A note on the initial motion and break-up of a two-dimensional air bubble in water

Shorter Communications

me

(b) high degree of supersaturation is required before condensation takes place. The analysis presented will not be applicable to group (a). Little information is available on systems contained in group (b). Work by FRIEDLANDERand LEVINE 161 using steam and glycerol have found that the loci of points where condensation occurred. could be represented on a straight line on a plot of Log m vs. l/T but the slope of this line was smaller than that of the equilibrium curve. It follows that under these conditions the effect of condensation in enhancing the vaporization rate will be smaller. For such systems equation (18) may be useful for predicting the upper bounds of the possible vaporization rates. A. W. D. HILW J. SZEKELY Department of Metallurgy Imperial College London S. W.1

m.m

Am .I 4.I 4 T AT Y CP

Mass flux density of evaporation species Volume rate of condensation of evaporation species within the boundary layer Mass fraction difference (mu, - me) Heat flux density Volume rate of heat liberation within the boundary layer ‘Temperature Temperature difference (T, - To) Local position coordinate normal to the evaporating surface Specific heat in gas phase

Dimensionless Groups Nu

Nusselt number $

Sh

hDL Sherwood number pa

NOTATION

Roman Letters B, C, Constants in the vapour pressure relation D Diffusion coefficient of vapourizing species fi Function relating m and T G’ H h ha k L m

Local mass flux vector Heat of vaporization Heat transfer coefficient Mass transfer coefficient Thermal conductivity Characteristic length Mass fraction of evaporation

G,‘eek Letters I’ Exchange coefficient 5 Conserved property p

= $ = Dp

= T+ ( Local density in gas phase

Suffixes s Apertaining the gas w Apertaining

species

(

H C,

.

1 m

to values in the free stream or bulk of to values at the evaporating

surface

REFERENCES Toop G. W., Ph.D. Thesis, University of London 1963. e.g. ECKERTE. R. G. and DRAKE R. M., Heat and Mass Transfer. McGraw Hill, New York 1959. POTTER O., Trans. Inst. Chem. Engrs, 1958 36 414. SPALDINGD. B., Znt. J. Heat Mass Transfer, 1960 1 192. e.g. SCHLICHLINGH., Boundary Layer Theory, McGraw Hill, New York 1955. FRIEDLANDERS. K. and LEVINED. G., Chem. Engng. Sci. 1960,13 49.

A note on the initial motion and break-up of a two-dimensional air bubble in water (Received 27 July 1963) 0.05 set intervals. The bubble is approximately circular at the time the air is cut off but quickly develops a narrow spout from its base which grows until it almost touches the roof. Irregularities develop in the initially fairly smooth upper margin and these are quite marked at about 0.20 sec. The bubble subsequently recovers its circular leading edge as satellite bubbles break off and general turbulence develops in its wake. The characteristic stable shape is achieved about 0.4 set after the start of the injection. Figure 2 is a further series of photographs taken under identical conditions to the above but where the boundary irregularities lead to a different result. After 0.20 set, similar irregularities appear but grow into a distinct indentation at about the forward stagnation point. This grows into a blunt finger that eventually divides the bubble in two. Presumably mutual interference prevents each bubble from achieving a

EARLY stages in the motion of a two-dimensional air bubble injected into water have been followed by high speed tine ohotograohy. The apparatus was made from two vertical sheets-of Perspex 0.58 in. apart, closed at the bottom and two sides so as to contain a vertical slab of water 12 in. wide by 19 in. deep by 0.58 in. thick. ‘A hole & in. dia. through which air could be injected was drilled through the rear Persoex sheet 6 in. from the bottom and near the centre line. Injection was controlled by a splenoid valve connected to a comparatively large reservoir charged with air to 30 lb/in2 (gauge). The valve could be opened for O-062 set and then closed abruutlv. The behaviour of the admitted air which initially formed a circular bubble of about 4 in. dia., was then followed by photographing the front face of the slab with a Vinten 35 mm camera run at about 220 frames/set. Figure 1 is a series of photographs showing events at

