Influence of boundary conditions and confinement on nonlocal effects in flows of wormlike micellar systems

In this paper we report on the influence of different geometric and boundary constraints on nonlocal (spatially inhomogeneous) effects in wormlike micellar systems. In a previous paper, nonlocal effects were observable by measuring the local rheological flow curves of micelles flowing in a microchannel under different pressure drops, which appeared to differ from the flow curve measured using conventional rheometry. Here we show that both the confinement and the boundary conditions can influence those nonlocal effects. The role of the nature of the surface is analyzed in detail using a simple scalar model that incorporates inhomogeneities, which captures the flow behavior in both wide and confined geometries. This leads to an estimate for the nonlocal “diffusion” coefficient (i.e., the shear curvature viscosity) which corresponds to a characteristic length from 1 to 10 μm.

Figure 1

Schematic view of the canyon geometry, $L$ is the length of the microchannel and $2e$ its width. Bottom: schematic temporal acquisition of the images using the PIV device, where $t_a$ is the acquisition time of each image and $\delta t$ the time interval between two frames. Above are shown two schematic frames with tracers in the $x\text{−}z$ plane and the corresponding velocity vectors $\text{d}x$ estimated using standard cross-correlation algorithms.

Figure 2

Electronic microscopy pictures of the surfaces. (a) Rough glass surface R1; (b) rough glass surface R2; (c) smooth coated film of PDMS on glass (S2); and (d) smooth glass S1 from a capillary tube (composite metal service). The size of bar is $20\mu \text{m}$.

Figure 3

Real $\left(G^{\prime }\right)$ and imaginary $\left(G^{″}\right)$ linear rheological response as a function of the frequency $f$, for a 6% CPCl-Sal mixture in brine solution. The sample is seeded with $0.001\text{wt}\mathrm{\%}$ latex beads. The lines correspond to the Maxwell model. The relaxation time ${\tau }_r=4\text{s}$ and the elastic modulus $G_0=70\text{Pa}$ are extracted from the fit of the data to this model.

Figure 4

Global flow curve from a rheometer using a cone-and-plate cell, with shear stress $\sigma$ and the shear rate $\dot{\gamma }$ defined by Eqs. (1, 2). The stress plateau occurs at ${\sigma }^{\star }=54\text{Pa}$. The sample is seeded with $0.001\text{wt}\mathrm{\%}$ latex beads. The line corresponds to the flow curve predicted in wide plate geometry by the nonlocal model (see Sec. 4).

Figure 5

Velocity profiles at different $\Delta P$ (wall stresses ${\sigma }_w=46,65,74,78,83,87\text{Pa}$ from bottom to top) for channel width $2e=120\mu \text{m}$ and rough surface R2. Solid lines are calculated using the nonlocal model described Sec. 5 with the same fitting parameters ${\eta }_1=20.6\text{Pa}\text{s}$, ${\eta }_2=0.35\text{Pa}\text{s}$, $A=44\text{Pa}$, ${\gamma }_1=5.6{\text{s}}^{-1}$, $\kappa ={10}^{-10}\text{Pa}\text{s}{\text{m}}^2$.

Figure 6

Slip velocity $V_s$ as a function of the wall shear stress ${\sigma }_w$ for R2 surfaces for two microchannels with different widths (◻ to R2 $2e=200\mu \text{m}$, ◇ to R2 $2e=120\mu \text{m}$). The line corresponds to the equation $V_s=0.000045{\sigma }_w^{3.5}$.

Figure 7

Solid line: global flow curve from a rheometer using a cone-and-plate cell, with shear stress $\sigma$ and the shear rate $\dot{\gamma }$ defined by Eqs. (1, 2). Points (◇): local flow curve deduced from the velocity profiles for different wall shear stress $\left({\sigma }_w=46,65,74,78,83,87\text{Pa}\right)$. The local shear rate is the derivative of the velocity profile and the local shear stress is calculated using Eq. (3). The velocity profiles are measured in a $2e=120\mu \text{m}$ thick microchannel made with rough R2 surfaces.

