Power-law Rheology Research Articles

A time-shift correction for extraction of viscoelastic parameters from ramp-hold AFM experiments

The stress relaxation and creep measurements are commonly implemented in AFM indentation experiments to analyze the viscoelastic properties of a sample in the time domain. The assumption of an instantaneous step load is often applied which can lead to erroneous estimation of viscoelastic parameters from the experimental data. A viscoelastic solution for the ramp-hold indentation profile might be used but requires higher computational costs. Here, we employed simulations for the ramp-hold indentations for the power-law rheology (PLR) and standard linear solid models. The error coming from the instantaneous load assumption was analyzed and a simple empirical correction for the finite ramp time was employed as a time shift of the hold region. The correction was tested in experiments on cells, in which the data from stress relaxation, creep, and force volume experiments were compared. The correction showed good applicability for the extraction of the PLR viscoelastic parameters.

Felsic Tesserae on Venus Permitted by Lithospheric Deformation Models

AbstractVenus tessera terrain contains the oldest rocks preserved on the surface of the planet. The composition of the tesserae is unknown, but recent observations of near‐infrared emissivity from orbit suggest that they are silica rich. This contrasts with previous interpretation that silica‐rich rheologies are too weak to produce tessera folds. We examine the range of compositions consistent with the observed folds in the tessera. We map folds in Tellus Regio tessera using the highest resolution Magellan radar images (∼100 m) and stereo topography (∼1 km). We use fold spacings as input into a buckling analysis that simulates power law rheology using contemporary dry‐mineral flow laws and modeled igneous rock compositions, consistent with a predicted dry lithosphere for Venus in the recent era. The folds have dominant wavelengths of 6–19 km with amplitudes of 500–1,000 m, consistent with low strains (1%–12%). We confirm that observed folds require heat flows of ≥40 K/km and strain rates faster than 10−16s−1, higher than today. These low strain folds are unlikely to have significantly modified the preexisting tessera surface, thus older impact craters must have been removed by an earlier geologic process. We demonstrate that the folds can form in dry crust with a SiO2 content ranging from 48% to 72%. Dry felsic rheologies could explain low emissivity signatures of tesserae that are attributed to a low Fe2+. If felsic, tessera materials may have formed in a prior era via partial melting of hydrous materials derived from recycling of surface waters into the Venus mantle.

Transient Deformation and Stress Patterns Induced by the 2010 Maule Earthquake in the Illapel Segment

Evaluating the transfer of stresses from megathrust earthquakes to adjacent segments is fundamental to assess seismic hazard. Here, we use a 3D forward model as well as GPS and seismic data to investigate the transient deformation and Coulomb Failure Stresses (CFS) changes induced by the 2010 Maule earthquake in its northern segment, where the Mw 8.3 Illapel earthquake occurred in 2015. The 3D model incorporates the coseismically instantaneous, elastic response, and time-dependent afterslip and viscoelastic relaxation processes in the postseismic period. We particularly examine the impact of linear and power-law rheology on the resulting postseismic deformation and CFS changes that may have triggered the Illapel earthquake. At the Illapel hypocenter, our model results in CFS changes of ∼0.06 bar due to the coseismic and postseismic deformation, where the coseismic deformation accounts for ∼85% of the total CFS changes. This is below the assumed triggering threshold of 0.1 bar and, compared to the annual loading rate of the plate interface, represents a clock advance of approximately only 2 months. However, we find that sixteen events with Mw ≥ 5 in the southern region occurred in regions of CFS changes > 0.1 bar, indicating a potential triggering by the Maule event. Interestingly, while the power-law rheology model increases the positive coseismic CFS changes, the linear rheology reduces them. This is due to the opposite polarity of the postseismic displacements resulting from the rheology model choice. The power-law rheology model generates surface displacements that fit better to the GPS-observed landward displacement pattern.

