Experimental investigation of viscoelastic drop deformation in Newtonian matrix at high capillary number under simple shear flow

The steady state deformation of a viscoelastic drop (Boger fluid) in a Newtonian liquid at high capillary number under simple shear flow is investigated by direct visualization using a specially designed Couette apparatus which enables visualization from two perpendicular directions. Two drop deformation modes are found: (1) Mode I – drop deformation in the flow direction and (2) Mode II – drop deformation in the vorticity direction. The drop deformation mode depends on the relative strength of the elastic contribution to viscous contribution. If the elastic contribution is weak compared to the viscous contribution, the drop elongates in the flow direction via Mode I. If the elastic contribution is strong, the drop elongates in the vorticity direction via Mode II. The drop size also affects the drop deformation. At the same capillary number, bigger drops have larger deformations than smaller drops.     

In situ observations of unusual drop deformation and wobbling in simple shear flow

Deformation and wobbling of a liquid drop immersed in a liquid matrix were studied under mild shear conditions for various viscosity ratios. In situ visualization experiments were conducted on a homemade transparent Couette cell incorporated to the Paar Physica MCR500 shear rheometer. The effect of drop or matrix elasticity was examined and was found to play a major role in both deformation and wobbling processes. Experimental results were compared to Jackson and Tucker (J Rheol 47:659–682, 2003), Maffettone and Minale (J Non-Newton Fluid Mech 78:227–241, 1998) and Yu and Bousmina (J Rheol 47:1011–1039, 2003) ellipsoidal models. It was found that the agreement between the Newtonian models and the experimental results required an increase in the drop viscosity. Such increment in viscosity was found to scale with the first normal stress difference.     

Break-up of a Newtonian drop in a viscoelastic matrix under simple shear flow

The effect of matrix elasticity on the break-up of an isolated Newtonian drop under step shear flow is herein presented. Constant-viscosity, elastic polymer solutions (Boger fluids) were used as matrix phase. Newtonian silicon oils were used as drop phase. Three viscosity ratios were explored (drop/matrix), i.e. 2, 0.6 and 0.04. Following the theoretical analysis of Greco [Greco F (2002) J Non-Newtonian Fluid Mech 107:111–131], the role of elasticity on drop fluid dynamics was quantified according to the value of the parameter p=/_em, where is a constitutive relaxation time of the matrix fluid and _em is the emulsion time. Different fluids were prepared in order to have p ranging from 0.1 to 10. At all the viscosity ratios explored, break-up was hindered by matrix elasticity. The start-up transient of drop deformation, at high, but sub-critical capillary numbers, showed an overshoot, during which the drop enhanced its orientation toward the flow direction. Both phenomena increase if the p parameter increases. Finally, the non-dimensional pinch-off length and break-up time were also found to increase with p.This paper was presented at the first Annual European Rheology Conference (AERC) held in Guimarães, Portugal, September 11-13, 2003.     

Analysis of simple shear endochronic equations for finite plastic deformation

IntroductionThestress_strainbehaviorofmaterialswithfiniteplasticdeformationisaninterestingissue ,onwhichsignificantprogresshasbeenmadethroughboththephenomenologicalandphysicalapproaches.Thephenomenologicalapproachisbasedoncontinuummechanicsofplasticity .Ithasitsadvantageinsolvingcomplicatedproblemsbecauseofitssimplicity .Mostofphenomenologicaltheoriesareinvolvedintheconceptofcorotationalrates.Thematerialderivativeofstresswasnotobjectiveunderfinitedeformation .TheJaumannratewasusuallyusedbefo…     

Effects of viscosity ratio on deformation of a viscoelastic drop in a Newtonian matrix under steady shear

