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Effect of spreading of material on the surface of a fluid—an exact solution
Int. J. Non-Linear
Mechanics.
F’ol. 6, pp. 255-262.
Pcrgamon Press 1971.
Printed in Great Britain
EFFECT OF SPREADING OF MATERIAL ON THE SURFACE OF A FLUID-AN EXACT SOLUTION CHANG YI WANG Department of Mathematics, Michigan State University, East Lansing, Michigan 48823, U.S.A. Ah&raft--A material of small density is being supplied at constant rate at a point on the surface of a quiescent fluid. The material may be an immiscible, lighter fluid of higher viscosity, or may be composed of a stream of solid particles. After some time the material spreads out with constant thickness and a steady state is reached. We would like to know the effect on the fluid, due to the spreading of surface material. The problem is governed by the non-dimensional parameter B/v where Yis the viscosity and B/r is the velocity imparted by the spreading material to the fluid underneath. Depending on the range of B/v, the Navier-Stokes equation in spherical coordinates admits one exact solution for positive B, and three different exact solutions for negative B. An exact boundary layer solution is found for B/v + 1. A uniformly valid solution is then constructed. The Stokes solution for IB/v ( $ 1 is also determined. Both approximations are compared with the exact solution.
1. INTRODUCTION
THE present paper is motivated by the frequent accidental spillings of oil on the surface of water. Due to the balance of gravity and surface tension, the oil spreads with more or less constant thickness covering a large area on the water surface. Since the oil is much more viscous than water, we assume the water has little dynamic effect on the layer of oil. Our problem thus considers only the effect of the spreading oil on the water underneath. The model consists of an infinite layer of oil spreading out from a constant source. After all transients have died out, we assume the thickness h of the spreading oil is constant. Then the velocity imparted by the spreading oil to the fluid below is Q/2nhr where Q is the volume rate of discharge, and r is the distance from the source. When velocities are inversely proportional to radius, the Navier-Stokes equations admit exact solutions in spherical polar coordinates. Landau [l] found the exact solution of a momentum jet in an infinite fluid. Following Landau, Yatseyev [Z] obtained the general solution in terms of hypergeometric functions, and applied the solution to the flow due to a semi-infinite line source. Independently but later, Squire [3] arrived at the same solution obtained by Landau [l]. Squire [4] extended the solution to a momentum jet emerging from a plane wall, albeit the boundary conditions on the wall are not satisfied. Squire [S] considered conical jets, but the solution requires infinite velocities on the axis of symmetry. In this paper we shall study the exact solution applied to the steady spreading of oil on water. All the boundary conditions of the model will be satisfied. 2. THE EXACT SOLUTIONS
The Navier-Stokes
equation in spherical polar coordinates (p, 8, cp)is 1
1
_ a( !I’, E2 Y’)
vE41p + ~ p2sine
a(p,fq 255
=O
(2.1)
CHANG YI WANG
256
where
a p2 ae'
cot e
&E+i~_-_
ap2 pzaez
(2.2)
p is the distance from the origin, and 8 is the angle from the axis of symmetry. The stream function Y is related to the velocity components by -1
P v=pZsin
ay
av
i
(2.3)
ve = psin-
If we let Y = vp g(q), q = cos 8, (2.1) becomes (1 - q”) g”” - 419” + 3g’g” + gg”’ = 0.
(2.4)
This non-linear equation can be integrated thrice to give the Riccati equation (1 - rq)g’ + 2qg + $gs’ = c&
+ c,q +
(2.5)
c3.
If we let g = 2(1 - q2) u’/u, (2.5) becomes linear 2(1 - $)2 11”= (c,#
+ c,?j +
(2.6)
c3)u
and is solvable in terms of hypergeometric functions, see Yatseyev [2]. If the velocities are to remain finite on the axis rl = 1, the right-hand side of (2.5) must necessarily be of the form c1q2 + c,rj +
c3 =
(2k2 - $(l
- 1)2,
(2.7)
where k may be a real or a purely imaginary constant. Then solving (2.6) and substituting for g, we have
r (l
_
J1 + 2k)(l + q)2k+ c(1 - 2k) (1 + t#k
k#O
+ c
2 ln(l + r~)+ c
1
k = 0.