Phenomenological model of viscoelasticity for systems undergoing sol–gel transition

A material undergoing sol–gel transition evolves from the pre-gel (sol) state to the post-gel state through the critical gel state. It is well-known that critical gels exhibit power-law rheology. The faster decay of the relaxation modulus in the pre-gel state can be empirically described by modifying this power-law decay with a stretched exponential factor. A phenomenological analytical expression for the relaxation modulus in the post-gel state is proposed by invoking the symmetry associated with the evolution of the relaxation time on either side of the critical gel state and by accounting for natural constraints. This expression, which depends on the extent of cross-linking, can be suitably transformed to obtain analytical expressions for the dynamic moduli and the continuous relaxation time spectrum. Thus, the proposed model facilitates a comprehensive description of viscoelastic evolution from the pre-gel to the post-gel states. It is validated by carrying out experiments on a model colloidal gel-forming system and by considering other diverse gel-forming systems studied in the literature. After calibrating the parameters of the phenomenological model, it is found to be in excellent agreement with experimental data. Such a well-calibrated phenomenological model can be used to determine any linear viscoelastic response over a wide range of frequencies and extents of cross-linking encompassing the entire sol–gel transition.

Thickness regimes of power law liquids dip coated onto permeable substrates

Coating thin liquid films with complex rheological behaviour on permeable substrates is often an important requirement in several applications such as contact lenses, textiles, and paper-based electronics. Here, we extend the classical Landau-Levich problem of dip coating of Newtonian liquids on rigid substrates to liquids of power-law rheology on permeable substrates. Our results suggest distinct deviation from the classical Landau-Levich relation through exhibition of different regimes of varying dependence of coating film thickness on withdrawal speed. A process map is presented depicting these coating thickness regimes for a wide range of operating parameters such as the substrate permeability factor, power-law exponent of the liquid, and a rescaled capillary number.

Controlling the flow and heat transfer characteristics of power-law fluids in T-junctions using a rotating cylinder

In this work, the combined influence of the power-law rheology and isothermal rotating cylinder has been investigated numerically on the flow and heat transfer characteristics in a T-channel. The cylinder placed at the T-junction imitates the functioning of a rotating valve which controls the flow rate and the enthalpy distribution of the exiting streams in two branches. The range of parameters considered in this work is as: Reynolds number, 1 < Re ≤ 50, power-law index, 0.2 ≤ n ≤ 1, Prandtl number, 10 ≤ Pr ≤ 100 and non-dimensional circumferential velocity of the cylinder, −5 ≤ α ≤ 5. Results suggest that the rotating cylinder can be used as a technique to create and/or reduce the formation of momentum and thermal boundary layers in the flow domain. The rotational velocity (α) and power-law flow index (n) are seen to have a strong effect on the critical Reynolds number for both the main and side branches. The hydrodynamic forces acting over the cylinder surface exerted by the fluid in the flow and perpendicular to the flow direction show a strong relationship with the cylinder rotation. For a particular combination of the studied parameters, these forces are also seen to be negative. Further, it is interesting to note that as the power-law index increases, the flow split ratio decreases while it shows an opposite trend with the Reynolds number. It can also be varied by switching the rotation direction. The dimensionless exiting temperature and enthalpy distribution are seen to have a strong relationship with the direction of rotation, fluid power-law index, Reynolds and Prandtl numbers. The mean values of the Nusselt number are seen to increase and/or decrease with the rotational velocity. For the anticlockwise rotation, the maximum suppression in the heat transfer is seen to be 79% while 62% for the clockwise rotation for the Newtonian fluid behaviour at low Prandtl number and high Reynolds numbers.

