Interpretation of spatially dependent convective derivative

[Mzzed]

<Moderator's note: Moved from the homework forum.>

Homework Statement

The following equation is one of a few equations that describe a plasma model. The left hand side is the part I am having trouble with in that I can't seem to visualise what (V dot delV) actually does. My physics teacher has described the situation in comparison to a flowing river. He said to imagine a frozen snap shot in time so that as you look further and further up the river, the velocity will change depending on location, not time. eg a section of the river with a waterfall with have large velocity whilst a flat section of the river will have low velocity. This helps describe how there is a velocity gradient (not acceleration) but then how does the dot product with velocity work to create this scenario?

I am also confused as to the order of operations within the brackets as the gradient of v is a vector and then the dot product of v with the gradient of v would produce a scalar. OR if you choose the other order, the divergence of v is a scalar and then multiplied with the vector v, the result is a vector. Obviously something I have just said is wrong but I am unsure what.

Thanks for your help!

Homework Equations

The Attempt at a Solution

There is no real solution to my question, I just need to find a way to imagine the above mentioned section of this equation and to understand which order to use so that I can better understand what it is actually doing.

 

[fresh_42]

$\rho \left( (\vec{v} \cdot \nabla )\vec{v} \right)$ would be the better notation.
Here's an example: https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/
(about the middle of the page above Navier-Stokes)

 

[Chestermiller]

For a steady state flow of a fluid (nothing changing with time at at each location in the fluid) , the term in parenthesis can be interpreted as the local acceleration vector of the fluid parcels, and the left hand side can be interpreted as the mass times acceleration per unit volume (locally).

The mathematical form written by fresh_42 is the more conventional (and easier to understand) form of the term in question, but both forms are acceptable. In the case of the original form, the gradient of velocity is a second order tensor (called the velocity gradient tensor) which, when dotted with the velocity vector, gives the acceleration vector for a steady state flow.

 

[Mzzed]

Thankyou! So to make sure I understand, are you saying that the left hand side of the equation is essentially describing the local force vectors within the plasma so that at any point in the plasma, this term will describe that point's force vector? I say force because you mentioned mass times local acceleration but I'm not really sure if I've interpreted this correctly?

The other thing I am stuck with then is why the part in brackets is the local acceleration vector? from my limited understanding of vector calculus del dot v gives the divergence. Why then does this multiply by the velocity to become acceleration?

 

[Chestermiller]

Thankyou! So to make sure I understand, are you saying that the left hand side of the equation is essentially describing the local force vectors within the plasma so that at any point in the plasma, this term will describe that point's force vector? I say force because you mentioned mass times local acceleration but I'm not really sure if I've interpreted this correctly?

The other thing I am stuck with then is why the part in brackets is the local acceleration vector? from my limited understanding of vector calculus del dot v gives the divergence. Why then does this multiply by the velocity to become acceleration?

The left hand side is the net force per unit volume.

Regarding your second question, the velocity v is a function of time and position: $$\mathbf{v}=\mathbf{v}(t,\mathbf{x})$$So the time derivative of the velocity, following the motion of the material is (formally, from the chain rule), $$\mathbf{a}=\frac{\partial \mathbf{v}}{dt}+\frac{\partial \mathbf{v}}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=\frac{\partial \mathbf{v}}{dt}+\mathbf{v}\frac{\partial \mathbf{v}}{\partial \mathbf{x}}$$When this latter equation is expressed in proper mathematical form (not formally), it becomes$$\mathbf{a}=\frac{\partial \mathbf{v}}{dt}+\mathbf{v}\centerdot (\nabla \mathbf{v})$$
The last term is not the divergence of the velocity vector. There is no dot in this term. It is a second order tensor called the velocity gradient tensor.

 

[vanhees71]

The last term in the last equation should read $(\boldsymbol{v} \cdot \boldsymbol{\nabla}) \boldsymbol{v}$. I don't understand the derivation at all since $\boldsymbol{x}$ is independent of $t$ to begin with. I switch to the arrow notation for vectors from now on.

The point is that you want a "material time derivative", i.e., the derivative of the change of a fluid element consisting of one and the same particles. This is most easily achieved, using the Euler description of fluid mechanics. For that purpose you define the trajectories of material fluid elements as
$$\vec{x}=\vec{x}(t,\vec{x}_0), \quad \vec{x}(t=0,\vec{x}_0)=\vec{x}_0.$$
You define the fluid element by the particles within a macroscopically small volume that are located at positions $\vec{x}_0$ at the initial time $t=0$. Then $\vec{x}(t,\vec{x}_0)$ is the location of these specific particles at any later time.

Now in the Lagrangian description, which is the more convenient one, you consider quantities of the fluid in the sense of a field theory, i.e., you look at a fixed point $\vec{x}$, what are the properties (velocities, accelerations, etc.) of the particles present there at time $t$. Now if you have given such a quantity you have function $f(t,\vec{x})$. But if you want "material time derivatives", describing the temporal change of the quantity along the trajectory of the fixed individual particles you have to define
$$\mathrm{D}_t f(t,\vec{x})= \frac{\mathrm{d}}{\mathrm{d} t} f[t,\vec{x}(t,\vec{x}_0)]=\partial_t f[t,\vec{x}(t,\vec{x}_0)]+\partial_t \vec{x}(t,\vec{x}_0) \cdot \vec{\nabla} f[t,\vec{x}(t,\vec{x}_0).$$
Now obviously
$$\vec{v}(t,\vec{x}_0)=\partial_t \vec{x}(t,\vec{x}_0) \equiv \vec{v}(t,\vec{x})$$
is the fluid-velocity field in the usual Lagrangian description, and thus you get
$$\mathrm{D}_t f(t,\vec{x})=\partial_t f(t,\vec{x}) + \vec{v}(t,\vec{x}) \cdot \vec{\nabla} f(t,\vec{x}).$$
Now if you want Newton's Law you need the acceleration in the sense of the material time derivative of the velocity, i.e., the accerlation field reads
$$\vec{a}(t,\vec{x})=\mathrm{D}_t \vec{v}(t,\vec{x})=\partial_t \vec{v}t,\vec{x}) + [\vec{v}(t,\vec{x}) \cdot \vec{\nabla}]\vec{v}(t,\vec{x}),$$
i.e., you apply the material time derivative to each of the Cartesian components of the velocity field. In Cartesian Ricci notation this reads
$$a_j=\partial_t v_j + v_k \partial_k v_j,$$
where the Einstein summation convention is applied.