OF THE SECOND-ORDER BOUNDARY-LAYER EQUATIONS MICHAEL Virginia
Polytechnic
Institute,
J. WERLEt Engineering
and
R. THOMAS
Mechanics
Department,
DAVIS Blacksburg,
Va. 24061,
U.S.A
Abstract- Approximate solutions to the second-order boundary-layer equations for incompressible flow are considered. A simplified form of the integral momentum equation for two-dimensional and axisymmetric flow is presented. Solutions to the problem of shear flow over a flat plate are obtained for five different polynomial approximations to the second-order boundary-layer velocity profile. It is found that the results obtained depend rather erratically on the polynomial degree employed.
INTRODUCTION THE
PURPOSE of this note is to bring to light results of an investigation concerned with the approximate solution of the integral form of the second-order boundary-layer equations for laminar incompressible flow. In particular, attention will be centered on the sensitivity of the method to the order of the polynomial used to represent the firstand second-order boundary-layer velocity profiles. The insensitivity of the first-order solutions to the polynomial order (see Schlichting, [ 11) certainly gives impetus to the present approach because it indicates that it is more important to satisfy the integral momentum equation than to be concerned with the details of the velocity profile employed. It is not immediately apparent that this latter fact is directly applicable to the second-order solutions and it is felt that an investigation of this point offers one means of testing the validity of the polynomial approximations.
GOVERNING
EQUATIONS
The most general form of the integrated boundary-layer equations for two-dimensional or axisymmetric incompressible how was presented in reference [2]. Subsequent study (see reference [3]) has shown that considerable simplification is achieved through a simple redefinition of terms involved. For the sake of completeness, the general problem is briefly restated here. Thus, the first-order (or classical) integrated boundarylayer equation may be stated as+:
*Former address: Applied Aerodynamics Division, U.S. Naval Ordnance Laboratory, Silver Spring. Md. 20910, U.S.A. Sunless, otherwise noted, the nomenclature used here is identical to that of reference [2] with one added simplification, i.e. the inviscid flow properties evaluated at the surface will be written without their arguments. Thus, for example, U, is in reality, C/, (s, 0). 151
and. as was shown
in reference
[3]. the second-order
equation
can be written
as
In equation (2). the second-order contributions due to vorticity interaction can be identified by the coefficient B’(O), the displacement speed effect by iJz, the longitudinal curvature by K, and the transverse curvature by u& The origins of the new boundarylayer thicknesses, & and 6,. introduced in equation (2) are best understood in terms of the classical concepts of the displacement and momentum thicknesses. Simply stated. the displacement thickness is that amount by which a body must be thickened in order to correctly calculate the inviscid flow field about that shape. The appropriate asymptotic expansion for large Reynolds numbers is then given as; 6-
c6, +E’&+.
. ..
(3)
where 1 E-S
(4)
P
and R, is the flow’s characteristic Reynolds number. The related to the second-order displacement thickness i& through
new thickness the relation
8, is then
(5) These
two thicknesses
are defined
by the integrals
(6)
(7)
In an analogous
manner,
one can define a momentum
o- Eo,+&&+. where 0, is related
to H, through
thickness, ...
0 such that: (8)
the relation
(9) and
(10)
Second-order
boundary-layer
equations
153
with
TEST
PROBLEM
The test problem considered here is that of a semi-infinite flat plate placed in a stream which has a uniform (but symmetric) shear. The exact solution for such was first given by Murray [4] and was later discussed by Van Dyke [S]. Devan and Oberai [6] found that they could match these exact solutions within a few percent through use of fourth-order polynomial approximations to both the first- and second-order velocity profiles in conjunction with the appropriate integral momentum equations. In the present work the integral equations will be solved for six different polynomial approximations to the second-order velocity profile in order to test the validity of the concept being applied. For a free-stream velocity distribution of the form U(--3c,_v) = I $_oJ\>I
(12)
where o is constant. it has been shown by Van Dyke[5] that the solution of the firstorder boundary-layer equations is identical to that forw = 0, and that there is no secondorder correction to the inviscid flow. The vorticity interaction produces the only second-order viscous effect and the integral equation reduces to simply de, -72,.-w, ds where 6, is given by (see equation 8x-J;:
(13)
( I I ))
(1~,--wN)(I-2rr,)dN-t2~J;;
(I-u,)NdN,
( 14)
and
For the first-order profile, the Karman so that withr) = N/l, = N/C,.s”‘.
