Upper-convected derivative

The upper-convected derivative of a contravariant tensor is $\stackrel{▽}{\mathbf{C}}=D\mathbf{C}/Dt-(\nabla \mathbf{u})\mathbf{C}-\mathbf{C}(\nabla \mathbf{u}{)}^T$.

For azimuthal velocity $v(r)e_{\theta }$, the velocity gradient in the rotating polar basis is $\left(\begin{array}{cc}0& -v/r\\ v^{\prime }& 0\end{array}\right)$. The basis rotates along a particle at angular speed $v/r$. Accounting for that rotation in the material derivative replaces $L$ by $L-(v/r)\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ in the upper-convected derivative. Its remaining shear rate is $v^{\prime }-v/r$, so the local viscoelastic normal-stress difference follows the same constitutive algebra as simple shear flow.

Differentiate the Kirchhoff stress tensor relation $\mathbf{\tau }=FS{F}^T$, using $\dot{F}=LF$ for the velocity gradient. Subtracting the two deformation terms gives the displayed upper-convected derivative. This removes the transport generated solely by deformation of a fixed material stress tensor.

The upper-convected Maxwell model evolves viscoelastic stress according toIt has constant shear viscosity and predicts a divergent extensional viscosity at a finite extension rate.

The Giesekus model adds the nonlinear anisotropic-drag term $\alpha \lambda {\mathbf{\tau }}^2/\eta$ to the Upper-convected Maxwell model. Positive dimensionless mobility factor $\alpha$ produces shear thinning and a nonzero second normal-stress difference.