Shear-thinning Viscoelastic Fluid Research Articles

Elasto-inertial instability in torsional flows of shear-thinning viscoelastic fluids

It is well known that inertia-free shearing flows of a viscoelastic fluid with curved streamlines, such as the torsional flow between a rotating cone and plate or the flow in a Taylor–Couette geometry, can become unstable to a three-dimensional time-dependent instability at conditions exceeding a critical Weissenberg ( $Wi$ ) number. However, the combined effects of fluid elasticity, shear thinning and finite inertia (as quantified by the Reynolds number $Re$ ) on the onset of elasto-inertial instabilities are not fully understood. Using a set of cone–plate geometries, we experimentally explore the entire $Wi$ – $Re$ phase space for a series of nonlinear viscoelastic fluids (with the dependence on shear rate $\dot{\gamma}$ quantified using a shear-thinning parameter $\beta _P(\dot {\gamma })$ ). We tune $\beta _P(\dot {\gamma })$ by varying the dissolved polymer concentration in solution. This progressively reduces shear thinning but leads to finite inertial effects before the onset of elastic instability, and thus naturally results in elasto-inertial coupling. Time-resolved rheometric measurements and flow visualization experiments allow us to investigate the effects of flow geometry, and document the combined effects of varying $Wi, Re$ and $\beta _P(\dot {\gamma })$ on the emergence of secondary motions at the onset of instability. The resulting critical state diagram quantitatively depicts the competition between the stabilizing effects of shear thinning and the destabilizing effects of inertia. We extend the curved streamline instability criterion of Pakdel & McKinley (Phys. Rev. Lett., vol. 77, no. 12, 1996, p. 2459) for the onset of purely elastic instability in curvilinear geometries by using scaling arguments to incorporate shear thinning and finite inertial effects. The augmented condition facilitates predictions of the onset of instability over a broader range of flow conditions, and thus bridges the gap between purely elastic and elasto-inertial curved streamline instabilities.

Numerical Investigation on the Conical Flow of Viscoelastic Fluids

The conical problem represents one of the important problems in the various fields of industrial as the automotive industry and the aerospace industry, due to their great role in controlling the flow of liquids and gases. This article covers a computational investigation of incompressible the Phan–Thien/Tanner shear-thinning viscoelastic fluid flow through a conical converging channel. Here, we select hybrid finite element/finite volume algorithm as a first time to treat such problem. This method consists of the combination of a Taylor-Galerkin/pressure correction finite element method (TGPC-FEM) and a cell-vertex finite volume approach (CV-FEA) to solve the system of partial differential equations that govern the fluid flow. The TGPC-FEM is employed to the momentum and mass conservation models, while the stress constitutive models are treated by finite volume implementation. The findings of current study are concerned with stress response, deformation rate, and pressure drop under variations in Weissenberg number and EPTT parameters. The effect of shear-thinning behaviour with the EPTT representation is also considered.

Numerical simulations of suspensions of rigid spheres in shear-thinning viscoelastic fluids

In multiphase flows, accurately modeling the interaction between the liquid phase of complex fluids and a porous medium of solid spheres poses a fundamental challenge. The dynamics of moderately dense non-colloidal suspensions constituted by static random arrays of mono-disperse spherical particles in non-linear viscoelastic fluids is studied numerically. This numerical study consists of about 9000 different systems, in which the volume fraction ϕ (0.04≤ϕ≤0.2) of the dispersed solid phase, the Reynolds number Re(5≤Re≤50), the solvent viscosity ratio β(0.05≤β≤0.9), the Weissenberg number Wi(0.5≤Wi≤4), and the mobility parameter of the Giesekus model α (0.1≤α≤0.5) were varied to understand the particle's interactions with the viscoelastic suspending fluid. We aim to investigate the relationship between the volume fraction of the dispersed solid phase and the non-linear rheology of shear-thinning viscoelastic fluids with the normalized average drag force ⟨F⟩. In addition, by assessing the flow patterns predicted numerically, we were able to provide a characterization of the velocity and stress fields as a function of the simulation parameters.

