The effect of elastic walls on suspension flow

TL;DR

This study demonstrates that elastic walls in suspension flows cause shear thinning by inducing particle migration and altering effective viscosity, providing a new rheological model for such systems.

Contribution

It introduces a novel understanding of how elastic wall deformability influences suspension rheology and offers a closure model for predicting viscosity in elastic channels.

Findings

Elastic walls induce shear thinning in suspension flow.

Deformable walls cause particles to migrate towards the channel center.

A closure model for suspension viscosity in elastic channels is proposed.

Abstract

We study suspensions of rigid particles in a plane Couette flow with deformable elastic walls. We find that, in the limit of vanishing inertia, the elastic walls induce shear thinning of the suspension flow such that the effective viscosity decreases as the wall deformability increases. This shear-thinning behavior originates from the interactions between rigid particles, soft wall and carrier fluid; an asymmetric wall deformation induces a net lift force acting on the particles which therefore migrate towards the bulk of the channel. Based on our observations, we provide a closure for the suspension viscosity which can be used to model the rheology of suspensions with arbitrary volume fraction in elastic channels.

Equations

$$\frac{\mu }{{\mu }_0}={\left[1+B\frac{\Phi }{1-\Phi /{\Phi }_m}\right]}^2\text{/},$$

$$\frac{{\delta }_{-}}{R}\approx {\Phi }^{c_1}\text{mbox}{Ca}^{c_2}{\left(\frac{h^e}{R}\right)}^{c_3},$$

$${\Phi }_e=\mathcal{N}\frac{\frac{4}{3}\pi R^3}{{\mathcal{V}}_{tot}^e},$$

Full text

Understanding how elastic structures interact with fluid flows is a problem attracting a great deal of attention in different fields of science and technology, ranging from biological applications grodzinsky_lipshitz_glimcher_1978a ; abkarian_lartigue_viallat_2002a ; fish_lauder_2006a ; greene_banquy_lee_lowrey_yu_israelachvili_2011a ; freund_2014a to energy harvesting mckinney_delaurier_1981a ; boragno_festa_mazzino_2012a . In this context we consider a fluid-structure interaction problem particularly relevant to understand biological flows: we study the rheology of suspensions in the presence of walls which are allowed to deform elastically.

The study of rheology is motivated by the many fluids in nature and industrial applications which exhibit a non-Newtonian behavior, i.e., a non-linear relation between the shear stress and the shear rate, such as shear thinning, shear thickening, yield stress, thixotropy, shear banding and viscoelastic behaviors. The relation between these macroscopic behaviors and the microstructure is often studied assuming suspensions of objects in a Newtonian solvent with dynamic viscosity $\mu_{0}$ and density $\rho$. In the simplest case of suspensions of rigid spheres in a Newtonian fluid, Einstein einstein_1956a showed in his pioneering work that in the limit of vanishing inertia and for dilute suspensions (i.e., $\Phi\rightarrow 0$), the relative increase in effective viscosity $\mu_{\rm e}$ is a linear function of the particle volume fraction $\Phi$. However, at present there is no theory that allows us to calculate $\mu_{\rm e}$ for any given $\Phi$, thus different empirical formulas have been proposed to provide a good description to the existing experimental and numerical results ferrini_ercolani_de-cindio_nicodemo_nicolais_ranaudo_1979a ; singh_nott_2003a ; kulkarni_morris_2008a . Among those, we consider here the Eilers formula stickel_powell_2005a ; mewis_wagner_2012a ,

[TABLE]

which well fits experimental and numerical data zarraga_hill_leighton-jr_2000a ; singh_nott_2003a for both low and high values of $\Phi$, up to about $0.6$. In the expression above, $\Phi_{m}$ is the geometrical maximum packing fraction and $B$ a constant; fits to the data are usually obtained for $\Phi_{m}=0.58-0.63$ and $B=1.25-1.7$.

In this letter we add a further complexity to this problem, one that is particularly relevant to understand rheology of biological flows: we allow the walls to deform elastically. The problem of a single object interacting with a soft wall has been the object of several recent works mani_gopinath_mahadevan_2012a ; beaucourt_biben_misbah_2004a ; skotheim_mahadevan_2004a ; salez_mahadevan_2015a ; saintyves_jules_salez_mahadevan_2016a ; rallabandi_saintyves_jules_salez_schoenecker_mahadevan_stone_2017a ; davies_debarre_el-amri_verdier_richter_bureau_2018a ; rallabandi_oppenheimer_zion_stone_2018a ; on the other hand, here we focus for the first time on particle suspensions interacting with soft walls. In particular, we model the walls as two viscoelastic layers with an elastic shear–modulus $G$ and viscosity $\mu_{m}$. Thus, we introduce two new dimensionless parameters in the problem: the capillary number $\mbox{Ca}\equiv\mu_{0}\dot{\gamma}/G$ and the solid to fluid viscosity ratio $K\equiv\mu_{m}/\mu_{0}$, the latter fixed to $1$ for simplicity. We aim to describe for the first time the non linear effects of the wall elasticity on the rheology of a suspension. Furthermore, we will provide a simple approach able to accurately model the presence of deformable walls, without the necessity to solve the complex nonlinear interactions between the multiphase flow and the structure dynamics.