The constants c and k are to be determined through the boundary vg = 0; q = 0, v, = B/p where B = Q/(2nh). The solution for the stream function when B is positive is (4k2 - 1) [(1 + v)‘~ - l]
y = “(l
- ‘) (2k - 1) (1 + q)2k+ (2k + 1)
(2.8)
(2.9) conditions
q = 0,
(2.10)
where k=
When the surface material is drawn off from a point, B is negative. The solutions are then (1 - 4k2) [(l + q)2k- l] ’ = “(l
- ‘)(l
- 2k)(l + q)2k- (1 + 2k)
for--i<;<0
(2.11)
Effect of spreading of material on the surface of a fluiban
exact soluiion
251
where k=
Y = vp(l
ln(l + q) - 2
1’
(2.12)
and Y = vp(1 -
(1 + 4b2) ‘) 1 - 2b cot [b In (1 + q)]
for - cc < $ c - ;,
(2.13)
where k = ib,
The exact solutions (2.10H2.13) depend on the non-dimensional parameter B/v, which represents the rate of discharge or suction of surface material. In particular, equation (2.13) is exactly the solution found by Squire [4], where it is applied to a momentum jet emerging from a plane wall. As we noted before in that case the boundary conditions cannot be satisfied since the solution require up = B/p on the wall. Thus Squire’s application would be physically meaningful only if the solution were insensitive to the boundary conditions on the wall. However, the boundary conditions are important since the flow (2.13) is not due to a momentum jet, but is entirely caused by the non-zero boundary conditions. Therefore, the appropriate application of the exact solution (2.13) is probably the suction of surface material above a viscous fluid, and not the jet. In this paper we shall concentrate on the spreading of surface material, where B > 0.
3. THE BOUNDARY
LAYER
SOLUTION
Let us compare our exact solution (2.10) with the boundary layer approximation. We shall normalize lengths by L, velocities by B/L, pressure by B’p’/L? where p’ is the density of the fluid and L is an appropriate length scale. The Navier-Stokes equations in cylindrical polar coordinates become,
uw,+ww,=
-pz+2
(f-4, +
(
1
w,,+-w,+w,, r
w, = 0,
, >
(3.2)
(3.3)
where U, w are the velocity components in the direction r, z respectively, and 6’ = v/B. The boundary conditions at z = 0 are, w = 0, u = l/r; and at z = co are w = 0, u = 0.
Expanding the dependent variables in a power series in e*f3.lff3.2) shows that the outer fiow is potential to the first two orders. Inside the boundary layer we set z =tzq u = U()-t- EU1+ . . . w = ewe + p = po + tpi -I- . . . where 4 is the new stretched variable. Noting that the zero-th order outer flow is identically zero, the boundary-layer is
(3.4) equation
We define an inner stream function !P
= - j war dr = J uor dq,
(3.6)
and seek similarity of the form Ye) = rf([)
(3.7)
where c = QY/F. Then (3.5) reduces to an ordinary differential equation f”‘+f’“+ff”=O*
(3.8)
The boundary conditions are that f(U) = 0, f’(O) = 1 and f(ao) is finite. Equation (3.8) can be integrated to give an exact boundary-layer solution: f = (J2)tanh The fact that f(a)
i ( J2 > .
(3.9)
= ,/2 signifies an influx at the outer edge of the boundary layer.
Since the outer flow is potential, the outer stream function is governed by f3.10) Together with the boundary conditions I = Q, Fe5 = 0; z = 4 !F@”= (42) r; p2 = r2 f zz = to, velocities = 0; we obtain P”
= (42) [J(z2 + r2) - 21.
(3.11)
A uniformly valid composite solution can then be constructed: yI =1 eyly”’+ &Y(O)- coinmon part = e{(JBrtanh($)
+ (,/2)[J(z2
+ r2) - .r] - (,/2)r~ + o(&.
