VPMechanics10

Repetitive Diving and the VPM

Copyright 1999-2001 Eric Maiken

This notebook takes a look at some of the issues affecting repetitive diving from the perspective of the VPM

How are the affected by the equilibrium radial sizes ?

Following a dive, the bubble radii that delineate growing and shrinking bubbles in the various compartments presumably increase, and can be described by a set , with j corresponding to individual compartments. A set of  ,corresponding to the , are
= +
Substitution of the constants γ = 17.9 dyne/cm and = 257 dyne/cm, with appropriate conversion of  units,  gives:
=+  0.07 ΔP
This equation is closely represented by the 635-min compartment's , shown as the lowest purple-hatched surface in the figure below. For large , the limiting form of the rapidly approaches the value for all compartments.

The following figure shows 16 surfaces for , corresponding to the ZHL16 halftime set. The surfaces were calculated for non-decompression diving on air, at a descent rate of 3 atm/min, and an ascent rate of 1 atm/min, as described in VPMechanics9.
The range of r0 extends from 0.7 to 20 μm. For ΔP >~ 8 atm, and r0 >~ 10 μm, all but the fastest surface have converged to the limiting form: ~ 0.07 ΔP.

To emphasize the insensitivity of the present formulation to large radius nuclei, a different view of the surfaces is shown below.

One way to set up a VPM calculation for a repetitive dive

This section looks at how the VPM treats a dive starting with a set of initial compartment tensions that are greater than the surface saturation value. In other words, a dive initiated with residual nitrogen in the compartments that affects both the allowed ascent gradients and also increases compartment tensions over what they would be for a non-repetitive dive. Both of these factors serve to restrict the repetitive dive compared to a non-repetitive dive.

Following the methods and notation of VPM7,  the following expression describes compartment tensions during descent on a repetitive dive:

= + (t - )  + ( + ) exp(- t), 0 <= t <=

The zero of time is taken at the end of the surface interval separating dives. are the set of compartment tensions at the end of the surface interval, with >= + δ.

As in VPM7, the set of repetitive are determined for a constant descent.

= + ,
with = ΔP(1 - ) +( )(1 - exp(- ) ).

In the limit of a fast descent,→0, and   -> ΔP + () ~~ ΔP, and
= + .
So, for repetitive dives, the are restricted by the increased radii . Within this framework, the problem then becomes how to determine the .
For <~ 5 μm, the gradients are reduced as the increase, but,  there isn't much change in the for radii greater than a few microns. In this case, have the assumptions underlying the VPM broken down? In particular, Laplacian skin tension may be an ineffective constraint for limiting supersaturation and diffusive growth of large bubbles.