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

In summary, we derived the relationship $$1/V (dV/dt) = \nabla \cdot \vec{v}$$ by analyzing linear deformations for a fluid volume. However, the professor mentioned that the $d/dt$ operator represents the material derivative instead of the common derivative, which has caused confusion. There are two ways to describe fluid motion, the Lagrangian formulation and the Eulerian formulation, depending on which point or fluid element is used as a reference. The question remains as to which formulation was used in the derivation and which one the professor is using.
  • #1
SebastianRM
39
4
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.

Thank you for your time.
 
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  • #2
There are two ways of describing fluid motion.

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

The Eulerian formulation uses [itex]\mathbf{x}[/itex] 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 [itex]\mathbf{x}[/itex] at time [itex]t + \delta t[/itex] is not the same element as the one which occupied that position at time [itex]t[/itex], 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?
 
  • #3
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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  • #4
pasmith said:
There are two ways of describing fluid motion.

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

The Eulerian formulation uses [itex]\mathbf{x}[/itex] 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 [itex]\mathbf{x}[/itex] at time [itex]t + \delta t[/itex] is not the same element as the one which occupied that position at time [itex]t[/itex], 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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Related to Relating volumetric dilatation rate to the divergence for a fluid-volume

1. How do you define volumetric dilatation rate?

Volumetric dilatation rate is a measure of the rate at which a fluid volume is expanding or contracting. It is typically expressed in units of volume per unit time, such as cubic meters per second.

2. What is the relationship between volumetric dilatation rate and divergence?

Volumetric dilatation rate and divergence are closely related, as they both describe changes in fluid volume. Specifically, divergence is a measure of how much a fluid is spreading out or converging at a given point, while volumetric dilatation rate is a measure of the rate at which this change in volume is occurring.

3. How is volumetric dilatation rate calculated?

Volumetric dilatation rate can be calculated by taking the divergence of the velocity field of a fluid. This involves taking the partial derivatives of the velocity components with respect to each spatial coordinate and summing them together.

4. What factors can affect the volumetric dilatation rate of a fluid?

The volumetric dilatation rate of a fluid can be affected by a variety of factors, including the velocity and direction of fluid flow, the density and compressibility of the fluid, and the presence of any external forces or boundaries.

5. Why is it important to study the relationship between volumetric dilatation rate and divergence?

Understanding the relationship between volumetric dilatation rate and divergence is crucial in various fields of science and engineering, such as fluid dynamics, meteorology, and oceanography. This relationship can help us better understand and predict the behavior of fluids in different scenarios, such as in turbulent flows or in the presence of obstacles.

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