Figure 8

Shear stress at the interface between the two bands and $z_{int}/z^{\ast }$ as a function of the shear stress at the wall for various geometries and surfaces. The line corresponds to the plateau value ${\sigma }^{\ast }=54\text{Pa}$ measured in cone-and-plate geometry (◻ to R2 $2e=200\mu \text{m}$, ◇ to R2 $2e=120\mu \text{m}$).

Figure 9

Velocity profiles for different $\Delta P$ (wall stresses ${\sigma }_w=51,58,66,73,80,87\text{Pa}$ from bottom to top), for channel width $2e=200\mu \text{m}$ with rough surface R2. Solid lines are calculated using the nonlocal model described in Sec. 5 with ${\eta }_1=20.6\text{Pa}\text{s}$, ${\eta }_2=0.35\text{Pa}\text{s}$, $A=44\text{Pa}$, ${\gamma }_1=5.6{\text{s}}^{-1}$, $\kappa ={10}^{-10}\text{Pa}\text{s}{\text{m}}^2$.

Figure 10

Global flow curve from a rheometer using a cone-and-plate cell (◻), with shear stress $\sigma$ and the shear rate $\dot{\gamma }$ defined by Eqs. (1, 2). (○): local flow curve deduced from the velocity profiles for a shear stress at the wall equal to ${\sigma }_w=80\text{Pa}$. $(+)$: local flow curve deduced from the velocity profiles for a shear stress at the wall equal to ${\sigma }_w=94\text{Pa}$. The local shear rate is the derivative of the velocity profile and the local shear stress is calculated using Eq. (3). Solid lines are calculated using the nonlocal model described in Sec. 5 with ${\eta }_1=20.6\text{Pa}\text{s}$, ${\eta }_2=0.35\text{Pa}\text{s}$, $A=44\text{Pa}$, ${\gamma }_1=5.6{\text{s}}^{-1}$, $\kappa ={10}^{-10}\text{Pa}\text{s}{\text{m}}^2$. The velocity profiles are measured in a $2e=200\mu \text{m}$ thick microchannel made with rough R2 surfaces.

Figure 11

Velocity profiles corrected by the slip velocities for different $\Delta P$ for various geometries. Solid lines are calculated using the nonlocal model described Sec. 5 with the same fitting parameters ${\eta }_1=20.6\text{Pa}\text{s}$, ${\eta }_2=0.35\text{Pa}\text{s}$, $A=44\text{Pa}$, ${\gamma }_1=5.6{\text{s}}^{-1}$, $\kappa ={10}^{-10}\text{Pa}\text{s}{\text{m}}^2$. Top left: $2e=200\mu \text{m}$ thick microchannel with rough surfaces R1. Wall shear stresses (bottom to top) ${\sigma }_w=33,42,50,58,66\text{Pa}$. Top right: $2e=200\mu \text{m}$ thick microchannel with smooth PDMS surfaces (S2). Wall shear stresses (bottom to top) ${\sigma }_w=38,54,62,70,77.5\text{Pa}$, Bottom left: $2e=200\mu \text{m}$ thick microchannel with smooth glass surfaces (S1). Wall shear stresses (bottom to top) ${\sigma }_w=55,60,65,70,75\text{Pa}$. Bottom right: $2e=200\mu \text{m}$ thick microchannel with rough surfaces (R2). Wall shear stresses (bottom to top) ${\sigma }_w=51,58,66,73,80\text{Pa}$.

Figure 12

Shear stress at the interface between the two bands and $z_{int}/z^{\ast }$ as a function of the shear stress at the wall for various geometries and surfaces. The line corresponds to the plateau value ${\sigma }^{\ast }=54\text{Pa}$ measured in cone-and-plate geometry. (○ correspond to R1, ◻ to R2 $2e=200\mu \text{m}$, ◇ to R2 $2e=120\mu \text{m}$, $◁$ to S2, $▷$ to S1).

