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Numerical solution of the singular integrodifferential equation of contact hydrodynamics
Short
communications
189
REFERENCES
1.
2.
G.I. Methods
MARCHUK, sibirsk,
CHARAKHCH’YAN,
Nauk SSSR, Moscow,
A.A. An approach
SAMARSKII,
A.A. and SOBOL,
CHARAKHCH’YAN, In: Dynamics
5.
1974.
BRODE,
H.L.
BRODE,
to the calculation
vychislitel’noi
matematiki),
“Nauka”,
Novo-
of the transfer
More
equation
izluchayushchego
for problems
of the dynamics
gaza), No. 2, 16-35,
V’Ts Akad.
I.M. Examples
A.A. Calculation
about
of the nonstationary
nuclear
Novoe v zarubezhnoi
H.L. Gas-dynamic
of the numerical
calculation
of temperature
waves. Zh. @hisl.
1963.
of a radiating gas (Dinamika
Moscow,
(Mekhanika. 6.
(Metody
1976.
Mat. mat. Fiz., 3, 4, 702-719, 4.
mathematics
gas. In: Dynamics of a radiating gas (Dinamika
of a radiating
3.
of numerical
1973.
explosions
nauke),
motion
spherical
izluchayushchego
gaza),
in subterranean
No. 3.68-103,
with radiation.
symmetric
cavities.
“Mir”, Moscow,
A general
numerical
flows of a radiating
No. 1, 54-74,
VTs Akad.
In: Mechanics.
grey gas.
Nauk
New foreign
SSSR, science
1975. method.
Astronaut.
Acta, 14, 5, 433-
444.1969.
U.S.S.R. Comput.Maths @Pergamon
Math. Pbys. Vol. 18, pp. 189-191
Press Ltd. 1978. Printed
NUMERICAL
in
0041-5553/78/0301-0189
$7.5010
Great Britain.
SOLUTION OF THE SINGULAR INTEGRODIFFERENTIAL EQUATION OF CONTACT HYDRODYNAMICS* M.A. GALAKHOV,
ICI. ZAPPAROV
and A.G. PATRAKOV
Moscow (Received 29 September
1976)
A NUMERICAL solution is given of the isothermal equations of the elastohydrodynamic theory of lubrication in a linearized formulation. It is shown that linearization preserves the main features of the problem. The equations of the isothermal one-dimensional problem of the elastohydrodynamic theory of lubrication can be written in the form [ 11
(1)
$
(2)
Bo(h-i)==
[
h'exp(-Qp)$ 1 39 - c2 + -
==-=, 0
2
=
x s a
c-t P(t)ln ,t_-s, dt,
where p(x) > 0 and h(x) > 0 are unknown functions, V > 0, Q > 0, a < - 1 are specified parameters, a 0 and c are unknown constants. The boundary conditions (3)
P(a)=P(c)+c)=o.
(4)
=5. Jap(t)& c
We linearize (l), having frozen the coefficients
* Zb. vjkbisl. Mat. mat. Fiz., 18, 2, 504-506,
1978.
