New formulation of the Navier–Stokes equations for liquid flows

Massimiliano Giona; Giuseppe Procopio; Alessandra Adrover; Roberto Mauri

doi:10.1515/jnet-2022-0095

Published by De Gruyter December 16, 2022

New formulation of the Navier–Stokes equations for liquid flows

Massimiliano Giona , Giuseppe Procopio , Alessandra Adrover and Roberto Mauri

From the journal Journal of Non-Equilibrium Thermodynamics

https://doi.org/10.1515/jnet-2022-0095

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Abstract

For isothermal liquid flows, the condition of incompressibility provides a useful simplification for describing their mechanical properties. Nevertheless, it overlooks acoustic effects, and it provides the unpleasant shortcoming of infinite propagation speed of velocity perturbations, no matter the type of constitutive equation for the shear stresses is adopted. In this paper, we provide a derivation of a new formulation of the Navier–Stokes equations for liquid flows that overcomes the above issues. The pressure looses its ancillary status of mere gauge variable (or equivalently Lagrange multiplier of the incompressibility condition) enforcing the solenoidal nature of the velocity field, and attains the proper physical meaning of hydrodynamic field variable characterized by its own spatiotemporal evolution. From the experimental evidence of sound attenuation, related to the occurrence of a non-vanishing bulk viscosity, the evolution equation for pressure in out-of-equilibrium conditions is derived without introducing any adjustable parameters. The connection between compressibility and memory effects in the propagation of internal stresses is established. Normal mode analysis and some preliminary simulations are also discussed.

Keywords: compressibility; finite propagation velocity; Navier–Stokes equations; pressure dynamics

Corresponding author: Massimiliano Giona, Faculty of Industrial Engineering, La Sapienza Università di Roma, DICMA, Rome, Italy, E-mail: massimiliano.giona@uniroma1.it

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: Derivation of Eqs. (15) and (16)

Consider the second Eq. (10), where

(58) d d t = ∂ ∂ t + v ⋅ ∇

By taking the divergence in this equation,

(59) ρ 0 ∇ ⋅ d v d t = ρ 0 ∂ ∇ ⋅ v ∂ t + ρ 0 ∇ ⋅ v ⋅ ∇ v = μ ∇ ⋅ ∇ 2 v + μ 3 ∇ ⋅ ∇ 2 v − ∇ 2 P = 4 3 μ ∇ 2 ∇ ⋅ v − ∇ 2 P

In the limit of small Reynolds number (linearized theory), the nonlinear contributions, such as ∇ ⋅ v ⋅ ∇ v , can be neglected, so that

(60) ρ 0 ∇ ⋅ d v d t ≃ ρ 0 ∂ ∇ ⋅ v ∂ t ≃ ρ 0 d ∇ ⋅ v d t = 4 3 μ ∇ 2 ∇ ⋅ v − ∇ 2 P

that corresponds to Eq. (15). Substituting this result into Eq. (14), and neglecting the nonlinear term v ⋅∇P in the expression for dP/dt, it follows from Eqs. (59) and (60) and from the first Eq. (10) that

(61) d P d t ≃ ∂ P ∂ t = − c s 2 ρ 0 ζ − κ ρ 0 4 3 μ ∇ 2 ζ − ∇ 2 P

which is Eq. (16).

Appendix B: A simple model highlighting the Stokesian propagation paradox

Parabolic scalar (diffusion equation) and vectorial (incompressible unsteady Stokes equation) suffer of the problem of infinite-velocity propagation, albeit either its origin or the phenomenology are completely different. To show this difference, consider the two vector fields u(x, t), v(x, t) defined for x ∈ Q L = [ 0 , L ] × [ 0 , L ] ⊆ R 2 , x = (x, y), by the parabolic evolutions

(62) ∂ u ( x , t ) ∂ t = ν ∇ 2 u ( x , t )

