Lagrangian for Navier-Stokes equations of motion: SDPD approach

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.

💡 Research Summary

The paper investigates whether the Smoothed Dissipative Particle Dynamics (SDPD) model, a particle‑based discretisation of the Navier‑Stokes (NS) equations, admits a Lagrangian formulation that can be used with the classical Euler‑Lagrange variational principle. The authors employ the Helmholtz conditions, a set of necessary and sufficient criteria for a system of second‑order differential equations to be derivable from a Lagrangian. These conditions involve symmetry of the second‑derivative terms, specific relationships between partial derivatives with respect to velocities, and a compatibility condition linking position and velocity derivatives.

First, the SDPD equations of motion are rewritten in the generic form H_i(t,q_i, ẋ_i, ¨q_i)=0, where the generalized coordinates q_i correspond to the Cartesian components of particle positions. The authors derive explicit expressions for H_i that contain kernel‑based interaction terms, distance vectors r_ij, velocity differences v_ij, and material parameters such as shear viscosity η, bulk viscosity ξ, and thermal conductivity κ. By substituting these H_i into the Helmholtz conditions, they find that the first condition (symmetry of the acceleration coefficients) is trivially satisfied because each particle’s acceleration depends only on its own coordinates. However, the second and third conditions generate non‑zero residual terms that involve the kernel function W, its derivative F, and combinations of the particle distances and velocities. These residuals are expressed as Q_xij, Q_xyij, and similar quantities, which are clearly not zero for a finite number of particles.

Consequently, for any finite particle count N, the SDPD system does not satisfy the Helmholtz criteria, implying that a conventional Lagrangian cannot be constructed and the equations cannot be obtained from the Euler‑Lagrange formalism. This result aligns with the intuition that dissipative, non‑conservative forces (viscous stresses, heat conduction) break the variational structure in a discrete setting.

The authors then turn to the macroscopic limit N → ∞, where SDPD is known to converge to the continuous NS equations. They analyse how the kernel support radius h must shrink with increasing N to keep a fixed number of neighbours K₀ inside the kernel sphere, leading to the scaling h ∝ N^{‑1/3}. Using the Lucy kernel, they estimate particle number density, inter‑particle spacing, and the scaling of the residual terms. Under the assumption that distances and velocity differences scale as N^{‑1/3}, the offending terms in the second and third Helmholtz conditions decay as negative powers of N (e.g., N^{‑7/3}, N^{‑2}). Therefore, in the limit of infinitely many particles, all Helmholtz conditions are satisfied, establishing a conceptual possibility of deriving the continuous NS equations from a Lagrangian via the principle of least action.

To corroborate the analytical scaling, the authors perform numerical SDPD simulations using the LAMMPS package with a water‑like fluid model. They keep physical parameters (viscosity, sound speed, density, temperature) constant while varying N from 10³ to 10⁵. The computed magnitudes of the Helmholtz residuals decrease systematically with N, confirming the theoretical prediction that the discrepancies vanish in the macroscopic limit. They also verify that the kernel cutoff radius h = 0.18 µm is sufficiently large for convergence of the summed kernel contributions.

In the discussion, the paper emphasizes two main conclusions: (1) finite‑N SDPD lacks a Lagrangian representation due to violation of Helmholtz conditions, and (2) the continuous NS equations admit a Lagrangian in the infinite‑particle limit, thereby reconciling dissipative fluid dynamics with variational principles at the macroscopic scale. The authors acknowledge that their analysis relies on simplifying assumptions—constant temperature, constant viscosities, and neglect of the entropy equation—and suggest that extending the framework to include variable thermodynamic fields, anisotropic viscosities, and stochastic SDPD terms would be a valuable direction for future work.

Overall, the study provides a rigorous mathematical bridge between particle‑based mesoscopic fluid models and continuum variational formulations, offering theoretical support for the development of multiscale simulation techniques that exploit Lagrangian mechanics even in the presence of dissipation.