To address this problem, we perform direct numerical simulations of suspensions of rigid spheres, simulated by an immersed boundary method izbassarov_rosti_niazi-ardekani_sarabian_hormozi_brandt_tammisola_2018a , flowing in a plane channel Couette flow. Two viscoelastic layers are attached to the moving rigid walls, simulated with a pseudo volume of fluid approach rosti_brandt_2017a , see Fig. (1) for a sketch of the geometry considered note1 . We cover a wide range of volume fractions $\Phi$ (up to $30\%$) and capillary numbers Ca (up to $8$) and calculate the effective viscosity of the suspension $\mu_{\rm e}=\mathcal{F}\left(\Phi,\mbox{Ca}\right)$. This is displayed in Fig. (2) for different volume fractions $\Phi$ and for various level of wall elasticity, i.e., for various capillary numbers Ca. Also, three different thicknesses $h^{e}$ of the elastic walls are considered: $0.5R$, $R$ and $1.5R$. We observe that, the suspension viscosity increases with the volume fraction $\Phi$ but decreases with the wall deformability (increasing capillary number). The effect of the wall deformability is more pronounced as the volume fraction increases, where we observe larger differences from the reference case (rigid walls), indicated by the Eilers fit in the figure (solid black line). In particular, we find a reduction of up to $25\%$ in the effective viscosity at $\Phi\approx 30\%$ for the largest capillary number considered in this study. Note that, the reduction of the suspension viscosity with the capillary number can be interpreted as a shear-thinning rheological behavior of the system, see Fig. (2), where the effective viscosity $\mu_{\rm e}$ is shown as a function of Ca. Interestingly, a similar behaviour was observed in Ref. rosti_brandt_mitra_2018a for suspensions of deformable particles. It is worth noticing that, the presence of the elastic wall has no effect on the fluid rheology in the absence of particles, i.e., $\mu_{e}=\mu_{0}$ when $\Phi=0$. Thus, the observed shear thinning is the result of the interaction of the elastic walls and the particles. This behaviour is affected by the thickness of the elastic layer $h^{e}$: when the size of the layer is increased, the effect on the suspension is enhanced and the effective viscosity further reduces.

To understand the mechanism that generates the shear-thinning behavior of the suspension, we study the mean particle concentration $\Psi(y)$ across the channel, shown in the left panel of Fig. (3). The particles concentration $\Psi$ is null inside the elastic layer ($y\geq h$), rapidly grows in the near wall region ($h/2\lesssim y\lesssim h$) and finally reaches an approximately uniform value in the middle of the channel ($h\lesssim h/2$). Unlike the case of rigid walls (grey line), on average there are no particles in contact to the deformable wall as these are lifted towards the channel center. As the total volume fraction $\Phi$ increases, the concentration distribution grows faster close to the wall and reaches higher values at the bulk of the channel to ensure the imposed total volume fraction $\Phi$. The effect of the wall elasticity is shown by the shaded colored areas in the graph: as Ca increases, the particles are displaced further away from the wall and concentrate more in the bulk of the channel. This effect is present for all the volume fractions $\Phi$ and capillary numbers Ca that we studied, but is more evident at low volume fractions than at high ones due to the larger available space for the particles to migrate. Note that, despite this, the effect on the effective viscosity of the suspension increases with the volume fraction, as shown in Fig. (2). The mean displacement of the particles from the wall is quantified in the right panel of Fig. (3), where the wall-normal distance ($h-y$) at which the concentration $\Psi$ is equal to $1\%$ of the nominal value is reported as a function of the capillary number Ca. The wall-normal distance where $\Psi=1\%$ increases for all the volume fractions with the capillary number Ca, i.e., as the wall becomes more deformable, and decreases for increasing total volume fraction $\Phi$.

The displacement of the particles from the walls to the center of the channel can be related to the non-zero wall-normal velocity fluctuations $v^{\prime}$ at the elastic walls ($y=h$), although the mean wall-normal velocity is zero for incompressible materials. These fluctuations are shown in Fig. (4), where $v^{\prime}$ is reported as a function of the capillary number Ca for all the volume fractions $\Phi$ considered. We observe that $v^{\prime}$ grows from zero as the capillary number is increased, i.e., the wall is allowed to deform; the fluctuation growth rate is high for low capillary numbers before we observe a tendency to saturate. Also, the decrease of the growth rate at high capillary numbers is faster for high volume factions, in accordance with what already observed in Fig. (3). The wall-normal velocity fluctuations are induced by the deformation of the walls due to the interaction with the rigid particles, as sketched in Fig. (5). A particle approaching the elastic wall deforms it generating a wall-normal flow; in addition, the particle moves along the wall due to the applied shear. The combination of these two effects induces an asymmetric wall deformation which leads to the generation of a net lift force acting on the particle, which therefore migrates away from the wall. These results are consistent with and confirm previous theoretical works salez_mahadevan_2015a ; rallabandi_saintyves_jules_salez_schoenecker_mahadevan_stone_2017a which considered the deformations induced by a single rigid sphere translating parallel to a soft wall in a viscous fluid; in these previous studies, lubrication theory is shown to predict a lift force acting on the object, as observed experimentally saintyves_jules_salez_mahadevan_2016a .

