I am reading this paper and trying to understand on page 14-15,

First it rewrites Navier-Stokes in terms of a ''zero order problem'', and where $u_i = \mathscr{U}_i + u_i'$ , where $U_i$ is the base flow vector and $u_i'$ is the disturbance (perturbation):

$$\begin{array}{c}{\frac{\partial\mathscr{U}_{i}}{\partial t}+\mathscr{U}_{j} \frac{\partial \mathscr{U}_{j}}{\partial x_{j}}=-\frac{1}{\rho} \frac{\partial p^{(0)}}{\partial x_{i}}+\nu \frac{\partial^{2} \mathscr{U}_{i}}{\partial x_{j}} \frac{\partial}{\partial x_{j}}+\mathbb{F}_{i}^{(0)}} \\ {\partial \mathscr{U}_{j} / \partial x_{j}=0}\end{array}\,\,(1)$$

Then we choose the basic velocity field so its linear in the space coordinates $x_i$, ie.

$$\mathscr{U}_i (x_j,t= \sigma_{ij}(t)x_j+\mathscr{U}_i^0(t) \,\,(2)$$

Then define $$P\left(x_{j}, t\right) \equiv-\int^{x_{i}} \mathbb{F}_{j}^{(0)} \mathrm{d} x_{j}+\frac{p^{(0)}}{\rho}+\left(\frac{\partial \mathscr{U}_{i}^{(0)}}{\partial t}+\mathscr{U}_{j}^{(0)} \sigma_{i j}\right) x_{i}\,\,(3)$$

where $\mathscr{U}_i^{(0)}$ denotes the instantaneous velocity of the basic flow at the origin of coordinates and perhaps i could have just replaced this with $\mathscr{U}_i$ in this question just to simplify things..

My question though, is how does it follow from Eqn (1) and (2) that

$$\mathrm{d} \sigma_{i k} / \mathrm{d} t+\sigma_{j k} \sigma_{i j}=-\partial^{2} P / \partial x_{i} \partial x_{k}, \quad \sigma_{j j}=0 \,\,(4)$$

where $\sigma$ is stress tensor?

Second, what does (3) say about the flow ? What constraint does defining (2) and (3) have on the flow?


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