Extensional rheometry of mobile fluids. Part I: OUBER, an optimized uniaxial and biaxial extensional rheometer

Numerical optimization of a “six-arm cross-slot” device yields several three-dimensional shapes of fluidic channels that impose close approximations to an ideal uniaxial (biaxial) stagnation point extensional flow under the constraints of having four inlets and two outlets (two inlets and four outlets) and for Newtonian creeping flow. One of the numerically designed geometries is considered suitable for fabrication at the microscale, and numerical simulations with the Oldroyd-B and Phan-Thien and Tanner models confirm that the optimal flow fields are observed in the geometry for both constant viscosity and shear thinning viscoelastic fluids. The geometry, named the optimized uniaxial and biaxial extensional rheometer (OUBER), is microfabricated with high precision by selective laser-induced etching of a fused-silica substrate. Employing a refractive index-matched viscous Newtonian fluid, microtomographic-particle image velocimetry enables the measurement of the flow field in a substantial volume around the stagnation point. The flow velocimetry, performed at low Reynolds number (<0.1), confirms the accurate imposition of the desired and predicted flows, with a pure extensional flow at an essentially uniform deformation rate being applied over a wide region around the stagnation point. In Part II of this paper [Haward et al., J. Rheol. 67, 1011–1030 (2023)], pressure drop measurements in the OUBER geometry are used to assess the uniaxial and biaxial extensional rheometry of dilute polymeric solutions, in comparison to measurements made in planar extension using an optimized-shape cross-slot extensional rheometer [OSCER, Haward et al., Phys. Rev. Lett. 109, 128301 (2012)].

Analysis of Interaction and Flow Pattern of Multiple Bubbles in Shear-Thinning Viscoelastic Fluids

A numerical study was conducted on the interaction of bubbles with different diameters and arrangements in shear-thinning viscoelastic fluids using OpenFOAM. The Volume of Fluid (VOF) method combined with the surface tension model was used to track the gas–liquid interface, and the rheological properties of the fluid were characterized with the Giesekus model. The numerical results are corresponded with the previous references, verifying the correctness of the simulation method. The influences of the initial bubble diameter, horizontal spacing, and arrangement on the motion state of three parallel bubbles were studied in detail. The flow pattern of the bubble rising was analyzed and compared with the motion state of parallel unequal double bubbles. As the diameter of the bubbles increases, the interaction among three equal size bubbles is changed from coalescence to detachment. Changing the diameter of one of the bubbles will significantly affect the movement of the larger diameter bubble, which is due to the enhancement in kinetic energy. The final state of some arrangement ways is consistent with the phenomenon of unequal double bubbles. The shear thinning effect, the velocity difference between bubbles, and the flow field around bubbles are considered the main reasons that decide the interaction between bubbles.

A generalized model for field assessment of particle settling velocity in a viscoelastic fluid

The accurate prediction of particle settling velocity in viscoelastic fluids is important for the design, analysis, and optimization of various oil-field operations including hole cleaning and cuttings transport in oil and gas well drilling, proppant transport in hydraulic fracturing operations. An experimental study was, therefore, conducted to investigate the effect of fluid viscoelastic properties on the particle settling velocity. The main objectives of the study were to understand the factors that influence the particle settling velocity in shear-thinning viscoelastic fluids and develop a generalized model for the field assessment of particle settling velocity in shear-thinning viscoelastic fluids.We prepared ten different fluids, which were divided into two groups based on their shear viscosity values. In each group, five fluids were having similar shear viscosity and variable elasticity values. Nineteen different spherical particles were used to conduct particle settling experiments with a density range from 2700 kg/m3 to 6000 kg/m3 and a diameter range from 1 mm to 4 mm. Rheological characterizations of the fluids have been conducted by using funnel viscometer, API Rotational viscometer, controlled shear rate, and amplitude sweep test measurements. The fluid elasticity was quantified by using the concept of energy dissipation, which is determined by using standard field measurements of the apparent viscosity (AV) and the funnel viscosity.Finally, based on the experimental results and theory of the particle settling in non-Newtonian fluids, we proposed a model that can be used for predicting particle settling velocity in shear thinning viscoelastic fluids. The statistical analyses have shown that the root means square error and mean absolute percentage error of model predictions were 0.0032 m/s and 4.1%, respectively.

