{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

phys documents (dragged) 36

phys documents (dragged) 36 - Chapter 9 Transport phenomena...

This preview shows page 1. Sign up to view the full content.

Chapter 9 Transport phenomena 9.1 Mathematical introduction An important relation is: if X is a quantity of a volume element which travels from position r to r + dr in a time dt , the total differential dX is then given by: dX = X x dx + X y dy + X z dz + X t dt dX dt = X x v x + X y v y + X z v z + X t This results in general to: dX dt = X t + ( v · ) X . From this follows that also holds: d dt Xd 3 V = t Xd 3 V + X ( v · n ) d 2 A where the volume V is surrounded by surface A . Some properties of the operator are: div( φ v ) = φ div v + grad φ · v rot( φ v ) = φ rot v + (grad φ ) × v rot grad φ = 0 div( u × v ) = v · (rot u ) - u · (rot v ) rot rot v = grad div v - ∇ 2 v div rotv = 0 div grad φ = 2 φ 2 v ( 2 v 1 , 2 v 2 , 2 v 3 ) Here, v is an arbitrary vector field and φ an arbitrary scalar field. Some important integral theorems are: Gauss: ( v
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern