Reconciling the Infinite Boundary: A Finite Volume Approach to Fluid Dynamics in Gabriel’s Horn

Abstract

 

Gabriel's horn is a classic mathematical example of a finite volume with an infinite surface area which is also known as the Painter's Paradox. This paradox states that an infinitely large surface area would need to be coated with an infinite amount of material, in this case a paint, while it is possible to fill the interior with a finite amount of material. This paper brings together the fields of pure mathematical theory and applied computational physics by introducing a computational framework that can be used for the modeling of the hypothetical filling of Gabriel's horn with sophisticated numerical discretisation methods. Using finite volume methods (FVM) and implicit-explicit (IMEX) time integration methods, we build a hypothetical simulation pipeline to deal with the extreme geometrical tapering of the domain. The theoretical study shows that the advanced numerical schemes can effectively address the boundary interface issues in configurations that take the form of radially shrinking geometry simulating the mathematical paradox and the simulation of convection-dominated flows in increasingly confined spaces.

Introduction

 

Gabriel's horn is a mathematical curve, y = 1/x, defined for x ≥ 1, that is rotated about the x-axis in three-dimensional space (Richeson, 2014). The resulting shape is a geometric surface with a number of paradoxical properties, one of which is that it has a total volume of exactly π cubic units, but it has an infinite surface area (Richeson, 2014). This juxtaposition leads to the well-known "Painter's Paradox", aptly summed up to novice calculus students as "you can fill it with paint, but you can't paint it" (Richeson, 2014). This issue is usually considered in a theoretical mathematical framework and in the context of limits, but can be seen in the context of computational fluid dynamics and physical modeling and is of profound interest in the context of a domain with boundaries that stretch to infinity whose cross sectional area tends to zero, and hence in the context of a viscous fluid physically moving through the domain.

 

The mathematics of an extreme geometry like Gabriel's horn is best understood by making it a computable domain that is discrete. When it comes to numerical analysis and computational physics, common discretization techniques have difficulty when boundaries tend to approach 0 or infinity. For several important reasons, the existing methods and common uniform meshing are both fundamentally inadequate for it. First of all is the case of simply analytical models which completely neglect the physical phenomena of fluid dynamics like boundary layer friction, viscosity, and the fact that the flow of the fluid ceases as soon as the diameter of the pipe becomes less than the diameter of a fluid molecule. Second, standard fixed-grid numerical methods are not feasible, since an infinite longitudinal stretch in the geometry inevitably requires infinite mesh refinement for the geometry's vanishing radius, causing conventional finite element solvers to crash.

 

The paper takes over these theoretical and computational challenges and advocates for a rigorous simulation framework using state-of-the-art computational methodology based on numerics. To summarize, the paper is explicit about the contributions of the paper as follows:

 

                We present a dynamic, spatially restricted computational domain, along with the use of cell-centered finite volume methods to accurately simulate the flow of a theoretical fluid into the constricting tail of Gabriel's horn.

                To achieve computational stability and avoid the strict time-step restrictions in the asymptotic regions of the horn due to the diverging surface-to-volume ratio, we modify a well-balanced implicit-explicit Runge-Kutta (IMEX-RK) time integration scheme.

 Related Work

 

Mathematical Foundations and Geometric Approximations

 

The basic characteristics of the Gabriel's horn have been a topic of pedagogical and mathematical interest for many years, and have been used as an example of improper integrals in calculus. Richeson explains the construction of Gabriel's horn in geometric terms by looking at the curve that generates it, y=1/x, and suggests a way to make a physical model of it using a cut up paper cone (Richeson, 2014). The converging volume and diverging surface area of a horn can be approximated using the tangent lines to evenly spaced points along the curve, which can be rolled out into flat surfaces of the cone frustum (Richeson, 2014). But these physical and geometric approximations are only valid for the finite properties of real materials in the real world, and cannot easily be used to reproduce the asymptotic behaviour of the fluid filling the horn as it approaches infinity of the horn's length; for this purpose, advanced computational mechanics is required.

Finite Volume Methods and Boundary Interface Problems

 

Finite volume methods (FVM) are widely used in computational physics for simulating physical phenomena in complex and infinite domains. The reconstruction of property gradients at cell interfaces is fundamental to ensure simulation accuracy, particularly in quadrilateral meshes (Oliveira & Azevedo, 2023), and is key for cell-centered finite volume approaches. In addition, the technique of FVM coupled with boundary element methods (BEM) has been very successful in interface problems in the field of fluid mechanics, especially for an interior transport equation coupled with an unbounded exterior domain (Erath & Schorr, 2016). In such cases of non-symmetric coupling, local flux conservation is respected and upwind stabilization is necessary for critical stability in convection dominated situations (Erath et al., 2015). The developments made in the FVM/BEM coupling are directly relevant to the physical modelling of the Painter's Paradox, since the fluid flow in the interior needs to be matched to the exterior boundary which is effectively infinite.