Numerical simulation of the behaviors of single bubble in shear-thinning viscoelastic fluids

In this paper, the behaviors of single bubble in shear-thinning viscoelastic fluids were investigated using OpenFOAM. The volume-of-fluid method was used to capture the interface, and the Giesekus model was adopted for describing the rheological behaviors of shear-thinning viscoelastic fluids. The bubble cusp, negative wake, and velocity jump phenomenons in viscoelastic fluids were obtained, and the effects of wall effect, mobility factor α, and Weissenberg number (Wi) on bubble behaviors were investigated. The results showed that the viscoelastic stress is the main reason for the formation of bubble cusp, the relaxation of polymer macromolecules leads to the formation of negative wake, and the negative wake may be the main reason for the velocity jump. The open angle θ of the negative wake decreases and the distribution region in the vertical direction increases with the increasing Reynolds numbers (Re). In addition, the magnitude of the maximum negative velocity increases with Re and appears further away from the bubble cusp. Moreover, decreasing the wall effect can accelerate the formation of viscoelastic stress and make the bubble cusp appear earlier. As the mobility factor α increases, the viscosity and viscoelastic stress of the fluid near the bubble decrease; this causes the terminal velocity of the bubble to increase. As the Wi increases, both the maximum velocity and terminal velocity of the bubble increase, and the time lag occur. The viscoelastic stress τyy has a longer linear distribution at the tail of the bubble with the increase in Wi.

Elasto-Inertial Focusing Mechanisms of Particles in Shear-Thinning Viscoelastic Fluid in Rectangular Microchannels.

Growth of the microfluidics field has triggered numerous advances in focusing and separating microparticles, with such systems rapidly finding applications in biomedical, chemical, and environmental fields. The use of shear-thinning viscoelastic fluids in microfluidic channels is leading to evolution of elasto-inertial focusing. Herein, we showed that the interplay between the elastic and shear-gradient lift forces, as well as the secondary flow transversal drag force that is caused by the non-zero second normal stress difference, lead to different particle focusing patterns in the elasto-inertial regime. Experiments and 3D simulations were performed to study the effects of flowrate, particle size, and the shear-thinning extent of the fluid on the focusing patterns. The Giesekus constitutive equation was used in the simulations to capture the shear-thinning and viscoelastic behaviors of the solution used in the experiments. At low flowrate, with Weissenberg number Wi ~ O(1), both the elastic force and secondary flow effects push particles towards the channel center. However, at a high flowrate, Wi ~ O(10), the elastic force direction is reversed in the central regions. This remarkable behavior of the elastic force, combined with the enhanced shear-gradient lift at the high flowrate, pushes particles away from the channel center. Additionally, a precise prediction of the focusing position can only be made when the shear-thinning extent of the fluid is correctly estimated in the modeling. The shear-thinning also gives rise to the unique behavior of the inertial forces near the channel walls which is linked with the 'warped' velocity profile in such fluids.

Dean flow velocity of shear-thickening SiO2 nanofluids in curved microchannels

We report the effects of a curvilinear microchannel width, height, and radius of curvature, as well as the kinematic viscosity and axial velocity of shear-thickening nanofluids, on the average Dean velocity (VDe) of the secondary flow in the microchannel. Manipulation of inertial and Dean drag forces in curvilinear microchannels has enabled high-throughput and high-resolution size-based separation of microparticles and cells in various biomedical applications. VDe plays a deterministic role in the estimation of the Dean drag force and the design of these microfluidic devices. Despite the previous numerical and experimental studies on VDe of Newtonian and shear-thinning viscoelastic fluids, VDe of shear-thickening metallic nanofluids, such as SiO2 nanoparticles in water, in curved microchannels is still unknown. Such shear-thickening fluids are being used in thermal microsystem applications and are on the verge of entering the field of inertial microfluidics for particle and cell sorting. Our investigations have shown that VDe of shear-thickening SiO2–water nanofluids scales directly with the channel width and the fluid axial velocity, while being inversely proportional with the SiO2 concentration and the channel radius of curvature. Our non-dimensional analysis has led to the development of an empirical correlation that relates VDe-based Reynolds number of the nanofluid to the Dean number and the normalized kinematic viscosity of the nanofluid. It provides a significant accuracy in estimating VDe of shear-thickening fluids, compared to application of Newtonian or shear-thinning equations in the literature, which could be useful toward future design of particle and cell sorting and washing microdevices.

