1291:: the pressure depends not just on the balloon diameter, but also on the manner in which inflation took place and on the initial direction of change. For instance, the pressure during inflation is always greater than the pressure during subsequent deflation at a given radius. One consequence is that equilibrium will generally be obtained with a lesser change in diameter than would have occurred in the ideal case. The system has been modeled by a number of authors, for example to produce
886:
1198:
force Fe which is proportional to pressure (P=Fe/S) plus air pressure in small balloon is greater than air pressure in big balloon. So, when the valve is opened, the smaller balloon pushes air into the larger balloon. It becomes smaller, and the larger balloon becomes larger. The air flow ceases when the two balloons have equal pressure, with one on the left branch of the pressure curve (
20:
1334:
had very different elasticities or airway resistance, there could be large discrepancies in the amount of air delivered. They argued that this might be seen as an example of the two-balloon experiment, with the two sets of lungs playing the role of the two balloons: "The 'two-balloon effect' (Merritt
1197:
When the valve is released, air will flow from the balloon at higher pressure to the balloon at lower pressure. The lower pressure balloon will expand. Figure 2 (above left) shows a typical initial configuration: The smaller balloon has the higher pressure because of the sum of pressure of elastic
42:
Two identical balloons are inflated to different diameters and connected by means of a tube. The flow of air through the tube is controlled by a valve or clamp. The clamp is then released, allowing air to flow between the balloons. For many starting conditions, the smaller balloon then gets smaller
58:
The key to understanding the behavior of the balloons is understanding how the pressure inside a balloon varies with the balloon's diameter. The simplest way to do this is to imagine that the balloon is made up of a large number of small rubber patches, and to analyze how the size of a patch is
889:
Fig. 2. Pressure curve for an ideal rubber balloon. When air is first added to the balloon, the pressure rises rapidly to a peak. Adding more air causes the pressure to drop. The two points show typical initial conditions for the experiment. When the valve is opened, the balloons move in the
406:
is the relative extension. In the case of a thin-walled spherical shell, all the force which acts to stretch the rubber is directed tangentially to the surface. The radial force (i.e., the force acting to compress the shell wall) can therefore be set equal to zero, so that
1064:
1282:
and the like. As a result, if the two balloons are initially very extended, other outcomes of the two-balloon experiment are possible, and this makes the behavior of rubber balloons more complex than, say, interconnected
727:
649:
374:
1164:
490:
1236:, defined as the number of molecules in both balloons if they both sit at the peak of the pressure curve, then both balloons settle down to the left of the pressure peak with the same radius,
880:
271:
Suppose that the balloon is composed of many such interconnected patches, which deform in a similar way as the balloon expands. Because rubber strongly resists volume changes, the volume
900:
574:
196:
43:
and the balloon with the larger diameter inflates even more. This result is surprising, since most people assume that the two balloons will have equal sizes after exchanging air.
1618:
1258:, the only possible equilibrium state is the one described above, with one balloon on the left of the peak and one on the right. Equilibria in which both balloons are on the
1173:
increases. This behavior is well known to anyone who has blown up a balloon: a large force is required at the start, but after the balloon expands (to a radius larger than
765:
524:
1510:
1716:
1262:
of the pressure peak also exist but are unstable. This is easy to verify by squeezing the air back and forth between two interconnected balloons.
1490:
660:
1754:
585:
23:
Fig. 1. Two balloons are connected via a hollow tube. When the valve is opened, the smaller balloon shrinks and the larger balloon expands.
1335:
and
Weinhaus 1978) could possibly have contributed to this volume discrepancy, and the inclusion of one-way valves could possibly help."
1330:, it has been proposed that one ventilator could be shared between two patients. However Tronstad et al. found that when the two sets of
1641:
1319:. Bio-physical models suggest that this process is effectively similar to the behavior of the balloons in the two-balloon experiment
1433:
1227:
Equilibria are also possible in which both balloons have the same size. If the total quantity of air in both balloons is less than
281:
1575:
1773:
Tronstad, C.; Martinsen, T.; Olsen, M. (2020), "Splitting one ventilator for multiple patients -- a technical assessment",
1089:
413:
1359:
1186:
1814:
776:
1274:
balloon once again goes up. This is due to a number of physical effects that were ignored in the James/Guth theory:
1059:{\displaystyle P_{\mathrm {in} }-P_{\mathrm {out} }\equiv P={\frac {f_{t}}{\pi r^{2}}}={\frac {C}{r_{0}^{2}r}}\left}
1567:
1541:
539:
79:
1311:
grow, others shrink, and fluid flow between interconnected cells causes the shrinking (smaller) cell to undergo
1189:
will occur, since the pressure in both balloons will drop when some air flows from one balloon into the other.