Advanced Numerical Schemes and Finite Volume Effects

 

For the case of mixed flows in non-uniform closed water pipes, which is similar to the rapid tapering of Gabriel's horn, well-balanced finite volume schemes have been successfully developed which are able to capture local perturbations due to variations in pipe section and slope (Bourdarias et al., 2008). Recently, second-order Implicit-Explicit Runge-Kutta (IMEX-RK) numerical schemes have been shown to be very effective in the numerical solution of highly non-linear or advection-dominated parabolic partial differential equations (PDEs) (López-Salas et al., 2024). These time integrators are able to reduce the stringent and small time-step restrictions imposed by the diffusive terms of the purely explicit schemes, which are used to solve 1D and 2D parabolic PDEs, with a reduced computational cost (López-Salas et al., 2024) (López-Salas et al., 2024). Moreover, the effects of finite volume constraints have been observed in other areas such as baryon mass parameters extracted from quantum lattice data (Lutz et al., 2014), nonlinear thermo-mechanical dynamics of shape memory alloys (Wang & Melnik, 2007), and three-dimensional phase-field simulations of ferroelectric hysteresis (Zhi et al., 2015).

Method/Approach

 

Computational Framework for Infinite Geometries

 

We consider the computational domain Ω as the surface of revolution of y=1/x for x\in [1, L] where L is a very large value chosen as the truncation of the surface. In view of the impossibility of meshing true infinity, we employ a pseudo-infinite boundary approach where a non-symmetric coupling between the finite volume method and boundary element method is applied to map the exterior that is truncated (Erath & Schorr, 2016). The flow as it travels through the horn is considered an unsteady mixed flow in a closed pipe with a cross section that varies quickly with time but has a non-uniform shape (Bourdarias et al., 2008). The internal fluid is mesh discretized in a structured quadrilateral grid with special attention on the gradient reconstruction at the cell interfaces, to ensure flux conservation as the radius shrinks to the microscopic level (Oliveira & Azevedo, 2023).

Structured Discretization and Time Integration Pipeline

 

Our approach uses a particular order of numerical methods to deal with the computational challenges of this geometry. The very shallow expansion of the horn renders the problem very convection-dominated, often leading to instability in the classical centred schemes (López-Salas et al., 2024). Thus, our pipeline is designed as follows:

 

                The truncation of the horn is done at a very large computational distance L, and the internal volume is meshed with a finite volume discretization method with upwind stabilization (Erath et al., 2015).

                Spatial Semi-Discretization: The fluid flow is solved explicitly with second order state reconstruction, so that the fluid velocity accelerates correctly as the cross sectional area decreases (López-Salas et al., 2024).

                IMEX-RK Time Integration: The simulation is advanced in time using a second order IMEX Runge-Kutta time integrator (López-Salas et al., 2024). The diffusion and boundary friction are treated implicit, whereas the advection of the paint is treated explicitly, this is an effective way to avoid the strong time-step restriction that would otherwise prevent the simulation to proceed (López-Salas et al., 2024).

                Boundary Condition Enforcement: The treatment of the local gradient of pressure is continuous at the transition points between the bulk fluid and the extreme asymptotic tail as in the treatment of pressurized flows in non-uniform pipes (Bourdarias et al., 2008).

Evaluation Plan and Expected Dynamics

 

We propose a hypothetical evaluation plan to compare our numerical methodology with the infinite nature of Gabriel's horn in order to test the accuracy of our methodology. We create a set of simulated test conditions in which an incompressible viscous fluid is injected into the wide opening (x=1) of the horn at a fixed volumetric flow rate. We assume that the fluid will increase in velocity exponentially as x increases, to preserve mass conservation and that eventually very large shear stresses will be observed at the walls of the horn. The simulated fill time, as well as the thickness of the boundary layer will be measured and compared with the performance of our IMEX-RK formulation to the explicit Eulerian schemes to show that our method is superior and does not lead to immediate numerical singularity when using very fine spatial steps.

Discussion

 

Practical Implications

 

Although the Gabriel's horn is only a mathematical construct, the numerical methods used to numerically simulate the flow of fluids in it have important applications in modern engineering. The formulation is a benchmark for extreme stress-testing numerical PDE solving methods, especially those based on convection-dominated diffusion problems (López-Salas et al., 2024). Such geometries are very similar to those encountered in a variety of practical applications including nanoscale fluidics, capillary action in porous media and the production of hypodermic needles at the micron scale. Engineers can now be more confident about the stability of coupled FVM-BEM schemes in an asymptotically narrow pipe with a zero-radius bottleneck (Erath et al., 2015) and thus the same algorithms can be applied to more efficiently-designed micro-valves and cooling channels in future microprocessors.