Droplet deformation and breakup in shear-thinning viscoelastic fluid under simple shear flow

A three-dimensional lattice Boltzmann method, which couples the color-gradient model for two-phase fluid dynamics with a lattice diffusion-advection scheme for the elastic stress tensor, is developed to study the deformation and breakup of a Newtonian droplet in the Giesekus fluid matrix under simple shear flow. This method is first validated by the simulation of the single-phase Giesekus fluid in a steady shear flow and the droplet deformation in two different viscoelastic fluid systems. It is then used to investigate the effect of Deborah number De, mobility parameter α, and solvent viscosity ratio β on steady-state droplet deformation. We find for 0.025<α<0.5 that as De increases, the steady-state droplet deformation decreases until eventually approaching the one in the pure Newtonian case with the viscosity ratio of 1/β, which is attributed to the strong shear-thinning effect at high De. While for lower α, the droplet deformation exhibits a complex nonmonotonic variation with De. Under constant De, the droplet deformation decreases monotonically with α but increases with β. Force analysis shows that De modifies the droplet deformation by altering the normal viscous and elastic stresses at both poles and equators of the droplet, while α mainly alters the normal stresses at the poles. Finally, we explore the roles of De and α on the critical capillary number Cacr of the droplet breakup. By establishing both Ca–De and Ca–α phase diagrams, we find that the critical capillary number increases with De or α except that a plateau critical capillary number is observed in Ca–De phase diagram.

Data-driven closure model for the drag coefficient of the creeping flow past a translating sphere in a shear-thinning viscoelastic fluid

A drag force closure model for particle-laden viscoelastic fluid flows is the key to describing the ensemble-averaged behavior of the mixture. The effects of fluid rheological properties on the flow dynamics of a spherical particle in viscoelastic fluids in the creeping flow regime are parameterized using the Giesekus rheological model. Direct numerical simulations are performed within a large range of Deborah number(0−10).The drag force of a sphere in unbounded Giesekus fluids decreases monotonically with the increase of Deborah number. The negative wake may occur when the viscosity ratio is larger than 0.6 in Giesekus fluids but is absent in Oldroyd-B fluids in all conditions. An explicit closure model for the drag coefficient of a sphere in Giesekus fluids is established using the backpropagation artificial neural network. The drag closure model provides a method for explicitly formulating the momentum exchange model for dilute suspensions of solid particles in shear-thinning viscoelastic fluids.

Emergent patterns and stable interfaces during radial displacement of a viscoelastic fluid

We explore the interfacial instability that results when a Newtonian fluid (a glycerol-water mixture, inner fluid) displaces a viscoelastic fluid (a dense cornstarch suspension, outer fluid) in a radial Hele-Shaw cell. As the ratio of viscosities of the inner and outer fluids is increased, side branched interfacial patterns are replaced by more stable interfaces that display proportionate growth and finger coalescence. We correlate the average finger spacing with the most dominant wavelength of interfacial instability, computed using a mathematical model that accounts for viscous fingering in miscible Hele-Shaw displacements. The model predictions on the role of viscosity ratio on finger spacing are in close agreement with the experimental observations. Our study lends insight into the significant contribution of the viscoelasticity of the outer fluid on the morphology and growth of interfacial patterns.

On peculiar behaviours at critical volumes of a three-dimensional bubble rising in viscoelastic fluids

In this paper, we carried out an investigation of the dynamics of an air bubble rising in viscoelastic liquids with increasing volumes via volume-of-fluid based direct numerical simulations (DNS) with local adaptive mesh refinement techniques. The exponential Phan-Thien Tanner (EPTT) model is adopted for describing the rheological behaviours of the shear-thinning viscoelastic fluid. The well-known jump discontinuity in the terminal rise velocity at a critical bubble volume is captured, which agrees qualitatively with the experimental observation in the literature. On this basis, the numerical simulations have been performed for different rheological properties in wider flow regimes, providing new insights into the local flow field and the stress distribution around the deformable rising bubbles. A characteristic dimensionless regime map has also been proposed to distinguish the peculiar behaviours of a bubble rising in viscoelastic fluids at different Galilei (Ga) and Eötvös (Eo) numbers. We found that at low Ga and low Eo numbers, the large elastic stresses concentrate in the small region near the bubble’s trailing end, leading to the formation of bubble cusp. The appearance of the negative wake is the main reason for the velocity jump to occur. However, for higher Ga of 10, the viscous forces become less dominating. The fluctuations of the polymeric stresses are substantially intensified with increasing bubble volume, and large bubbles experience a pulsating rising velocity with oscillation shapes. On the other hand, at intermediate Ga and high Eo of 10, the modest surface tension induces a cap with a thin skirt trailing the bubble main body at early times. Due to the polymer stretching, the strong extensional flow behind the small bubble eventually forces the bubble tail to break up. The above analyses on the bubble dynamics at constant Morton (Mo) numbers facilitate the extrapolation of the volume effects at different flow regimes and, therefore, will guide the applications in the future.