The method to quantify cell elasticity based on the precise measurement of pressure inducing cell deformation in microfluidic channels

The cell elasticity has attracted extensive research interests since it not only provides new insights into cell biology but also is an emerging mechanical marker for the diagnosis of some diseases. This paper reports the method for the precise measurement of mechanical properties of single cells deformed to a large extent using a novel microfluidic system integrated with a pressure feedback system and small particle separation unit. The particle separation system was employed to avoid the blockage of the cell deformation channel to enhance the measurement throughput. This system is of remarkable application potential in the precise evaluation of cell mechanical properties. In brief, this paper reports:•The manufacturing of the chip using standard soft lithography;•The methods to deform single cells in a microchannel and measure the relevant pressure drop using a pressure sensor connecting to the microfluidic chip;•Calculation of the mechanical properties including stiffness and fluidity of each cell based on a power-law rheology model describing the viscoelastic behaviors of cells;•Automatic and real-time measurement of the mechanical properties using video processing software.

Effects of energy metabolism on the mechanical properties of breast cancer cells

Tumorigenesis induces actin cortex remodeling, which makes cancerous cells softer. Cell deformability is largely determined by myosin-driven cortical tension and actin fiber architecture at the cell cortex. However, it is still unclear what the weight of each contribution is, and how these contributions change during cancer development. Moreover, little attention has been paid to the effect of energy metabolism on this phenomenon and its reprogramming in cancer. Here, we perform precise two-dimensional mechanical phenotyping based on power-law rheology to unveil the contributions of myosin II, actin fiber architecture and energy metabolism to the deformability of healthy (MCF-10A), noninvasive cancerous (MCF-7), and metastatic (MDA-MB-231) human breast epithelial cells. Contrary to the perception that the actin cortex is a passive structure that provides mechanical resistance to the cell, we find that this is only true when the actin cortex is activated by metabolic processes. The results show marked differences in the nature of the active processes that build up cell stiffness, namely that healthy cells use ATP-driven actin polymerization whereas metastatic cells use myosin II activity. Noninvasive cancerous cells exhibit an anomalous behavior, as their stiffness is not as affected by the lack of nutrients and ATP, suggesting that energy metabolism reprogramming is used to sustain active processes at the actin cortex.

Laminar Forced Convection in Viscous Shear-Thinning Liquid Flows Inside Circular Pipes: Case for a Modified Power-Law Rheology

Abstract Laminar forced convection in viscous, non-Newtonian polymeric liquids that exhibit pseudoplastic or shear-thinning behavior is characterized. The fluid rheology is characterized by a new asymptotic power-law (APL) model, which appropriately represents extensive data for apparent viscosity variation with shear rate—from the low-shear constant-viscosity plateau to shear thinning at high shear rates. This is contrasted with the traditional Ostwald-de-Waele or power-law (PL) model that invariably over-extends the pseudoplasticity in the very low shear-rate region. The latter's limitations are demonstrated by computationally obtaining frictional loss and convective heat transfer results for fully developed laminar flows in a circular pipe maintained at uniform heat flux. The Fanning friction factor and Nusselt number, as would be anticipated from the rheology map of pseudoplastic fluids, are functions of flow rate with the APL model unlike the Newtonian-like constant value obtained with the PL model. Comparisons of the two sets of results highlight the extent of errors inherent in the PL rheology model, which range from 23% to 68% for frictional loss and 3.8% to 13.7% for heat transfer. The new APL rheology model is thus shown to be the more precise characterization of viscous shear-thinning fluids for their thermal processing applications with convective heat transfer.

Nonlinear steady states of hyperelastic membrane tubes conveying a viscous non-Newtonian fluid

We study possible steady states of an infinitely long tube made of a hyperelastic membrane and conveying either an inviscid, or a viscous fluid with power-law rheology. The tube model is geometrically and physically nonlinear; the fluid model is limited to smooth changes in the tube’s radius. For the inviscid case, we analyse the tube’s stretch and flow velocity range at which standing solitary waves of both the swelling and the necking type exist. For the viscous case, we first analyse the tube’s upstream and downstream limit states that are balanced by infinitely growing upstream (and decreasing downstream) fluid pressure and axial stress caused by fluid viscosity. Then we investigate conditions that can connect these limit states by a single solution. We show that such a solution exists only for sufficiently small flow speeds and that it has a form of a kink wave; solitary waves do not exist. For the case of a semi-infinite tube (infinite either upstream or downstream), there exist both kink and solitary wave solutions. For finite-length tubes, there exist solutions of any kind, i.e. in the form of pieces of kink waves, solitary waves, and periodic waves.

