Linear and second-order response of the differential pom-pom model ($G_{\mathrm{rel}}(t)=\frac{G}{3}{e}^{-t/{\tau }_1},{\mu }_0=\frac{G{\tau }_1}{3},{\psi }_1=\frac{2G{\tau }_1^2}{3},{\psi }_2=0$)

Linearizing the differential pom-pom model about its relaxed state leaves no first-order stretch perturbation in incompressible flow. Its anisotropic orientation obeys $\dot{a}+a/{\tau }_1=A_1$, producing the displayed relaxation modulus. In a slow-flow expansion, $A=I+{\tau }_1{A}_1+{\tau }_1^2(2A_1^2-A_2)+O(3)$. Trace normalization and the first stretch correction affect only isotropic stress at this order, giving the displayed second-order fluid coefficients. Linear small-amplitude response allows arbitrary frequency, whereas this local second-order expansion also requires slow time dependence.