1349:
1275:
1667:
1525:
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1403:
253:
36:
1482:
1295:
specifying under what conditions the small balloon can inflate the larger, or the other way round.
66:
264:
is the volume of the sample. Thus, the force consists of two parts: the first one (caused by the
1774:
1735:
1591:
Dreyer, W.; Müller, I.; Strehlow, P. (1982), "A Study of
Equilibria of Interconnected Balloons",
228:
1461:
46:
The behavior of the balloons in the two-balloon experiment was first explained theoretically by
1693:
1571:
1486:
1327:
750:
1725:
1683:
1675:
1658:
Levin, Y.; de
Silveira, F. L. (2003), "Two rubber balloons: Phase diagram of air transfer",
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1411:
1354:
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1271:
509:
70:
1637:
1434:"Simple presentation of network theory of rubber, with a discussion of other theories"
235:
is a constant related to the number of possible network configurations of the sample,
1808:
1739:
1560:
1391:
1387:
1292:
47:
894:
Integrating the internal air pressure over one hemisphere of the balloon then gives
885:
1304:
506:
refer to the initial and final thicknesses, respectively. For a balloon of radius
275:
can be considered constant. This allows the stress-strain relation to be written
1730:
1429:
1284:
63:
1711:
1679:
1457:
1795:
1759:
1323:
1288:
1604:
268:) gives a tendency to contract, while the second gives a tendency to expand.
1344:
1312:
1308:
1755:"Desperate Hospitals May Put Two Patients on One Ventilator. That's Risky."
1697:
1316:
257:
1688:
1799:
1079:
This equation is plotted in the figure at left. The internal pressure
722:{\displaystyle p={\frac {1}{C_{2}}}\left({\frac {r_{0}}{r}}\right)^{4}}
32:
644:{\displaystyle {\frac {t}{t_{0}}}=\left({\frac {r_{0}}{r}}\right)^{2}}
1537:
1415:
19:
1779:
1303:
The two-balloon instability may play a role in the early stages of
884:
18:
1511:"On the Equilibrium States of Interconnected Bubbles or Balloons"
1331:
1185:
When both balloons are initially inflated to the peak pressure,
1249:. On the other hand, if the total number of molecules exceeds
1476:
369:{\displaystyle f_{i}=(C_{1}/L_{i})(\lambda _{i}^{2}-C_{2}p)}
1593:
Quarterly
Journal of Mechanics and Applied Mathematics
35:. It is used in physics classes as a demonstration of
1159:{\displaystyle r=r_{p}=7^{1/6}r_{0}\approx 1.38r_{0}}
1092:
903:
779:
753:
663:
588:
542:
512:
485:{\displaystyle \lambda _{r}^{2}=(t/t_{0})^{2}=C_{2}p}
416:
284:
82:
1559:
1158:
1058:
874:
759:
721:
643:
568:
518:
484:
368:
190:
1278:, imperfect flexibility of the molecular chains,
1182:), less force is needed for continued inflation.
16:Physics experiment used to demonstrate elasticity
875:{\displaystyle f_{t}\propto (r/r_{0}^{2})\left.}
1562:Elasticity, Plasticity and Structure of Matter
8:
1753:Gabrielson, R.; Edwards, K. (May 26, 2020),
1315:while the larger cell eventually becomes an
1710:Chan, C. J.; Hirashima, T. (May 21, 2022),
1270:At large extensions, the pressure inside a
569:{\displaystyle t\propto {\frac {1}{r^{2}}}}
191:{\displaystyle f_{i}={1 \over L_{i}}\left.}
1717:Seminars in Cell and Developmental Biology
31:is an experiment involving interconnected
1778:
1729:
1687:
1509:Weinhaus, F.; Barker, W. (October 1978),
1392:"The Pressure Curve for a Rubber Balloon"
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165:
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105:
96:
87:
81:
1619:"Numerical analysis of rubber balloons"
1370:
1287:. In addition, natural rubber exhibits
210:is the externally applied force in the
1558:Houwink, R.; de Decker, H. K. (1971).
