Let $x = f(t)$ and $\phi = f(t)$ and $r = f(t)$ and $\dot{x} = u = f(x, r,\phi,t)$ where $x$ is the particle displacement, $\phi$ is the angular rotation, $r$ is the radius, $u$ is the particle velocity, and $t$ is the time.
One of the terms in the Navier-Stokes system is $\dfrac{\partial u }{\partial r}\bigg|_{\phi, x, t}$
If we want to convert to displacement formulation, we could replace $u$ with $\dot{x}$
$\dfrac{\partial u }{\partial r}\bigg|_{\phi, x, t} = \dfrac{\partial \dot{x} }{\partial r}\bigg|_{\phi, x, t}$
When is it legal to move the partials and how should this be handled when you take partials holding certain variables constant and you want to maintain this notation? This is important in thermodynamics.
$ \dfrac{\partial \dot{x} }{\partial r}\bigg|_{\phi, x, t} = \dfrac{\partial}{\partial r}\bigg(\dot{x}\bigg)\bigg|_{\phi, x, t} = \dfrac{\partial}{\partial r}\bigg(\dfrac{\partial x}{\partial t}\bigg)\bigg|_{\phi, x, t} = \dfrac{\partial x}{\partial r \partial t}\bigg|_{\phi, x, t} $