Mechanics of fluids

  • #1
660
16

Homework Statement


Small part of fluid which in the moment ##t=0## was at the point ##(X_1,X_2,X_3)## in some other moment is in the point ##(x_1,x_2,x_3)##. Where
[tex]x_1(t)=X_1[/tex]
[tex]x_2(t)=X_2+\sin \pi t\sin \pi X_1[/tex]
[tex] x_3(t)=X_3[/tex]
We know that the piece of the fluid is in the t=0 in line which connects points ##(0,0,0)## i ##(1,0,0)##. Plot piece of the fluid at the moment ##t=\frac{1}{2},t=1## and ##t=\frac{3}{2}##.


Homework Equations


As far as I undestand this is Lagrange formalism.


The Attempt at a Solution


First of all I want to try to understand the problem. So that small part is one of the point in the line? Right? And every point in the line which connects points ##(0,0,0)## i ##(1,0,0)## changes with equations
[tex]x_1(t)=X_1[/tex]
[tex]x_2(t)=X_2+\sin \pi t\sin \pi X_1[/tex]
[tex] x_3(t)=X_3[/tex]?
Right?
 

Answers and Replies

  • #2
21,155
4,671
Imagine a vector ΔS joining the point (0,0,0) to the point (1,0,0) at time t = 0. As time progresses the material points at the two ends of the vector move to new locations, so ΔS changes. You want to find the vector ΔS joining these same two material points at later times. Note that the material point at the origin stays at the origin. So you only need to determine where the other material point is located as a function of time. You need to plot this, and show the vector joining the two material points at the different times. This will illustrate visually a little bit about how the fluid is deforming.

Chet
 

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