Analytic Theory for The Determination of Velocity 

and Stability of Bubbles in a Hele-Shaw Cell 

Part 11: Stability 

S. Tanveer' 

Mat hemat ics Department 

Virginia Polytechnic Institute & State University 

Blacksburg, VA 24061 

Abstract 

Here, we extend the analysis of part I to determine the linear stability of a bubble in 

a Hele-Shaw cell analytically. Only the solution branch corresponding to largest possible 

bubble velocity U for given surface tension is found to be stable, while all the others are 

unstable, in accordance with earlier numerical results. 

This research reported here was supported by the National Aeronautics and Space Administration under 

NASA Contract No. NAS1-18605 and was carried out while the author was in residence at ICASE, NASA 

Langley Research Center, Hampton, VA 23665. 

1 
1. Mathematical formulation and regular perturbation for the modes 

Now we consider the problem of linear stability of a bubble for disturbances that 

are symmetric about the channel centerline. For the class of disturbances considered, 

the flux of fluid at infinity is assumed to be the same &B in the steady state. This is a 

realistic assumption for most experiments where the pressure gradient is held fixed. From 

incompressibility of fluid flow, the bubble area is time independent. Our formulation here 

is an extension of earlier method for the analytical determination of the zero surface tension 

symmetric stability modes as shown in section I11 of Tanveer & Saffman(l987). 

Without any loss of generality we set the channel half width a to unity and the fluid 

velocity at 00 , V to unity since this is equivalent to nondimensionalization of all relevant 

variables using combinations of these two quantities. In the frame of the steady bubble 

translating with constant velocity U , the pressure boundary condition on the bubble 

boundary is 

+ constant 

b2T 1 

12p R r$+ux=-- 

where the variables have the same physical meaning as the first part of the paper. The 

constant appearing on the right hand side of (1) can now depend on time. The kinematic 

boundary condition that a point on the bubble boundary moves along with the velocity of 

the fluid at that point can be expressed as 

(2) 4X2 + 4°F" + Ft = O on F(x, y, t) = o 

where F(x,y,t) = 0 determines the bubble boundary (x(t),y(t)) at time t . As for 

the steady problem (Tanveer, 1986), we consider the mapping function z(c, t) from the 

interior of the unit < semi-circle into the fluid flow region exterior of the bubble on one 

side of the channel centerline (z = x + iy ) such that c = fl are mapped to the rear and 

front stagnation points on the bubble respectively; < = f& are mapped to z = TOO .The semi-circular arc corresponds to the bubble boundary, while the real diameter in the 

ti depends 

on time and has to be determined. In the linear stability problem, it is convenient to 

decompose 

-plane corresponds to the wall y = 1 and the channel centerline y = 0 . 

(3) &=a + a1 

where a is time independent and is specified as a parameter characterizing the size of the 

steady bubble whose stability is being studied. It is convenient to decompose the mapping 

function into 

(4) 

2 
where j is some analytic function within the unit circle that has to be determined and 

zo is given by 

(5) 

and k, is a constant given by 

(6) 

We find 

(7) 

where 

2a 

TU k, = -[U(l+ 2) - 24 

dz 

d< 

- = k8(j’ + vi) 

&[U(l+ 2) - 221 

CY[U(l+ a2) - 2a2] ??= 

and the function is given by 

(9) ~ U(1+ti2) -2 

U(1+ &2) - 22 where p2 = i= (1 - YC2) 

(52 - 2) (1 - 2<2) 

We decompose the complex velocity potential W 4 + i $J into 

where 

On substituting (10) and (5) into the dynamic boundary condition (1) on the bubble, we 

find that it is equivalent to 

on < = e;” where v is real and in the interval [0, 7r] and 

(13) 

b2T 

7= 12pUk3 

as in the first part of the paper. Since for symmetric disturbances, the streamlines coincide 

both with the cell walls and the channel centerline, 

(14) Imw = 0 

3 
on the real 5 diameter. The geometric condition that real 5 axis in the interval 

