# Relating volumetric dilatation rate to the divergence for a fluid-volume

SebastianRM
Homework Statement:
In class we derive a relationship through the analysis of infinitesimal displacement.
Relevant Equations:
$$\frac{1}{V}\frac{dV}{dt}= \nabla \cdot \vec{v}$$
in class we derived the following relationship:
$$\frac{1}{V}\frac{dV}{dt}= \nabla \cdot \vec{v}$$
This was derived though the analysis of linear deformation for a fluid-volume, where:
$$dV = dV_x +dV_y + dV_z$$
I understood the derived relation as: 1/V * (derivative wrt time) = div (velocity).
However, my professor recently told me that the $d/dt$ operator before V, stood for the material derivative and not the common derivative. I am very confused as to how is that the case, given that we did an infinitesimal analysis of linear deformation, in a way I could call analogous to any other infinitesimal analysis that results in the common derivative.

I also tried deriving the equation by taking the material derivative of V, and dividing by V. But I was unable to reach derive the result.

I hope you can help me understand guys.

Homework Helper
2022 Award
There are two ways of describing fluid motion.

The Lagrangian formulation uses $\mathbf{x}$ to identify a particular fluid element (the one which was initially at $\mathbf{x}$). Here partial derivatives with respect to time are derivatives following the fluid.

The Eulerian formulation uses $\mathbf{x}$ to identify a particular point in space, past which the fluid moves. Here partial derivatives with respect to time are derivatives at a fixed point in space. Typically the fluid element at $\mathbf{x}$ at time $t + \delta t$ is not the same element as the one which occupied that position at time $t$, and to follow the same fluid element you have to use the material derivative.

So the question is: What formulation did you use to derive your equation, and what formulation is your professor using?

Gold Member
My interpretation: What you are solving is conservation of mass in terms of the relative volume v which is 1/density. It is also known as the continuity equation.

Think of an infinitesimal box which does not move or deform. For a given direction, on one side of the cube, you have an incoming mass flux, F. On the opposite side you have an outgoing flux, F+dF. Repeat for the other directions. The change in mass is the difference between the sums of the incoming and outgoing fluxes.

See if this helps.

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SebastianRM
There are two ways of describing fluid motion.

The Lagrangian formulation uses $\mathbf{x}$ to identify a particular fluid element (the one which was initially at $\mathbf{x}$). Here partial derivatives with respect to time are derivatives following the fluid.

The Eulerian formulation uses $\mathbf{x}$ to identify a particular point in space, past which the fluid moves. Here partial derivatives with respect to time are derivatives at a fixed point in space. Typically the fluid element at $\mathbf{x}$ at time $t + \delta t$ is not the same element as the one which occupied that position at time $t$, and to follow the same fluid element you have to use the material derivative.

So the question is: What formulation did you use to derive your equation, and what formulation is your professor using?
We looked at linear deformations in the x,y and z direction, as infinitesimal displacement, then by rearranging terms we got
$$\frac{1}{V} \frac{dV}{dt} = \nabla \cdot \vec{v}$$
We talked about infinitesimal displacements and we rearranged them (by treating them like ##\Delta x_i##), so we ended up with that relationship. Usually through this analysis I have only ever seen ##\Delta x_i/\Delta x_j \approx dx_i/dx_j ## where that is just the common derivative.

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