Analytical Solutions and Analyses of the Displacement Separating Point in Diffusers

The main purpose of this study is to develop an understanding of the turbulent boundary layer calculation to analyze displacement of the separating point in diffusers. An approximate method has been used which is based on an analogy with the rheological power law and is applied in the study of non-linear viscous flows. At first, the method has been validated with the experimental data in the same experimental cases study. An appropriate geometric with and without the Nano-fluids in the straight-wall and curved-wall conical diffusers has been investigated. Its analyses output was compared with the results obtained by a numerical code. Also, the proposed method is more practical and can be used in diffuser design procedures.

Impact of power-law rheology on the viscoelastic relaxation pattern and afterslip distribution following the 2010 Mw 8.8 Maule earthquake

After large earthquakes at subduction zones, the plate interface continues moving due to mostly frictional afterslip or simply afterslip processes. Below approximately 60 km depth, the seismic moment release at the plate interface is quite small indicating that the shear strength is low and stable sliding is the prevailing process. This agrees with the lack of significant interseismic locking at deeper segments (>60 km) resulting from the inversion of geodetic data and thus low afterslip can be expected. However, inversion models that employ linear viscoelastic mantle rheology and an elastic crust result in significant afterslip at depths >60 km. In this paper, we present a combination of a 3D forward geomechanical model with power-law rheology that simulates postseismic relaxation with dislocation creep processes in the crust and upper mantle and an afterslip inversion. We estimate the cumulative viscoelastic relaxation and the afterslip distribution for the first six years following the 2010 Mw 8.8 Maule earthquake in Chile. The cumulative afterslip distribution is obtained from the inversion of the residual surface displacements between the observed displacements from the continuous GPS (cGPS) and the ones from the forward modelling. We investigate five simulations, four with different dislocation creep parameters for the crust, slab, and upper mantle and one with elastic properties for the crust and slab, and a linear viscoelastic upper mantle for comparison. Our preferred simulation considers a weak crust since it shows the best fit to the cumulative cGPS postseismic displacements, a good fit to the time-series, and, in particular, a good spatial correlation between afterslip and aftershock activity. In this simulation, most of the viscoelastic relaxation occurs in the continental lower crust beneath the volcanic arc due to dislocation creep processes. The resulting afterslip pattern from the inversion is reduced at depths >60 km, which correlates to the low cumulative seismic moment that is released from aftershocks at these depths. Furthermore, the cumulative afterslip moment release from this simulation corresponds to 10% of the main shock in six years, which is approximately half of the moment release that results from models with an elastic crust and linear viscosity in the upper mantle. We conclude that an integrated analysis by considering power-rheology with dislocation creep processes in the continental crust and upper mantle along with aftershock activity may be used to constrain location and magnitude postseismic relaxation processes better.

Condition for the Incipient Motion of Non-Cohesive Particles Due to Laminar Flows of Power-Law Fluids in Closed Conduits

The results of an experimental study on the condition of incipient transport of non-cohesive particles due to the flow of a power-law fluid in a rectangular pipe are presented in this article. The pipe can change its inclination, and experiments were carried out with positive and negative slopes. From a dimensional analysis, the parameters that define the condition of incipient motion were found and validated with experimental data. Thus, the threshold condition is well defined by a particle Reynolds number and a Galileo number, properly modified to take into account the power-law rheology of the fluid. The experimental data are also presented in a standard Shields diagram, including the data obtained in other studies carried out in open-channel laminar flows of Newtonian and power-law fluids.

