The discussion centers on the application of the convective derivative in fluid dynamics, particularly in the context of plasma physics. A fluid element is defined as a small volume where properties can be considered concentrated at a point, allowing for integration over a larger volume. Two frames of reference are utilized: absolute coordinates for global analysis and local coordinates for analyzing individual fluid elements. The convective derivative is expressed mathematically as Df/Dt, which accounts for changes in properties as fluid elements move through space and time.
PREREQUISITESThis discussion is beneficial for physicists, engineers, and students specializing in fluid dynamics, plasma physics, and magnetohydrodynamics, as well as anyone interested in the mathematical modeling of fluid behavior.
Wht do we need to work in a frame relative to the fluid?Studiot said:Good morning Aniket,
I will try to answer both your posts together here. You should really ask a mod to combine them.
A fluid element is another word for the same concept as particle in dynamics or mechanics.
That is it is small enough for all the mass or other property of interest to be considered concentrated at one point. Then we can integrate this property over a volume by summing it for all the elements - possibly numerically possibly by formula if we can find one.
It's dimensions are non zero but small enough for its internal structure to be unimportant to the problem in hand.
As regards frames of reference; two are used in fluid mechanics.
We can use absolute coordinates by setting an XYZ coordinate system and tracking the progress of a given fluid elelement. For example an element that enters a pipe, passes down the pipe and exits the other end with the pipe axis aligned in the x direction. In your case the pipe will have magnetic walls of course. Such analyses use streamlines or streamtubes and produce global results for the overall fluid. Global analysis equations are characterised by partial derivatives with respect to the global XYZand time axes.
This however is of little help in calculating local effects so we also employ a local coordinate system, based on the individual fluid element. We call this method of analysis 'following the fluid' and the derivatives are given special symbols (capital D) to mark the difference.
So if f(x,y,z,t) is a function of interest, say density or a component velocity, then the rate of change of this quantity at a fixed point in space as the fluid flows past (a succession of different element) is ∂f/∂t.
If, however we want to know what happens to a particular fluid element we take the rate of change 'following the fluid'
\frac{{Df}}{{Dt}} = \frac{d}{{dt}}f\left\{ {x(t),y(t),z(t),t} \right\}
Where dx/dt = u, dy/dt = v, dz/dt = w are the components of the local velocity vector u
Since by chain rule
\begin{array}{l}<br /> \frac{{Df}}{{Dt}} = \frac{{\partial f}}{{\partial x}}\frac{{dx}}{{dt}} + \frac{{\partial f}}{{\partial y}}\frac{{dy}}{{dt}} + \frac{{\partial f}}{{\partial z}}\frac{{dz}}{{dt}} + \frac{{\partial f}}{{\partial t}} \\ <br /> \frac{{Df}}{{Dt}} = \frac{{\partial f}}{{\partial t}} + u\frac{{\partial f}}{{\partial x}} + v\frac{{\partial f}}{{\partial y}} + w\frac{{\partial f}}{{\partial z}} = \frac{{\partial f}}{{\partial t}} + ({\bf{u}}.\nabla )f \\ <br /> \end{array}
This is your convective derivative.
Does this help?
Wht do we need to work in a frame relative to the fluid?
Aniket1 said:Wht do we need to work in a frame relative to the fluid?
wiki said:Warning: In solid state physics and fluid mechanics, the stress tensor is defined to be the spatial components of the stress–energy tensor in the comoving frame of reference. In other words, the stress energy tensor in engineering differs from the stress energy tensor here by a momentum convective term.
Aniket1 said:I am studying plasma physics. While treating the plasma as a fluid, we replace the generla time derivative of velocity of any particle by the convective derivative.
I would like to know the reason and also what is the frame of reference when we work with fluid dynamics.
Studiot said:You did ask about applications.
It would be rather unfair to suggest the the D derivatives are of little or no value. They would not have been introduced if that were so.
Nor would if be correct to say it is an 'either or' situation. We can naturally move between analysis methods as convenient.
The equations I gave before lead, quite naturally to Eulers momentum equation
\frac{{D{\bf{u}}}}{{Dt}} = - \frac{1}{\rho }\nabla p + g
This in turn leads back to the momentum equation in its other guise
\frac{{\partial {\bf{u}}}}{{\partial t}} + (\nabla x{\bf{u}})x{\bf{u}} = - \nabla \left( {\frac{p}{\rho } + \frac{{{{\bf{u}}^2}}}{2} + \chi } \right)
Either lead to the equations of small amplitude sound waves.
The vorticity equation can also be used in either format
\frac{{D{\bf{\omega }}}}{{Dt}} = \left( {{\bf{\omega }}{\bf{.}}\nabla } \right){\bf{u}}or
\frac{{\partial {\bf{\omega }}}}{{\partial t}} + \left( {{\bf{u}}{\bf{.}}\nabla } \right){\bf{\omega }} = \left( {{\bf{\omega }}{\bf{.}}\nabla } \right){\bf{u}}
The D derivative also leads to Reynolds Transport Theorem and Kelvins Circulation theorem.
For the LaTex code, I suggest using the × and ⋅, i.e. \times and \cdot commands for the cross and scalar product, respectively