# Application of a Coupled Eulerian-Lagrangian Approach to the Shape and Force of Scientific Balloons
## 1 Introduction  
Near-space holds tremendous value for both scientific research and commercial applications. It has broad prospects and significant potential development in fields such as space science, communications, observation, and remote sensing. However, the altitude is too low for satellites and too high for aircraft. The flight speed of sounding rockets is relatively fast, limiting the effective detection space that they can cover. Scientific balloonsare particularly well suited for operations in this area and are among the very few types of vehicles capable of stable operations in the stratosphere [1]. With continuous development, the flight capabilities of scientific balloons have evolved from traditional short-duration flights to missions lasting several weeks or months, progressing towards long-endurance flight, ultra-high altitudes, heavy payloads, trajectory control, area surveillance, and planetary exploration. The balloon platform can work seamlessly with satellites, manned spacecraft, and other platforms, complementing each other effectively. It enables the rapid execution of innovative scientific experiments, leading to discoveries and results of significant scientific value. Additionally, it provides an efficient and low-cost approach to validating scientific concepts, instrument principles, and technical methods for large-scale space missions [2; 3]. For instance, the unique polar environment of Antarctica offers low electromagnetic interference and stable airflow conditions, enabling scientific balloons to achieve more precise observations with minimal interference. In such an environment, balloons can carry highly sensitive detection equipment and maintain prolonged, stable flight, significantly improving the efficiency of cosmic ray detection. These observations allow scientists to delve deeper into the origin, composition, and properties of cosmic rays, providing critical data to uncover the origins and evolution of the universe. Moreover, such studies offer vital clues in searching for dark matter, as cosmic rays may contain signals generated by the annihilation or decay of dark matter particles. Scientific research conducted in Antarctica not only advances the fields of astronomy and particle physics but also accumulates valuable experience in the efficient utilization of scientific balloons, driving further innovation in flexible aerospace technologies [4; 5]. Scientific balloons are categorized into zero-pressure balloons and super-pressure balloons. Zero-pressure balloons have vents at the bottom that connect to the atmosphere, resulting in no pressure difference between the inside and outside of the balloon. Although they are limited by short flight durations, low pressure resistance, and a lack of propulsion, zero-pressure balloons possess unique advantages [6]. Firstly, they excel in terms of their high altitude and heavy payload capabilities, maintaining a leading edge in both aspects. Zero-pressure balloons are also cost-effective, offering a very high cost-performance ratio in completing various scientific experiments or missions--an advantage unmatched by other types of aerostats, including airships. Moreover, zero-pressure balloons can carry large payloads, and their design allows for flexibility. Payloads are connected to the balloon via cables and parachutes, which imposes minimal restrictions on the payload size, shape, installation position, or installation method. Since the balloon drifts with the wind, there are no aerodynamic shape requirements, making it possible to carry large, complex, open payloads like scientific telescopes--an advantage unmatched by other aerospace platforms. To date, the largest zero-pressure balloons have reached volumes of up to 1.7 million cubic meters and payload capacities of 3.6 tons. Figure 1 shows the 1.14 million cubic meter \"work horse\" zero-pressure balloon frequently used by NASA/CSBF, which has been launched in multiple long-duration flight experiments over Antarctica [7].