Dynamics of rigid particles in a confined flow of viscoelastic and strongly shear-thinning fluid at very small Reynolds numbers

Despite growing interest in the focusing and manipulation of particles in non-Newtonian fluids in confined flows, the combined effect of viscoelastic and shear-thinning effects on particle dynamics is not well understood. Herein, we report the dynamics of rigid microparticles in confined flows of strongly shear-thinning viscoelastic (STVE) fluids at very low Reynolds numbers. Our experiments with different STVE fluids reveal five different regimes: original streamline, bimodal, center migration, defocusing, and wall migration (WM), depending upon the fluid properties and flow rates. It is found that the occurrence of the different regimes depends on the STVE parameter (ψ) and average strain rate (γ̇¯). We find that the dynamics of particles in the different regimes is underpinned by the synergy between viscoelastic lift force (FVE) and shear-thinning lift force (FST). Numerical simulation results of strain rate and viscosity profiles at different ψ and γ̇¯ enable estimation of the forces and explaining the dynamics observed. We expect that our study will find relevance in applications involving positioning and manipulation of particles in confined flows of STVE fluids.

A lattice Boltzmann modeling of the bubble velocity discontinuity (BVD) in shear-thinning viscoelastic fluids

The bubble velocity discontinuity (BVD), when single bubble rising in shear-thinning viscoelastic fluids, is studied numerically. Our three-dimensional numerical scheme employs a phase-field lattice Boltzmann method together with a lattice Boltzmann advection-diffusion scheme, the former to model the macroscopic hydrodynamic equations for multiphase fluids, and the latter to describe the polymer dynamics modeled by the exponential Phan–Thien–Tanner (ePTT) constitutive model. An adaptive mesh refinement technique is implemented to reduce computational cost. The multiphase solver is validated by simulating the buoyant rise of single bubble in a Newtonian fluid. The critical bubble size for the existence of the BVD and the velocity-increasing factor of the BVD are accurately predicted, and the results are consistent with the available experiments. Bubbles of different sizes are characterized as subcritical (smaller than critical size) and supercritical (larger than critical size) according to their transient rising velocity behaviors, and the polymeric stress evolution affecting the local flow pattern and bubble deformation is discussed. Pseudo-supercritical bubbles are observed with transition behaviors in bubble velocity, and their sizes are smaller than the critical value. The formation of bubble cusp and the existence of negative wake are observed for both the pseudo-supercritical and the supercritical bubbles. For the supercritical bubble, the trailing edge cusp and the negative wake arise much earlier. The link between the BVD, the bubble cusp, and the negative wake is discussed, and the mechanism of the BVD is explained.

Experimental investigation of the solid-liquid separation in a stirred tank owing to viscoelasticity

In this study, we investigated the motion of solid particles dispersed in highly viscous complex fluids agitated in a stirred vessel. We used a refractive index matching method and a combination of planar laser- induced fluorescence (PLIF), particle image velocimetry (PIV), and particle-tracking velocimetry (PTV) techniques to measure the velocity fields of the solid and fluid phases simultaneously along with the spatiotemporal distribution of the solids in the tank. The experimental data show that in a Newtonian ambient fluid, particles disperse uniformly in the plane of measurement, while in a strongly shear-thinning viscoelastic ambient fluid they tend to accumulate in the core of the vortices formed in the flow domain. We found that the solids migrate to the core of the vortices also when the ambient fluid is a Boger fluid, i.e., viscoelastic but not shear thinning. The effect of the first normal stress difference, N1, on the vortex sizes and circulation intensities was also examined, with both properties decreasing for increasing N1. Finally, we observed that the clustering time of the solids in the vessel for viscoelasticity-induced migration was at least three orders of magnitude lower than that obtained from the literature for inertia-induced migration.

Elastic instabilities in the electroosmotic flow of non-Newtonian fluids through T-shaped microchannels.

Electroosmotic flow (EOF) has been widely used to transport fluids and samples in micro- and nanofluidic channels for lab-on-a-chip applications. This essentially surface-driven plug-like flow is, however, sensitive to both the fluid and wall properties, of which any inhomogeneity may draw disturbances to the flow and even instabilities. Existing studies on EOF instabilities have been focused primarily upon Newtonian fluids though many of the chemical and biological solutions are actually non-Newtonian. We carry out a systematic experimental investigation of the fluid rheological effects on the elastic instability in the EOF of phosphate buffer-based polymer solutions through T-shaped microchannels. We find that electro-elastic instabilities can be induced in shear thinning polyacrylamide (PAA) and xanthan gum (XG) solutions if the applied direct current voltage is above a threshold value. However, no instabilities are observed in Newtonian or weakly shear thinning viscoelastic fluids including polyethylene oxide (PEO), polyvinylpyrrolidone (PVP), and hyaluronic acid (HA) solutions. We also perform a quantitative analysis of the wave parameters for the observed elasto-elastic instabilities.

