Horn-Torus Equilibrium Bubble

• Horn-torus-shaped equilibrium bubble is a nonspherical gas domain with a unique pinched torus geometry arising under steady rotational fluid flow.
• The bubble's equilibrium is achieved by balancing gas pressure and surface tension across a curved interface, with curvature and rotational dynamics playing key roles.
• Advanced computational techniques like Physics-Informed Neural Networks efficiently approximate the bubble profile, validating theoretical predictions of its topological and energetic properties.

A horn-torus-shaped equilibrium bubble is a stationary, nonspherically symmetric gaseous domain arising as a solution to the spatial uniform gas pressure (isobaric) approximate model for a bubble immersed in an incompressible fluid under surface tension, where the equilibrium shape assumes the topological form of a horn torus—characterized by coincident major and minor radii and a “pinched” inner region. These equilibria emerge specifically when the ambient fluid exhibits nontrivial steady rotational flow with mild spatial decay, in contrast to classical spherical equilibria requiring strong far-field decay in the liquid velocity. Analytical construction, mathematical formulation, the influence of curvature and topological properties, and recent computational approaches—such as Physics-Informed Neural Networks—collectively underpin current understanding of horn-torus-shaped equilibrium bubbles in continuum fluid mechanics.

1. Geometric Definition of the Horn-Torus Bubble

A horn torus is defined geometrically as the surface generated by revolving a circle of radius $C$ with its center passing through the axis of revolution, yielding a singular “pinched” topology at the torus’ core. In spherical coordinates $(r, \theta, \phi)$, the bubble surface admits the explicit parametrization

$$r = R(\theta) \qquad\text{with}\qquad R(\theta) = C \sin\theta,$$

where $C > 0$ is fixed by the combined constraints of bubble mass, pressure, and temperature. The surface $r = C\sin\theta$ encapsulates the key geometric property: unlike a standard torus, whose local Gaussian curvature varies smoothly, the horn torus possesses nontrivial curvature characteristics at its inner “neck.” Such features exert significant influence on the energetic and dynamical properties of a confined gas bubble. The volume of the gaseous region is given by $|*| = (\pi^2/4)C^3$.

2. Analytical Equilibrium Model and Boundary Conditions

The equilibrium is addressed in the context of the inviscid (zero viscosity) approximation to the momentum equation for an incompressible fluid with free boundaries. For axisymmetric equilibria and under rotational flow, the liquid velocity field is assumed to possess only a non-zero azimuthal (toroidal) component: $$\boldsymbol{v}_\ell = v_\phi(r,\theta)\,\boldsymbol{e}_\phi,\qquad v_r = v_\theta = 0,$$ with the external pressure tending to a finite value as $|x| \to \infty$ and not fulfilling strong spatial decay. The condition for bulk equilibrium at the gas–liquid interface is enforced by the Laplace–Young law: $p_g - p_\ell = \sigma \kappa,$ where $p_g$ is the interior gas pressure, $p_\ell$ is the local liquid pressure at the interface, $\sigma$ is the surface tension, and $\kappa$ is twice the mean curvature. On the horn-torus interface, the vector surface divergence yields

$$\kappa = \nabla_{S}\cdot\boldsymbol{n} = \frac{1}{C}\left(\frac{1}{\sin^2\theta} - 4\right).$$

Solving the radial stress balance equation under these ansätze, and with the liquid pressure expressed as $p_\ell(r,\theta) = p_\infty + g(r\sin\theta)$, produces a closed-form relating the system’s physical and geometric parameters. Mass conservation, via the ideal gas law and the known surface volume, imposes the constraint

$$p_\infty C^3 - 4C^2 - \frac{4RT_\infty M}{\pi^2} = 0,$$

with $C > 4/p_\infty$, linking the horn-torus scale to the system’s total mass $M$, far-field pressure $p_\infty$, temperature $T_\infty$, and gas constant $R$.

