Viscoelastic Taylor-Couette Instability of Shear Banded Flow

We study numerically shear banded flow in planar and curved Couette geometries. Our aim is to capture two recent observations in shear banding systems of roll cells stacked in the vorticity direction, associated with an undulation of the interface between the bands. Depending on the degree of cell curvature and on the material’s constitutive properties, we find either (i) an instability of the interface between the bands driven by a jump in second normal stress across it or (ii) a bulk viscoelastic Taylor-Couette instability in the high shear band driven by a large first normal stress within it. Both lead to roll cells and interfacial undulations but with a different signature in each case, thereby suggesting that the roll cells in each of the recent experiments are different in origin.

Figure 1

Bulk flow curve of 1D base states for $q=0.0$ (left), $q=0.16$ (right). Symbols (i, ii, I, II) for reference in Figs. 2, 3.

Figure 2

Left: Maximum growth rate for an unbanded base state just into the high shear branch versus the first normal stress scaling variable $1/(1-a)$ for flat and curved cells: $\mathbf{(}q,\dot{\gamma }(1-a^2{)}^{1/2}\mathbf{)}=(0.0,6.87),(0.16,8.58)$. Bulk VTC instability is seen in the curved case at high $N_{1h}$. Right: Same for an applied shear rate $\dot{\gamma }(1-a^2{)}^{1/2}=1.91$ in the banding regime in a flat geometry, showing interfacial instability. Also shown is the jump in second normal stress (dotted line).

Figure 3

Top left: Maximum growth rate for an applied shear rate $\dot{\gamma }(1-a^2{)}^{1/2}=1.91$ in the banding regime versus the first normal stress scaling variable $N_{1h}\sim 1/(1-a)$ for three different cell curvatures $q$. Top right: Same versus cell curvature at fixed $1/(1-a)=416$. Inset shows the wave vector. Bottom: Corresponding phase diagram in the plane of curvature versus first normal stress scaling variable. Dashed lines: Power $-1$.

Figure 4

Gray-scale snapshots of ${\Sigma }_{xx}$ on the ultimate attractor in the $p\mathrm{\text{−}}z$ (horizontal-vertical) plane for a shear rate in the banding regime $\dot{\gamma }\sqrt{1-a^2}=1.91$. Left: Interfacial instability in a flat cell $q=0$ with $1/(1-a)=1.43$. Others left to right: Bulk instability of the VTC kind in the high shear band for a large value of the first normal stress scaling variable $1/(1-a)=416$, for curvatures $q=0.115,0.13,0.16$.