Physical meaning of material derivative

[utab]

Dear all,

For my Ph.D research. I have to use the material derivative concept. I reviewed some of my previous continuum mechanics course notes but this topic was superficial in our course. I am reading the book "A first course in continuum mechanics" by Fung. I also noted from some books in the library that the material derivative is the "time rate change measured by an observer moving with the specific particles under study". I can understand the mathematical concept from Fung's book but I have difficulty in visualizing it in my mind.

Can someone kindly explain the physical meaning of the material derivative with some physical examples?

Regards,

Umut

 

[momentum_waves]

It is the Total Derivative that mathematicians use, with variables t,x,y,z.

Sometimes called the Substantial Derivative - in fluid mechanics - as used in the derivation of the Navier-Stokes equations.

 

[charng]

The material derivative is a concept used in fluid mechanics and continuum mechanics to describe the change in a physical property of a material as it moves through space and time. It takes into account both the convective (due to the movement of the material) and the local (due to internal changes in the material) effects on the property being studied.

To understand the physical meaning of the material derivative, let's consider an example of a fluid flowing through a pipe. The fluid has a property, such as velocity or temperature, that we want to track as it moves through the pipe. The material derivative of this property would describe how it changes with time as the fluid moves along the pipe.

One way to think about the material derivative is to imagine yourself as an observer moving with the fluid particles. As you move along with the particles, you can measure the change in the property you are interested in. This is the convective effect. However, the fluid particles themselves may also be changing internally, for example, due to mixing or chemical reactions. This is the local effect. The material derivative takes into account both of these effects to give the overall change in the property.

In this example, the material derivative would be important for understanding how the flow of the fluid affects the property being studied. It can also be used in other areas of mechanics, such as in the study of elastic materials, to understand how the material deforms and changes over time.

I hope this explanation helps you to better visualize and understand the physical meaning of the material derivative. Good luck with your research!