Bubble Stability and Flow Conditions in BS-Fusion
Bubble integrity is a critical concern in BS-Fusion, especially during acoustic transport through molten salt. The expected flow regime is laminar, not turbulent. This is largely due to the high viscosity of the working fluid, a modified Yb:FLiBe mixture.
Although the term “FLiBe” typically refers to a 2:1 molar mixture of LiF:BeF₂ (which has a water-like viscosity), variations in composition allow for significant tuning of physical properties. For example, a 1:1 mixture is eutectic and melts at a lower temperature, but has a viscosity nearly 10× greater. The increased viscosity arises from the formation of BeF₂ polymer chains, which grow with higher BeF₂ concentration and are terminated by LiF.
In the current model, a high-viscosity (μ = 3 Pa·s) coolant was assumed. While this would be challenging for high-throughput heat exchange, it may be suitable for localized, slow, precision flow scenarios like acoustic bubble positioning.
Flow Regime: Laminar vs. Turbulent
The Reynolds number (Re) determines whether flow is laminar or turbulent:
Where:
- ρ = 2000 kg/m³
- v = 0.08 m/s (fluid velocity)
- D = 1 m (characteristic length, e.g., pipe diameter)
- μ = 3 Pa·s (dynamic viscosity)
Result:
Re= \frac{2000 \cdot 0.08 \cdot 1}{3} ≈ 53
This is well below the turbulence threshold (Re > 4000), indicating laminar flow.
Bubble Integrity: Weber Number (We)
The Weber number compares inertial forces to surface tension:
Using the following parameters for a 20 mg DT bubble at 750 K and 5 atm:
- v = 0.09 m/s (relative velocity)
- L = 0.046 m (bubble diameter)
- σ = 0.19 N/m (surface tension)
We=\frac{2000 \cdot (0.09)^2 \cdot 0.046}{0.19} ≈ 3.9
This value suggests moderate deformation (1 < We < 10), but not fragmentation.
Bubble integrity can be improved by reducing either the velocity or bubble size (via lower fuel mass or higher ambient pressure). A 20 mg DT bubble is about 100× more massive than a standard NIF capsule.
Lower-Weber Scenario
For a smaller, slower-moving bubble:
- v = 0.0254 m/s (1 inch/s)
- L = 0.0254 m (1 inch diameter)
We= \frac{2000 \cdot (0.0254)^2 \cdot 0.0254}{0.19} ≈0.17
This value falls well below the breakup threshold (We < 1), indicating a stable bubble even under acceleration.
Additional Dimensionless Numbers: Ga and Eo
Two other important figures-of-merit for bubble motion under gravity are:
- Galilei number (Ga): gravitational vs. viscous forces
- Eötvös number (Eo): gravitational vs. surface tension
Definitions:
For a 1-inch bubble (R = 0.0127 m), under 1 g (9.81 m/s²):
Ga≈ \frac{2000 \cdot \sqrt{9.81} \cdot (0.0127)^{3/2}}{3} ≈ 3.0
Eo≈ \frac{2000 \cdot 9.81 \cdot (0.0127)^2}{0.19} ≈ 16.7
According to phase plots (e.g., Nature Communications, 2015), these values (Ga ≈ 3, Eo ≈ 17) fall within Region I, associated with stable ellipsoidal bubbles. These bubbles maintain shape while accelerating to terminal velocity under laminar conditions.
Conclusion
Under plausible conditions in BS-Fusion—modest fluid velocities, high viscosity, and appropriately sized DT bubbles—laminar flow and bubble integrity are both maintainable. A 1-inch DT bubble moving at 1 inch/second remains stable (We = 0.17) and tractable. Bubble size and ambient pressure can be tuned to further enhance robustness during acoustic positioning.
Future work will continue refining models of bubble deformation and resonance, especially under combined acoustic and convective forces.