Viscous-elastic dynamics of power-law fluids within an elastic cylinder

Evgeniy Boyko, Moran Bercovici, and Amir D. Gat

Phys. Rev. Fluids 2, 073301 – Published 12 July 2017

Abstract

In a wide range of applications, microfluidic channels are implemented in soft substrates. In such configurations, where fluidic inertia and compressibility are negligible, the propagation of fluids in channels is governed by a balance between fluid viscosity and elasticity of the surrounding solid. The viscous-elastic interactions between elastic substrates and non-Newtonian fluids are particularly of interest due to the dependence of viscosity on the state of the system. In this work, we study the fluid-structure interaction dynamics between an incompressible non-Newtonian fluid and a slender linearly elastic cylinder under the creeping flow regime. Considering power-law fluids and applying the thin shell approximation for the elastic cylinder, we obtain a nonhomogeneous p-Laplacian equation governing the viscous-elastic dynamics. We present exact solutions for the pressure and deformation fields for various initial and boundary conditions for both shear-thinning and shear-thickening fluids. We show that in contrast to Stokes' problem where a compactly supported front is obtained for shear-thickening fluids, here the role of viscosity is inversed and such fronts are obtained for shear-thinning fluids. Furthermore, we demonstrate that for the case of a step in inlet pressure, the propagation rate of the front has a $t^{\frac{n}{n + 1}}$ dependence on time $(t)$ , suggesting the ability to indirectly measure the power-law index $(n)$ of shear-thinning liquids through measurements of elastic deformation.

Received 23 March 2017

DOI:https://doi.org/10.1103/PhysRevFluids.2.073301

Physics Subject Headings (PhySH)

Microfluidics Non-Newtonian fluids Shear thickening Shear thinning

Fluid Dynamics

Authors & Affiliations

Evgeniy Boyko, Moran Bercovici^*, and Amir D. Gat^†

Faculty of Mechanical Engineering, Technion–Israel Institute of Technology, Haifa, Israel

^*mberco@technion.ac.il
^†amirgat@technion.ac.il

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Vol. 2, Iss. 7 — July 2017

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Images

Figure 1
Schematic illustration of the examined configuration, showing the coordinate system and relevant physical parameters. An elastic cylinder of inner radius ${\tilde{r}}_{c}$ and wall thickness ${\tilde{h}}_{m}$ is clamped at $\tilde{z} = 0$ and contains a non-Newtonian power-law fluid. Viscous-elastic interaction between the fluid and the elastic structure, induced by externally applied pressures ${\tilde{p}}_{inlet}$ and ${\tilde{p}}_{e}$ at the inlet and the outer boundary respectively, leads to radial, ${\tilde{d}}_{r}$ , and axial, ${\tilde{d}}_{z}$ , displacement fields of the elastic cylinder.
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Figure 2
Propagation of a power-law fluid in an elastic cylinder due to instantaneous injection of mass at $z = 0$ . (a, b) The pressure distribution and radial deformation, respectively, as a function of the axial coordinate, both of which are characterized by the decay rate of $t^{- n / 2}$ . (c, d) The axial deformation and volume flux, respectively, as a function of the axial coordinate, both of which have a decay rate that is independent of $n$ . Black and gray lines correspond to $t = 0.1$ and $t = 1.0$ , respectively. Dotted lines represent a shear-thickening fluid $(n = 1.5)$ , dashed lines represent a Newtonian fluid $(n = 1)$ , and solid lines represent a shear-thinning fluid $(n = 0.5)$ . At early times the shear-thickening fluid is more localized (but not compactly supported) than the Newtonian and shear-thinning fluids, while for longer times the shear-thinning fluid has the lowest propagation rate remaining compactly supported. Black and gray dots indicate the location of the front of shear-thinning fluid, $z_{front} (t) = 2.99 t^{1 / 4}$ .
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Figure 3
Viscous-elastic dynamics of power-law fluids within an elastic cylinder due to a step increase in inlet pressure. (a, b, c) The pressure distribution and the radial and axial deformations, respectively, as a function of the axial coordinate. While for sufficiently long times, $t \to \infty$ , the radial deformation approaches a constant value, $d_{r} \to 0.85$ , the axial deformation is a linear function of $z$ with the slope of 0.2, $d_{z} / z \to 0.2$ , independently of $n$ . (d) Volume flux versus axial coordinate, characterized by the decay rate of $t^{- 1 / (n + 1)}$ . Black and gray lines correspond to $t = 0.1$ and $t = 1.0$ , respectively. Dotted lines represent a shear-thickening fluid $(n = 1.5)$ , dashed lines represent a Newtonian fluid $(n = 1)$ , and solid lines represent a shear-thinning fluid $(n = 0.5)$ . Notably, self-similar solutions exhibit a compactly supported propagation front (black and gray dots), $z_{front} (t) = 2.62 t^{1 / 3}$ , solely for shear-thinning fluids. All calculations were performed using $p_{0} = 1, p_{i} = 0$ , and $ν = 0.3$ .
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Figure 4
FFT analysis results showing active frequencies due to imposed frequency $ω = 2 π$ . (a) Resulting frequencies from FFT analysis applied directly to the source term of (72). (b, c) Resulting frequencies from FFT analysis applied on the numerical solution of (28) and corresponding to shear-thinning and shear-thickening fluids, respectively. Clearly, the source term itself predicts well the governing frequencies of the solution, though of course cannot predict the amplitudes.
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Figure 5
Numerical results showing the steady-state pressure distribution resulting from viscous-elastic interaction induced by oscillating inlet pressure with $ω = 2 π$ . (a, b, c) Pressure versus axial coordinate for one time period, corresponding to shear thinning, Newtonian, and shear-thickening fluids, respectively. Dotted lines represent the envelope of the pressure decay, and indicate the most rapid decay for the shear-thinning fluid. (d) Comparison of numerical results (solid lines) and approximate solutions (dashed lines), (77), for pressure distributions at $z = 1.4$ as a function of time and corresponding to $n = 0.5, 1, 1.5$ . While Newtonian fluid (dark gray line) is characterized by a sinusoidal waveform, shear-thinning (light gray lines) and shear-thickening (black lines) fluids exhibit nonsinusoidal “saw-tooth” and “shark-tooth” waveforms, respectively.
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