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Lagrangian for Navier-Stokes equations of motion: SDPD approach

Tatyana Kornilova, Anna Shokhina, Timothy Nerukh, Dmitry Nerukh

Abstract

The conditions necessary and sufficient for the Smoothed Dissipative Particle Dynamics (SDPD) equations of motion to have a Lagrangian that can be used for deriving these equations of motion, the Helmholtz conditions, are obtained and analysed. They show that for a finite number of SDPD particles the conditions are not satisfied; hence, the SDPD equations of motion can not be obtained using the classical Euler-Lagrange equation approach. However, when the macroscopic limit is considered, that is when the number of particles tends to infinity, the conditions are satisfied, thus providing the conceptual possibility of obtaining the Navier-Stokes equations from the principle of least action.

Lagrangian for Navier-Stokes equations of motion: SDPD approach

Abstract

The conditions necessary and sufficient for the Smoothed Dissipative Particle Dynamics (SDPD) equations of motion to have a Lagrangian that can be used for deriving these equations of motion, the Helmholtz conditions, are obtained and analysed. They show that for a finite number of SDPD particles the conditions are not satisfied; hence, the SDPD equations of motion can not be obtained using the classical Euler-Lagrange equation approach. However, when the macroscopic limit is considered, that is when the number of particles tends to infinity, the conditions are satisfied, thus providing the conceptual possibility of obtaining the Navier-Stokes equations from the principle of least action.
Paper Structure (13 sections, 36 equations, 7 figures, 1 table)

This paper contains 13 sections, 36 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theory
  3. First Helmholtz condition
  4. Second Helmholtz condition
  5. Third Helmholtz condition
  6. The macroscopic limit leading to continuous Navier-Stokes system
  7. How particles converge to hydrodynamic fields
  8. The Lucy kernel function and particle's number density
  9. The macroscopic limit
  10. Simulation details
  11. Numerical results and discussion
  12. Conclusions
  13. Expressions for the third Helmholtz condition

Figures (7)

  • Figure 1: The form of the function $W_1$
  • Figure 2: The linear dependence of function $Q$ on $K$
  • Figure 3: Second Helmholtz condition residue as a function of the number of particles in the system, equation (\ref{['eq:HC2_1']}); the graphs for $x$, $y$, and $z$ coordinates overlap completely
  • Figure 4: Second Helmholtz condition residue as a function of the number of particles, equation (\ref{['eq:HC2_2']}); the graphs for $x,y$ (blue), $x,z$ (orange), and $y,z$ (green) combinations of the coordinates are shown
  • Figure 5: Second Helmholtz condition residue as a function of the distance between two particles, equation (\ref{['eq:HC2_3']}); the values for different number of particles in the system are shown
  • ...and 2 more figures