Optimised multi-stream microfluidic designs for controlled extensional deformation

In this study, we optimise two types of multi-stream configurations (a T-junction and a flow-focusing design) to generate a homogeneous extensional flow within a well-defined region. The former is used to generate a stagnation point flow allowing molecules to accumulate significant strain, which has been found very useful for performing elongational studies. The latter relies on the presence of opposing lateral streams to shape a main stream and generate a strong region of extension in which the shearing effects of fluid–wall interactions are reduced near the region of interest. The optimisations are performed in two (2D) and three dimensions (3D) under creeping flow conditions for Newtonian fluid flow. It is demonstrated that in contrast with the classical-shaped geometries, the optimised designs are able to generate a well-defined region of homogeneous extension. The operational limits of the obtained 3D optimised configurations are investigated in terms of Weissenberg number for both constant viscosity and shear-thinning viscoelastic fluids. Additionally, for the 3D optimised flow-focusing device, the operational limits are investigated in terms of increasing Reynolds number and for a range of velocity ratios between the opposing lateral streams and the main stream. For all obtained 3D optimised multi-stream configurations, we perform the experimental validation considering a Newtonian fluid flow. Our results show good agreement with the numerical study, reproducing the desired kinematics for which the designs are optimised.

Linear instability of shear thinning pressure driven channel flow

We study theoretically pressure driven planar channel flow of shear thinning viscoelastic fluids. Combining linear stability analysis and full nonlinear simulation, we study the instability of an initially one-dimensional base state to the growth of two-dimensional perturbations with wavevector in the flow direction. We do so within three widely used constitutive models: the microscopically motivated Rolie-Poly model, and the phenomenological Johnson-Segalman and White-Metzner models. In each model, we find instability when the degree of shear thinning exceeds some level characterised by the logarithmic slope of the flow curve at its shallowest point, n=dlogΣ/dlogγ˙|min. Specifically, we find instability for n < n*, with n* ≈ 0.21, 0.11 and 0.30 in the Rolie-Poly, Johnson-Segalman and White-Metzner models respectively. Within each model, we show that the critical pressure drop for the onset of instability obeys a criterion expressed in terms of this degree of shear thinning, n, together with the derivative of the first normal stress with respect to shear stress. Both shear thinning and rapid variations in first normal stress across the channel are therefore key ingredients driving the instability. In the Rolie-Poly and Johnson-Segalman models, the underlying mechanism appears to involve the destabilisation of a quasi-interface that exists in each half of the channel, across which the normal stress varies rapidly. (The flow is not however shear banded in any parameter regime that we consider.) In the White-Metzner model, no such quasi-interface exists, but the criterion for instability nonetheless appears to follow the same form as in the Rolie-Poly and Johnson-Segalman models. This presents an outstanding puzzle concerning any possibly generic nature of the instability mechanism. We finally make some brief comments on the Giesekus model, which is rather different in its predictions from the other three.

Study of bovine sperm motility in shear-thinning viscoelastic fluids

To elucidate the process whereby sperm arrive at an egg in the female reproductive organs, it is essential to investigate how rheological properties of the fluid around mammalian spermatozoa affect their motility. We examined the motility and flagellar waveform of bovine sperm swimming in a fluid with similar rheological properties as mammalian cervical mucus. The results indicated that the surrounding rheological properties largely affected the flagellar waveform of mammalian spermatozoa; in particular, shear-thinning viscoelastic fluid increased the progressive motility of the sperm. To investigate the influence of flagellar waveform on sperm motility in more detail, the waveform was expressed as a function and the progressive thrust of the sperm was calculated based on the empirical resistive force theory. The results of this study showed that the progressive thrust increased with the curvature of the flagellar tip. Moreover, we calculated the thrust efficiency of motile sperm. Results showed that the thrust efficiency in shear-thinning viscoelastic fluids was larger than that in Newtonian fluids, regardless of viscosity. This suggests that motile sperm in cervical mucus move efficiently by means of a motion mechanism that is suited to their surrounding environment.