[-1,1] corresponds to the wall and the channel centerline is equivalnet to 

Imj = o 

on the real < axis in [-1, 11. Since = 1 corresponds to the bubble boundary at all 

times, the kinematic boundary condition (2) is equivalent to the following equation on the 

unit < semi-circular arc (see Tanveer & Saffman (1987) for details) 

where * indicates complex conjugate and subscript t denotes time derivative. On 

substituting for z from (4) and W from (10) and using (11), one finds that (16) is 

equivalent to the following boundary condition on 5 = ei" u in the interval [0, 7r] : 

where in the above prime on quantitites w and f denotes derivatives with respect to 

and subscript T indicates derivative with respect to T , where r is scaled time given 

by 

For linear stability analysis, we decompose 

(19) f=f+F 

where f satisfies the steady state equations (1) and (2) in the first part of the paper. 

Assuming F and a1 small compared to f and a , linearization of (12) and (17) about 

the steady state gives us the following boundary conditions on = ei" : 

d F' + Hiai d F' + Hiai 

(20) 

7 {Re5z( f'+h ) -Re [ l+c-Zn(f'+h) de 

If' + hl 

Re (w+F) = - 

where subscript a denotes derivative with respect to a. 

4 
and 

SaU(1 - 52) - 4a 

U(1+ a2)(1- <2) - 2(a2 - p) 

2a + + 

(23) 

(1 - P2S2) 

(p - a2)(1- a2p) 

HI = 

p - a2 1 - a252 

For 

(1) in Part I implies 

real in (-1, l), in addition to (14) substitution of (19) into (15) and use of equation 

(24) ImF =O 

Equations (21), (22), (14) and (24) are the linear stability equations determining the 

analytic functions w and F and the time dependent constant a1 . For purposes 

of analysis of these equations, it is convenient to define analytic functions k , k and 

? within the unit circle with continuity at the boundary such that on the real diameter 

(25) Imk = 0 

(26) Imk = 0 

(27) Im?= o 

and on < = ei" , 

(29) Re k = Re [c*(h* + f")?] 

(30) Re ? = Re [<*f'*F] 

Each of k , k , ? so defined are analytic functions of < in 1st 5 1 because of smoothness 

of the right hand side of (28), (29) and (30) on < = eiv and from the application of 

Schwarz reflection principle for continuation to the lower half < semitircle. Substituting 

for h from (22) into (20) and using the definitions of k , and f' it follows that on 

< = eiv 

5 
By subtracting off the simples poles of the expression within the square parantheses in 

(32) and adding on regular terms with the same real part, one finds that (32) is equivalent 

to 

I (33) Reg = 0 

~ 

where 

(34) 

41 - as) (s + 4(1+ as) 

B = -sw'+ Fr (4 

(1 - ,2<2)($2 - a2) 

~ 

Clearly Im s = 0 for s real in [-1, 11 because of (14), (24), (26) and (27). Therefore, 

(33) implies that ,? = 0 for 5 1 and so 

Again the function within the parantheses on the left hand side of (31) is an anlaytic 

fucntion in 111 5 0 with vanishing imaginary part on real < axis in [-1,1] in view of 

~ 

(14), (24) and (25). Thus for any with Is1 5 1 , 

Equations (35) and (36) hold for any s with > 1 as well when one obtains proper 

analytic continuation of each of rc , k and defined originally in 5 1 . On 

substituting for w from (36) into (35) one obtains 

On evaluation of the above equation at 5 = 0 , we find that 

[&(a) - F4-41 + R(0) + k(O)a1, = 0 (a2 - P2) 

241 - a4) 

and this determines tyl . One can use (38) back into (37) to obtain 

(39) 

where 

k2 - P2> F, = Rs, + yrc"' F' + (1 - ,252) (52 - a2) 

