Exploring Advective Acceleration in Fluid Mechanics

[aaaa202]

I am just starting on this subject and I am trying to get a feeling for the concepts. One thing that confuses me a lot is the idea of an advective acceleration. We have that the rate of change of the velocity in a fluid is given by the material time derivative:

Dv/Dt = ∂v/∂t + (v⋅∇)v

Now in undergraduate mechanics an expression like this did not exist. You had a body subjected to some forces and could then calculate acceleration, velocity and position from that using Newtons 2nd law.
You can still use Newtons 2nd law for the above equation if you identify the material time derivative of v with the acceleration and equate this to a force density. But is this valid to assume and to what body exactly do you apply Newtons 2nd law? It is certainly not each particle in the fluid since this is subject to all kinds of fluctuations.
Also which type of forces are included in the effective force density? For example if you have a fluid flowing through a pipe and the cross-section becomes smaller, then the fluid will accelerate, i.e. there is an effective force acting on it. Is this force contained in the above equation?
For me this is all very confusing, since I am only used to undergraduate mechanics, where you have given that some force acts on a body and start from there. What do you know from the start in problems with fluid mechanics?

I hope you understand my confusion.. It seems there is a big step to fluid mechanics from undergraduate mechanics.

 

[Chestermiller]

I am just starting on this subject and I am trying to get a feeling for the concepts. One thing that confuses me a lot is the idea of an advective acceleration. We have that the rate of change of the velocity in a fluid is given by the material time derivative:

Dv/Dt = ∂v/∂t + (v⋅∇)v

Now in undergraduate mechanics an expression like this did not exist. You had a body subjected to some forces and could then calculate acceleration, velocity and position from that using Newtons 2nd law.
You can still use Newtons 2nd law for the above equation if you identify the material time derivative of v with the acceleration and equate this to a force density. But is this valid to assume and to what body exactly do you apply Newtons 2nd law? It is certainly not each particle in the fluid since this is subject to all kinds of fluctuations.

This equation assumes that the body is a continuum, so you can't take it down to the molecular level (since, as you note, you would have to consider fluctuations). But, it works down to the "small fluid parcel" level. Of course, you couldn't take a "rigid body" dv/dt all the way down to the molecular level either, and still use dv/dt of the rigid body to describe the kinematics of the individual molecules.

Also which type of forces are included in the effective force density? For example if you have a fluid flowing through a pipe and the cross-section becomes smaller, then the fluid will accelerate, i.e. there is an effective force acting on it. Is this force contained in the above equation?

There are two sides to the equation ma = F_net. The above equation describes the left side of this equation. The right side of this equation for a fluid includes the effects of pressure forces, gravitational forces, and viscous stresses.

For me this is all very confusing, since I am only used to undergraduate mechanics, where you have given that some force acts on a body and start from there. What do you know from the start in problems with fluid mechanics?

The best way to answer this question is to work some problems in fluid mechanics and see how it all plays out.

Chet