Ridged geometries induce axial flow vortices in Couette systems

Abstract

A yield stress fluid has a critical stress above which the material starts to flow. Typically, the yield stress behaviour is captured in the Herschel-Bulkley (HB) model, which assumes a constant yield stress as material parameter. It is not clear whether the simultaneous superposition of a flow in an orthogonal direction to the main flow, that displays HB behaviour, affects the yield stress and will make the yield stress either flow rate- or field-dependent. Therefore, it is important to understand how the presence of flow in two orthogonal directions affects the yielding behaviour of the fluid in general. In this work, we showed that wall patterning can be used to generate flow in two orthogonal directions simultaneously. We find that these orthogonal flows measurably affected each other. We induced spatially varying secondary flows by shearing a standard Newtonian fluid and two common yield stress fluids in a rheometer using a concentric cylinder geometry with angled ridges. We measured the normal force as a function of rotation rates for different angled ridges from conventional rheological measurements. We also imaged the flow fields by employing rheo-MRI, to directly measure the penetration depth of the fluids into the rough boundary of the ridged geometry. Finally, we relate the penetration depth to the axial flow for different geometries, at different imposed rotation rates and the fluid type, to show that two flow directions in the yield stress fluids are indeed significantly related to each other.

Figures (13)

• Figure 1: a) Angled ridged geometry used in the rheological experiments, indicating the ridges aligned at an angle $\theta$ to the vertical axis. b) Angled ridged geometries used in the rheo-MRI experiments, with ridges at an angle $\theta$ relative to the vertical axis. Curved arrows in a and b indicate the direction of rotation. Note the different orientations of the ridges in the two geometries in a and b. c) Schematic representation of the fluid rotation rate (in the azimuthal direction) as a function of position in the gap. The grey dashed line indicates the tip of the ridge. Between the ridges, fluid moves with the geometry, while Taylor-Couette flow is observed outside the ridges. $\delta$ is the penetration depth, that we measure from this profile. d) Schematic representation of the mean axial velocity as a function of position in the gap. The grey dashed line indicates the tip of the ridge, with fluid being pushed downward (or upward if the rotation direction is reversed) between the ridges, and the fluid outside the ridges moving upward to maintain the overall volume. Negative axial velocities point in the direction of gravity.
• Figure 2: Torque as a function of rotation rate for a) castor oil and b) Carbopol, measured using geometries with ridges at various $\theta$ (indicated in the legend). Torque scales linearly with rotation rate for castor oil (slope = 1), while for Carbopol, the torque follows a Herschel-Bulkley model relation with rotation rate with $n \approx 0.5$. Insets in a and b show the variation of torque with rotation rate on a linear scale, where the geometric variations are more noticeable. Normal force as a function of rotation rate for c) castor oil and d) Carbopol, measured using geometries with ridges at various $\theta$ (indicated in the legend). For castor oil, normal force exhibits linear scaling with rotation rate (slope = 1), whereas in the case of Carbopol, normal force scales non-linearly with rotation rate in a Herschel-Bulkley fashion with $n \approx 0.5$. In all cases, $F>0$, indicating a force against gravity hence the downward displacement of the fluid by the geometry. The solid lines represent linear fits (in a, c) and the Herschel-Bulkley fits (in b, d), respectively.
• Figure 3: Rotational velocity (a--c), normalised with respect to the imposed rotation rates and axial velocity (d--f) as a function of position in the gap for castor oil, measured at different imposed rotation rates. Each panel presents velocities in geometries with ridges at three different angles: purple for 10$^\circ$, green for 20$^\circ$, and brown for 45$^\circ$. Legends denote the specific imposed rotation rates, $\Omega_\textnormal{o}$ in rps, for each curve, while the dashed line marks the tip of the ridge. The solid lines are fits to a sigmoidal function (Equation (\ref{['ch3:eq:sigmoidalfunction']})), from which the penetration depths are obtained.
• Figure 4: Rotational velocity (a--c), normalised with respect to the imposed rotation rates and axial velocity (d--f) as a function of position in the gap for Carbopol, measured at different imposed rotation rates. Each panel presents velocities in geometries with ridges at three different angles: purple for 10$^\circ$, green for 20$^\circ$, and brown for 45$^\circ$. Legends denote the specific imposed rotation rates, $\Omega_\textnormal{o}$ in rps, for each curve, while the dashed line marks the tip of the ridge. The solid lines are fits to a sigmoidal function (Equation (\ref{['ch3:eq:sigmoidalfunction']})), from which the penetration depths are obtained.
• Figure 5: Rotational velocity (a--c), normalised with respect to the imposed rotation rates and axial velocity (d--f) as a function of position in the gap for castor oil-in-water emulsion, measured at different imposed rotation rates. Each panel presents velocities in geometries with ridges at three different angles: purple for 10$^\circ$, green for 20$^\circ$, and brown for 45$^\circ$. Legends denote the specific imposed rotation rates, $\Omega_\textnormal{o}$ in rps, for each curve, while the dashed line marks the tip of the ridge. The solid lines are fits to a sigmoidal function (Equation (\ref{['ch3:eq:sigmoidalfunction']})), from which the penetration depths are obtained.