(63) ∂ v ( x , t ) ∂ t = ν ∇ 2 v ( x , t ) , ∇ ⋅ v ( x , t ) = 0

with the same value of the diffusivity ν. In Eq. (63), ∇² = −∇ × ∇ × , (as v(x, t) is solenoidal), and for this reason no gradient gauge, corresponding to the pressure gradient contribution −∇P is required. Albeit u(x, t) is vector-valued, Eq. (62) corresponds to the free diffusional propagation of u(x, t), where the two entries u ₁ and u ₂ of the field u evolve independently of each other. Therefore, it represents a vector-valued scalar diffusion problem. The dynamics of the field v(x, t) is identical, but a constraint on its divergence (solenoidal nature of v) has been added, determining a coupling between the entries v ₁ and v ₂. These problems are equipped with periodic boundary conditions in Q _L, and identical initial conditions

(64) u ( x , 0 ) = f 1 δ ( x − x c ) , v ( x , 0 ) = Proj sol [ f 1 δ ( x − x c ) ]

where f ₁ = (1, 1) and x _c = (x _c, y _c) = (L/2, L/2) is the mid-point of Q _L, and Proj_sol is the projection operator onto the solenoidal subspace of the functional space L vec 2 ( Q L ) of square-summable vector-valued functions in Q _L. The two problems are identical apart from the solenoidal constraint on v(x, t), that acts also on the initial condition. Let L = 1 [a.u.], and ν = 10⁻² [a.u.], and consider the dynamics for values of t ≪ L ²/ν, so that the influence of the periodic boundary conditions is negligible, thus simulating the free-space propagation. These two problems are amenable to a simple closed-form solution using spectral methods (Fourier series). Let k m , n = 2 π ( m , n ) = k m , n 1 , k m , n 2 , with m , n ∈ Z and k _m,n = |k _m,n|.

The expression for u(x, t) reads

(65) u ( x , t ) = ∑ m , n = − ∞ ∞ f 1 e − ν | k m , n | 2 t e i k m , n ⋅ ( x − x c )

while v(x, t) attains the form

(66) v ( x , t ) = ∑ m , n = − ∞ ∞ k m , n 2 − k m , n 1 k m , n ⋅ f 0 | k m , n | 2 e − ν | k m , n | 2 t e i k m , n ⋅ ( x − x c )

Figure 6 depicts the profiles of the first entry of the two fields u and v at y = y _c = 1/2, evaluated at the same time instants. While the scalar problem, namely the dynamics of u(x, t) (panel a) shows the typical features of the propagation paradox occurring in diffusion: (i) Gaussian concentration profiles, centered at the location of the impulsive initial condition (ii) rapidly decaying with the distance from it, but (iii) nevertheless possessing unbounded support, the behavior of v(x, t), i.e., the solution of an incompressible unsteady Stokes problem, is radically different as depicted in panel (b). For any time t > 0, no matter how small, the profiles of v(x, t) are characterized by a persistent power-law tail v _h(x, t) ∼ 1/|x−x _c|^α, h = 1, 2, (with α = 2, since the problem is two-dimensional), possessing an invariant behavior at short-time scale and in the far-field.

Figure 6:

Profiles of u ₁(x, y _c, t) (panel a) and v ₁(x, y _c, t) (panel b) versus x of the two fields u(x, t) and v(x, t), at different time instants. The arrows indicate increasing values of t = 5 × 10⁻⁴, 10⁻³, 2 ⋅ 10⁻³, 3 ⋅ 10⁻³, 4 ⋅ 10⁻³, 5 ⋅ 10⁻³, 10⁻².

For this reason, the Stokesian propagation paradox is drastically different from the homonymous problem in scalar diffusion. To describe, pictorially, the phenomenology depicted in Figure 6 panel (b), the concept of “teleportation of fluid elements” has been used in [11], just to indicate the convective propagation at unbounded velocity characterizing incompressible flows and their velocity profiles.

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Received: 2022-11-25

Accepted: 2022-11-30

Published Online: 2022-12-16

Published in Print: 2023-04-28

New formulation of the Navier–Stokes equations for liquid flows

Abstract

Appendix A: Derivation of Eqs. (15) and (16)

Appendix B: A simple model highlighting the Stokesian propagation paradox

References

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