(3.12)
The approximate solution (3.12) is compared with the exact solution (2.10) in Fig. I for B/v = 50. The difference is small inside the boundary layer while in the outer Bow (3.12) over estimates the effects of inertia.
We nuts that if B were negative9 (3-g) does not admit ~un~-~ay~ suhztions with the boundary conditions j(U) = 4 f’f@) = -1 and S(clc) finite ~h~s~~~~y in this case vorticity is not conBrmed to the vicinity of the surf$~, even if 1B/Y 1 is very large. 1x1that case the exact solution (Nl), (2.12) or (2.13) should then be used. The streamlines for B negative look similar to those for B positive.
4. THE
STOKES APPBOXIMATIOTI’
For very small B/v, (2.1) reduce to the Stokes equation E4!P = 0.
(4.1)
The general sohrtion is (4.0
The Stokes solution (4.2) is compared with the exact solution (2.10) in Fig. 2 for B/v m 04. The inertial effects are underestimated. Figure 3 shows the steamhnes for B/v = 5, where neither the Stokes solution nor the boundary-layer solution is adequate to describe the flow. They serve as the upper and lower bounds of the exact solution. Equations (3.11) and (4.2) can also be obtained by taking limits on the exact sohrtion (ZUO). Care must be taken, however, in deducing the bo~dary-mayor suhrtion (3.7) from (210) in which both 19and I@ are smalf.
260
WANG
YI WANG
I
I \ \\ I\ \
~~ JI
I1
0.05
yr0.0,0.02
FIG. 2. Streamlines for E/v = 0.5 exact solution (2.10) -- Stokes solution (4.2).
FIG. 3. Comparison of streamlines, B/v = 5.0 (2.10);---(4.2). -(3.12);
-
Effect of spreading of material on the surface of ajlui&an
exact solution
261
5. D~CUSSIO~S
The present solutions may also be applied to a steady stream of light solid particles spreading on a fluid, provided the particles behave as a continuous layer. Less emphasis is placed on negative B/v, since most surface layers are unstable when suction is applied at a point. We note that for the surface of the fluid to remain plane the Froude number B’/(Cg) must be small compared to unity. The solutions are also restricted to radial distances large compared to the diameter of the source at the origin. The streamline patterns show a characteristic sharp turn near the origin, which differ from say, the flow towards a plate. This is important since the fluid acts as an effective centrifuge for suspended heavier particles in the fluid. The effect is more pronounced for high values of B/v. In the limit of infinite B/v any ~homogeneity of larger density than the fluid shall be deposited inertially on the surface material. In such a case the elhciency of capture or collection becomes one hundred per cent.
REFERENCES
PI L. D. LANDAU,A new exact solution of the Navier-Stokes equation. (in English) Computes I ‘Academic des Sciences de I’URSS Nouvelle Serie (International
Render de
edition of Dokl. Nauk. SSSR) 43, 286-288
(1944). Pksp. teor. Fiz. 20 (in Russian), 1031-1034. Translated as NACA Tech. Memo. No. 1349 (1950). [31 H. B. SQUIRE,The round laminar jet. #& J. up@. Me&. Math. 4,321-329 (1951). 141 H. 3. SQ~JIRE,Some viscous fluid flow problems I: Jet emerging from a hoie in a plane wall. Phil. Mug. 7th Series 43, 942-945 (1952). 151H. B. SQUIRE,Radial jets. In: 50 Jahre Gre~zsch~ck~r~chung, edited by H. Gortler. Braunschweig, 47-54
PI V. I. YATSEYE\‘,On a class of exact solutions of the equations of motion of a viscous fluid. Zh.
