Some Results on Modifications of Three-Dimensional Navier—Stokes Equations

Abstract

In the mid-1960s I suggested a number of modifications to the Navier—Stokes equations (MNS) for the description of the dynamics of viscous fluids when velocity gradients are large [1], see also [2]–[4]. Here we are going to consider those which have the following form:

$$Lv\left( t \right) \equiv {v_t}\left( t \right) + {v_k}\left( t \right){v_{xk}}\left( t \right) - divT\left( {\hat v\left( t \right)} \right) = - \nabla p\left( t \right) + f\left( t \right)$$

(7.0.1)

where v = (v ₁,…, v _n ) is the velocity field, $\hat v = \left( {{v_{ij}}} \right)$, i, j = 1,^- n, is the matrix with elements v _ij = v _ixj + v _jxi ), $T\left( {\hat v} \right) = \left( {{T_{ij}}\left( {\hat v} \right)} \right)$ is a symmetric stress tensor and p is the pressure. The space variable x = (x ₁,…, x _n ) (which was not shown above explicitly in (7.0.1)) is changing in domain Ω of the Euclidian space ℝⁿ (n = 2 or 3) and t ∈ ℝ⁺ = [0, ∞).