Renovating shear flow ($S(a)=\left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right),a=\Lambda T$)

A renovating shear flow holds a planar shear flow fixed for an interval $T$, then chooses a fresh orientation independently of preceding intervals. Each interval acts on a material line by a rotated copy of the displayed deformation gradient. The orientation law must be specified: independent uniformly distributed orientations define the isotropic model, whereas a biased orientation law generally gives another Lyapunov exponent.

For independent uniform orientations in a renovating shear flow, the incident angle relative to each new shear is uniform even after earlier deformations. One interval stretches a unit material line by $\ell (\theta {)}^2=1+a\sin 2\theta +a^2{\sin }^2\theta$. Write this as $A+B\cos (2\theta -\delta )$, where $A=1+a^2/2$ and $A^2-B^2=1$. The angular logarithmic integral gives $⟨\log \ell ⟩=\frac{1}{2}\log [(A+1)/2]$. Add these independent logarithmic increments to get the displayed rate. At fixed shear rate, small $T$ gives $\mu \sim {\Lambda }^{2}T/8$ and large $T$ gives $\mu \sim \log (|\Lambda |T/2)/T$. The unique positive maximum is at $|\Lambda |T\approx 3.9605826$, obtained from $\log z=2(1-z^{-1})$ with $z=1+a^2/4$.

The logarithm of the stretch after $N$ independent uniform renovations is a sum of independent identically distributed bounded increments. For singular stretching $e^{\gamma }$, $\gamma =\mathrm{arsinh}(|a|/2)$, the increment has Fourier expansion $r+{\sum }_{n\ge 1}(-1{)}^{n+1}{q}^n\cos (2n\theta )/n$, with $q=\tanh \gamma$ and $r=\frac{1}{2}\log (1+a^2/4)$. Orthogonality proves the displayed variance. The central limit theorem gives an approximate central log-normal distribution of ${\lambda }_N$, not an exact description of extreme tails. For $|a|\ll 1$, both the mean and variance of $\log {\lambda }_N$ are $N{a}^2/8$ to leading order. Direct expansion of fixed-power moments gives $⟨{\lambda }_N^p⟩=\exp [Np(p+2)a^2/16+O(N{a}^4)]$; in particular mean stretch grows faster than typical stretch.