1504:
1502:
1382:
1380:
1378:
1376:
1374:
732:The equation for the tangential force
654:and the radial force equation becomes
526:, a fixed volume of rubber means that
7:
1076:is the balloon's uninflated radius.
59:affected by the force acting on it.
1796:"Two balloons – physics experiment"
1712:"Tissue hydraulics in reproduction"
1193:Why does the larger balloon expand?
1617:Verron, E.; Marckmann, G. (2003),
934:
931:
928:
913:
910:
890:direction indicated by the arrows.
14:
1390:; Weinhaus, F. (October 1978),
1211:) and one on the right branch (
73:of ideal rubber can be written
855:
833:
819:
793:
457:
435:
363:
329:
326:
298:
1:
1638:10.1016/S0263-8231(03)00023-5
1360:Spontaneous symmetry breaking
1187:spontaneous symmetry breaking
533:is constant, or equivalently
248:is an unstretched dimension,
239:is the absolute temperature,
1731:10.1016/j.semcdb.2022.05.008
1518:American Journal of Physics
1478:Applied Mechanics of Solids
1396:American Journal of Physics
50:and Fred Weinhaus in 1978.
1831:
1680:10.1103/PhysRevE.69.051108
1568:Cambridge University Press
1458:10.1002/pol.1949.120040206
1438:Journal of Polymer Science
54:Theoretical pressure curve
1475:Bower, Allan F. (2009).
760:{\displaystyle \propto }
223:is a linear dimension,
1626:Thin-Walled Structures
1605:10.1093/qjmam/35.3.419
1350:Artificial ventilation
1160:
1083:reaches a maximum for
1060:
891:
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723:
645:
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520:
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370:
192:
67:stress-strain relation
29:two-balloon experiment
24:
1322:Due to a shortage of
1169:and drops to zero as
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22:
1483:Taylor & Francis
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414:
282:
80:
1815:Physics experiments
1672:2004PhRvE..69e1108L
1530:1978AmJPh..46..978W
1450:1949JPoSc...4..153J
1408:1978AmJPh..46..976M
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1266:Non-ideal balloons
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229:Boltzmann constant
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25:
1660:Physical Review E
1492:978-1-4398-0247-2
1328:COVID-19 pandemic
1280:steric hindrances
1039:
1006:
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685:
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519:{\displaystyle r}
252:is the internal (
159:
111:
1822:
1784:
1783:
1782:
1770:
1764:
1763:
1750:
1744:
1742:
1733:
1707:
1701:
1700:
1691:
1655:
1649:
1648:
1646:
1640:, archived from
1623:
1614:
1608:
1607:
1588:
1582:
1581:
1565:
1555:
1549:
1548:
1546:
1540:, archived from
1515:
1506:
1497:
1496:
1472:
1466:
1465:
1460:, archived from
1425:
1419:
1418:
1384:
1355:Laplace pressure
1240: <
1215: >
1202: <
1165:
1163:
1162:
1157:
1155:
1154:
1139:
1138:
1129:
1128:
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1007:
1005:
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982:
977:
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973:
960:
959:
950:
939:
938:
937:
918:
917:
916:
881:
879:
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868:
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862:
850:
845:
844:
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803:
789:
788:
766:
764:
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728:
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712:
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491:
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97:
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1830:
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1805:
1804:
1792:
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1709:
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1652:
1644:
1621:
1616:
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1611:
1590:
1589:
1585:
1578:
1557:
1556:
1552:
1544:
1538:10.1119/1.11487
1524:(10): 978–982,
1513:
1508:
1507:
1500:
1493:
1474:
1473:
1469:
1427:
1426:
1422:
1416:10.1119/1.11486
1402:(10): 976–978,
1386:
1385:
1372:
1368:
1341:
1301:
1276:crystallization
1268:
1257:
1248:
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1223:
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1099:
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961:
951:
922:
904:
899:
898:
854:
836:
826:
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780:
775:
774:
770:) then becomes
749:
748:
747:
738:
694:
688:
687:
675:
659:
658:
616:
610:
609:
594:
584:
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266:polymer network
247:
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134:
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117:
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101:
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78:
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56:
17:
12:
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1790:External links
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1632:(8): 731–746,
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1599:(3): 419–440,
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1467:
1444:(2): 153–182,
1432:(April 1949),
1428:James, H. M.;
1420:
1388:Merritt, D. R.
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1272:natural rubber
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1666:(5): 051108,
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1647:on 2012-04-02
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