Why the derivative of the coordinate of a volume control is not zero? When deducing the Navier-Stokes equation, for conservation of momentum, in an Eulerian frame (a control volume)

the derivative of fluid velocity $U_{(t)}$ is calculated
$$\frac{\mathrm{dU} }{\mathrm{d} t}=\frac{\partial U}{\partial t}+\frac{\partial U}{\partial x}\cdot\frac{\partial x}{\partial t}$$
Then $\frac{\partial x}{\partial t}$ is replaced with $\frac{\partial x}{\partial t}=U_x$, where $U_x$ is the $x$ component of $U_{(t)}$
I do not understand how $\frac{\partial x}{\partial t}=U_x$.
$x$ is a coordinate of a volume control, which is static in time. It does not moves, so $x$ cannot be a function of time $t$, therefore, it should be $\frac{\partial x}{\partial t}=0$
How can the variation of the coordinate $x$ of the control volume be equal to the velocity $U_x$ of the fluid?
 A: There are two different position vectors at work here, and it is easy to mix them up.
The control volume has a fixed position in space, its size and shape do not change. In your derivation, the control volume is a rectangular cuboid with edge lengths of $\operatorname{d}x$, $\operatorname{d}y$ and $\operatorname{d}z$ along the $x$, $y$ and $z$ axis respectively. Let $\mathbf{X}_C$ be the control volume's position (the point C), which is constant in time.
Now consider a fluid element (infinitesimal mass of fluid) that has a velocity vector, a temperature, a density and a pressure. The current location of that fluid element is denoted by $\mathbf{x}$ and it changes with time. One can describe the current location $\mathbf{x}$ of the fluid element as a function of its initial position $\mathbf{X}_0$ at the initial time $t=t_0$ and the current time $t$. The initial position is constant. The current location of that fluid element is then $\mathbf{x}\left(\mathbf{X}_0, t\right)$ and a fluid property like the density of that very fluid element is then given by $\rho \left( \mathbf{X}_0, t\right)$ or even $\rho \left( \mathbf{X}_0\left(\mathbf{x}, t\right), t\right)$.
Looking at specific fluid elements (Lagrange formalism) to derive the equations of motion is inconvenient, because their neighbouring elements change all the time, one has to track the path of each fluid element from its initial position $\mathbf{X}_0$ to its current position $\mathbf{x}$ and the element continuously changes its shape. Instead, one looks at the fluid that is inside the control volume at the current time $t$ and position $\mathbf{X}$. Notice that the fluid in this volume can change and that we are no longer considering only one specific fluid element, but a fixed volume in space. The fluid properties at that position are then given by, for example, $\rho \left( \mathbf{X}, t \right)$.
To derive the equations of motion, one has to look at the changes of properties with respect to time. But we want to look at the very fluid element that is inside our control volume at the current time $t$. Thus, we use the description $\rho \left( \mathbf{x}\left(t\right), t \right)$ of the properties of that fluid element, where $\mathbf{x}$ is the current location of the fluid element and not the location of the control volume. To get the fluid element's change of properties with respect to time at the location $\mathbf{X}_C$ of the control volume, we calculate
$$\left.\frac{\operatorname{d}\rho \left( \mathbf{x}\left(t\right), t \right)}{\operatorname{d}t}\right|_{\mathbf{x}=\mathbf{X}_C} = \left.\frac{\partial \rho \left( \mathbf{x}\left(t\right), t \right)}{\partial t}\right|_{\mathbf{x}=\mathbf{X}_C} + \left.\frac{\partial \rho \left( \mathbf{x}\left(t\right), t \right)}{\partial \mathbf{x}}\right|_{\mathbf{x}=\mathbf{X}_C} \cdot \left.\frac{\partial \mathbf{x}\left(t\right)}{\partial t}\right|_{\mathbf{x}=\mathbf{X}_C}$$
Here, $\partial \mathbf{x}/\partial t$ is the change of the position of the fluid element with respect to time, or just its current velocity vector $\mathbf{u}$. The term $\partial \rho /\partial t$ is the change of the density of the fluid element with respect to time, as it changes its position. And the term $\partial \rho /\partial \mathbf{x}$ is the change of the fluid property with respect to space (more specifically its gradient). The whole equation is called the material or substantial derivative and the "subscript" $\mathbf{x}=\mathbf{X}_C$ is often left out.
In your derivation, you erroneously considered $\mathbf{x}$ to be the position of the control volume, while it correctly is the current position of the fluid element under consideration. The material derivative is usually evaluated at the position of the control volume, and then $\mathbf{x}=\mathbf{X}_C$ holds.
The governing equations balance the rates of change of mass, momentum and energy. The Lagrangian formulation $\rho \left( \mathbf{X}_0, t\right)$ tracks each and every material element as it changes with time. No mass moves through the element boundaries, thus neighbouring material elements do not change and the rate of change of a property in the material element is just $\frac{\partial \rho \left( \mathbf{X}_0, t\right)}{\partial t}$ as $\mathbf{X}_0$ is constant. The Lagrangian formulation is used in solid mechanics where the displacement of each mass element is a result of a simulation. It is unsuitable for fluids, because neighbouring material elements change all the time. Thus, usually the Eulerian formulation $\rho \left( \mathbf{x} \left(t\right), t\right)$ is used. In this formulation, one looks at a control volume and mass can flow through its boundaries. At one instant in time, the position $\mathbf{x} \left(t\right)$ of a material element is equal to the location $\mathbf{X}_C$ of the control volume, but the material may have a non-zero velocity and thus be somewhere else in the next second. Thus, the rate of change of a material property inside the control volume consists of two contributions: First, the rate of change of the property of this very fluid element with time, which can happen with zero flow velocity. Second, the rate of change of the property due to flow velocity: fluid with a different value of the property being transported into the control volume and the fluid being in the control volume right now flowing out of it. This can happen without the fluid element itself changing its value, e.g. cold water flowing in a pipe that is followed by warmer water: The cold fluid element does not get warmer, but the temperature at one fixed position in the pipe rises. The rate of change of a property in the Eulerian formalism (the observer is fixed to the pipe) is the material/substantial derivative above. In the Lagrangian formalism the observer is fixed to the material element.
A: Given a (scalar) quantity $\varphi(\mathbf x,\,t)$ that exists in a continuum and has a macroscopic velocity represented by the vector field $\mathbf{u}(\mathbf{x},\,t)$. Then via the chain rule,
$$\frac{\mathrm{d}\varphi}{\mathrm{d}t}=\frac{\partial\varphi}{\partial t}+\dot{\mathbf{x}}\cdot\nabla\varphi.$$
Here, $\dot{\mathbf{x}}\equiv\mathrm{d}\mathbf{x}/\mathrm{d}t$ represents the time derivative of a chosen path in space, $\mathbf{x}(t)$, and, at this point, we have two seemingly obvious choices on the path:

*

*$\dot{\mathbf{x}}=0$.

*$\dot{\mathbf{x}}=\mathbf{u}$ (where $\mathbf{u}$ is the fluid velocity).

In the first choice, the total time derivative is equal to the partial time derivative. While in the second choice, we end up with the material derivative, in which the path of interest follows the fluid velocity field--this latter option, of course, is more useful for analyzing fluid flows and so it is the common choice.