81

Shorter Communications truly circular leading edge but, as before, the stable shape is reached after about 0.4 sec. These observations can be compared with the results of the elegant experiment of WALTERS and DAVIDSON[l] where the initial motion of a two-dimensional bubble was observed without the general fluid disturbance that follows sudden air injection. WALTERSand DAVIDSON’Sbubbles preserved a smooth circular upper margin throughout their development, the liquid spout from the base was much thicker than shown here and grew in such a way that it cut off a symmetrical pair of daughter bubbles that formed a wake to the parent. It seems therefore that the break-up of a comparatively large bubble may be caused by irregularities in the liquid

flow around its upper surface. If irregularities in the flow, such as those produced by the sudden formation of the bubble in the liquid, develop near the front stagnation point, splitting of the bubble may follow. Acknowledgement-The photographs were taken by Mr. L. J. HEW~IT’SPhotographic Section at A.E.R.E., Harwell. P. N. Rows B. A. PARTRIDGE Chemical Engineering Division U.K.A.E.A. Research Group Atomic Energy Research Establishment HarweN

REFERENCE [l] WALTERSJ. K. and DAVIDSONJ. F., J. Fluid Mech. 1962 12 408.

On the inversion of Laplace transforms

by linear programming

(Received 19 August 1963; in revised form 14 October 1963) 1.

An alternative procedure is to represent f(t) by its value at N chosen points 0, to evaluate the right-hand side of equation (1) numerically treating the fj = f(tt) as N unknowns (making special allowance for t + co) and to solve the resulting N simultaneous linear algebraic equations for ft,

INTRODUCTION

many elaborate analytic solutions to the problem of inverse Laplace transformation have been given [l, 2, 31, the problem of obtaining physically significant inverse functions from experimentally measured transforms remains [4, 51. If we write THOUGH

“e-pt

F(P) = s

f(t) dt,

(1)

where the constants Cij depend on the tt and the particular method of numerical integration employed. As before, the difficulty with this method is that several of the f3 turn out to be negative; the problem then becomes one of choosing suitable tj. As numerical experiment soon showed, the exact solution of a determinate set of equations (3) or (5) was critically dependent on the numerical values Fi -equivalently, the matrices G3r or Cij proved to be highly ill-conditioned. In other words, by taking F(p) to be the known transform of a function f(t) satisfying (2), small errors in Ft could be shown to lead to inverses (CAjgj or fj) that were highly oscillatory. ORCUTT[5] “resolved” this difficulty by graphical smoothing of the inverse function he obtained by a highly refined method. Our first attempts to produce an automatic and convergent method for approximating to the values Ff by means of an everywhere positive sum of inverses gj or positive values fj relied on minimizing the mean-square deviation of the calculated F(pl) from the measured Fr, a method traditional since the days of Gauss. However, the elaboration required in a computer programme to do this persuaded us to use the simpler minimizing criterion of Linear Programming [6], which can be arranged to minimize the mean modulus of deviation. Because both our own IBM 1620 computer and the larger University Mathematical Laboratory EDSAC II computer libraries contained Linear Programming tapes, it was a relatively simple matter to obtain optimal solutions based on an initially chosen number of functions gf or points t3 (j= 1,2, . . M), where here M is not necessarily equal to N but can be as large as permitted by the size of the computer.

0

then we shall call F(p) the transformed and f(t) the inverse functions respectively. In our problem F(p) is given as a set of numerical values measured at a finite number N of points pi (i = 1, 2, . . . N). We require an approximate representation off(t), subject to the condition f(t) 3 0

for

t 3 0.

(2)

Furthermore, we want the method of inversion to be as simple as possible. One procedure is to replace the measured “function” fi = F@c) by a sum of analytic functions, having simple inverses, that coincide in value with Fi at the points pi. In general any set of N distinct functions Gj(p), j = 1, 2, . . . N, leads to a unique decomposition given by the solution of the N equations N

Fi = c AjGjd

(i = 1, 2, . . . N)

(3)

j=l

for the N coefficients AI, where Gjc = G&t). If we write the known inverses of Gf(p) as g&), then we obtain the “solution”

f(t) = f Am(t).

(4)

j=l

The trouble is that this usually leads to an f(t) that is not everywhere positive, and so the problem becomes one of choosing suitable functions G&). 82