6 
where 

We note that k(0) # 0 is a necessary condition for the above to be valid. This conditions 

is satisfied when 7 = 0 for in that case k = ko , where from (5),(22) and (29), ko is 

determined from the boundary condition 

on s = eiu and on the real diameter the imaginary part of vanishes. It is not 

difficult to see that 

ko(s) = 1 - 2 c2(s2 - $1 1 (a2 - P2) + rrrk, [ (52 - a2)(1 - a2s2) [-(s2 - a2) (1 - a2s2)(1 - cy2 P) 

(44) 

and so 

1 +"')+ 

a2 - p2 1 - a4 (1 - a4)2(1 - a2p2) 

which is clearly non zero. For small y , f is small and therefore one could argue that 

k(0) # 0 for sufficiciently small 7 . We will assume that is indeed the case. Now we 

assume that F has a cut time dependence then the mode corresponding to Q , we can 

try a regular perturbation expansion of the form 

(46) F= FO(<,a,U,a) + yFl(s,Q,U,a), + 1.0 

where for convenience of regular perturbation expansion, U is being treated as a parameter 

together with a , though the steady state selection implies that U is a function of 7 and 

a! . Tanveer & Saffman (1987) calculated the eigenmode Fo . Using their procedure, it is 

straighforward though laborious to find the next term Fl and one finds no restriction on 

cr . We assume that this is indeed the case to every order in y . 

7 
2. Analytical continuation and reduction to an ODE 

We now proceed to extract the leading order transcendentally small terms by analytic 

continuation of the boundary conditions to the unphysical region 151 > 1 . The procedure 

is in principle very much like the first part of this paper. This illustrates the power of this 

formalism, which otherwise may appear rather complicated compared to the CombescotDombre analysis. The complication here compared to the first part of this paper is that 

we have two analytic functions f and w rather than one. We now proceed with the 

analytic continuation of the equations. Rather than work on the linearized set of equations 

just derived, it is convenient to analytically continue the full nonlinear equation and carry 

out the linearization at a later stage. 

We can write equation (12) as 

(47) 

where 5 is an analytic function in the unit semicircle defined by the boundary conditions 

and on the real diameter, 

Using Poisson's integral formula relating a harmonic function and its conjugate to the 

value of the harmonic function on it's boundary, by using the definition of , we find that 

(48) implies 

(50) 

where 

and 

-2 2 

(52) R(1) = v(1 - P 1 ) + (Sa - G2)(1 - h212)71(1) 

Analytic continuation outside the unit circle gives us 

8 
Now, we notice that the quantity within the paranthesis in (47) is an analytic function of 

s with simple poles at s = f& and has vanishing imaginary parts on the real diameter. 

By subtracting off the simple poles and adding on terms with the same real parts on the 

unit circle, it is easy to conclude that (47) is equivalent to 

(54) 

Equation (54) is valid everywhere in the complex plane. For > 1 we must use 

the expression (53) rather than (50) for fj . Using the decomposition (3) and (19) and 

assuming that each of 01 , F and w are small, we expand to linear order to find 

9 
and 

(63) 

where 

L(5,a) = {L+ a U(1+ a2) - 2a2 ] C2k2 - P2) 2a(U - 2) 

i 2Ua - 2a(U(l+ a2) - 2)(U - 2) 

U(1+ a2) - 2a2 (U(l+ a2) - 2a2)2 -52 [ 

1 1 

5 

-241 - a252)f'($ - 2a52(52 - d)f'(-) 

1 24U - 2) } (1 - p252) 21(5,4 = { -+ a! U(1+ a!2) - 2a2 

1 2a(U(1+ a2) - 2)(U - 2) 

(U(l+ a2) - 242 - 2ua 

-52 { U(l + a2) - 2a2 

22 -2a(l - a2s2)f'(s) - 2as (s - a2)f1(c) 

Linearizing (54) and using (53) and subtracting off the steady state, we find 

L1 F" + (Pi + Q1)F' + -(F + W) = R2 

7 (66) 

where 

where 

where g is as defined in the first part of the paper and so from equation (20) of Part I, 