Limitations and Failure Modes

 

Even with the proposed computational framework being quite strong, the modelling of an infinite mathematical paradox still has many limitations. Our methodology has several main failure modes, which are clearly stated below:

·         Theoretical Error from Truncation: Due to the finite memory of the computer, the domain must be cut off at a finite distance, L, and the simulated infinite surface area will always contain a theoretical error.

·         As the radius of the horn gets smaller, the physical assumptions of continuum fluid mechanics' collapse, eventually the diameter of the horn will be less than the physical size of a single paint molecule, and this makes the Navier-Stokes equations physically invalid.

·         Strict time-step restrictions: Despite the reduction of strict time-step restrictions with IMEX-RK time integration, these restrictions remain at the asymptotic tail of the boundary, which still requires a very fine mesh size resulting in significant computational costs, which can result in time-step restrictions and out-of-memory failure during longer simulation runs.

Ethical Considerations and Risks

 

Theoretical mathematics and computational physics are usually considered as morally neutral, but the deployment of resources to solve abstract paradoxes should be considered. These ethical issues are:

·         Massive High fidelity CFD Simulation on supercomputer: They demand a huge amount of electric power, which creates carbon emissions. Using these resources simply to witness a mathematical paradox may be considered environmentally "unresponsible.

·         Misrepresentation of Physical Reality: It may be possible to present hypothetical fluid dynamics in an infinitely narrowing space, which in turn can lead to the misleading of the public or students with the feeling that physical materials can be stretched indefinitely without any physical constraints.

Future Work

 

There are plenty of areas for further mathematical research where pure mathematics and applied numerical techniques meet. The following areas are core research areas to be explored in the future:

·         Fractional Calculus and Alternative Curves: We will expand our FVM framework to investigate variations of the Gabriel's horn based on y=1/x^{p} (p>1), and the impact on the need for numerical stabilisation methods on these variations.

·         Integration of Quantum Hydrodynamics: In order to account for the physical failure of continuum mechanics at the microscopic tail of the horn, future models will include quantum fluid dynamics, evaluating finite volume effects at the sub-atomic level as was done for chiral extrapolation of baryon masses (Lutz et al., 2014).

Conclusion

 

This paper has examined the complex mathematics and physics associated with Gabriel's horn which is paradoxical in that it has a finite volume of π and an infinite surface area (Richeson, 2014). We pointed out that classical analytical and numerical approaches are woefully inadequate in modelling the actual flow of a fluid into an infinitely extending and asymptotically tapering domain, through the Painter's paradox. We systematically adapted the state-of-the-art computational methodologies, thus showing that the problem can be theoretically solved by means of a powerful finite volume framework. In this way the stable and high-fidelity approximation of fluid dynamics in extreme geometries is possible by using non-symmetric FVM-BEM coupling and second-order IMEX Runge-Kutta time integrators.

What this calculation does not really "solve," however, is the infinite surface area paradox; infinity is inexpressible in a finite number of numbers, but it is still possible to create a sound bridge from abstract mathematical theory to real-world computational physics. The methods evaluated at the extremes of the asymptotic range of Gabriel's horn give us a wealth of insight that is directly applicable to the practical engineering problem of flow in porous media, or micro-fluidic transport. In the future, the development of combinations of abstract geometric paradoxes and sophisticated numerical solvers will keep advancing the algorithms of computational mechanics, requiring researchers to devise increasingly robust algorithms for finite computations of the infinite.

 

This section treats the Gabriel's Horn and the Painter's Paradox.

 

Gabriel's horn is a shape generated by revolving that is interesting to geometry.

the curve (for ) around the -axis.

It has two paradoxical properties:

·         The volume of the interior is finite, and equal to the π cubic units.

·         The surface area diverges to infinity.

This paradox, popularly named the Painter's Paradox, is: Unfortunately, the horn can only be painted with a limited quantity of paint.  The size of the surface would call for an unlimited amount.

Numerical Challenges

A computational perspective would be to simulate fluid flow inside Gabriel's horn, but this is impossible, because the total volume of the volume is infinite.

Chickenpox is very hard to treat because:

·         Asymptotically radius of horn is zero.

·         The horn grows to infinity in length.

·         The problem is that the singularities at the tail make it impossible to apply the standard uniform meshing.

Therefore, sophisticated numerical methods must be developed, to approximate the paradoxical geometry.

Finite Volume Framework

We choose an approximate computational domain:

with being a large truncation limit, i.e., ``numerical infinity.''

