One question professor. how does one mathematically measure the velocity in the Eulerian description exactly? If we're looking at/fixated on one point in space (no dx,dy,dz)? Doesn't the Eulerian inherently depend on the Lagrangian to measure velocity?
May be not because Eulerian aims on a particular area or point where fluid particles enters and exits It basically works on conservation of mass... but lagrangian process is like tracking a single fluid particle on its entire journey.
When you are looking at a fixated point in space the acceleration would be 0. Using the Eulerian equation if you made acceleration 0 you are left with Pressure + Specific Weight(Depth)= constant. Which is the formula for a fluid at rest.
Because every component of the acceleration vector can depend on any of those variables. That is the whole point of writing something like V=V(x,y,z,t)=(Ux,Uy,Uz) That is telling you: the velocity vector V depends on x,y,z and t. Since V is composed of the elements Ux, Uy, Uz, any one of those can depend on x,y,z,t. Which is why when you try to calculate something like the total derivative of V with respect to t, you can use the chain rule to write out how time changes with respect to all the elements V depends on. Doing that gives you the material acceleration, as shown in the video.
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One question professor. how does one mathematically measure the velocity in the Eulerian description exactly? If we're looking at/fixated on one point in space (no dx,dy,dz)? Doesn't the Eulerian inherently depend on the Lagrangian to measure velocity?
My guess is that is where the Material derivative comes in.
May be not because Eulerian aims on a particular area or point where fluid particles enters and exits It basically works on conservation of mass... but lagrangian process is like tracking a single fluid particle on its entire journey.
No,
We apply the Eulerian equation to calculate the velocity of the fluid.
When you are looking at a fixated point in space the acceleration would be 0. Using the Eulerian equation if you made acceleration 0 you are left with Pressure + Specific Weight(Depth)= constant. Which is the formula for a fluid at rest.
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If (small u) is velocity component in x direction than why we still represent it as function of (x,y,z,t) ?
Because every component of the acceleration vector can depend on any of those variables. That is the whole point of writing something like V=V(x,y,z,t)=(Ux,Uy,Uz) That is telling you: the velocity vector V depends on x,y,z and t. Since V is composed of the elements Ux, Uy, Uz, any one of those can depend on x,y,z,t. Which is why when you try to calculate something like the total derivative of V with respect to t, you can use the chain rule to write out how time changes with respect to all the elements V depends on. Doing that gives you the material acceleration, as shown in the video.
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Great, a lecture without any content whatsoever...