The physical meaning of the PDE?

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Homework Help Overview

The discussion revolves around understanding the physical meaning of a partial differential equation (PDE) related to fluid dynamics. Participants are exploring the implications of the equation, its variables, and the context in which it is applied.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants question the physical interpretation of the variables involved, particularly the units of distance and velocity. There is a focus on understanding the background of the equation and the significance of boundary conditions. Some participants suggest that the equation may not be a PDE but rather an ordinary differential equation (ODE).

Discussion Status

The discussion is ongoing, with participants providing hints and insights into the interpretation of the equation. There is recognition of the need to clarify the meaning of the derivatives and the physical context of the fluid's motion. Multiple interpretations are being explored, particularly regarding the relationship between the variables and their units.

Contextual Notes

Participants note the specific units for velocity and distance, as well as the domain of the problem. There is an emphasis on the complexity of fluid dynamics and the importance of understanding the reasoning behind the differential equation.

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the physical meaning of the PDE?!

Homework Statement



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Homework Equations



How can I know the physical meaning of the following partial differential equation?!

​
 
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You cannot, It can refer to anything.
In physics the first thing is the units. What is the unit of x and u?
And what is the background of the equation? Why the boundary conditions have been choosen like this?
After you have the answer to these questions, you can start wondering what would the equation means.
 


You can't. Mathematics is not physics and equations may have physical applications but separate from the application, there is no "physical" meaning. The crucial point is the meaning or interpretation of u and x themselves. As magwas suggested, you can learn something about the that by looking at the units x and u have.

By the way, this is NOT a "partial" differential equation. Since there is only one independent variable, it is an ordinary differential equation.
 


magwas said:
You cannot, It can refer to anything.
In physics the first thing is the units. What is the unit of x and u?
And what is the background of the equation? Why the boundary conditions have been choosen like this?
After you have the answer to these questions, you can start wondering what would the equation means.


thank you for the hint.

u: velocity of a fluid m/sec
x: Distance m
[-1,1] : the Domain.

Can you help me now wondering what would the equation means?!​
 


HallsofIvy said:
You can't. Mathematics is not physics and equations may have physical applications but separate from the application, there is no "physical" meaning. The crucial point is the meaning or interpretation of u and x themselves. As magwas suggested, you can learn something about the that by looking at the units x and u have.

By the way, this is NOT a "partial" differential equation. Since there is only one independent variable, it is an ordinary differential equation.



I'm sorry :blushing: ...that's right... it is an ODE.​
 


Well, let's see. We have a fluid. It moves relative to our 2m wide/long something.
It would be helpful to know the direction of the speed relative to the something.
If the sopeed is parallel to the something (I could think of a pipe which have the same cross area at -1 and 1 (but fluid dynamics is much more complicated than that)), then we have u'(x) with the units of 1/s, some kind of frequency.
If it is not paralell, then u'(x) have units of \frac{m_{y}}{s m_{x}} (just not to confuse length in one direction to length in other direction.
Similarly u''(x) is either 1/ms or \frac{m_{y}}{s m_{x}^2}.
Anyway, we should have a magical constant A of units m^2, thus the diff equation is really
A u''(x) = u(x)

Now we should try to figure out the physical meaning of u''(x). It is the change of change of speed according to distance, which I honestly don't know what could mean. Maybe looking up equations of fluid dynamics or knowing more about the reasoning which led to this diff equation would help to understand more.
 


magwas said:
Well, let's see. We have a fluid. It moves relative to our 2m wide/long something.
It would be helpful to know the direction of the speed relative to the something.
If the sopeed is parallel to the something (I could think of a pipe which have the same cross area at -1 and 1 (but fluid dynamics is much more complicated than that)), then we have u'(x) with the units of 1/s, some kind of frequency.
If it is not paralell, then u'(x) have units of \frac{m_{y}}{s m_{x}} (just not to confuse length in one direction to length in other direction.
Similarly u''(x) is either 1/ms or \frac{m_{y}}{s m_{x}^2}.
Anyway, we should have a magical constant A of units m^2, thus the diff equation is really
A u''(x) = u(x)

Now we should try to figure out the physical meaning of u''(x). It is the change of change of speed according to distance, which I honestly don't know what could mean. Maybe looking up equations of fluid dynamics or knowing more about the reasoning which led to this diff equation would help to understand more.



Thank you for this explanation
and I will try to find more about "change of change of speed according to distance".​
 

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