On the motion of ν-fluids

Abstract

The paper is concerned with a class of non-Newtonian fluids, ν-fluids, all of whose properties are determined by a single dimensional constant of the same dimensions as a viscosity. A regular nth-order ν-fluid is then defined to be one whose nth order time derivative of stress is a regular function of the local stress and flow fields and any of their space and time derivatives. The regularity condition determines the constitutive relation of such a fluid completely in terms of a finite set of non-dimensional constants which define the fluid.An obvious property of these fluids is that their motions obey the same principles of Reynolds number similarity as those of a Newtonian fluid, and the primary aim of the paper is to examine the extent to which their flow properties are the same as those of the mean turbulent flow of a Newtonian fluid.It is shown that a third-order fluid is the simplest ν-fluid which shares enough properties with turbulent motion to be worth further consideration in this context. At infinite Reynolds number, the constitutive relation for such a fluid reduces to the form \[ AS\ddot{S} + B\dot{S}^2 + CS^2 S^{\prime\prime} + DSS^{\prime 2} + Eu^{\prime 2}S^2 = 0, \] where A,B, …,E, are isotropic tensor constants of the fluid, S is the stress tensor, u’ is the total rate of strain tensor, dots denote total time derivatives, and primes denote space derivatives. A number of illustrative examples of the properties of such a constitutive relation are then considered, representing the decay of a homogenous stress field, the effect of rigid-body rotation on such a decay, the structure of the equilibrium stress field in the presence of homogeneous rate of strain, both with and without vorticity, and the nature of flow near a plane boundary. In all cases, the results appear to be consistent with known properties of turbulent motion, to the extent that the analysis is taken.Finally, the effect of finite Reynolds number on the decay of an isotropic and homogeneous stress field is shown to be consistent with observations on the decay of isotropic turbulence.