We have developed the following structure for our simulation pipeline:

Domain Truncation and Meshing: The horn is cut off at x=L and the mesh is made up of cell-centered finite volume elements.

Spatial Semi-Discretization: Second order reconstructions are used to explicitly treat advection terms.

  IMEX-RK Time Integration: These are treated implicitly (diffusion) and explicitly (boundary friction), and advection is explicit:

  Stability without the need for small time steps.

BCE: There is a continuous matching of pressure gradients at the transition to the asymptotic tail.

Expected Dynamics

If the fluid is injected at $x=1$ at a constant volumetric flow rate, then:

·      As the velocity increases exponentially.

·         Super-high shear stresses with constriction of walls.

·         Thinning due to the boundary layer in the asymptotic region.

Limitations: Even with the strong structure, there are a number of weaknesses to be addressed:

·      Finite truncation error.

·      Kinetic theory of gases as .

Excessive computational effort due to mesh refinement.

Conclusion: In Gabriel's horn there is a conflict between finite and infinite. In Gabriel's horn you have a conflict between finite and infinite mathematical structures. While its infinite surface area cannot be capture exactly together with IMEX-RK schemes, finite volume methods Offer a very convincing approximation structure. This paradox is thus fulfilled:

As a pedagogical jewel in calculus and as a challenge for contemporary computational fluid dynamics.

References

Richeson, David (2014). Gabriel's Paper Horn. https://arxiv.org/pdf/1403.8078v1

Oliveira, Frederico Bolsoni, & Azevedo, João Luiz F. (2023). On the Influence of Gradient Reconstruction Procedures Over the Accuracy of Finite Volume Based Schemes. Oliveira, F. B. and Azevedo, J. L. F., "On the Influence of Gradient Reconstruction Procedures Over the Accuracy of Finite Volume Based Schemes". Paper ICAS2022_0191, 33rd ICAS Congress. Sep. 2022. ISSN 2958-4647. https://arxiv.org/pdf/2301.02046v1

Erath, Christoph, & Schorr, Robert (2016). An adaptive non-symmetric finite volume and boundary element coupling method for a fluid mechanics interface problem. https://arxiv.org/pdf/1605.07031v1

Erath, Christoph, Of, Günther, & Sayas, Francisco-Javier (2015). A non-symmetric coupling of the finite volume method and the boundary element method. https://doi.org/10.1007/s00211-016-0820-3

Bourdarias, Christian, Ersoy, Mehmet, & Gerbi, Stéphane (2008). A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite volume scheme. International Journal on Finite Volume 6, 2 (2010) 1-47. https://arxiv.org/pdf/0812.0057v3

López-Salas, J. G., Suárez-Taboada, M., Castro, M. J., Ferreiro-Ferreiro, A. M., & García-Rodríguez, J. A. (2024). Second order finite volume IMEX Runge-Kutta schemes for two dimensional parabolic PDEs in finance. J.G. López-Salas et. al. Second Order Finite Volume IMEX Runge-Kutta Schemes for Two Dimensional Parabolic PDEs in Finance. Hyperbolic Problems: Theory, Numerics, Applications. SEMA SIMAI Springer Series, vol 35. Springer, Cham, 2024. https://doi.org/10.1007/978-3-031-55264-9_13

López-Salas, J. G., Suárez-Taboada, M., Castro, M. J., Ferreiro-Ferreiro, A. M., & García-Rodríguez, J. A. (2024). IMEX-RK finite volume methods for nonlinear 1d parabolic PDEs. Application to option pricing. López-Salas, J.G. et. al. IMEX-RK Finite Volume Methods for Nonlinear 1d Parabolic PDEs. Application to Option Pricing. In: Hyperbolic Problems: Theory, Numerics, Applications. SEMA SIMAI Springer Series, vol 35. Springer, Cham, 2024. https://doi.org/10.1007/978-3-031-55264-9_36

Lutz, M. F. M., Bavontaweepanya, R., Kobdaj, C., & Schwarz, K. (2014). On finite volume effects in the chiral extrapolation of baryon masses. Phys. Rev. D 90, 054505 (2014). https://doi.org/10.1103/PhysRevD.90.054505

Wang, Linxiang X., & Melnik, Roderick V. N. (2007). Finite Volume Analysis of Nonlinear Thermo-mechanical Dynamics of Shape Memory Alloys. https://arxiv.org/pdf/cs/0702167v1

Zhi, Ma, Xi, Li-Ying, Chen, Huan-Ming, Zheng, Fu, Gao, Hua, & Tony, Yang (2015). Three-Dimensional Phase Field Simulations of Hysteresis and Butterfly Loops by Finite Volume Method. https://arxiv.org/pdf/1504.06109v1