Longitudinal strain rate derivation

[enc08]

Hi,

The attachment below is about strain rate in fluids*. It shows how the strain rate $$d\phi/dt$$ is related to the velocity field derivative $$du/dx$$ when you stretch the element in x (i.e. longitudinal strain).

It has no intermediate steps, and I can't see how the angle has been related to the velocity field (or where the 0.5 factor comes from).

Thanks very much.

*From website: http://www.see.ed.ac.uk/~johnc/teaching/fluidmechanics4/2003-04/fluids5/stress.html

Sorry about the small picture - that's how it is on the website.

 

[enc08]

I would assume there's a simple trigonometric identity to apply. What makes this difficult for me is that phi is present in both the x and y directions, whilst the equation somehow describes it through the x-axis.