Some Results on Modifications of Three-Dimensional Navier—Stokes Equations
Some Results on Modifications of Three-Dimensional Navier—Stokes Equations
复制标题
三维纳维-斯托克斯方程修正的一些结果
DOI：
10.1007/978-1-4612-2196-8_7
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发表时间：
1998
影响因子：
2.4
通讯作者：
O. Ladyzhenskaya
中科院分区：
文献类型：
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作者：
O. Ladyzhenskaya
在 20 世纪 60 年代中期，我建议对纳维-斯托克斯方程 (MNS) 进行一些修改，以描述速度梯度较大时粘性流体的动力学 [1]，另请参阅 [2]–[4]。在这里，我们将考虑具有以下形式的那些：
$$Lv\left( t \right) \equiv {v_t}\left( t \right) + {v_k}\left( t \right){v_{xk}}\left( t \right) - divT\left( {\hat v\left( t \right)} \right) = - \nabla p\left( t \right) + f\左( t \右)$$
(7.0.1)
其中 v = (v 1,…, v n ) 是速度场， \(\hat v = \left( {{v_{ij}}} \right)\), i, j = 1,- n, 是元素为 v ij = v ixj + v jxi 的矩阵， \(T\left( {\hat v} \right) = \left( {{T_{ij}}\left( {\hat v} \right)} \right)\) 是对称应力张量，p 是压力。空间变量 x = (x 1,…, x n ) （在上面的 (7.0.1) 中没有明确显示）在欧几里得空间 ℝ n (n = 2 或 3) 和 t ∈ ℝ+ = [0, ∞) 的域 Ω 中发生变化。
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 ℝ n (n = 2 or 3) and t ∈ ℝ+ = [0, ∞).