as follows:
M.A. Gaiukhov, K.1. Zupparov and A.G. Patrakov
190
po(x)=(1-2*)“~9(1-t2), (5) ho(s)-lW0-‘{Jsl
(zz--l)‘~*--Lo
[~x~+{;t~-~)“‘J~a(~~--I),
8 is the Heaviside function. This linearization is meaningful at least for Q 9 1, since (5) and c = 1 satisfy (11, (2) asymptotically for 1x1< 1 and x: < - 1. Substituting (5) into (l), we obtain a linear singular ~t~o~~ferential equation
(6)
~oaW~xpt-Q~o(o)l$ 0
The solution of (6) is constructed by the use of first-order spline functions [2]. La V, Q, a, Ho and c be fixed. We introduce on [a, cl the nonuniform mesh {xi) i = 1, 2, ,.., N, x1 = a, xN = c. We represent the unknown function p(x) by spline-functions of the fist order at the mesh nodes xi (piecewise-linear approximation): Xii-X (7)
p’(X)=Pi-I-
X--lf-i
+pi-,
&-W-l
5i-;GI_$
where The unknowns will be the values of p’&) -pi. For the ZE[Zi_-fr Zi], i=2, 3, *. . , N. differential term in Eq. (6) we write down a homogeneous conservative difference scheme. The quadrature formula for calculating the singular integral at the mesh nodes is obtained from (7) [2]. Therefore, (6) with the first two boundary conditions of (3) reduces to the system of linear algebraic equations (8)
2 Zt+t-Xi-i WI
f
~os~(~r+i+~i)/21
@r+i-pr)
t6xp[Qp,((xi+i+rl)/2)l["i+,-x{)-
(Pi-Pf-1)
(s*-,+zr)/21
-exp[QP0((5i-*+5i)/2)1{5i-tt--!f
I -
)I 2v =t+i=f‘----li( -z (pi;% -pr.4 4” pix1-x1--1 Hc? ~lnzkl-2(+i
+
_
PA
(
XA-i-xi
xh-a--i
311 -
i-2,.
. . , N,
p,=px=o.
The system (8) was solved by Gauss’s method with the choice of the principal element. In order to satisfy the last condition of (3) and condition (4) the following approximations were introduced:
e
N
37 ‘pz=---
2
@Ji-Z&i). 0
h-e2
The search for Ho and c was organized so as to minimize (b = peated for each pair Ho and c. The minimum Q was found by formed with a successive increase in the number of modal practical convergence of the method. The maximum number was chosen to be more dense close to x = c, so that
#I ’ + 42 ‘. The solution of (8) was rethe algorithm of [ 3 1, Calculations perpoints of the mesh demonstrated the N of nodal points was 100. The mesh
191
Short communications
min(zf--Ii-j)i
3*fW,
max(8i-tt_1)== 1
&W2.
The results of calculations for three values of the parameters V and Q are shown in the graph. Curve 1 corresponds ro Q = IO, V = 3, Ho = 2.037, c = 1.062, curve 2 corresponds to Q = 10, V = 0.1, Ho = 0.264, c = 1.041, curve 3 corresponds to Q = 10, V = 0.01, Ho = 0.060, c = 1.023. The dashed line is the graph of the function po (x) close to which the system (1) (2) was linearized.
Fig. 1. By approxi~t~g the results of the calculation of 15 versions in the band 0.001 G V G 3, 5 Q Q Q 30 the formula Ho = 0.53 ?f@*61Q0*3 was obtained with a maximum error of about 6%. The solution of the non-linear equations (l)--(4) gives values of Ho sufficiently close to the result of solving Eq. (6). It is noteworthy that the linearized system (l)--(4) permits the main features of the non-linear problem to be preserved I41 . In particular, p(x) may have two characteristics of the maximum. The authors thank V.L. Smetanina for assistance in performing the calculations.
REFERENCES CALAKHOV, M.A., GOLUBKIN, V.N_ and SHIROBOKOV, V.V. Rheological models of a fluid in exnemal conditions and elastohydrodynamics. In: Numetical methods in the mechanics of a continlrous medium (Ghisl. metody
v mekhan. sploshnoi sredy). Vol. 7, No. 3, 49-54,
GABDULKHAEV,
VTs SO Akad. Nauk SSSR, Novosibirsk, 1976.
B.G. Spiine methods of solving a class of singular inte~diffe~e~tial
equations. Izv. WBO~.
Matematika, No. 6, 14-24,197s.
MITROFANOV, V.B. On an algorithm of multidimensional
random search. Preptint ZPM Akad. NauR SSSR,
No. 118,1974.
GALAKHOV. M.A., ZAPPAROV, #I. and SMETANIN, V.L. Numerical solution of a plane isothermal contacthydrodynamic
problem. Subjects of reports. All-Union Conference ‘~Contact~y~odyn~~c in technology” (Vses. konf. “Kontakmo-gidrodinamich.
tion and its practical application
praktich. primenenie v tekhnike”), 8, KuAS, Kuibyshev, 1976.
theory of hbrica-
teoriya smazki i ee
communications
189
REFERENCES
1.
