Math and Physics / 数学・物理

What is the material derivative?

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Basic of fluid mechanics

 

 

 

 

 

Questions about the material derivative

 

Unlike rigid-body dynamics, fluid mechanics analyzes the movement of fluid at some fixed positions at a certain time like the figure below. In other words, a velocity is a function of time in rigid-body dynamics, while a velocity in fluid mechanics is expressed as a function of time and position.

 

 

So, we can partially differentiate a velocity with respect to time in fluid mechanics but what does the partial derivative mean? Does it mean an acceleration?

 

 

 

They say that the material derivative of a velocity of fluid is an acceleration of the fluid, but why can we say so? What is the difference from an ordinary derivative in rigid-body dynamics?

 

 

 

If you have those kinds of questions, this article will help you understand what the material derivative is. We are going into a topic about what the partial derivative of a velocity with respect to time means first.

 

 

Does the partial derivative of a velocity with respect to time mean the acceleration?

 

The figure below clearly shows what the partial derivative of a velocity with respect to time is. The time partial derivative means the time rate of change of the velocity of fluid A and the velocity of fluid B, which came into the place where fluid A left. Since the time partial derivative is the time rate of change in velocities of “different materials”, the partial derivative does NOT express the acceleration. The partial derivative is the time rate of change in velocities at a certain position.

 

 

 

Why does the material derivative of a velocity mean the acceleration?

 

As I have already told in this article, a velocity in fluid mechanics is expressed as a function of time and position like in the figure below. It’s because fluid mechanics observes many fluid particle at the same time at a certain time.

 

 

 

Then, how can we know the acceleration of fluid in fluid mechanics expression. We need to observe the movement of one fluid particle like what we find an acceleration in rigid-body dynamics. Since we observe the movement of one fluid particle, we can’t think that time and position are indipendent parametes. We need to think that time and position are related with each oter. In other words, we need to think that position vector is a function of time.

 

 

 

Now you can easily find the acceleration of fluid. Please take a look at the figure below.

 

 

 

I’d like you to pay attention to only x component since there is no space on the screen. In order to differentiate it, we can apply “chain rule” to it.

 

 

 

Did you realize something? Actually, the last expression is the material derivative.

 

 

 

I hope this article helped you understand the reason why the material derivative of a velocity means the acceleration.

 

 

 

 

 

257 thoughts on “What is the material derivative?”

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  245. I urge you stay away from this site. The experience I had with it has been nothing but dismay as well as suspicion of fraudulent activities. Proceed with extreme caution, or even better, find an honest platform for your needs.

    返信

  246. I urge you to avoid this platform. My personal experience with it was purely frustration and suspicion of deceptive behavior. Be extremely cautious, or better yet, seek out an honest platform for your needs.

    返信

  247. I urge you steer clear of this platform. My own encounter with it was only disappointment and suspicion of fraudulent activities. Be extremely cautious, or even better, look for a more reputable platform for your needs.

    返信

  249. I highly advise steer clear of this site. The experience I had with it was only frustration as well as doubts about deceptive behavior. Be extremely cautious, or alternatively, look for an honest platform to meet your needs.

    返信

  250. I highly advise stay away from this platform. The experience I had with it has been nothing but frustration and suspicion of deceptive behavior. Proceed with extreme caution, or better yet, look for a trustworthy site to meet your needs.

    返信

  251. I highly advise stay away from this site. My own encounter with it has been purely dismay as well as doubts about deceptive behavior. Be extremely cautious, or even better, find a trustworthy platform for your needs.

    返信

  252. I urge you to avoid this platform. My personal experience with it has been nothing but dismay and concerns regarding scamming practices. Proceed with extreme caution, or better yet, find a trustworthy platform to fulfill your requirements.

    返信

  253. I highly advise to avoid this platform. My own encounter with it has been only disappointment along with suspicion of scamming practices. Proceed with extreme caution, or even better, seek out a trustworthy platform to meet your needs.

    返信

  254. I highly advise stay away from this platform. My personal experience with it has been purely disappointment along with suspicion of fraudulent activities. Proceed with extreme caution, or even better, look for an honest site to fulfill your requirements.

    返信

  255. I highly advise stay away from this platform. My personal experience with it was only frustration along with concerns regarding deceptive behavior. Be extremely cautious, or better yet, find an honest site for your needs.

    返信

  256. I highly advise to avoid this platform. The experience I had with it was purely frustration as well as doubts about fraudulent activities. Proceed with extreme caution, or alternatively, look for a more reputable service for your needs.

    返信

  257. I urge you to avoid this platform. My personal experience with it has been nothing but dismay and suspicion of scamming practices. Exercise extreme caution, or even better, find a more reputable site for your needs.

    返信

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