Understanding Linear Distortion and Divergence in Thermodynamics

[student85]

In my thermodynamics class, I'm seeing that when a body gets linearly distorted we use the divergence of the velocity to find how much the body gets distorted and if the divergence is 0 then the body is incompressible.

Div v = (nabla) (v)

Please! Can someone explain to me why this works just shortly? What is divergence?

I had never seen divergence in my life so I'm very confused. Thanks a lot and hope my translation of all the mentioned terms is correct, since my class is in spanish.

 

[Crosson]

What you are missing is a course in vector calculus, where the nabla operator (also called del) and the divergence are discussed in detail.

In my opinion, the best concise treatment is in the preliminary chapter of an Electromagnetism textbook, Introduction to Electrodynamics by David Grifiths.

 

[Mentz114]

Have you learned what a vector field is ? Divergence is a property of such a field. It tells us about the way the field 'flows'. It is used a lot in fluid dynamics and electromagnetism.

 

[student85]

Yeah, I know I need a vector calculus course. However, can anyone give me an idea of what a divergence of a velocity mean? How is the idea of the way the field flows relate with a velocity?

 

[vanesch]

Yeah, I know I need a vector calculus course. However, can anyone give me an idea of what a divergence of a velocity mean? How is the idea of the way the field flows relate with a velocity?

Divergence is the local equivalent of a small surface integral over a closed surface (say a small sphere) of said vector field. Now, the surface integral over a closed surface of a vector field gives you the balance of how much "goes in" and how much "goes out". If the integral is positive (divergence is positive), then there's more "coming out" of the sphere than "going in". On the other hand, if the divergence is negative, then there's more "going in" than "going out".
And, if the divergence is zero, then, there's just as much "going into the sphere", than "coming out of the sphere". This is typically what you want for the flow of an incompressible fluid: in a given, static small volume, with as boundary a closed surface (say, a sphere), you want there to flow as much fluid INTO the volume than fluid OUT OF the volume, so that the amount of fluid in the (fixed) volume remains the same (and hence its density remains constant).

 

[student85]

omg vanesch THANK YOU very much. You diserve a medal... or something