On the application of simplified rheological models of fluid in the hydraulic fracture problems

In this paper a problem of a hydraulic fracture driven by a non-Newtonian shear-thinning fluid is analysed. For the PKN fracture geometry three different rheological models of fluid are considered: (i) the Carreau fluid, (ii) the truncated power-law fluid, (iii) the power-law fluid. For each of these models a number of simulations are performed. The results are post-processed and compared with each other in order to find decisive factors for similarities/dissimilarities. It is shown that under certain conditions even the basic power-law rheology can be a good substitute for the Carreau characteristics. Although for a particular fluid such a conclusion cannot be made a priori, post-processing based on average values of the fluid shear rates is a very good tool to verify credibility of the results obtained for simplified rheological models. The truncated power-law rheology is a good alternative for the Carreau model. It always produces results that are very similar to those obtained with the equivalent Carreau fluid and simultaneously provides a relative ease of numerical implementation.

Multiscale analysis of viscoelastic properties, topography and internal structure of a biodegradable thermo-responsive shape memory polyurethane

In this paper, the effect of the temperature on viscoelastic properties of the thermo-responsive biodegradable shape memory polyurethane (SMPU) is presented. Comprehensive micro and nano mechanical and morphological studies were performed to assess the behavior of the considered material in the form of 3D-printed fibers. For this purpose, specimen's topography, nano-mechanical and macro-mechanical characterization of the material were carried out. Data obtained from the temperature-dependent AFM indentation test was used to quantify nanoscale viscoelastic properties of the investigated material. Power-law rheology (PLR) was chosen as constitutive viscoelastic relaxation model describing behavior of the investigated material at different temperatures. Topography and phase contrast studies done in the nanoscale exposed bi-phasic structure of heated SMPU polymer - soft and hard domains corresponding with structural composition of the material were detected. The applied fitting procedure enabled for the identification of the viscoelastic properties of the revealed phases identified as the hard and the soft domains.

Augmentation and diminution of non-Boussinesq effects due to non-Newtonian power-law behavior in natural convection

This study investigates the extent and significance of augmentation and diminution of non-Oberbeck–Boussinesq (NOB) effects due to power-law rheology in natural convection. The relative significance of each temperature dependent thermophysical property of a non-Newtonian power-law fluid was first evaluated using an order of magnitude analysis. The limiting criteria were derived for a class of fluids having an Arrhenius-type thermodependent consistency index, and a constant power-law index. Significant fluid property dependencies were identified, incorporated into the conservation equations, and solved numerically. The benchmark problem of natural convection around an isothermal vertical flat plate, immersed in a quiescent power-law fluid, was re-investigated from the viewpoint of NOB effects. It was found that power-law rheology significantly augmented or diminished the acceleration of flow caused by NOB effects. Compared to the Newtonian case, strong shear thinning behavior more than doubled the NOB acceleration of the flow field, and shear thickening inhibited the acceleration by nearly half. Numerical solutions facilitated the visualization of velocity and temperature distributions, and thus provided insights to the underlying physics. It is demonstrated that the cumulative effect of the power-law behavior and temperature dependence of properties is more than a mere superposition of the individual effects. Important contributions from the present work include modifications to the OB approximation’s limiting criteria applicable to power-law fluids, insights into the effect of temperature dependent properties on the flow field and consequent heat transfer, and their correspondence to the results of the order of magnitude analysis. The present investigation of the underlying physics of augmentation and diminution facilitate a better understanding for future studies of NOB effects combined with non-Newtonian behavior.