(1955). (Received 10 April 1970)
R&mn&-On projette un mattriau de faible densite a debit constant a la surface d’un fluide au repos. Le mattriau peut &re un fluide non miscible plus leger et de plus grande viscosite ou bien un flot de particules solides. Apres quelques temps le materiau s’ttend avec une epaisseur constante et on atteint un &at permanent. On se propose d’ittudier l’effet sur le fluide du depot d’un materiau en surface. Le probleme est r&i par le parambtre sans dimension B/v on v est la viscosite et B/r la vitesse donn6e par le materiau depose au fluide sous-jacent. Selon l’ordre de grandeur de B/v l’bquation de Navier-Stokes en coordonn&.es sphtiques admet une solution exacte pour B positif et trois solutions exactes differentes pour B negatif. On trouve une solution exacte dans la couche limite pour B/v + 1. On construit alors une solution un~orrn~rn~t valabfe. On determine egalement la solution de Stokes pour /B/v\ < 1. Les deux approximations sent compar&s avec la solution exacte.
Zusammenfaasung--Eine Substanz geringer Dichte wird mit einer konstanten Rate in einem Punkt auf der Oberfliiche einer ruhenden Fliissigkeit erzeugt. Die Substanz selber kann eine nicht mischbare, leichtere Fltissigkeit hbherer Viskositat sein oder kamr aus einem Strom fester Teilchen bestehen. Nach einiger Zeit hat sich die Substanz mit gleichmassiger Dicke verteilt, und ein stationarer &stand ist erreicht. Wir machten den durch die Ausbreitung der Oberflachensubstanz auf die Flilssigkeit ausgetibten Effekt wissen. Das Problem wird durch den nichtdimensionalen Parameter (E/o) beherrscht, wobei u die Viskositat ist und (B/u) die Geschwingigkeit, die von dem sich ausbreitenden Material auf die darunterliegende Fltissigkeit iibertragen wird. Abhangig von der Grossenordnung von (B/u), erlaubt die Navier-Stokes-Gleichung in spharischen Koordinaten eine exakte Lijsung fiir positive
C&SANGYI WANG
262
B und drei verschiedene exakte Losungen ftir negative 8. Eine exakte Grenzschichtldsung wird ftir (B/u) 9 1 gefunden. Damit wird eine gieichmissig gtihige L&sung konstruiert. Die Stokes-Losung fiir 1 B/v I< 1 wird ebenfalls bestimmt. Beide NLherungen werden mit der exakten Losung verghchen.
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Mechanics.
F’ol. 6, pp. 255-262.
Pcrgamon Press 1971.
Printed in Great Britain
EFFECT OF SPREADING OF MATERIAL ON THE SURFACE OF A FLUID-AN EXACT SOLUTION CHANG YI WANG Department of Mathematics, Michigan State University, East Lansing, Michigan 48823, U.S.A. Ah&raft--A material of small density is being supplied at constant rate at a point on the surface of a quiescent fluid. The material may be an immiscible, lighter fluid of higher viscosity, or may be composed of a stream of solid particles. After some time the material spreads out with constant thickness and a steady state is reached. We would like to know the effect on the fluid, due to the spreading of surface material. The problem is governed by the non-dimensional parameter B/v where Yis the viscosity and B/r is the velocity imparted by the spreading material to the fluid underneath. Depending on the range of B/v, the Navier-Stokes equation in spherical coordinates admits one exact solution for positive B, and three different exact solutions for negative B. An exact boundary layer solution is found for B/v + 1. A uniformly valid solution is then constructed. The Stokes solution for IB/v ( $ 1 is also determined. Both approximations are compared with the exact solution.
