Recently, Professor Xinliang Tian's research group from the School of Ocean and Civil Engineering at Shanghai Jiao Tong University published a significant paper in the well-known journal of Journal of Fluid Mechanics titled "Multiple stable postures of a falling object in fluids". The paper explores the multi-solution phenomenon of objects freely falling in fluids and proposes a method to quantify and predict the distribution of these stable postures.

Figure 1. Can a single leaf fall in multiple ways?

The study of free-falling objects in fluids has long intrigued scientists, with examples like shooting stars, falling leaves, or sinking sediment in water. These phenomena are not only widespread in nature but also prevalent in engineering, such as predicting the landing point of re-entry capsules, the descent of deep-sea submersibles, or the fall of ship and aircraft wreckage into water. Over 400 years ago, Italian physicist Galileo revolutionized our understanding of free-falling objects through meticulously designed experiments, laying the foundation for classical mechanics and dispelling Aristotle's erroneous view that an object’s fall speed was proportional to its weight. Since then, countless scientists have been captivated by this question, refining our understanding of what factors influence the falling posture of objects. Today, we know that the shape, density, and inertia of an object, as well as the properties of the fluid it falls through, all impact its falling behavior. However, few have focused on a more fundamental question: "If all these factors remain constant, is an object's falling posture unique?" Or, in simpler terms, "Can a single leaf fall in multiple ways?" Unfortunately, nature does not grant us the chance to observe the same object's fall twice, making this phenomenon elusive.

Figure 2. The force balance relationship of a stably falling object.

To answer this question, Tian's group designed a simple yet effective model to study the falling behavior of two-dimensional objects under low Reynolds numbers. Under these conditions, an object’s falling posture tends to stabilize. By analyzing the lift and drag characteristics of stationary objects at various angles of attack and Reynolds numbers, and combining these with the balance of gravity, buoyancy, and fluid dynamics, the team was able to determine the stable postures and trajectories of falling objects.

Figure 3. The process of deriving stable solutions for an elliptical object:  (a) Two primary families of stable solutions;  (b) The single stable region (Single Stable) and bistable region (Bistable) obtained from the overlap of the two primary families.

The team initially examined elliptical, arbitrary triangular, and L-shaped objects through a combination of static derivation, numerical simulations, and physical experiments, all of which yielded consistent results. They found that a single object's stable falling posture is not unique, and the number of solutions depends on the shape and center of gravity of the object. For example, as shown in their elliptical object study, when the center of gravity shifts, two primary families of stable solutions emerge. In the overlapping region of these families, there exists a bistable zone.

Figure 4. Free-fall experiments of two-dimensional objects: Elliptical shape (left), triangular shape (middle), and L-shape (right).

Further, they demonstrated through carefully designed free-fall experiments with three objects—an ellipse, a triangle, and an L-shape—that each object can exhibit two distinct falling postures and trajectories.

Figure 5. Examples of typical falling postures of "SJTU"-shaped objects (arrows indicate the falling direction, and the dots represent the center of gravity of the objects).

The team discovered that this phenomenon applies not only to simple shapes but also to irregular ones. Using the initials of Shanghai Jiao Tong University (SJTU) as examples, they showcased multiple stable postures and trajectories for S-shaped, J-shaped, T-shaped, and U-shaped objects. Notably, this multiplicity of solutions arises not from geometric symmetry but from the irregularity of the object’s shape. Given that most objects in the natural world are irregular, this multi-solution phenomenon in falling objects could be quite common.

The research not only uncovers a multi-solution phenomenon in a simple system but also proposes a novel method for calculating the number and distribution of these stable solutions. This discovery deepens our understanding of the free movement of objects and organisms in air and water and holds significant theoretical value for designing and controlling the stability and trajectory of flying vehicles. It is worth noting that the study was conducted at relatively low Reynolds numbers, and the results at higher Reynolds numbers remain to be explored.

The first author of the paper is Shuyue Sun, a PhD student in the Department of Naval Architecture and Ocean Engineering. Professor Xinliang Tian is the corresponding author, and Shanghai Jiao Tong University is the sole affiliation for the paper. This research was supported by the National Key Research and Development Program and the National Natural Science Foundation of China.

[Link to the paper: https://doi.org/10.1017/jfm.2024.557]