3. Role of Rotational Fluid Flow

The existence of nonspherical, horn-torus-shaped equilibria critically depends on relaxing the spatial decay of the liquid flow. Classical results, which assume strong decay (such as $v_\ell(x) = O(|x|^{-2})$), enforce the uniqueness of spherical equilibrium solutions. In contrast, permitting merely $v_\ell(x) \to 0$ at infinity admits rotationally symmetric, non-vanishing azimuthal liquid velocity. The solution’s explicit form, $v_\phi = \sqrt{\sigma/(\rho_\ell r \sin\theta)}$, generates a nonzero vorticity

$$\operatorname{curl} \,\boldsymbol{v}_\ell = \frac{\cot\theta}{r^2}\,\boldsymbol{e}_r \neq 0,$$

breaking spherical symmetry and sustaining the horn-torus shape. This highlights the profound connection between fluid kinematics—specifically vorticity and decay rates—and the rich class of possible gas–liquid interface morphologies.

4. Influence of Curvature and Confinement

The surface energy of a bubble within a curved environment is modulated by local and global geometric features. For a bubble confined between curved plates,

$E = E_0 - E_1 G(q),$

where $G(q)$ is the local Gaussian curvature, $E_0$ the reference (zero-curvature) energy, and $E_1$ a curvature-dependent correction. Positive curvature lowers the energetic cost of forming the bubble, and negative curvature raises it. In toroidal or horn-torus confinements, numerical and analytical results confirm that the local curvature acts as a geometric potential guiding bubbles from regions of negative $G$ (outer torus) to positive $G$ (inner torus), effectively shaping both their steady-state location and morphology (Mughal et al., 2016). This geometric potential is strictly tied to the topological and differential geometric properties of the confining boundaries and the bubble surface itself.

5. Computation via Physics-Informed Neural Networks

The governing equations for the horn-torus equilibrium surface—especially the Laplace–Young equation—are highly nonlinear and, for general confinement, analytically intractable. A computational approach employing Physics-Informed Neural Networks (PINNs) is implemented to approximate the profile $R(\theta)$. The PINN represents $R_{\mathrm{NN}}(\theta; w)$ with parameters $w$ and uses a softplus activation to guarantee positivity and a symmetric architecture to encode the relation $R(\theta) = R(\pi - \theta)$. The composite loss functional incorporates:

• A stress-balance term $\mathcal{L}_{SB}(w)$ measuring the residual of the Laplace–Young condition;
• A volume constraint term $\mathcal{L}_V(w)$ enforcing $$V = (2\pi/3)\int_{0}^{\pi} [R(\theta)]^3 \sin\theta\, d\theta$$ matches the prescribed bubble mass;
• Additional terms $\mathcal{L}_B(w),\, \mathcal{L}_S(w)$ maintaining endpoint smoothness and centering;
• Adam optimizer ($\text{lr}\sim 10^{-4}$), automatic differentiation, and extensive training ($\sim 10^4$ epochs).

Quantitative evaluation finds the PINN profile’s relative root mean square error (rRMSE) near $5.48\times10^{-2}$ relative to the closed-form $R(\theta) = C\sin\theta$, validating the network’s robustness for this class of equilibrium. Visualization in 3D and 2D coordinate projections confirms the encapsulation of the horn-torus geometry.

6. Topological, Energetic, and Dynamical Implications

The interplay of surface tension, enclosed mass, and ambient rotational liquid motion accesses a broader set of equilibrium morphologies with profoundly nontrivial topology compared to classical spherical cases. The horn torus, with singular curvature at its core, induces complex energetic landscapes for the trapped bubble, as local curvature directly governs both surface energy and migration under geometric-driven potential gradients. In toroidal Hele-Shaw cells, analogous movement of bubbles from negative to positive curvature regions is observed (Mughal et al., 2016). A plausible implication is the existence of metastable or dynamically driven bubble states whose topological nature is intricately linked to both the global symmetries and local curvature of the confining medium.

7. Outlook and Extensions

The construction and realization of horn-torus-shaped equilibrium bubbles under mild decay and nontrivial vorticity substantially generalize the landscape of possible steady-state solutions in continuum multiphase flows. Open problems include the detailed exploration of stability under perturbations, the influence of additional physical effects (e.g., viscosity, inhomogeneous boundary conditions), and the generalization to multibubble or nonaxisymmetric settings. Further research into the interplay between global topology, local curvature, and the kinetics of bubble interface evolution is suggested, especially for geometries where curvature-driven motion and topological constraints create complex energy landscapes (Mughal et al., 2016, Lai et al., 23 Jul 2025). These studies are relevant for bubble manipulation in microfluidic devices and fundamental investigations into geometric effects in fluid dynamics.