1. INTRODUCTION
THE present paper is motivated by the frequent accidental spillings of oil on the surface of water. Due to the balance of gravity and surface tension, the oil spreads with more or less constant thickness covering a large area on the water surface. Since the oil is much more viscous than water, we assume the water has little dynamic effect on the layer of oil. Our problem thus considers only the effect of the spreading oil on the water underneath. The model consists of an infinite layer of oil spreading out from a constant source. After all transients have died out, we assume the thickness h of the spreading oil is constant. Then the velocity imparted by the spreading oil to the fluid below is Q/2nhr where Q is the volume rate of discharge, and r is the distance from the source. When velocities are inversely proportional to radius, the Navier-Stokes equations admit exact solutions in spherical polar coordinates. Landau [l] found the exact solution of a momentum jet in an infinite fluid. Following Landau, Yatseyev [Z] obtained the general solution in terms of hypergeometric functions, and applied the solution to the flow due to a semi-infinite line source. Independently but later, Squire [3] arrived at the same solution obtained by Landau [l]. Squire [4] extended the solution to a momentum jet emerging from a plane wall, albeit the boundary conditions on the wall are not satisfied. Squire [S] considered conical jets, but the solution requires infinite velocities on the axis of symmetry. In this paper we shall study the exact solution applied to the steady spreading of oil on water. All the boundary conditions of the model will be satisfied. 2. THE EXACT SOLUTIONS
The Navier-Stokes
equation in spherical polar coordinates (p, 8, cp)is 1
1
_ a( !I’, E2 Y’)
vE41p + ~ p2sine
a(p,fq 255
=O
(2.1)
CHANG YI WANG
256
where
a p2 ae'
cot e
&E+i~_-_
ap2 pzaez
(2.2)
p is the distance from the origin, and 8 is the angle from the axis of symmetry. The stream function Y is related to the velocity components by -1
P v=pZsin
ay
av
i
(2.3)
ve = psin-
If we let Y = vp g(q), q = cos 8, (2.1) becomes (1 - q”) g”” - 419” + 3g’g” + gg”’ = 0.
(2.4)
This non-linear equation can be integrated thrice to give the Riccati equation (1 - rq)g’ + 2qg + $gs’ = c&
+ c,q +
(2.5)
c3.
If we let g = 2(1 - q2) u’/u, (2.5) becomes linear 2(1 - $)2 11”= (c,#
+ c,?j +
(2.6)
c3)u
and is solvable in terms of hypergeometric functions, see Yatseyev [2]. If the velocities are to remain finite on the axis rl = 1, the right-hand side of (2.5) must necessarily be of the form c1q2 + c,rj +
c3 =
(2k2 - $(l
- 1)2,
(2.7)
where k may be a real or a purely imaginary constant. Then solving (2.6) and substituting for g, we have
r (l
_
J1 + 2k)(l + q)2k+ c(1 - 2k) (1 + t#k
k#O
+ c
2 ln(l + r~)+ c
1
k = 0.
The constants c and k are to be determined through the boundary vg = 0; q = 0, v, = B/p where B = Q/(2nh). The solution for the stream function when B is positive is (4k2 - 1) [(1 + v)‘~ - l]
y = “(l
- ‘) (2k - 1) (1 + q)2k+ (2k + 1)
(2.8)
(2.9) conditions
q = 0,
(2.10)
where k=
When the surface material is drawn off from a point, B is negative. The solutions are then (1 - 4k2) [(l + q)2k- l] ’ = “(l
- ‘)(l
- 2k)(l + q)2k- (1 + 2k)
for--i<;<0
(2.11)
Effect of spreading of material on the surface of a fluiban
exact soluiion
251
where k=
Y = vp(l
ln(l + q) - 2
1’
(2.12)
and Y = vp(1 -
(1 + 4b2) ‘) 1 - 2b cot [b In (1 + q)]
for - cc < $ c - ;,
(2.13)
where k = ib,
The exact solutions (2.10H2.13) depend on the non-dimensional parameter B/v, which represents the rate of discharge or suction of surface material. In particular, equation (2.13) is exactly the solution found by Squire [4], where it is applied to a momentum jet emerging from a plane wall. As we noted before in that case the boundary conditions cannot be satisfied since the solution require up = B/p on the wall. Thus Squire’s application would be physically meaningful only if the solution were insensitive to the boundary conditions on the wall. However, the boundary conditions are important since the flow (2.13) is not due to a momentum jet, but is entirely caused by the non-zero boundary conditions. Therefore, the appropriate application of the exact solution (2.13) is probably the suction of surface material above a viscous fluid, and not the jet. In this paper we shall concentrate on the spreading of surface material, where B > 0.
