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Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

Can you please explain how to solve streamlines, pathlines and streaklines please. I would like to know the process of each one and what steps/ rules I should follow.

Expert Solution

Step 1

Streamlines, pathlines, streaklines of a fluid can be solved by the differential equations of the respective lines.

If a fluid particle is following a particular path then the line drawn by its path is known as path line. So if we consider a change in position of the path of the fluid as:

$\to d \vec{r} = d x \hat{i} + d y \hat{j} + d z \hat{k} . . . . . . . . . . . . . e q u a t i o n 1$

where dx, dy and dz are the change in the position of the respective directions.

Now differentiating the equation 1 with respect to time to find the velocity component in different direction:

$\to \frac{d \vec{r}}{d t} = \frac{d x}{d t} \hat{i} + \frac{d y}{d t} \hat{j} + \frac{d z}{d t} \hat{k} \to \vec{V} = u \hat{i} + v \hat{j} + w \hat{k}$

So the differential equation of the path lines will be:

$\to \frac{d x}{d t} = u; \frac{d y}{d t} = v; \frac{d z}{d t} = w . . . . . . . . . . . . . . e q u a t i o n 2$

And by this differential equation we can find out the path of the fluid particle.

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