MHB Motion of an Element: Understand Rheology Notes

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The discussion revolves around understanding a complex equation in rheology related to the motion of fluid elements. The initial equation for the velocity component at point Q is straightforward, derived from neighboring point P's velocity components. However, the subsequent equation introduces additional terms that seem to involve a combination of derivatives, raising confusion about its derivation and purpose. One participant suggests that the complexity may stem from the author's attempt to express the equation in a different form, possibly related to a cross product. Clarification on the derivation and context of this equation is sought for better comprehension.
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I'm working through my rheology notes and there is a fairly basic part that I don't 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 don't 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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renlok said:
I'm working through my rheology notes and there is a fairly basic part that I don't 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 don't 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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