Instability of viscous flow over a deformable two-layered gel: Experiments and theory

The instability of the flow of a viscous fluid past a soft, two-layered gel is probed using experiments, and the observations are compared with results from a linear stability analysis. The experimental system consists of the rotating top plate of a rheometer and its stationary bottom plate on which the two-layer gel is placed. When the flow between the top plate and the two-layer gel is viscometric (i.e., laminar), the viscosity obtained from the rheometer is a measure of the material property of the fluid. However, after a critical shear stress, there is a sudden increase in apparent viscosity, indicating that the flow has undergone an instability due to the deformable nature of the two-layer gel. Experiments are carried out to quantify how the critical value of fluid shear stress required to destabilize the flow varies as a function of ratio of solid to fluid layer thickness, and the ratio of the shear moduli of the two gels. A linear stability analysis is carried out for plane Couette flow of a Newtonian fluid past the two-layered gel, by assuming the two solid layers to be elastic neo-Hookean materials. In order to compare the experimental and theoretical results, the effective shear modulus ($G_{\mathrm{eff}}$, defined by $H/G_{\mathrm{eff}}=H_1/G_1+H_2/G_2$) of the two-layer gel is found to be useful, where $H=H_1+H_2$. Here, $H_i$ and $G_i$ ($i=1,2$), respectively, denote the thickness and shear modulus of each layer. Results for the nondimensional parameter ${\Gamma }_{\mathrm{eff}}=\eta V/\left(d{G}_{\mathrm{eff}}\right)$ ($V$ is the velocity of the top plate; $\eta$ is fluid viscosity, $d$ is the fluid thickness) as a function of solid to fluid thickness $H/d$ obtained from the stability analysis agree well with experimental observations, without any fitting parameters. In general, we find that the flow is more unstable if the softer gel is adjacent to the fluid flow compared to the case when it is not. This suggests that the instability is more interfacial in nature and is crucially dependent on the relative placement of the two layers, and not just on the effective modulus of the two-layer gel. We further show that the theoretical and experimental data for two-layer gels can be suitably collapsed onto the results obtained for a single-gel layer.

Figure 1

Schematic diagram of the experimental setup: The standard configuration of the rheometer is modified for studying flow past a soft two-layer gel. The radius of the rotating top plate is $R$, and the fluid thickness between the rotating top plate and the two-layer gel is $d$. The gel (of total thickness $H$) consisting of two layers (bottom layer thickness $H_2$ and top layer thickness $H_1$) is placed over the stationary bottom plate of the rheometer.

Figure 2

Viscosity of the PEO solution as a function of shear stress. Data demonstrate that the fluid shows negligible shear thinning and remains Newtonian in the range of stresses probed in this study. Measurement was carried out at ${25}^{\circ }\mathrm{C}$ using the parallel plate geometry.

Figure 3

Preparation of the two-layer gel. (a) The microscopic glass slides are pasted on the sides of a glass plate, and this template forms a well. Small square-shaped glass slides were pasted on the four corners of the well to ensure that the thickness of the bottom layer of the two-layer gel is uniform. (b) The constituents of the polymer gel are dispensed in the above template. (c) A glass plate is placed on the small square-shaped glass slides, and we allowed 2 h for the solution to get cross-linked. (d, e) The glass plate is carefully removed and the constituents of the polyacrylamide gel are added. (f) A larger glass plate is kept on the template, and the constituents are allowed to cross-link.

Figure 4

Characterization of the polyacrylamide gels. (a) Storage modulus ($G^{\prime }$) of polyacrylamide gel as function of frequency. $G^{\prime }$ is approximately constant for the frequency range for different monomer concentrations: $5\%$ (open circle), $6\%$ (open triangular up), $7\%$ (open triangular right), and $8\%$ (open square), and the plateau value increases with increasing monomer concentration. (b) Gel viscosity ${\mu }_g=G^{\prime \prime }/\omega$ as a function of frequency. The experiments are carried out using 2.5 cm diameter parallel plate geometry of Anton Paar rheometer (MCR502) at ${25}^{\circ }\mathrm{C}$. The gel thickness is 4 mm.

Figure 5

Characterization of the PDMS gels. (a) Storage modulus ($G^{\prime }$) of PDMS gel as function of frequency. Two different concentrations of the silicone oil (viscosity 1 Pa s) concentration in the gel mixture is used here, viz., $85\%$ (open square) and $87.5\%$ (open triangular). The $G^{\prime }$ is nearly constant for the frequency range. (b) Gel viscosity as a function of frequency. The experiments are carried out using 2.5 cm diameter parallel plate geometry of Anton Paar rheometer (MCR502) at ${25}^{\circ }\mathrm{C}$. The gel thickness is 4 mm.