Pohlhausen
fourth-order
profile
is employed
11, = 27) - 2$ - n4. Solution of equation ( I) using equations momentum equation now becomes
Before
d
m
-[I ds
0
(16)
(6) and ( 16) gives C, = 5.84. The integral
(l-++4$+2r)4)(~~X-~N)dN
considering
solutions
of equation
(I 7) via polynomial
approximations
to
153
hl. J. WERLE
Us. it is fruitful to first consider be employed here are
and R. I’. DAVIS
an appropriate
Np(S,
set of boundary
conditions.
The ones to
0)
=
0,
0)
=
-+y
12)
=
cfJLJ.
(20)
&Js,
1,)
=
w.
(21)
L4,,v,,(s,
12)
=
0,
(22)
u,,v,v(s,
u,(s,
( IX)
( 19)
and
where I, represents the value of N at which the second-order quantities take on theii limit values (note C, is a constant resulting from the first-order solutions, C, = 1.752). With respect to these conditions. equation ( 18) represents the no-slip condition, equation (19) is required to satisfy the boundary-layer equation at the wall, equation (20) represents the outer matching condition and equations (21) and (22) are required to satisfy the boundary-layer equations at N = I,?. While admittedly, one could generate an infinite number of boundary conditions by successive differentiation of the secondorder boundary-layer equation: those presented above represent a natural set. That second-order profile which satisfies all of the conditions of equations (I 8-22), plus the integral momentum equation, will be the second-order counterpart of the first-order Karman Pohlhausen solutions. Since both 14, and uq appear within the integral of equation (I 7). it will be necessary to investigate the relationship between I, and i2. The possibility that 1, > I, has already been eliminated and it remains to consider the two distinct cases of I, = I, or I, < I,. This investigation begins by considering I, = I2 and employs polynomial profiles in 5 = N/l, = 7, taking the general second-order velocity profile to be of the form
(23) With equation
(23), the integral
momentum
equation
(equation
(I 7)) reduces
to
(24) where I = I,~f’(~)[l-4~+4~3-2~“] The second-order
d&
(25)
wall shear is given by
tThese conditions are based on the assumption that the first-order velocity goes to its limit value at or before the second-order velocity does so. This assumption appears a logical one since the first-order quantities in fact appear as coefficients in the second-order equations and therefore serve to drive the second-order velocity to its limit value.
Second-order
boundary-layer
Equation (24) has been solved for five separate in equation (25) and the results are given in Table conditions which have been used.
Boundary
Shear stress - ~~,,,/w
(IX). (20) (181, (20). (21)
-i)
l-5)’ Ic,GY+t)+t515(1-5)’
(18-21) (18-22)
[C,(31+l)+f515(1-i)” ioriginally
The exact solution
conditions
(18)
Cl5 c‘,i(l C,C(
polynomial representations off(c) 1, along with a list of the boundary
I
Table Profile-J‘(c)
due to Devan
and Oberia,
was given by Murray, T
1.55
equations
1.337 2.209 3.510 3.022t 3.734
(6).
141, as
pUz= 3.126 w,
(27)
which was predicted quite accurately using the fourth-order profile above. The rather erratic dependence of the solutions on the profile order certainly does little to build one’s confidence in this approach. As a final test of the polynomial approach a solution has also been obtained taking I, > I,, and a velocity distribution of the form (note that 77no longer equals 5)
Fitting a fourth-order (18-22)) yields
polynomial
f(5)
in 5 to all of the boundary
=85(1-L)“.
conditions
(equations
(29)
and the shear stress at the wall as rq(. --
2.414~.
(30)
Certainly this is a poor approximation to the exact result (equation (27)). While the above results serve as a test only of the vorticity interaction solution using the polynomial approximations, the conclusions reached above can be extended to some extent with little difficulty. As discussed in reference [7], the linearity of the second-order equations produces a breakdown of the general problem into four separate contributions: the displacement speed effect. the vorticity interaction effect, the effects due to longitudinal curvature, and those effects due to the presence of transverse curvature. As was shown in reference [3], the latter three of these problems all have velocity profiles of the same general form. Since the polynomial approach failed for the vorticity problem, and both the longitudinal and transverse curvature profiles are of the same form, (but with different slopes as N + m) it is reasonable to assume equally poor results will be obtained for the latter as well.
I”/,
kl..l.
and
wL-Kl.t~
I<. I.
