Improvements in the Variational Principle for Fluid Dynamics

The realized motion of a system minimizes the action. This is called the variational principle and consideredas a guiding principle in various fields of the physics. Using this principle, we can formulate thedynamics of a system even if it has complicated constraints. Various variational principles for the perfect fluid have been proposed for a long time, while Onsager’s variational principle has been useful in formulating the dissipative dynamics in the soft matter physics. However, they have several open problems. My research proposes a general framework to solve them, and provides the associated Hamiltonian formulation as follows.

1. The variational principle for the perfect fluid

There are two ways to describe the dynamics of fluid. The first way is the Lagrangian description, where we track path line. The second way is the Eulerian description, where we observe the time evolution at spatially fixed points. It is known that the equation of motion for the perfect fluid can be derived in terms of the variational principle in the Lagrangian description, as in the mechanics of mass particles. On the other hand, the variational principle in the Eulerian description requires some auxiliary fields, called Clebsch potentials, to derive rotational velocity field on the isentropic condition. However the physical meaning of the potentials has been obscure. We showed that Clebsch potentials are required to fix the endpoints of each path line. Here, the mass conservation law and adiabatic condition are holonomic constraints. Thus we can incorporate them into the action by means of the method of undetermined multiplier.

2. The variational for a dissipative system

In a dissipative fluid, entropy is produced along the path line. It gives a non-holonomic constraint, to which the above method cannot be applicable. However, we can minimize the action under the nonholonomic constraint because it is expressed in terms of differential forms. Our formulation yields the whole equation of momentum balance for a viscous fluid, although Onsager's variational principle yields only its linear part. We showed that our formulation can be also applied to viscoelastic fluid and polymer solution and also pointed out the case that dissipation is caused by diffusion.

3. Hamiltonian formulations

In the control theory, the optimized input minimizes the cost functional, and derives the Hamilton’s equations as a pair of conjugate equations. We applied this theory to fluid dynamics, where the input is the velocity field. The state variables in the Lagrangian and Eulerian descriptions are respectively given by the position of fluid particles and Clebsch potentials. The resultant Hamiltonian equation for perfect fluid is canonical. In a viscous fluid, dissipative force is added to the equation. The associated symmetries are related to the conservation laws.

Doctoral dissertation

Hiroki Fukagawa "Improvements in the Variational Principle for Fluid Dynamics" Graduate School of Science and Technology, Keio University, Tokyo, Japan 2012. https://iroha.scitech.lib.keio.ac.jp:8080/sigma/bitstream/handle/10721/2623/document.pdf?sequence=4

Research Interest

Differential geometry

Category theory

Variational principle

Relativistic theory

Thermodynamics

Soft-matter

Conjugate gradient method

GPGPU

Publications

1)Hiroki Fukagawa and Youhei Fujitani, A Variational Principle for Dissipative Fluid Dynamics, Prog. Theor. Phys. 127, 921-935 (2012).

2)Hiroki Fukagawa, Youhei Fujitani, The variational principle with using dissipative functions, RIMS Kôkyûroku, 1776 (2012) in Japanese.

3)Hiroki Fukagawa, Youhei Fujitani, "Clebsch potential in the variational principle for a perfect fluid", RIMS Kôkyûroku, 1749 (2011) in Japanese.

4)Hiroki Fukagawa, Youhei Fujitani, Clebsch Potentials in the Variational Principle for a Perfect Fluid, Prog. Theor. Phys. 124, 517-531 (2010).

Presentations & Posters:

Conference-

1)Hiroki Fukagawa, ”Variational principles for dissipative systems"，The 4th International Symposia on Leading-Edge Research Activities for Global World GCOE International Symposium, Yokohama, Japan, February, 2012.

2)Hiroki Fukagawa, "Deriving the equation of motion for dissipative fluids via the variational principle", Society of Geomagnetism and Earth, Planetary and Space Sciences, Fukui, Japan, December, 2011.

3)Hiroki Fukagawa and Youhei Fujitani, "Deriving the equation of motion for dissipative systems via the variational prinicple”, 10th Kanto soft-matter society, Tokyo, Japan, November, 2011.

4)Hiroki Fukagawa, Youhei Fujitani, "Variational principles for perfect and viscous fluids"，8th Liquid Matter Conference, Wein, Austria,　September, 2011.

5)Hiroki Fukagawa, Youhei Fujitani, "The derivation of Navier-Stokes equations from the variational principles with nonholonomic constraints", Fall Meetings of physical Society of Japan, Toyama, Japan, September, 2011.

6)Hiroki Fukagawa, Youhei Fujitani, "The derivation of equation of motion for viscous fluids from the variational principles with using inner energy nad dissipative function", YITP Workshop 2011: Physics of Non-equilibrium Systems -A Bridge between Micro and Macro-, Kyoto, Japan, August, 2011.

7)Hiroki Fukagawa, Youhei Fujitani, "The variational principle with using dissipative functions",Mathematical analysis of the Euler equation : 100 years of the Kaman vortex street and unsteady vortex motion. RIMS, Kyoto Japan, July, 2011.

8)Hiroki Fukagawa, Youhei Fujitani, "Reconsideration of the Eulerian Variational Principle for a Perfect Fluid", Fall Meetings of physical Society of Japan, Osaka, Japan, September, 2010.

9)Hiroki Fukagawa, Youhei Fujitani, "Clebsch potential in the variational principle for a perfect fluid",Mathematical analysis of the Euler equation : 100 years of the Kaman vortex street and unsteady vortex motion. RIMS , Kyoto, Japan, July, 2010.

10)Hiroki Fukagawa, Youhei Fujitani, "Variational Derivation of the Equations of Motion for a Relativistic Perfect Fluid with Vorticity", Annual meetings of physical Society of Japan", Tokyo, Japan March, 2009.

11)Hiroki Fukagawa, Youhei Fujitani, ”Lagrange-coordinate and Eulerian-coordinate Lagrangians in Perfect Fluid Dynamics”, Fall Meetings of physical Society of Japan, Iwate, Japan、September, 2008.　

Seminar-

1）Hiroki Fukagawa, "A variational principle for a complex fluid", International research networks for non-equilibrium dynamics of soft matter, Kyoto University, July, 2012.

2）Hiroki Fukagawa, "Deriving the equation of motion from the dissipative function and inner energy via the variational principle", Many Body System Group, Nagoya University,November, 2011.