Why do we take convective derivative in fluid dynamics

[Aniket1]

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.

 

[Aniket1]

What is a fluid element in fluid dynamics

What do you mean by a fluid element? What is its velocity?
What is your frame of reference while dealing with fluid mechanical problems?

 

[Studiot]

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


$$\begin{array}{l} \frac{{Df}}{{Dt}} = \frac{{\partial f}}{{\partial x}}\frac{{dx}}{{dt}} + \frac{{\partial f}}{{\partial y}}\frac{{dy}}{{dt}} + \frac{{\partial f}}{{\partial z}}\frac{{dz}}{{dt}} + \frac{{\partial f}}{{\partial t}} \\ \frac{{Df}}{{Dt}} = \frac{{\partial f}}{{\partial t}} + u\frac{{\partial f}}{{\partial x}} + v\frac{{\partial f}}{{\partial y}} + w\frac{{\partial f}}{{\partial z}} = \frac{{\partial f}}{{\partial t}} + ({\bf{u}}.\nabla )f \\ \end{array}$$

This is your convective derivative.

Does this help?

 

[vanhees71]

From a modern point of view, fluid dynamics is an effective theory for a many-body system. In many situations, for a macroscopic situation of very many interacting particles, one realizes that there are different scales for correlations, described by many-body Green's functions, which allow for an effective description of collective observables. This is the case when changes of such collective observables with time and space are "slow" compared to the rapid fluctuations of microscopic variables.

To discribe the dynamics of the collective observables, then one can average over the microscopic observables over time intervals and volume elements large compared to the typical time scale and spatial scale over which the microscopic observables change but that are small compared the typical space-time scales over which the collective observables change considerably.

Formally this coarse graining is achieved by doing a Fourier transformation of the one-body Green's functions with respect to the spact-time difference of its coordinates. The resulting function is a function of the average space-time position of its arguments and four-momentum from the Fourier transform. This function is also known as the Wigner function, and it obeys a quantum-master equation known as the Kadanoff-Baym equations. These are still full quantum equations and very hard to solve numerically as they are. Only for very simple toy-model systems like $\phi^4$ theory this has been achieved in the recent years.

Now the coarse graining comes into the game: if there is a micro-macro separation of scales as described above, the Wigner functions are slowly varying functions with respect to its space-time and four-momentum arguments. Then you can perform a gradient expansion, which leads from the Kadanoff-Baym equation to semi-classical transport equations, very similar to the Blotzmann transport equation, which is derived in similar ways from classical many-body theory. Another approximation that is often applicable is the quasi-particle approximation. Then the imaginary part of the retarded Green's function, which is the spectral function, is very sharply peaked as a function of energy and momentum. Then one can approximate it by $\delta[p^0-E(x,\vec{p})]$, i.e., one has a sharp energy-momentum dispersion relation similar to free particles in the vacuum. This finally makes a complete analogy with classical transport equations. The only difference to the Boltzmann equation is that in the collision term there appear the Bose enhancement and Pauli blocking factors, leading to the corresponding Bose-Einstein and Fermi-Dirac distributions instead of the classical Boltzmann approximation in the equilibrium limit. At this level the corresponding transport equations are called Boltzmann-Uehling-Uhlenbeck (BUU) equations.

This allows for an interpretation of the gradient expanded Wigner function as a phase-space distribution (particularly only after coarse graining it is positive semidefinite everywhere when the initial conditions are chosen appropriately).

Sometimes even this description is still too complicated. Now, fortunately, it often happens that one has it to do with strongly interacting quasi particles, where the system is locally, i.e., in each fluid cell, very rapidly in (local) thermal equilibrium again. Then the BUU equation can be approximated by using the approximation that the phase-space distribution is given by the appropriate (Bose or Fermi) equilibrium distributions with the temperature, and chemical potential, and average velocity (momentum) of the fluid element being functions of space and time. Then the BUU equations go over to the Euler equations of an ideal fluid, given by the conservation laws for energy, momentum, and conserved charges (or in the non-relativistic case particle numbers). This implies that the entropy stays constant (adiabatic change of the system) and is all the time in local thermal equilibrium.

This approximation can be systematically improved by taking into account small deviations from local thermal equilibrium, which leads in first-order approximation to the Navier-Stokes equation (viscous hydro dynamics), which however has its problems in the relativistic case and has to be extended to at least the second order, leading to equations very similar to the Israel-Stuart equations of relativistic viscous hydrodynamics. Very recently there have been extensions of these scheme to even higher orders and detailed comparisons to BUU-like cascade simulations in the description of the quark-gluon plasma in heavy-ion-collision physics.

If you are interested in more details about this ongoing research, I can give you references.

 

[arildno]

In the mathematical formulation, the "fluid element" is a point particle with associated mass that can be subject to forces according to Newton's laws of motion.
This is, of course, a simplification of reality.
Read the above comment from vanhees71 for a more reality-oriented approach.

 

[Aniket1]

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


$$\begin{array}{l} \frac{{Df}}{{Dt}} = \frac{{\partial f}}{{\partial x}}\frac{{dx}}{{dt}} + \frac{{\partial f}}{{\partial y}}\frac{{dy}}{{dt}} + \frac{{\partial f}}{{\partial z}}\frac{{dz}}{{dt}} + \frac{{\partial f}}{{\partial t}} \\ \frac{{Df}}{{Dt}} = \frac{{\partial f}}{{\partial t}} + u\frac{{\partial f}}{{\partial x}} + v\frac{{\partial f}}{{\partial y}} + w\frac{{\partial f}}{{\partial z}} = \frac{{\partial f}}{{\partial t}} + ({\bf{u}}.\nabla )f \\ \end{array}$$

This is your convective derivative.