Figure 13

Wall shear stress as a function of wall shear rate for various geometries and surfaces (○ correspond to R1, ◻ to R2 $2e=200\mu \text{m}$, ◇ to R2 $2e=120\mu \text{m}$, $◁$ to S2, $▷$ to S1). The line corresponds to the bulk rheology.

Figure 14

Slip velocity $V_s$ as a function of the wall shear stress ${\sigma }_w$ for various geometries and surfaces (○ correspond to R1, ◻ to R2 $2e=200\mu \text{m}$, ◇ to R2 $2e=120\mu \text{m}$, $◁$ to S2, $▷$ to S1). From top to bottom the lines correspond to the equation $V_s=0.0038{\sigma }_w^{2.7},V_s=0.0045{\sigma }_w^{2.6},V_s=0.000045{\sigma }_w^{3.5}$.

Figure 15

Homogeneous constitutive relation ${\sigma }_h\left(\dot{\gamma }\right)$ to represent the wormlike micelles. The parameters are given by ${\eta }_1=20.6\text{Pa}\text{s}$, ${\eta }_2=0.35\text{Pa}\text{s}$, $A=44\text{Pa}$, ${\gamma }_1=5.6{\text{s}}^{-1}$

Figure 16

Global flow curves in steady shear flow for various value of $N_l$ calculated following the procedure detailed in the text. We set ${\dot{\gamma }}_{\text{wall}}={\dot{\gamma }}_h\left({\sigma }_0\right)$, where ${\sigma }_0$ is the (homogeneous) selected shear stress at the wall, and choose the two walls to lie on different branches of the multivalued function. The two thinnest lines correspond to the ${\sigma }_h$ function. They describe homogeneous states, which do not depend upon $N_l$. The solid lines correspond to the heterogeneous states for (from thickest to the thinnest) $N_l=0.1,{10}^{-2},{10}^{-3}$ Right: zoom of the Plateau region.

Figure 17

Shear stress at the interface between the two bands and $z_{int}/z^{\ast }$ as a function of the shear stress at the wall in a Poiseuille flow. From thicker to thinner lines, $N_l$ is given by $N_l={10}^{-5},{10}^{-3},{10}^{-1}$. Near the wall, the fluid is assumed to be Newtonian with prescribed wall stress and viscosity ${\eta }_3=1\text{Pa}\text{s}$. The other parameters are ${\eta }_1=20.6\text{Pa}\text{s}$, ${\eta }_2=0.35\text{Pa}\text{s}$, $A=44\text{Pa}$, ${\gamma }_1=5.6{\text{s}}^{-1}$.

Figure 18

Shear stress at the interface between the two bands and $z_{int}/z^{\ast }$ as a function of the shear stress at the wall. The dashed lines correspond to the wall shear stress whereas the solid lines correspond to the stress at the interface. From thicker to thinner lines, $N_l$ is given by $N_l={10}^{-5},{10}^{-3},{10}^{-1}$. Near the wall, we assume a zero gradient boundary condition. The other parameters are given by ${\eta }_1=20.6\text{Pa}\text{s}$, ${\eta }_2=0.35\text{Pa}\text{s}$, $A=44\text{Pa}$, ${\gamma }_1=5.6{\text{s}}^{-1}$.

Figure 19

Velocity profile for ${\sigma }_w=73$, channel width $2e=200\mu \text{m}$ with rough surface R2. Solid lines are calculated using the nonlocal model described in Sec. 5. From top to bottom $\kappa ={310}^{-10},{1.2510}^{-10},{10}^{-10},{7.510}^{-10},{3.310}^{-11}$. The other parameters are given by ${\eta }_1=20.6\text{Pa}\text{s}$, ${\eta }_2=0.35\text{Pa}\text{s}$, $A=44\text{Pa}$, ${\gamma }_1=5.6{\text{s}}^{-1}$.