3. THE BOUNDARY
LAYER
SOLUTION
Let us compare our exact solution (2.10) with the boundary layer approximation. We shall normalize lengths by L, velocities by B/L, pressure by B’p’/L? where p’ is the density of the fluid and L is an appropriate length scale. The Navier-Stokes equations in cylindrical polar coordinates become,
uw,+ww,=
-pz+2
(f-4, +
(
1
w,,+-w,+w,, r
w, = 0,
, >
(3.2)
(3.3)
where U, w are the velocity components in the direction r, z respectively, and 6’ = v/B. The boundary conditions at z = 0 are, w = 0, u = l/r; and at z = co are w = 0, u = 0.
Expanding the dependent variables in a power series in e*f3.lff3.2) shows that the outer fiow is potential to the first two orders. Inside the boundary layer we set z =tzq u = U()-t- EU1+ . . . w = ewe + p = po + tpi -I- . . . where 4 is the new stretched variable. Noting that the zero-th order outer flow is identically zero, the boundary-layer is
(3.4) equation
We define an inner stream function !P
= - j war dr = J uor dq,
(3.6)
and seek similarity of the form Ye) = rf([)
(3.7)
where c = QY/F. Then (3.5) reduces to an ordinary differential equation f”‘+f’“+ff”=O*
(3.8)
The boundary conditions are that f(U) = 0, f’(O) = 1 and f(ao) is finite. Equation (3.8) can be integrated to give an exact boundary-layer solution: f = (J2)tanh The fact that f(a)
i ( J2 > .
(3.9)
= ,/2 signifies an influx at the outer edge of the boundary layer.
Since the outer flow is potential, the outer stream function is governed by f3.10) Together with the boundary conditions I = Q, Fe5 = 0; z = 4 !F@”= (42) r; p2 = r2 f zz = to, velocities = 0; we obtain P”
= (42) [J(z2 + r2) - 21.
(3.11)
A uniformly valid composite solution can then be constructed: yI =1 eyly”’+ &Y(O)- coinmon part = e{(JBrtanh($)
+ (,/2)[J(z2
+ r2) - .r] - (,/2)r~ + o(&.
(3.12)
The approximate solution (3.12) is compared with the exact solution (2.10) in Fig. I for B/v = 50. The difference is small inside the boundary layer while in the outer Bow (3.12) over estimates the effects of inertia.
We nuts that if B were negative9 (3-g) does not admit ~un~-~ay~ suhztions with the boundary conditions j(U) = 4 f’f@) = -1 and S(clc) finite ~h~s~~~~y in this case vorticity is not conBrmed to the vicinity of the surf$~, even if 1B/Y 1 is very large. 1x1that case the exact solution (Nl), (2.12) or (2.13) should then be used. The streamlines for B negative look similar to those for B positive.
4. THE
STOKES APPBOXIMATIOTI’
For very small B/v, (2.1) reduce to the Stokes equation E4!P = 0.
(4.1)
The general sohrtion is (4.0
The Stokes solution (4.2) is compared with the exact solution (2.10) in Fig. 2 for B/v m 04. The inertial effects are underestimated. Figure 3 shows the steamhnes for B/v = 5, where neither the Stokes solution nor the boundary-layer solution is adequate to describe the flow. They serve as the upper and lower bounds of the exact solution. Equations (3.11) and (4.2) can also be obtained by taking limits on the exact sohrtion (ZUO). Care must be taken, however, in deducing the bo~dary-mayor suhrtion (3.7) from (210) in which both 19and I@ are smalf.
260
WANG
YI WANG
I
I \ \\ I\ \
~~ JI
I1
0.05
yr0.0,0.02
FIG. 2. Streamlines for E/v = 0.5 exact solution (2.10) -- Stokes solution (4.2).
