The Physical vs. Mathematical Problem of Navier-Stokes Equations (NSE)

Abstract

The Navier-Stokes equations describing the motion of viscous/real fluids in Rn (n = 2 or 3) depend on a positive coefficient (the viscosity, ν) via the Reynolds number. The key of NSE problem is the Reynolds number, mathematically considered a simple small perturbation parameter without any physical explanation, or a vague physical Newtonian ratio of inertial to viscous forces, 𝑅𝑅𝑅𝑅=𝑈𝑈𝑈𝑈𝜈𝜈, in spite of its quantic physical meaning as the initial excitation to response ratio, at the beginning of motion (IC at t = 0). The paper deals with the thixotropic property of real viscosity which softens (ν ↓) when strained (Re↑), but it doesn’t tend to zero (ν → 0) as much as the Reynolds number increases, holding a finite value, corresponding to the new thermodynamic equilibrium state. The (ν → 0 for Re → ∞) false physical condition renders the NSE problem to a unique solution less one beyond a critical Reynolds number, Recr. The understanding of the wall-bounded viscous flows, at both small-scales (slow motion, small Re) and larger scale (turbulent motion, large Re) must be in conjunction with the more-subtle torsional buckling effect of the “wall” lag concept that the wall has on the inherent fluid dynamics during the starting phase. The limitations of the diathermal wall associated with the starting accelerations at the onset of motion, of the order of acr/g ≥ 2/3, create the physical conditions (thermomolecular changes) for the loss of the mathematical uniqueness of the NSE solutions. The physical limitations in conjunction with the validity area of NSE model are considered in the sequel. Because of the nonlinearity of the PDE differential equations, the variation of geometrical and physical properties can lead to bifurcations in the solution and thus, to multiple solutions. Considerations relative to laminar-turbulent transition as the main bifurcation source for the more complex structure of a solution, engendered by molecular structure changes of a flowing fluid in more or less contact with the walls, are given and illustrated for the canonical flows on flat plates and viscous decay of a starting/contact vortex (“vortex eye”).