(69) 

10 
and 

and 

and 

The linearized kinematic boundary condition is given by (35). We need the analytic continuation of each of ? and k outside the unit circle. For defined by (27) and (30), 

it follows from Posson's integral formula that for inside the unit circle 

For 111 > 1 , by contour deformation in the I' plane, we obtain the analytic continuation 

of (74) to be 

(75) r?. = w, p, c) + [w)p(d SI + cf'(c)F(;)] 

We noted before that k = ko given by (44) when 7 = 0 , i.e. f = 0 . So from (26) 

and (29) and Poisson integral formula it follows that for 111 5 1 , 

(76) = iio + W,F,I) 

where 

11 
Since equations (35) and (66) do not involve time T explicitly, we now assume a modal 

form with a ear time dependence. We note that in (66) the right hand side R2 involves 

values of F and w either on the boundary of the unit circle or inside where correction 

to the regular perturbation expansion 

for w and the corresponding series for F (given by (46)) for F are transcendentally small. 

Thus one can replace F and w on the right hand side of (66) by its regular perturbation 

expansion. In our case, as far as obtaining leading order transcendental correction, it is 

only necessary to retain the zeroth order approximation F = Fo , w = wo in evaluating 

R2 . Similarly we can substitute the regular perturbation expansion for F and w in the 

integral term 13 and I4(f, F, <) as well as for F(1/() and k given in (75) through 

(78). On carrying out this substitutions, (35) and (66) becomes a set of two ordinary 

differential equations for unknown modes F and w : Strictly speaking, we should use 

different notation for the modes since F and w were originally used to denote the general 

time dependent deviation from the steady state. However, from this point onward, we will 

only be talking about the modes and so in the interest of non- proliferation of symbols, we 

retain the same symbols for the modes. LFrom the two differential equations thus obtained 

from (35) and (66), we eliminate the variable F to obtain a single third order equation 

for w . 

I 

I 

I 

I 

where 

and 

where Rz , Rs , PI , Q1 and L1 are given by (67), (40), (68), (70) and (71) respectively. 

12 
3. Asymptotic solutions for small 7 

For small 7, dominant balance involving terms with factor of $ must give w = wo . 

Carrying this out this out to higher and higher precision only generates the perturbation 

expansion (79). It is clear that transcendental terms in surface tension must be generated 

by looking at the homogeneous part of (80) and trying a WKB approach. We find that 

the leading order transcendental correction away from the immediate neighborhood of the 

turning points ( = zkk must be linear combinations of 

(83) 

The above asymptotic expressions for two linearly independent solutions to the homogeneous part of (80) simplify considerably, once we realize that the steady state function 

f - 7f1 away from the immediate neighborhood of the critical points as pointed out in 

Part 1. Using the expressions for f1 , one finds that to the same order of approximation to 

which (83) is valid, the leading order transcendental correction are multiples of wH1 and 

wHl given by 

(84) 

where 

and 

a(1- a2p2) dl = 4a(l - a*) 

o(a2 - p2) 

4a(l - a') 4= 

as before in the case of the steady state. We notice the similarity of (84) with the expressions for 91 and 92 in part I (Equations 33,34). Each of WH~ and WH, is transcendentally 

small or large depending on the sign of Re P where, as in part I, 

Thus the Stokes lines picture of part I of this paper is critically relevant in determining 

the nature of transcendental correction. Our work in the 1st part of this paper has shown 

13 
that steady solutions exist only when a2 - p2 > 0 . Further, we found that at sufficiently 

small surface tension, U on any branch of solution approaches 2 from below and so in 

the limiting case of small 7 , one can assume that p2 is positive for fixed a . Thus only 

Fig. 1 in part I of the paper is relevant in the stability analysis in the limit of zero surface 

tension. 

On inclusion of the leading order transcendental correction, the leading order asymp 

totic behavior of w in sector I (see Fig. 1) is given by 

As mentioned in section 4 of Part I, Re P has a maximum value of -j1I2p in sector I. 

