2D Oldroyd-B model

The study of two-dimensional viscoelastic solutions will be addressed by means of the two-dimensional version of Oldroyd-B model (3.12-3.16), which is described by the equations:

$$\partial_t {\mbox{\boldmath$u$}} + ({\mbox{\boldmath$u$}}\cdot{\mbox{\boldmath$\nabla$}}) {\mbox{\boldmath$u$}} = -{\mbox{\boldmath$\nabla$} p} + \nu {\Delta} {\mbox{\boldmath$u$}... ...mbox{\boldmath$\sigma$}} - \alpha {\mbox{\boldmath$u$}} + {\mbox{\boldmath$f$}}$$ (4.1)

$$\partial_t {\mbox{\boldmath$\sigma$}} + ({\mbox{\boldmath$u$}}\cdot{\mbox{\boldmath$\nabla$}}) {\mbox{\boldmath$\sigma$}} = ({\mbox{\boldmath$\nabla$} \mbox{\boldmath$u$}})^T \cdot {\mbox{\... ...oldmath$u$}}) -{2 \over \tau}({\mbox{\boldmath$\sigma$}}-{\mbox{\boldmath$1$}})$$ (4.2)

The matrix ${\mbox{\boldmath$\sigma$}}$ is the conformation tensor of polymer molecules

$$\sigma_{ij} = R_0^{-2} \langle R_{i}R_{j} \rangle$$ (4.3)

and its trace $\textrm{tr}{\mbox{\boldmath$\sigma$}}$ is a measure of their square elongation. Because of its physical meaning the conformation tensor is symmetric and positive definite. The parameter $\tau$ is the (slowest) polymer relaxation time toward the equilibrium length $R_0$, therefore in absence of stretching the conformation tensor therefore relaxes to the the unit tensor ${\mbox{\boldmath$1$}}$. The matrix of velocity gradients which stretches the polymers is defined as $({\mbox{\boldmath$\nabla$} \mbox{\boldmath$u$}})_{ij}=\partial_i u_j$. The solvent viscosity is denoted by $\nu$ and $\eta$ is the zero-shear contribution of polymers to the total solution viscosity $\nu_{t}=\nu(1+\eta)$. The pressure term $-{\mbox{\boldmath$\nabla$}}p$ ensures incompressibility of the velocity field, which can be expressed in terms of the stream-function $\psi$ as ${\mbox{\boldmath$u$}} = (\partial_y \psi, -\partial_x \psi)$. The dissipative term $-\alpha {\mbox{\boldmath$u$}}$ models the mechanical friction between the thin layer of fluid and the surrounding environment, and plays a prominent role in the energy budget of Newtonian two-dimensional turbulence [71]. The energy source is provided by the large-scale forcing ${\mbox{\boldmath$f$}}$, which is Gaussian, statistically homogeneous and isotropic, $\delta$-correlated in time, with correlation length $L_f \approx 4$.

The numerical integration is performed by a fully dealiased pseudospectral code, with second-order Runge Kutta scheme, at different resolutions, $N^2 = 128^2, 256^2$ grid points, on a doubly periodic square box of size $L = 2\pi$. As customary, an artificial stress-diffusivity term $\kappa {\Delta} {\mbox{\boldmath$\sigma$}}$ is added to Eq.(4.2) to prevent numerical instabilities [72]. For the passive case we have adopted a Lagrangian code which explicitly which preserves the symmetries of the conformation tensor (see Appendix A).