Finally, we propose a model to easily include the effect of the elastic walls in the rheological description of suspensions. As mentioned above, $\mu_{e}$ is in general a non-linear function of the total volume fraction $\Phi$ and the capillary number of the walls Ca, i.e., $\mu_{e}=\mathcal{F}\left(\Phi,\mbox{Ca}\right)$. Below, we show that $\mu_{e}$ can be written as a function of a single variable $\Phi_{e}$, i.e., $\mu_{e}=\mathcal{G}\left(\Phi_{e}\right)$. Also, we will demonstrate that the nonlinear function $\mathcal{G}$ can be properly described by the Eilers fit reported in Eq. (1), similarly to the standard case of flow over rigid walls. The effective volume fraction $\Phi_{e}$ takes into account the fact that the particles feel a reduced confinement effect due to the wall flexibility: indeed, they can move beyond $y=h$ by inducing wall deformations. Thus, we introduce a reduced effective volume fraction, obtained by increasing the total volume available to the particles by a layer of thickness $\delta_{-}$, which is the amount of the penetration of the particles into the walls, or alternatively, the amplitude of the average negative wall deformation. We plot in the left panel of Fig. (6) the normalized amplitude of wall deformation $\delta_{-}$, which grows with both the wall elasticity Ca and the volume fraction $\Phi$. A simple fit of our data (shown with the solid lines in the plot) provides the following expression for $\delta_{-}$ as function of the volume fraction $\Phi$, the capillary number Ca and the elastic layer thickness $h^{e}$

[TABLE]

being $c_{i}$ fitting coefficients, $c_{1}\approx 0.9$, $c_{3}\approx 1.5$ and $c_{2}\approx 0.32$ for $\Phi=0.1$, $0.52$ for $\Phi=0.2$ and $0.66$ for $\Phi=0.29$. Note that, Fig. (6) reports also $\delta_{+}$, the amplitude of the positive wall deformation, which is always smaller than $\delta_{-}$ due to the aforementioned symmetry breaking at the deformable wall interface.

Once $\delta_{-}$ is known, one can compute the effective volume fraction as

[TABLE]

where $\mathcal{V}^{e}_{tot}$ is the sum of the total volume originally available to the particles $\mathcal{V}_{tot}=2h~{}L_{x}~{}L_{z}$ and the increase due to the wall elasticity $\mathcal{V}^{e}=\delta~{}L_{x}~{}L_{z}$, i.e., $\mathcal{V}^{e}_{tot}=\mathcal{V}_{tot}+\mathcal{V}^{e}=2\left(h+\delta_{-}\right)~{}L_{x}~{}L_{z}$. The right panel in Fig. (6) shows again the effective viscosity $\mu_{\rm e}$ now as a function of the effective volume fraction $\Phi_{\rm e}$. We observe that all the cases at different volume fractions $\Phi$, capillary numbers Ca and wall thickness $h^{e}$ collapse onto a single curve, which is well described by the Eilers formula. Thus, by using Eq. (2) and Eq. (1) we are able to effectively predict the suspension effective viscosity $\mu_{\rm e}$ in the presence of elastic walls, once the volume fraction $\Phi$, the capillary number Ca and the thickness of the elastic layers $h^{e}$ are known. Also, we explain the largest variations of the effective viscosity $\mu_{\rm e}$ at higher volume fractions by the large sensitivity of $\mu_{\rm e}$ to variations of $\Phi$ at higher concentrations (slope of the Eilers curve).

In conclusion, we have explored the rheological behaviour of a suspension of rigid particles in an elastic channel, for a wide range of solid volume fractions and wall elasticities. The main results of our analysis are the identification of a shear-thinning behavior of the suspension induced by the interaction of the elastic walls and the rigid particles, and its explanation in terms of a lift force acting on the particles which induces their migration towards the bulk of the channel. The lift force mechanism is robust enough to be effective also in relatively dense suspensions, whose behaviour is usually mainly determined by inter-particle interactions. Based on our observations and a simple mechanical model, we also provide a closure to effectively predict the rheological properties of suspensions in the presence of elastic walls. Our results extend to deformable walls the idea of using simple empirical fits, originally valid for inertialess suspensions of rigid spheres, such as the Eilers formula, to predict the rheology of suspensions with additional complexity embedded in the definition of an effective volume fraction, as previously done in Refs. picano_breugem_mitra_brandt_2013a , mueller_llewellin_mader_2010a and rosti_brandt_mitra_2018a for the cases of inertial effects, particles shape and deformability, respectively. The applicability of this scaling confirms that viscous dissipation is still the dominant mechanism at work in these flows.

Acknowledgment

The authors were supported by the ERC-2013-CoG-616186 TRITOS and by the VR 2014-5001 and acknowledge the computer time provided by SNIC (Swedish National Infrastructure for Computing).