Taylor, G. I. The Mechanics of Large Bubbles Rising through Extended Liquids and through Liquids in Tubes.pdf


Preview of PDF document untitled-pdf-document.pdf

Page 1...8 9 10111225

Text preview


1 PART I. DEDUCING THE RISE OF GAS BUBBLES
The observed values of (p0 − pθ ) / 12 ρU 2 , which may be equated to q 2 /U 2 , have
been taken from the faired curve for θm = 55◦ and 75◦ in figure 1.4, and their
values are given in the first row of table 1.1. In the next row are tabulated the
values of x/R = 1 − cos θ, where R is the radius of the spherical surface of the
lenticular body. Below these are given the values of q 2 /U 2 · (1 − cos θ) = q 2 R/U 2 x.
It will be seen that this ratio is nearly constant, its mean value being 3.28.
The condition (1.1) that bubbles of lenticular shape may have constant pressure over their spherical surfaces is satisfied if q 2 R/U 2 x = 3.28 is identical
with q 2 = 2gx. Eliminating x/q 2
U 2 = 2 · g · R/3.28 = 0.61 · g · R

or U = 0.78 ·

p

g·R

(1.3)

If the pressure had been exactly the same as the calculated pressure over a
complete sphere, the condition (1.1) could be satisfied over the portion near
the stagnation point only, for in that case q 2 = 94 U 2 sin2 θ and x = R · (1 − cos θ),
so that q 2 /x = 2g if


U2
8
1 − cos θ
= ·
.
g·R
9
sin2 θ
When θ is small, (1 − cos θ)/ sin2 θ → 12 , so that the pressure condition would be
satisfied near the stagnation point, i.e.
U2 =

4
·g·R
9

or U =

2 p
· g ·R.
3

(1.4)

1.3 COMPARISON WITH OBSERVATION
Fourteen bubbles, rising in nitrobenzene, were photographed. The results of
the measurement of the films are summarized in table 1.2, where the first
three columns give the volume V of the bubbles, the radius of curvature R
and the velocity of rise U . The fourth column gives the maximum transverse
dimension 2A, and the fifth column θm = sin−1 A/R. The sixth column gives
the drag coefficient, CD , of the bubble calculated from the equation
15◦

20◦

25◦

30◦

35◦

40◦

45◦

0.10

0.192

0.315

0.465

0.628

0.805

0.975

1 − cos θ

0.0341

0.0603

0.0937

0.1340

0.1808

0.2340

0.2929

q2
U 2 ·(1−cos θ)

2.94

3.18

3.25

3.44

3.47

3.44

3.33

θ
q2
U2

=

p0 −pθ
1
ρU 2
2

Mean value of

q2
U 2 ·(1−cos θ)

= 3.28

Table 1.1
Observed values of (p0 − pθ ) / 21 ρU 2 and values of x/R = 1 − cos θ

10