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Motion of an element

renlok

New member
Feb 24, 2012
1
I'm working through my rheology notes and there is a fairly basic part that I dont understand at all.

If you have two neighbouring points in a liquid P(x,y,z) and $$Q(x + \Delta{x}, y + \Delta{y}, z + \Delta{z})$$ now if the velocity components of P are given as (u, v, w) then $$u_Q = u_P + \frac{\delta{u}}{\delta{x}}\Delta{x} + \frac{\delta{u}}{\delta{y}}\Delta{y} + \frac{\delta{u}}{\delta{z}}\Delta{z}$$ which i understand its a quite simple derivation and we have similar equations of v & w

But then he just goes on to say

"Then $$u_Q = u_P + \frac{1}{2}(\frac{\delta{u}}{\delta{z}} - \frac{\delta{w}}{\delta{x}})\Delta{z} - \frac{1}{2}(\frac{\delta{v}}{\delta{x}}-\frac{\delta{u}}{\delta{y}})\Delta{y} + \frac{\delta{u}}{\delta{x}}\Delta{x} + \frac{1}{2}(\frac{ \delta {u}}{\delta {y}}+\frac{\delta {v}}{ \delta {x} })\Delta{y} + \frac{1}{2}(\frac{\delta{u}}{\delta {z}}+\frac{ \delta {w}}{\delta{x}})\Delta{z}$$"

which i dont understand what this is let alone how it was derived I also couldn't find it anywhere on the net

Cheers for any insight you may have
 
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Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,189
I'm working through my rheology notes and there is a fairly basic part that I dont understand at all.

If you have two neighbouring points in a liquid P(x,y,z) and $$Q(x + \Delta{x}, y + \Delta{y}, z + \Delta{z})$$ now if the velocity components of P are given as (u, v, w) then $$u_Q = u_P + \frac{\delta{u}}{\delta{x}}\Delta{x} + \frac{\delta{u}}{\delta{y}}\Delta{y} + \frac{\delta{u}}{\delta{z}}\Delta{z}$$ which i understand its a quite simple derivation and we have similar equations of v & w

But then he just goes on to say

"Then $$u_Q = u_P + \frac{1}{2}(\frac{\delta{u}}{\delta{z}} - \frac{\delta{w}}{\delta{x}})\Delta{z} - \frac{1}{2}(\frac{\delta{v}}{\delta{x}}-\frac{\delta{u}}{\delta{y}})\Delta{y} + \frac{\delta{u}}{\delta{x}}\Delta{x} + \frac{1}{2}(\frac{\delta {u}}{\delta {y}}+\frac{\delta {v}}{\delta {x}})\Delta{y} + \frac{1}{2}(\frac{\delta {u}}{\delta {z}}+\frac{\delta {w}}{\delta {x}})\Delta{z}$$"

which i dont understand what this is let alone how it was derived I also couldn't find it anywhere on the net

Cheers for any insight you may have
I'll take a crack at it. All the author has done to get this last line is to add and subtract various elements. If you simplify this last line, you get the earlier line. This step must be a setup: the author must be trying to recognize this more complicated expression as something else - it looks to me like a cross product, but without more context, I don't know what it is.
 
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