Thus in order that each of the multiples of and WH~ be transcendentally small in 

all of sector I, it is necessary to have 

(91) c2 = O(1) 

In sector I1 

In order that the transcendental corrections in all of sector I1 be small it is necessary that 

c4 e-T-l/aPm - (93) - O(1) 

(94) c, = O(1) 

In sector 111, the leading order behavior is given by 

Note that it is not possible to have a multiple of wH1 in each of (95) since it grows without 

bounds at < = -a inside the unit < semi-circle. In order that (95) be a transcendentally 

small correction everywhere in sector I11 of Fig.1, it is necessary that 

Now, we assume that a - p small with a of order unity, as assumed in Section 4b of 

part I of this paper. As before in part I, section 4b, we now introduce the local variables 

near < = -l/p by defining 

(97) 1+a< = €21 

14 
Then to the leading order in e , equation (80) transforms into 

where 

where 

and the function b(x1) is determined as described in section 4b of part I. For large 

21 with corresponding ( in sector I1 of Fig. 1, 

W(X1) - 21 +Al(P')Xl -s/a+~/(da) e-4,911/2 (+ 1 z1 314 3 +Az(Pl)sl -3/a+0/(4&) e 481 1/2 (1+~1)~/~/3 ol(4a)- # (102) 

and this matches with the expression for w in (92) in sector I1 provided 

where R is defined as in section 4b, part I (Equation 83). 

For large x1 with corresponding c in sector 111, 

u/(4a) - e + (PI) ,.. 3/8+0/ (la) e- 48' 'I' ( 1+ z 1) '/'/!I + ( p') z;3/8+o/ (4a) ,4/3' 'I2 ( l+ 2 1) 'I4 /3 

(104) 

w(x1) - Zl 

and this matches with (95) provided 

(105) A4 = 0 

and 

A~~P/~R 4 3/8~u/(~)-3/82-3/82-u/4~ 

We now move to the immediate neighborhood of ( = 2 Introduce local variables 

-4 ,-$ = ei3+a-3/4(1 

CS (106) 

(107) (1 - a() = €X2 

15 
where D2(52) is determined as described in section 4b of Part I. To the leading order in 

e , equation (80) becomes 

: 

For large values x2 , with corresponding c in sector 11, 

o/ (4a) (I+ za 'I4 /3 + A6(B)x2 + -3/8-o/(h) -i4B'/2(l+z2)'/4/3 A5 (3) 22 e 

ol(2a)- (110) 

4x2) - 52 

and this matches to behavior of solution given in (92) in sector I1 provided 

and this matches with (89) provided provided 

iFrom (94) and (103), it follows that A1 is transcendentally small compared to A2 since 

e7-'/apm is transcendentally small and much smaller that terms involvint p' since under 

the assumption made here, a! - p << 1 . Thus we can neglect A1 altogether. Then 

together with (105), this implies that we are interested in solution to.(99) so that for large 

z1 with corresponding < in sectors I1 and 111 in Fig. 1, the transcendental correction 

is small. This would determine a unique solution to (99), as will be shortly argued. As 

far as (log), we note from (93) and (111) that A6 is transcendentally small compared to 

Ab and will therefore be set to 0. This means that we should require that the solution to 

(109) contain no transcendentally large term for large x2 when the corresponding < is 

in sector 11. 

We now have to use the condition of smooth bubble back. When o is real, this would 

imply that the mode w(<,u) be real in some open interval containing c = 1 . From 

(80), it is clear that if u is real on such an open interval, then it must be real for in 

(a, l/a) as all the coefficients of the differential equation are real in that real < interval. 

This implies that the solution w(x2, a) of (121) must be real on the postive real x2 axis. 