2.
G.I. Methods
MARCHUK, sibirsk,
CHARAKHCH’YAN,
Nauk SSSR, Moscow,
A.A. An approach
SAMARSKII,
A.A. and SOBOL,
CHARAKHCH’YAN, In: Dynamics
5.
1974.
BRODE,
H.L.
BRODE,
to the calculation
vychislitel’noi
matematiki),
“Nauka”,
Novo-
of the transfer
More
equation
izluchayushchego
for problems
of the dynamics
gaza), No. 2, 16-35,
V’Ts Akad.
I.M. Examples
A.A. Calculation
about
of the nonstationary
nuclear
Novoe v zarubezhnoi
H.L. Gas-dynamic
of the numerical
calculation
of temperature
waves. Zh. @hisl.
1963.
of a radiating gas (Dinamika
Moscow,
(Mekhanika. 6.
(Metody
1976.
Mat. mat. Fiz., 3, 4, 702-719, 4.
mathematics
gas. In: Dynamics of a radiating gas (Dinamika
of a radiating
3.
of numerical
1973.
explosions
nauke),
motion
spherical
izluchayushchego
gaza),
in subterranean
No. 3.68-103,
with radiation.
symmetric
cavities.
“Mir”, Moscow,
A general
numerical
flows of a radiating
No. 1, 54-74,
VTs Akad.
In: Mechanics.
grey gas.
Nauk
New foreign
SSSR, science
1975. method.
Astronaut.
Acta, 14, 5, 433-
444.1969.
U.S.S.R. Comput.Maths @Pergamon
Math. Pbys. Vol. 18, pp. 189-191
Press Ltd. 1978. Printed
NUMERICAL
in
0041-5553/78/0301-0189
$7.5010
Great Britain.
SOLUTION OF THE SINGULAR INTEGRODIFFERENTIAL EQUATION OF CONTACT HYDRODYNAMICS* M.A. GALAKHOV,
ICI. ZAPPAROV
and A.G. PATRAKOV
Moscow (Received 29 September
1976)
A NUMERICAL solution is given of the isothermal equations of the elastohydrodynamic theory of lubrication in a linearized formulation. It is shown that linearization preserves the main features of the problem. The equations of the isothermal one-dimensional problem of the elastohydrodynamic theory of lubrication can be written in the form [ 11
(1)
$
(2)
Bo(h-i)==
[
h'exp(-Qp)$ 1 39 - c2 + -
==-=, 0
2
=
x s a
c-t P(t)ln ,t_-s, dt,
where p(x) > 0 and h(x) > 0 are unknown functions, V > 0, Q > 0, a < - 1 are specified parameters, a
P(a)=P(c)+c)=o.
(4)
=5. Jap(t)& c
We linearize (l), having frozen the coefficients
* Zb. vjkbisl. Mat. mat. Fiz., 18, 2, 504-506,
1978.
as follows:
M.A. Gaiukhov, K.1. Zupparov and A.G. Patrakov
190
po(x)=(1-2*)“~9(1-t2), (5) ho(s)-lW0-‘{Jsl
(zz--l)‘~*--Lo
[~x~+{;t~-~)“‘J~a(~~--I),
8 is the Heaviside function. This linearization is meaningful at least for Q 9 1, since (5) and c = 1 satisfy (11, (2) asymptotically for 1x1< 1 and x: < - 1. Substituting (5) into (l), we obtain a linear singular ~t~o~~ferential equation
(6)
~oaW~xpt-Q~o(o)l$ 0
The solution of (6) is constructed by the use of first-order spline functions [2]. La V, Q, a, Ho and c be fixed. We introduce on [a, cl the nonuniform mesh {xi) i = 1, 2, ,.., N, x1 = a, xN = c. We represent the unknown function p(x) by spline-functions of the fist order at the mesh nodes xi (piecewise-linear approximation): Xii-X (7)
p’(X)=Pi-I-
X--lf-i
+pi-,
&-W-l
5i-;GI_$
where The unknowns will be the values of p’&) -pi. For the ZE[Zi_-fr Zi], i=2, 3, *. . , N. differential term in Eq. (6) we write down a homogeneous conservative difference scheme. The quadrature formula for calculating the singular integral at the mesh nodes is obtained from (7) [2]. Therefore, (6) with the first two boundary conditions of (3) reduces to the system of linear algebraic equations (8)
2 Zt+t-Xi-i WI
f
~os~(~r+i+~i)/21
@r+i-pr)
t6xp[Qp,((xi+i+rl)/2)l["i+,-x{)-
(Pi-Pf-1)
(s*-,+zr)/21
-exp[QP0((5i-*+5i)/2)1{5i-tt--!f
I -
)I 2v =t+i=f‘----li( -z (pi;% -pr.4 4” pix1-x1--1 Hc? ~lnzkl-2(+i
+
_
PA
(
XA-i-xi
xh-a--i
311 -
i-2,.
