Interfacially Driven Instability in the Microchannel Flow of a Shear-Banding Fluid

Using microparticle image velocimetry, we resolve the spatial structure of the shear-banding flow of a wormlike micellar surfactant solution in a straight microchannel. We reveal an instability of the interface between the shear bands, associated with velocity modulations along the vorticity direction. We compare our results with a detailed theoretical study of the diffusive Johnson-Segalman model. The quantitative agreement obtained favors an instability scenario previously predicted theoretically but hitherto unobserved experimentally, driven by a normal stress jump across the interface between the bands.

Figure 1

(a) Sketch of the microchannel and coordinate system with a local 1D shear-banded flow profile. The grey rectangle represents a typical ($y$, $z$) area for a velocity imaging such as Fig. 2a. (b) Stress vs. strain rate relation at steady state. Squares: experimental from Ref. 7. Line: DJS model with $a=0.3$, $\eta =0.05$. Both are renormalized by the shear rate ${\dot{\gamma }}_c$ at the onset of shear banding, and by the value of the stress plateau ${\sigma }_c$. Correspondence is not expected at high shear rates (see main).

Figure 2

(a) Greyscale of the experimental velocity component $v_x(y,z)$ in a subpart of the channel cross section [see Fig. 1a], for ${\partial }_{x}p/G_b=1.3$ and $x=4\mathrm{cm}$, where the instability is fully developed. (b) Corresponding numerical snapshot of the fully developed pattern with $L_z/L_y=6.0$ and periodic boundaries in $z$. (c) Spatial evolution of the interface, progressively destabilized along the downstream direction $x$ for ${\partial }_{x}p/G_b=1.3$ at $x=1$, 2, 3, 3.5 and 4 cm, and obtained from experimental velocity measurements similar to (a).

Figure 3

Wave vector of the interfacial undulations ($L_y^{-1}$ units) as a function of applied pressure drop. Solid circles: experimental. Open squares: fully nonlinear state computed for $L_z/L_y=4.0$; the numerical wave vector is constrained by the cell length to be heavily quantized as $q_z^{\star }=2n\pi /L_z$. Open circles: linear stability analysis of the maximally unstable mode, which does not suffer quantization. Insert: the experimental amplitude $A$ in units of $L_y$.

Figure 4

(a) Experimental lateral components of the velocity $v_z(y,z)$ for ${\partial }_{x}p/G_b=1.3$ in a subsection of the channel cross section at $x=4\mathrm{cm}$. The maximal velocity represented is $7\mu \mathrm{m}·{\mathrm{s}}^{-1}$. The full line indicates the position of the interface between the two phases. (b) Numerically computed secondary flow field with $v_z$ and $v_y$ components in the whole cross section of the flow, for the same pressure gradient ${\partial }_{x}p/G_b=1.3$ and the same DJS model parameter values as for the numerics of Fig. 2b. (c) Cross correlation $R(z)$ between the interface position and the lateral velocities $v_z(y,z)$ at different heights $y/L_y$ inside the high shear phase, obtained from experiment data of (a) (symbols) and from simulation data of (b) (lines). Note that in the case of vorticies confined to the high shear phase, $R(z)$ would have opposite values for one of the considered height.