PX285 - H3 - mechanical perspective of fluids

fluid is a substance which cannot resist shear force without motion, and does not return to its initial state when stress is removed
stress can be defined similar to pressure, ie. force per unit area:

τ = \frac{F}{A}

considering a brick fixed on a surface, $x = 0$ , and an external force, $\vec{F}$ is applied to it
the stress deforms this solid object, where $X$ is the deformation due to the stress

defining strain, $e$ :

τ_{s o l i d} = G e

where, $G$ is a coefficient determined by the binding forces between molecules

in a fluid, there are no binding forces, and the strain, caused by stress, builds up in time

therefore, in fluids,

τ_{f u i l d} = μ \frac{d e}{d t}

where, $μ$ , the coefficient of proportionality is the viscosity

e = \frac{d X}{d y}

τ_{f l u i d} = μ \frac{d}{d t} (\frac{\partial X}{\partial y}) = μ \frac{\partial}{\partial y} (\frac{d X}{d t}) = μ \frac{\partial u_{x}}{\partial y}

where, $\frac{d X}{d t} = u_{x}$ is the horizontal velocity

in summary, as a response to shear forces, a solid deforms whereas a fluid flows
the relaxation time indicates how quickly the stress adjusts to the new sheared state
initially, $t^{'} < t < t_{r e l}$ , the stress is like in a solid
then, for $t > t_{r e l}$ , the fluid starts to flow, and the stress relaxes into a new sheared state