The convected terms are quadratic in the disturbance. Linear response requires small strain and small rate-based Weissenberg numbers, in particular
$$
\frac{\dot\gamma_0}{\omega}\ll1,
\qquad
\lambda_1\dot\gamma_0\ll1,
\qquad
\lambda_2\dot\gamma_0\ll1.
$$
The constitutive equation then reduces to
$$
\tau+\lambda_1\dot\tau
=\eta(\dot\gamma+\lambda_2\ddot\gamma).
$$
For $\dot\gamma=\dot\gamma_0\cos\omega t$, its periodic stress response is
$$
\tau(t)=\frac{\eta\dot\gamma_0}{1+\omega^2\lambda_1^2}
\left[
(1+\omega^2\lambda_1\lambda_2)\cos\omega t
+\omega(\lambda_1-\lambda_2)\sin\omega t
\right].
$$
Since $\gamma=(\dot\gamma_0/\omega)\sin\omega t$, comparison with $\tau=\gamma_0[G'\sin\omega t+G''\cos\omega t]$ gives
$$
\boxed{G'(\omega)=
\frac{\eta\omega^2(\lambda_1-\lambda_2)}{1+\omega^2\lambda_1^2},
\qquad
G''(\omega)=
\frac{\eta\omega(1+\omega^2\lambda_1\lambda_2)}{1+\omega^2\lambda_1^2}.}
$$
The condition $\lambda_1\geq\lambda_2$ makes the storage modulus nonnegative.


Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 352 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-352/2/b/solution