I).\\
IS
It hxwld appear then that the polynomial approach needs little further investigation. the rather accurate results given by Devan and Oberai [6] must. unfortunately. he viewed as being rather t’ortuitou% in light of the present study. ;tnd
K El’ EK t-:N(‘t:S
[ I ] 11.S<‘HLICHTING. 121 M. J. WER1.E 131M. J. WERLE,
Borrrrc/tr,?-Ltrv~,~ 7‘lr~or~. p. 205. Pergamon Pre\\ and R. T. DAVIS. /nt../. E/Is/?~.SC,~. 4. 423 (I 9hh). U.S. Ntr~:n/Ord. Lab.R~,J. No. TR 6%‘; I 196X).
[4] J. D. b1URRAY.J. 151 M. VAN DYKE,J. [6] [7]
(I’)551
FlrritlM~~~h. II. 309( I‘961). FluidMcvl~. 14.481 ( IYhZ).
I.. DEVAN and M. M. ORERAI.A/AA.//r/2. 1X38( IYh4). hl.V.AN I)YKF../.FI/A;~M~‘~./~.I~. Ihl (IYh2).
H&um6Les auteurs Ctudient des solutions approchkea des iquations du second ordre de la couche limite d’un Ccoulement de fluide incompressible et proposent une forme simplifiee de I’inttgrale de la quantitC de mouvement applicable aux Ccoulements :I deux dimensions ou de rCvolution. Ils obtiennent des solutions du problkme de la couche de choc sur une plaque plane pour cinq approximations diffkrentes de la rCpartition des vitesses dans la couche limite effectuees g I’aide d’un polynbme et trouvent que les rCsultats obtenuh dtpendent assez irrCgul&ement du degrC du polynBme employ6.
Zusammenfassung-
Ungefahre Liiungen fiir die Grenzschichtgleichungen zweiter Ordnung fiir inkompressible StrGmung werden untersucht. Eine vereinfachte Form der integralen Momentgleichung fiir zweidimensionale und achsensymmetrische Striimung wird vorgelegt. Liisungen fiir das Problem der ScherstrC mung iiber eine flache Platte werden fiir fiinf verschiedene polynomische Anngherungen an das Grenzschichtgeschwindigkeitsprofil zweiter Ordnung erhalten. Es wird gefunden, dass die erhaltenen Resultate errdtisch vom polynomischen Grad, der verwendet wird, abhgngen. Summarioconsiderano le aoluzioni approssimate delle equazioni di strata limite di seconda grandezra per un flusso incomprimibile. Si presenta una forma semplificata dell’equazione di moment0 integrato per un flusso bidimensionale ed assisimmetrico. Si ottengono soluzioni al problema di flusso su un piatto liscio per cinque approssimazioni polinomiali differenti al profilo di velocith dello strato limite di second’ordine. Si scoprc the i risultati ottenuti dipendono in manieraalquanto erratica dal grado polinomiale impiegato. A6CTpPKT-PPaCCMaTpuBaroTCrr IlOp!$LIKa n0
KOflWECTBJ
‘GlRaVH .IJfR
IIJISI HPA2KHMat?MOTO
n0
CKOPOCTA
OIlIM60~HOfi
MnororneHa.
i-PaHH’IHOrO
MHHMYto
PelIIeHMs
npenCTaBiWHa
LlBkGKKeHFiHLIJIR ABj’Xpa3MepHOrO
TtYEHHEO 06TeKafOLWMy
KOHTYPa
CKOpee
npu6nEixeHHble TeSCHBII.
ypaBHeHH8
yIlpOIIZHH%l
11 OCeCAMMeTPH’lHOrO
IIJlOCKyIO IIJIrlTy rIpEi IllTA CJlOIl
3aBNCF,MOCTb
“0
BTOpOrO
IIOpJlAKa.
nOJ,yYeHHblX
TplHHSHOMY
+OpMa
PaSJIA’IHbIX
Te’ieHPiB.
OT
YTO
CTeIleHA
BTOpOrO
ypaBH’ZHEiSl
nOJIj”faIoTCR
MHO,-O’WeHHblX
YCTaHaBJlHBaeTCSl, pe3yJIbTaTOB
CJIOH)
AHTerpaJIbHOrO
pWleHrtH
npFf6mmemnx CJIeAyeT
C’IATaTb
yrIOTpe6neHHOrO