Material Derivative (Convective Derivative Operator)

1. Mar 15, 2015

thehappypenguin

Hi,

I've learned that material derivative is equal to local derivative + convective derivative, but can't seem to find out which way the convective derivative acts, like for example in velocity fields:

The equation my teacher gave us was (with a and v all/both vectors):
Acceleration = material derivative of velocity = local acceleration + convective acceleration
∴ a = Dv/Dt = dv/dt + v⋅∇v

My question is whether the convective acceleration term (v⋅∇v) works like:
1. (v⋅∇)v, which in my understanding is the (v⋅∇) operator working on the vector v
2. v⋅(∇v), which I take as the grad operator working on the vector v, dotted with the vector v outside the brackets
3. Or is it that Options 1 and 2 are the same thing anyway?

[Side note: Sorry, I'm new to PF and don't know how to use the equation symbols or LaTeX.]

- TheHappyPenguin

2. Mar 16, 2015

joshmccraney

Hi thehappypeguin!

1) and 2) are equivalent expressions, so technically 3) is correct. I'm sure someone else here can easily send you to a link where you can figure out latex, if that's what you're trying to figure out.

3. Mar 16, 2015

dextercioby

Even if you can't write in LaTex, you can still use a writing to distinguish vectors from scalars. So write v for the velocity vector and v for its modulus (or projection onto a coordinate axis). :)

4. Mar 22, 2015

thehappypenguin

Thank you! I'll look up latex when I have time :)

About the Material Derivative, I read some more about it online and found that in Pgs 1-3 of http://www.chem.mtu.edu/~fmorriso/cm4650/lecture_6.pdf, ∇v is a tensor which would make 2) different from 1) as 1) is a vector. Is this accurate?

Thank you again!
- TheHappyPenguin :)

5. Mar 22, 2015

stevendaryl

Staff Emeritus
Yes, $\nabla \vec{v}$ is a tensor, but no, the two expressions are not different. The dot product of a vector with a tensor (of the right type) produces a vector. So both

$(\vec{v} \cdot \nabla) \vec{v}$ and $\vec{v} \cdot (\nabla \vec{v})$ produce vectors.