Skip to main content

An alternative to the concept of continuous medium


Discrete mechanics proposes an alternative formulation of the equations of mechanics where the Navier–Stokes and Navier–Lamé equations become approximations of the equation of discrete motion. It unifies the fields of fluid and solid mechanics by extending the fields of application of these equations to all space and time scales. This article presents the essential differences induced by the abandonment of the notion of continuous medium and global frame of reference. The results of the mechanics of continuous medium validated by fluid and solid observations are not questioned. The concept of continuous medium is not invalidated, and the discrete formulation proposed simply widens the spectrum of the applications of the classical equations. The discrete equation of motion introduces several important modifications, in particular the fundamental law of the dynamics on an element of volume becomes a law of conservation of the accelerations on an edge. The acceleration considered as an absolute quantity is written as a sum of two components, one solenoidal, the other irrotational, according to a local orthogonal Helmholtz–Hodge decomposition. The mass is abandoned and replaced by the compression and rotation energies represented by the scalar and vectorial potentials of the acceleration. The equation of motion and all the physical parameters are expressed only with two fundamental units, those of length and time. The essential differences between the two approaches are listed and some of them are discussed in depth. This is particularly the case with the known paradoxes of the Navier–Stokes equation or the importance of inertia for the Navier–Lamé equation.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3


  1. 1.

    Batchelor, G.: An Introduction to Fluid Mechanics. Cambridge University Press, Cambridge (1967)

    MATH  Google Scholar 

  2. 2.

    Buresti, G.: A note on Stokes is hypothesis. Acta Mechanica 226, 3555–3559 (2015).

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Caltagirone, J.P.: Discrete Mechanics, Concepts and Applications. ISTE John Wiley & Sons, London (2019).

    Book  Google Scholar 

  4. 4.

    Caltagirone, J.P.: Physique discrète et relativité. Annales de la Fondation Louis de Broglie 44, 1–13 (2019)

    Google Scholar 

  5. 5.

    Caltagirone, J.P.: Non-Fourier heat transfer at small scales of time and space. Int. J. Heat Mass Trans. 160, 120145 (2020).

    Article  Google Scholar 

  6. 6.

    Caltagirone, J.P.: On Helmholtz-Hodge decomposition of inertia on a discrete local frame of reference. Phys. Fluids 32, 083604 (2020).

    Article  Google Scholar 

  7. 7.

    Caltagirone, J.P.: Application of discrete mechanics model to jump conditions in two-phase flows. J. Comp. Phys. 432, 110151 (2021).

    MathSciNet  Article  Google Scholar 

  8. 8.

    Caltagirone, J.P.: On a reformulation of Navier–Stokes equations based on Helmholtz-Hodge decomposition. Phys. Fluids 33, 063605 (2021).

    Article  Google Scholar 

  9. 9.

    Caltagirone, J.P., Vincent, S.: On primitive formulation in fluid mechanics and fluid-structure interaction with constant piecewise properties in velocity-potentials of acceleration. Acta Mechanica 231(6), 2155–2171 (2020).

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Desbrun, M., Hirani, A., Leok, M., Marsden, J.: Discrete exterior calculus. arXiv:0508.341v2 pp. 1–53 (2005)

  11. 11.

    Gad-El-Hak, M.: Stokes hypothesis for a newtonian, isotropic fluid. J. Fluids Eng. 117(1), 3–5 (1995).

    Article  Google Scholar 

  12. 12.

    Guermond, J., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg. 195, 6011–6045 (2006).

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Hamman, C., Klewick, J., Kirby, R.: On the Lamb vector divergence in Navier–Stokes flows. J. Fluid Mech. 610, 261–284 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Happel, J., Brenner, H.: Low Reynolds Number Hydrodynamics. Kluwer Academic Publishers, Boston (1963)

    MATH  Google Scholar 

  15. 15.

    Harlow, F., Welch, J.: Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface. Phys. Fluids 8, 2182–2189 (1965).

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Holmes, M., Parker, N., Povey, M.: Temperature dependence of bulk viscosity in water using acoustic spectroscopy. J. Phys. Conf. Ser. (2011).

    Article  Google Scholar 

  17. 17.

    Kosmann-Schwarzbach, Y.: Noether Theorems. Invariance and Conservations Laws in the Twentieth Century. Springer, New York (2011).

    Book  MATH  Google Scholar 

  18. 18.

    Landau, L., Lifchitz, E.: Fluid Mechanics. Pergamon Press, London (1959)

    Google Scholar 

  19. 19.

    Landau, L., Lifchitz, E.: The Classical Theory of Fields. Third Revised English Edition. Pergamon Press Ltd, Oxford (1971)

    Google Scholar 

  20. 20.

    Rajagopal, K.: A new development and interpretation of the Navier–Stokes fluid which reveals why the Stokes assumption is inapt. Int. J. Non-Linear Mechan. 50, 141–151 (2013).

    Article  Google Scholar 

  21. 21.

    Shaskov, M.: Conservative Finite-Difference Methods on General Grids. CRC Press, Boca Raton (1996).

    Book  Google Scholar 

  22. 22.

    Sod, G.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978).

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Truesdell, C., Rosenhead, L.: The present status of the controversy regarding the bulk viscosity of fluids. Proc. R. Soc. Lond. A (1954).

    Article  MATH  Google Scholar 

  24. 24.

    Van Dyke, M.: Perturbation Methods in Fluids Mechanics. Academic Press, California University, California (1964)

    MATH  Google Scholar 

  25. 25.

    Yuan, G., Jiulin, S., Kaixing, Z., Xindao, H.: Determination of bulk viscosity of liquid water via pulse duration measurements in stimulated brillouin scattering. Chin. Opt. Lett. (2013).

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Jean-Paul Caltagirone.

Ethics declarations

Author contributions

J.-P. Caltagirone was responsible for physical modeling, conceptualization, methodology, research code, validation, writing—original draft preparation, reviewing and editing. The paper has been checked by a proofreader of English origin.

Conflict of interest

There are no conflicts of interest in this work.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Caltagirone, JP. An alternative to the concept of continuous medium. Acta Mech (2021).

Download citation

MSC codes:

  • 35Q30; 65N22; 74F10; 76A02; 76D05