# Application of a Coupled Eulerian-Lagrangian Approach to the Shape and Force of Scientific Balloons
## 4 Dynamic Fluid-Structure Coupling Shape and Force Analysis
### Dynamic Mushroom-Shaped Configuration Solving
As the velocity increases, the air resistance also increases, gradually hindering the balloon's upward movement. As shown in Figure 10, when the membrane in contact with the upper air reaches a certain speed, the upward velocity slows down, while the surrounding membrane continues to rise for a short time at its original speed. Afterward, due to air resistance, it decelerates, collapses downward, and  
Figure 9: Mushroom shape of a scientific balloon during ground launching. (**a**) Assembly of parts. (**b**) Bucket and sphere showing pressure gradient. expands outward. Since the balloon membrane is a flexible material, it cannot withstand shear forces and bending moments, and the external air pressure in the lower-middle part of the balloon is greater than the internal pressure. Under the combined effect of these factors, the lower-middle part of the balloon membrane collapses inward. At this moment, the overall shape of the balloon is slightly concave at the top, with outward expansion at the sides and inward contraction at the waist, forming a \"mushroom-like\" shape. As the balloon continues to rise, the mushroom head widens, and the bottom shortens, until the velocities of the top and bottom become nearly equal. At this point, the mushroom head rises again due to buoyancy and starts to contract inward, ceasing to expand further. The balloon as a whole maintains the concave shape at the top, with slight outward bulging at the head and middle and inward contraction at the bottom, moving upward. The simulated displacement, velocity, and force curves of the balloon are shown in Figures 11-16. Figure 11 presents the displacement at the top and bottom of the balloon, while Figure 12 shows the difference between these two displacements. The process where the bottom displacement surpasses the top displacement indicates the formation of the mushroom shape. Subsequently, the displacement difference decreases and rises uniformly, representing the balloon's upward movement after full expansion. Figure 11: Top and bottom displacement curves of the zero-pressure balloon. Figure 10: Balloon shape figures. Figure 14: Velocity difference curve between the top and bottom of the zero-pressure balloon. Figure 12: Displacement difference curve between the top and bottom of the zero-pressure balloon. Figure 13: Top and bottom velocity curves of the zero-pressure balloon. Figures 13 and 14 depict the velocity at the top and bottom and the difference between these velocities. Immediately after launching, the top velocity exhibits significantly larger amplitudes compared to the bottom velocity. Over time, both velocities oscillate within a range of 0 to 10. Figure 15 illustrates the bottom tensile force. Before launching, oscillations occur during the analysis step, and the force drops to zero immediately after launching. Figure 16 shows the stress at the top and bottom of the balloon. Before launching, the bottom stress is significant due to the tension required to stabilize the entire balloon, while the top stress remains close to zero. Upon launching, the bottom stress decreases to zero and then increases gradually, slightly lagging behind the top stress. With the oscillation of the internal helium, the top stress reaches its peak first, followed by the bottom stress reaching its maximum shortly after.