Figure 6

Schematic representation and coordinate system used in the theoretical analysis of plane Couette flow past a deformable two-layer gel.

Figure 7

The variation of ${\Gamma }_{\mathrm{top}}$ vs $k$ for specified values of $H_1/d=H_2/d=4$, and $G_r=0.14$ (hard-on-top) and $G_r=7.08$ (soft-on-top), the flipped configuration. The minimum of this curve is the critical nondimensional strain rate ${\Gamma }_c$ at which the flow becomes unstable.

Figure 8

Viscous instability for flow past single polyacrylamide gel layers: The dimensionless critical strain rate for instability ${\Gamma }_c$ as a function of $H/d$. Silicone oils with different viscosities [1 Pa s (open triangular), 0.33 Pa s (open square)] are used to carry out the experiments. The various polyacrylamide gels used in the experiments have shear moduli 0.9, 2.17, 4.21,and 6.38 kPa. For each of these shear moduli, gels of different thicknesses ($H/d$) were prepared. The dashed line is the theoretically predicted ${\Gamma }_c$ for various $H/d$ using a neo-Hookean model. The experiments are carried out using a 4 mm diameter parallel plate (DHR3 rheometer) at room temperature.

Figure 9

Viscous instability in the flow of PEO solutions past single-layer PDMS gels: The critical nondimensional strain rate ${\Gamma }_c$ as a function of $H/d$. The dashed line is the theoretical prediction (using neo-Hookean model) of ${\Gamma }_c$ for different values of $H/d$. The shear modulus of the PDMS gel used is 4.05 kPa (open triangular) and 1.67 (open square) kPa. The experiments are carried out using a 4 mm diameter parallel plate (DHR3 rheometer) at room temperature.

Figure 10

Onset of instability in the viscous flow past a two-layer gel as indicated by the apparent viscosity from the rheometer: The fluid used in the experiments is silicone oil with viscosity is 1 Pa s. The two-layer configuration consists of two gels with shear moduli $G^{\prime }=6375$ Pa and 900 Pa. (a) Results when the hard layer is on top (open triangles) with $G_r=0.41$; (b) results when the soft layer is on top (open circles), $G_r=7.08$. The dashed line denotes results from flow between rigid plates without the gel. The fluid and two-layer gel thicknesses used in the experiments are 0.3 and 2 mm, respectively. The experiment is carried out at $25{}^{\circ }\mathrm{C}$. The parallel plate geometry of DHR3 rheometer is used in the experiment, and the top plate diameter is 40 mm. The shear stress is increased at the rate of 4 Pa/s.

Figure 11

Instability of viscous flow past various two-layer gels: ${\Gamma }_{\mathrm{eff}}$ as a function of $H/d$ for different values of $G_r$ indicated in each panel at the upper right corner. (a)–(c) Results for flow of silicone oil over polyacrylamide two-layer gels; (d) results for flow of PEO solution over PDMS two-layer gels. Each panel shows the two configurations, viz., when the soft gel is on top and when hard gel is on top. Open squares represent data when the harder gel is on top, and the corresponding theoretical prediction is indicated by dotted lines. Open triangles represent the for the case where the gels are flipped (i.e., softer gel on top), and the dashed line represents the corresponding theoretical predictions. Two types of silicone oils (with viscosity 0.33 and 1 Pa s) are used as the fluid for each configuration in panels (a)–(c). The theoretical predictions are obtained using the neo-Hookean model for the gels.

Figure 12

Collapse of theoretical results for flow past two-layer gels with different $G_r$ values. (a) Theoretically predicted ${\Gamma }_{\mathrm{eff}}$ as a function of $H/d$ for different $G_r$. The dashed line indicates the theoretically predicted $\Gamma$ as a function of $H/d$ for a single solid layer. (b) The data of panel (a) collapse on to the single-layer results, if the results for two-layer gels with the hard layer on top are rescaled as $1.5{\Gamma }_{\mathrm{eff}}{G}_r^{0.6}$. The results for two-layer gels with the soft layer on top lie close to the single-layer curve without rescaling.

Figure 13

Collapse of experimental results (symbols) for ${\Gamma }_{\mathrm{eff}}$ for two-layer gels with different $G_r$ values onto the theoretical curve (dashed line) for a single solid layer. The results for two-layer gels with the harder gel on top were rescaled using $1.5{\Gamma }_{\mathrm{eff}}{G}_r^{0.6}$.