The motion of axis of symmetry of a spheroidal particle is given by the relation,

$p^{}=p.+\frac{k^2-1}{{}^2+1}(p.E-ppp$:$E)$

Where $p=$ orientation of the particle, $p^{}=$ angular velocity of the particle, $E=$ symmetric component of the velocity gradient tensor, and $=$ anti$-$symmetric component of the velocity gradient tensor. For a particle with an aspect ratio of $k=100,$ calculate the ratio of angular velocity of the particle when it is aligned in the flow direction $($x$-$direction$)$ and when it is aligned in the gradient direction $($y$-$direction$).$ The fluid flow is simple shear flow given by$,$

$v_x={}^{}y$

Where the shear rate, ${}^{}=1s^{-1}.$ Assume the rotary Brownian diffusivity of the particle to be negligible. $($Connect this ratio of angular velocity in the two orientations with the shear thinning behaviour of dilute particle suspension$)$

Consider it a two dinemensional problem

Given that E $=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

and omega $($anti$-$symmetric component of velocity$)=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$