Bi-velocity hydrodynamics: Single-component fluids

Acceptance of the Navier–Stokes–Fourier (NSF) equations as the fundamental equations of single-component continuum fluid mechanics for liquids and gases is noted to be inseparably linked to Euler’s implicit, but unproved, hypothesis that but a single-velocity field is required to characterize the four physically different, context-specific, velocities appearing in the mass, momentum, and energy equations. To test Euler’s hypothesis, velocity is added to the usual list of quantities requiring constitutive formulation – namely the heat flux q and viscous stress T – in order to effect closure of the mass, momentum, and energy equations. Establishment of this enlarged set of constitutive relations is effected by using conventional linear irreversible thermodynamics (LIT) principles governing the behavior of simple fluid continua, importantly including satisfaction of Onsager reciprocity as a fundamental continuum requirement. The resulting analysis shows that, in general, two velocities rather than one are required and, concomitantly, that additional driving forces must be added to each of the standard constitutive equations for the Fourier’s-law heat flux q = −k∇T   and the Newton’s-law viscous stress T=2η∇vm¯ (wherein the “mass velocity” vm is the context-specific velocity appearing in the continuity equation ∂ρ/∂t + ∇ · (ρvm) = 0). For the particular case of dilute gaseous continua explicit expressions are established for the phenomenological coefficients appearing in these additional constitutive contributions. Determination of these coefficients is effected using data derived from the Chapman–Enskog–Burnett constitutive expressions for q and T, the latter obtained by solving the Boltzmann equation at small Knudsen numbers, including so-called rarefied-gas contributions. These coefficients are found to be nonzero, confirming the conclusion, inter alia, that two velocities are constitutively required to quantify hydrodynamic behavior for gases and, by inference, for liquids too. Collectively, these velocity, heat flux, and stress constitutive findings collectively negate the current belief that the NSF equations fully describe the physics of viscous fluid continua. Rather, they do so only in limiting cases where the additional constitutive terms than we have found necessary for completeness are asymptotically small.