However, when a is complex, we cannot demand that each mode w(x2, a) should be 

16 
t 

! 

real on the real positive x2 axis. The condition of a smooth bubble back is equivalent to 

requiring that on the positive real x2 axis 

On formally taking the complex conjugate of equation (109) on the positive x2 axis and 

relating the corresponding solution to the condition (114), it is easily argued that the we 

must have the property 

U(XZ,U) = [w(x;, a*)]* (115) 

for any 52 not necessarily on the positive real axis. Thus the condition that for large 

x2 corresponding to f in sector 11, the solution to (109) does not contain any transcendentally large corrections implies from (115) that for large 52 with corresponding 5 in 

sector VI1 (which is a reflection of sector I1 in Fig. 1) also contain no trancendental large 

corrections. Indeed, this condition is equivalent to (115) since if we were to require that 

the asymptotic behavior for large 'x2 in sectors I1 and VI1 contain no transcendentally 

large terms, we will be assured that such solution will automatically satisfy the condition 

(114) and therefore (115) in view of the symmetries of (109). ' But the condition of no 

transcendentally large terms in two sectors will uniquely determine a solution to (109) as 

argued earlier for the steady problem in part 1. Note that in the special case when u is 

real, such a requirement is indeed equivalent to requiring that w(z2) be real on the real 

and positive 52 axis. The actual solution in (109) is not important as it is not needed 

for the leading order determination of the eigenvalue and the corresponding eigenmodes 

as the tip and the back problems decouple to the leading order. 

To ascertain that the tip of the bubble remains smooth under time dependent symmetric 

perturbation, we must require that W(C,T) as defined originally should be real in some 

open interval on the real axis containing -1. As before with the neighborhood of 

= +1 , it is easily argued from (80) that if indeed such an open interval exists then 

w(<, ~r) is analytic in the entire interval (-l/a, -a) . However, the same is only true 

for each of the modes w(f,u) if u is real. Generally, for complex u , the condition of 

smooth tip is equivalent to 

4% 0) = [w(x1, a')]' (116) 

on the positive real z1 axis, where w(zl, a) is the solution to (99) satisfying conditions 

in sectors I and I1 as mentioned earlier. It is easily argued that the condition (116) is 

equivalent to requiring that there exists solution to (99) with no transcendentally large 

term for large z1 with corresponding c in sectors I, 11, V and VI , where sec. t,ors V and 

VI are the reflections of sectors I and I1 on the part of the real 5 axis between a! and 

l/a! (see Fig. 1). From the figure, it is clear that this means that the solution to (99) 

17 
satisfying the smooth tip condition is equivalent to requiring that there be contain no 

transcendentally large term for large z1 in the entire interval (-T, ?r) . Note that such 

a requirement is also valid when u is purely real as can be seen by taking the complex 

conjugate of equation (99). 

We now relate the solution to (99) satisfying the above conditions to a problem which 

we already solved in the context of the linear stability of a finger (Tanveer, 1987~). We 

introduce the transformations 

Then equation (99) becomes 

- 

dSw 1 3- d2w 3 1 dw .u ,&MS + -(3 - -iP'MD) - + (1 - ?iPMD + ijtMs) -- + t- -w = 0 (120) dE3 € 2 dE2 c2dxi CY E3 

where 

The requirement of no transcendentally small terms for argument of z1 in the interval 

[-?r,~] is equivalent to requiring that in the plane for large argument, there be no 

transcendentally small terms for argument of in the interval [-T, 01 . This is precisely 

the problem that was encountered for the finger (Tanveer, 1987c) and numerical solutions 

were found for the cases when ,@ was of order unity. Only one branch of solutions for 

which a' was the smallest, i.e. U the largest was found to be stable and all the other 

branches unstable to tip breaking disturbances. Given the correspondence we have just 

established, the same must be true for the bubble. For large s) , ( 6 in the notation of 

the Tanveer paper), asymptotic expressions were found for Q and it was found that there 

were many unstable eigenvalues. 