FIG. 3. Comparison of streamlines, B/v = 5.0 (2.10);---(4.2). -(3.12);
-
Effect of spreading of material on the surface of ajlui&an
exact solution
261
5. D~CUSSIO~S
The present solutions may also be applied to a steady stream of light solid particles spreading on a fluid, provided the particles behave as a continuous layer. Less emphasis is placed on negative B/v, since most surface layers are unstable when suction is applied at a point. We note that for the surface of the fluid to remain plane the Froude number B’/(Cg) must be small compared to unity. The solutions are also restricted to radial distances large compared to the diameter of the source at the origin. The streamline patterns show a characteristic sharp turn near the origin, which differ from say, the flow towards a plate. This is important since the fluid acts as an effective centrifuge for suspended heavier particles in the fluid. The effect is more pronounced for high values of B/v. In the limit of infinite B/v any ~homogeneity of larger density than the fluid shall be deposited inertially on the surface material. In such a case the elhciency of capture or collection becomes one hundred per cent.
REFERENCES
PI L. D. LANDAU,A new exact solution of the Navier-Stokes equation. (in English) Computes I ‘Academic des Sciences de I’URSS Nouvelle Serie (International
Render de
edition of Dokl. Nauk. SSSR) 43, 286-288
(1944). Pksp. teor. Fiz. 20 (in Russian), 1031-1034. Translated as NACA Tech. Memo. No. 1349 (1950). [31 H. B. SQUIRE,The round laminar jet. #& J. up@. Me&. Math. 4,321-329 (1951). 141 H. 3. SQ~JIRE,Some viscous fluid flow problems I: Jet emerging from a hoie in a plane wall. Phil. Mug. 7th Series 43, 942-945 (1952). 151H. B. SQUIRE,Radial jets. In: 50 Jahre Gre~zsch~ck~r~chung, edited by H. Gortler. Braunschweig, 47-54
PI V. I. YATSEYE\‘,On a class of exact solutions of the equations of motion of a viscous fluid. Zh.
(1955). (Received 10 April 1970)
R&mn&-On projette un mattriau de faible densite a debit constant a la surface d’un fluide au repos. Le mattriau peut &re un fluide non miscible plus leger et de plus grande viscosite ou bien un flot de particules solides. Apres quelques temps le materiau s’ttend avec une epaisseur constante et on atteint un &at permanent. On se propose d’ittudier l’effet sur le fluide du depot d’un materiau en surface. Le probleme est r&i par le parambtre sans dimension B/v on v est la viscosite et B/r la vitesse donn6e par le materiau depose au fluide sous-jacent. Selon l’ordre de grandeur de B/v l’bquation de Navier-Stokes en coordonn&.es sphtiques admet une solution exacte pour B positif et trois solutions exactes differentes pour B negatif. On trouve une solution exacte dans la couche limite pour B/v + 1. On construit alors une solution un~orrn~rn~t valabfe. On determine egalement la solution de Stokes pour /B/v\ < 1. Les deux approximations sent compar&s avec la solution exacte.
Zusammenfaasung--Eine Substanz geringer Dichte wird mit einer konstanten Rate in einem Punkt auf der Oberfliiche einer ruhenden Fliissigkeit erzeugt. Die Substanz selber kann eine nicht mischbare, leichtere Fltissigkeit hbherer Viskositat sein oder kamr aus einem Strom fester Teilchen bestehen. Nach einiger Zeit hat sich die Substanz mit gleichmassiger Dicke verteilt, und ein stationarer &stand ist erreicht. Wir machten den durch die Ausbreitung der Oberflachensubstanz auf die Flilssigkeit ausgetibten Effekt wissen. Das Problem wird durch den nichtdimensionalen Parameter (E/o) beherrscht, wobei u die Viskositat ist und (B/u) die Geschwingigkeit, die von dem sich ausbreitenden Material auf die darunterliegende Fltissigkeit iibertragen wird. Abhangig von der Grossenordnung von (B/u), erlaubt die Navier-Stokes-Gleichung in spharischen Koordinaten eine exakte Lijsung fiir positive
C&SANGYI WANG
262
B und drei verschiedene exakte Losungen ftir negative 8. Eine exakte Grenzschichtldsung wird ftir (B/u) 9 1 gefunden. Damit wird eine gieichmissig gtihige L&sung konstruiert. Die Stokes-Losung fiir 1 B/v I< 1 wird ebenfalls bestimmt. Beide NLherungen werden mit der exakten Losung verghchen.
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