Does this help?

Wht do we need to work in a frame relative to the fluid?

 

[Studiot]

Wht do we need to work in a frame relative to the fluid?

I'm not sure I understand the question but the derivatives allow us to use take a small element of fluid and calculate the momentum, mass, energy fluxes into and out of it. This, in turn, allows us to calculate its interaction with neighbouring elements of fluid in terms of forces, shears, etc. These are both boundary and body effects.

In normal fluid mechanics we may consider the body force of gravity. In your case (MHD) you will be adding another body force due magnetic interaction. There may also be forces dues to electrostatic interaction.

 

[arildno]

Wht do we need to work in a frame relative to the fluid?

Usually, it is more practical, not the least for measurements.
You place a thermometer or velocimeter at ONE point, and measures whatever temperature or velocity happens to be there, as a function of time.
But, this will be a series of measurements of the different temperatures and velocities DIFFERENT "elements of the fluids" have, rather than the time series of a SINGLE element of the fluid.

However, Newton's laws are formulated as how forces affect SINGLE objects, and we have to re-express this in the equivalent field formulation in order to make the practical predictions, confirmations or falsifications.
---------------------------------------------------------
Particle-focused Lagrangian mechanics, opposed to the above Eulerian field formulation will in special cases be of more interest, but not generally.

 

[pervect]

I know that physicists define the stress energy tensor differently than fluid mechanics does, and the difference is convective terms.

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.

My take on this is that it isn't actually essential to use comoving frames , but it's convenient, especially for engineering work.

Just my $.02, I haven't thought about the matter deeply, but I thought I'd point out one field that does not use the convective terms.

 

[Chestermiller]

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.

In fluid mechanics, we treat the material as a continuum, rather than a collection of individual particles. The fluid velocity (and other parameters) are regarded as continuous functions of spatial position and time. Within this framework, it is particularly convenient to work using an Eulerian frame of reference which is fixed in space, and to let the fluid move and deform as it goes past. This makes solving fluid mechanics problems much easier than traveling along with the individual fluid elements (Lagrangian frame of reference). How come? Because the boundary conditions are usually applied at fixed spatial locations: e.g., the pressure at the pipe inlet, the pressure at the pipe outlet, the locations of rigid walls of a vessel. All these locations would be moving if we traveled with the fluid (and in different ways for different fluid elements), and would be much harder to work with mathematically. Using an Eulerian framework also simplifies the math in setting up the partial differential equations for the differential material and momentum balances: mass in minus mass out = accumulation. The convective derivative arises naturally from the Eulerian formulation, and represents the rate of change of a parameter with time following the fluid element. It is the same as the lagrangian derivative, and allows us to determine the lagrangian derivative even though we are using an eulerian framework.

 

[Chestermiller]

In fluid mechanics, we treat the fluid as a continuum. A fluid element is small parcel of fluid that is moving along with the flow.

We use an Eulerian frame of reference which is fixed spatially. This allows us to examine fluid movement as we usually would in the real world. We look at a stream, and see the flow going by. We do not have to jump into move along with the fluid in order to analyze the differential force and mass balances. We express these differential balances in terms of input - output = accumulation.

 

[Studiot]

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.

 

[Chestermiller]

This is a very interesting thread, and since some of the responders have brought up the subject of special relativity, I would like to comment a little on the relationship between all this and special relativity. In my judgement, Einstein must have had more than a nodding acquaintance with continuum and fluid mechanics when he developed special relativity.

The convected derivative (aka the material derivative) bears a close resemblance to the derivative of with respect to proper time. It is the same as the proper time derivative divided by the relativity factor.

The divergence of the stress energy tensor being equal to zero is equivalent to the fluid mechanics "continuity equation" (differential mass balance equation) in combination with the fluid mechanics "Euler momentum equation" (differential force balance equation). The time component of del dot T = 0 (multiplied by c²) gives the continuity equation, and the spatial components give the Euler equations.

Also, note what happens to the material derivative expression if you multiply the numerator and denominator of the partial derivative with respect to time by the speed of light c. Aside from a factor of γ, you get the 4 velocity vector dotted with the 4D del operator.

From my experience with fluid mechanics, one thing missing from the del dot T = 0 is that, in practice, the right hand side of the equation is not actually equal to zero. In fluid mechanics, the right hand side of the momentum equation is the divergence of the mechanical stress tensor. The stress tensor is related to the kinematics of the fluid (or solid) deformation through the rheological constitutive equation for the specific material. Examples of the rheological constitutive equation are Hooke's law for a solid (in tensorial form) and Newton's law of viscosity (in tensorial form). It is unfortunate that, in relativity, they chose to call T the stress energy tensor, since in this context, it has nothing to do with the stress tensor of mechanics. I have never studied the subject of relativistic rheology and am unfamiliar with the modifications that need to be made to the right hand side of the equation in order to make it compatible with special relativity. I'm guessing that some work must have been done in this area.

 

[dextercioby]

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 $\times$ and $\cdot$, i.e. \times and \cdot commands for the cross and scalar product, respectively.

 

[Studiot]

For the LaTex code, I suggest using the × and ⋅, i.e. \times and \cdot commands for the cross and scalar product, respectively

Kind of you to offer this advice, I am a tex ignoramus.

I can't even get your post to copy correctly into quotes.

I use Mathtype, a pretty poor program, but better than any other I've seen and take what little it gives me.
I'm feeling quite chuffed that I've just managed to work out how to embolden the vectors.