In addition to what has been reported in the Tanveer (1987~) about the limiting values 

of u as 7 + 0 , we note that in the context of the finger eigen values u for which 

Re CJ < 0 is not physically acceptable since they correspond to singularities in the tail of 

the finger. In the case of the bubble, these modes physically correspond to disturbances 

affecting the sides and the back of the bubble, as seen before numerically (Tanveer & 

Saffman, 1987b). Using the procedure described before (Tanveer, 1987c), we found two 

nonzero pairs of discrete eigenvalues u = -1.65 f0.004i and u = -4.1060f0.11294i for 

the Mclean-Saffman branch. 

4. Discussions and Conclusions 

18 
Here in this part of the paper, we have demonstrated how the calculation of transcendentally small terms in surface tension is crucial in the proper prediction of eigen values 

and corresponding eigenmodes for the linear stability of a bubble. We have shown that 

only one branch of bubble solutions is stable, while all the others are unstable, a result 

supported by earlier analytical and numerical work on the finger. The next step will be 

to calculate modal interactions in for calculation of non-linear interactions. It is to be 

pointed out that in this problem, the calculation of stability and the modal interactions in 

an analytical procedure is very important since all the usual numerical procedures break 

down because of inherent ill posedness of the time evolution problem as surface tension 

tends to zero. 

19 
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[24] Tanveer, S., 1987c, Analytic theory for the linear stability of Saffman-Taylor finger, 

Phys. Fluids 30, 2318 

[25] Tanveer, S. 8z Saffman, P.G., 1987, Stability of bubbles in a Hele-Shaw cell, Phys. 

Fluids 30, 2624. 

[26] Tanveer, S., 1988, Selection and Stability of Bubbles in a Hele-shaw cell, Submitted 

to the Proceedings of the 3rd Int’l Conference of Drops 8z Bubbles, Monterery, CA. 

[27] Taylor, G.I. & Saffman, P.G., 1959, A note on the motion of bubbles in a Hele-Shaw 

cell and porous medium, Q.J.Mech.Appl.Math, 12, 265 

[28] Vanden-Broeck, J.M., 1983, Phys. Fluids, 26, 2033. 

21 
22 
Report Documentation Page 

1. Report No. 

NASA CR-181837 

2. Government Accession No. 

ICASE Report No. 89-29 

4. Title and Subtitle 

20. Security Classif. (of this page) 

Unclassified 

21. No. of pages 

24 

19. Security Classif. (of this report) 

Uncl as si f ied 

ANALYTIC THEORY FOR THE DETERMINATION OF 

CELL. PART 11: STABILITY 

VELOCITY AND STABILITY OF BUBBLES IN A HELE-SHAW 

22. Price 

A0 3 

7. Author(s1 

Saleh Tanveer 

9. Performing Organization Name and Address 

Institute for Computer Applications in Science 

Mail Stop 132C, NASA Langley Research Center 

Hampton, VA 23665-5225 

2. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Langley Research Center 

Hampton, VA 23665-5225 

and Engineering 

15. Supplementary Notes 

3. Recipient's Catalog No. 

5. Report Date 

April 1989 

6. Performing Organization Code 

8. Performing Organization Report No. 

89-2 9 

10. Work Unit No 

505-90-21-01 

11. Contract or Grant No. 

NAS 1-1 8605 

13. Type of Report and Period Covered 

Zontrac tor Report 

14. Sponsoring Agency Code 

Langley Technical Monitor: 

Richard W. Barnwell 

The o ret ical and Com put a t ional 

Fluid Dynamics 

Final Report 

16. Abstract 

Here, we extend the analysis of part I to determine the linear stability of 

a bubble in a Hele-Shaw cell analytically. Only the solution branch 

corresponding to largest possible bubble velocity U for given surface tension is 

found to be stable, while all the others are unstable, in accordance with 

earlier numerical results. 

17. Key Words (Suggested by AuthorWl 18. Distribution Statement 

two phase flow, pattern formulation, 

asymptotic theory, stability 

67 - Theoretical Mathematics 

70 - General Physics 

Unclassified - Unlimited 

I I 

NASA FORM 1626 OCT 86 

NASA-Langley, 1989