. . , N,
p,=px=o.
The system (8) was solved by Gauss’s method with the choice of the principal element. In order to satisfy the last condition of (3) and condition (4) the following approximations were introduced:
e
N
37 ‘pz=---
2
@Ji-Z&i). 0
h-e2
The search for Ho and c was organized so as to minimize (b = peated for each pair Ho and c. The minimum Q was found by formed with a successive increase in the number of modal practical convergence of the method. The maximum number was chosen to be more dense close to x = c, so that
#I ’ + 42 ‘. The solution of (8) was rethe algorithm of [ 3 1, Calculations perpoints of the mesh demonstrated the N of nodal points was 100. The mesh
191
Short communications
min(zf--Ii-j)i
3*fW,
max(8i-tt_1)== 1
&W2.
The results of calculations for three values of the parameters V and Q are shown in the graph. Curve 1 corresponds ro Q = IO, V = 3, Ho = 2.037, c = 1.062, curve 2 corresponds to Q = 10, V = 0.1, Ho = 0.264, c = 1.041, curve 3 corresponds to Q = 10, V = 0.01, Ho = 0.060, c = 1.023. The dashed line is the graph of the function po (x) close to which the system (1) (2) was linearized.
Fig. 1. By approxi~t~g the results of the calculation of 15 versions in the band 0.001 G V G 3, 5 Q Q Q 30 the formula Ho = 0.53 ?f@*61Q0*3 was obtained with a maximum error of about 6%. The solution of the non-linear equations (l)--(4) gives values of Ho sufficiently close to the result of solving Eq. (6). It is noteworthy that the linearized system (l)--(4) permits the main features of the non-linear problem to be preserved I41 . In particular, p(x) may have two characteristics of the maximum. The authors thank V.L. Smetanina for assistance in performing the calculations.
REFERENCES CALAKHOV, M.A., GOLUBKIN, V.N_ and SHIROBOKOV, V.V. Rheological models of a fluid in exnemal conditions and elastohydrodynamics. In: Numetical methods in the mechanics of a continlrous medium (Ghisl. metody
v mekhan. sploshnoi sredy). Vol. 7, No. 3, 49-54,
GABDULKHAEV,
VTs SO Akad. Nauk SSSR, Novosibirsk, 1976.
B.G. Spiine methods of solving a class of singular inte~diffe~e~tial
equations. Izv. WBO~.
Matematika, No. 6, 14-24,197s.
MITROFANOV, V.B. On an algorithm of multidimensional
random search. Preptint ZPM Akad. NauR SSSR,
No. 118,1974.
GALAKHOV. M.A., ZAPPAROV, #I. and SMETANIN, V.L. Numerical solution of a plane isothermal contacthydrodynamic
problem. Subjects of reports. All-Union Conference ‘~Contact~y~odyn~~c in technology” (Vses. konf. “Kontakmo-gidrodinamich.
tion and its practical application
praktich. primenenie v tekhnike”), 8, KuAS, Kuibyshev, 1976.
theory of hbrica-
teoriya smazki i ee