Solitary waves in power-law deformable conduits with laminar or turbulent fluid flow

Fluid flow through pipe-like conduits embedded in viscously deformable material occurs in many natural systems, including magma transport in the Earth’s mantle and channelized water flow beneath glaciers. Here, we present and explore a model of fluid flow in viscously deformable conduits that unifies previously published models of magmatic and glacial systems. Previous results for magmatic systems have demonstrated the existence of solitary wave solutions for the case of laminar flow in Newtonian conduits. Here we extend these models to allow turbulent fluid flow in power-law materials consistent with models used in subglacial hydrology. The generalized model encompasses both laminar and turbulent fluid flow, and the solid matrix may deform according to any power-law rheology. A quasilinear approximation of the governing equations is introduced, along with an initial condition that develops into a perfect step shock. This initial condition is used in numerical solution of the full nonlinear system where a dispersive wave train forms at shock time. We show that solitary wave solutions exist for all parameters. Rheology-dependent flattening of the wave peaks is investigated. In the limit of a perfectly plastic matrix, the solitary waves approach square waves asymptotically. Motivated by subglacial hydrology models, we study the effect of discharge-dependent melting on evolution of the solitary waves. We find that melting focuses at the wave peaks, causing the waves to grow and accelerate over time.

Influence of non-Newtonian power law rheology on inertial migration of particles in channel flow.

In this paper, the inertial migration of particles in the channel flow of power-law fluid is numerically investigated. The effects of the power-law index (n), Reynolds number (Re), blockage ratio (k), and channel aspect ratio (AR) on the inertial migration of particles and equilibrium position are explored. The results show that there exist two stages of particle migration and four stable equilibrium positions for particles in the cross section of a square channel. The particle equilibrium positions in a rectangular channel are much different from those in a square channel. In shear-thinning fluids, the long channel face equilibrium position and two kinds of particle trajectories are found at low Re. With increasing Re, the short channel face equilibrium position turns to be stable, multiequilibrium positions, and three kinds of particle trajectories along the long wall start to form. Only two stable equilibrium positions exist in shear-thickening fluids. The equilibrium positions are getting closer to the channel centerline with increasing n and k and with decreasing Re. The inertial focusing length L 2 in the second stage of particle migration is much longer than inertial focusing length L 1 in the first stage. In the square channel, L 2 is decreased with increasing Re and k and with decreasing n. In the rectangular channel, L 2 is the shortest in the shear-thinning fluid.

Nanorheology of living cells measured by AFM-based force-distance curves.

Mechanobiology aims to establish functional relationships between the mechanical state of a living a cell and its physiology. The acquisition of force-distance curves with an AFM is by far the dominant method to characterize the nanomechanical properties of living cells. However, theoretical simulations have shown that the contact mechanics models used to determine the Young's modulus from a force-distance curve could be off by a factor 5 from its expected value. The semi-quantitative character arises from the lack of a theory that integrates the AFM data, a realistic viscoelastic model of a cell and its finite-thickness. Here, we develop a method to determine the mechanical response of a cell from a force-distance curve. The method incorporates bottom-effect corrections, a power-law rheology model and the deformation history of the cell. It transforms the experimental data into viscoelastic parameters of the cell as a function of the indentation frequency. The quantitative agreement obtained between the experiments performed on living fibroblast cells and the analytical theory supports the use of force-distance curves to measure the nanorheological properties of cells.

A unified rheological model for cells and cellularised materials.

The mechanical response of single cells and tissues exhibits a broad distribution of time-scales that often gives rise to a distinctive power-law rheology. Such complex behaviour cannot be easily captured by traditional rheological approaches, making material characterisation and predictive modelling very challenging. Here, we present a novel model combining conventional viscoelastic elements with fractional calculus that successfully captures the macroscopic relaxation response of epithelial monolayers. The parameters extracted from the fitting of the relaxation modulus allow prediction of the response of the same material to slow stretch and creep, indicating that the model captured intrinsic material properties. Two characteristic times, derived from the model parameters, delimit different regimes in the materials response. We compared the response of tissues with the behaviour of single cells as well as intra and extra-cellular components, and linked the power-law behaviour of the epithelium to the dynamics of the cell cortex. Such a unified model for the mechanical response of biological materials provides a novel and robust mathematical approach to consistently analyse experimental data and uncover similarities and differences in reported behaviour across experimental methods and research groups. It also sets the foundations for more accurate computational models of tissue mechanics.