Deformation of an Oldroyd B drop in a Newtonian matrix under steady shear is simulated using a front tracking finite difference method for varying viscosity ratio. For drop viscosity lower than that of the matrix, the long-time steady deformation behavior is similar to that of the viscosity matched system—the drop shows reduced deformation with increasing Deborah number due to the increased inhibiting viscoelastic normal stress inside the drop. However for higher viscosity ratio systems, the drop response is non-monotonic—the steady drop deformation first decreases with increasing Deborah number but above a critical Deborah number, it increases with further increase in Deborah number, reaching higher than the viscous case value for some viscosity ratios. We explain the increase in deformation with Deborah number by noting that at higher viscosity ratios, strain rate inside the drop is reduced, thereby reducing the inhibiting viscoelastic stress. Furthermore, similar to the viscosity matched system, the drop inclination angle increases with increasing Deborah number. A drop aligned more with the maximum stretching axis at 45 degree of the imposed shear, experiences increased viscous stretching. With increased ratio of polymeric viscosity to total drop viscosity, the drop deformation decreases and the inclination angle increases. Our simulation results compare favorably with a number of experimental and computational results from other researchers.     

A new simple third-order shear deformation theory of plates

An improved simple third-order shear deformation theory for the analysis of shear flexible plates is presented in this paper. This new plate theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the author; a system of 10th-order differential equilibrium equations in terms of the three generalized displacements of bending plates; five boundary conditions at each edge of plate boundaries. Although the resulting displacement field is the same as that proposed by Murthy, the variational consistent governing equations and the associated proper boundary conditions are derived and identified in this work for the first time in the literature. The applications and accuracy of the present shear deformation theory of plates are demonstrated by analytically solving the differential governing equations of a twisting plate, a bending beam and two bending plates to which the 3-D elasticity solutions are available, and excellent agreements are achieved even for the torsion of a plate with square cross-section as well the local effects of stresses at plate boundaries can be characterized accurately. These analytical solutions clearly show that the simple third-order shear deformation theory developed in this work indeed gives better results than the first-order shear deformation theories and other simple higher-order shear deformation theories, since the present third-order shear flexible theory is based on a more rigorous kinematics of displacements and consists of not only a system of variational consistent differential equations, but also a group of consistent boundary conditions associated with the differential equations. The present simple third-order shear deformation theory can easily be applied to the static and dynamic finite element analysis of laminated plates just like the applications of other popular shear flexible plate theories, and improved results could be obtained from the present simple third-order shear deformable theories of plates.     

Single bubble deformation and breakup in simple shear flow

Experiments in a parallel band apparatus and a transparent concentric cylinder device allow the observation of bubble deformation (shape and orientation) and breakup as a function of the viscosity ratio λ and the Capillary number Ca. For viscosity ratios between 3.1 × 10⁻⁷ and 6.7 × 10⁻⁸, critical Capillary numbers Ca _c for bubble breakup between 29 and 45 are found. It is furthermore shown that in the given parameter space no clear distinction between tip breakup and fracture can be made for bubbles. An erratum to this article can be found at     

Drop deformation under small-amplitude oscillatory shear flow

The deformation of an isolated drop in an immiscible liquid undergoing oscillatory shear flow is experimentally investigated as a function of frequency and up to moderate amplitudes. Oscillatory shear flow is generated by using a parallel plate apparatus. Drop shape is observed by video light microscopy along the vorticity direction of the shear flow. The two principal axes and the orientation of the drop in the plane of shear are measured by image analysis. In the small amplitude range, the time dependence of the axes is also harmonic, but not in phase with the applied strain, the phase difference being a decreasing function of the imposed frequency. The linear range (where the major axis is proportional to the amplitude) extends up to strains of 0.5. Good quantitative agreement was found with the Palierne linear viscoelastic model (Palierne, J. F., Linear rheology of viscoelastic emulsions with interfacial tension, Rheol. Acta, 29, 204–214, 1990), thus providing a further example of the good agreement between experiments and small deformation theory.     

On the non-oscillation criterion for multiplicative anisotropic plasticity at large simple shear deformation

The criterion for non-oscillatory stresses under monotonic large simple shear deformation in the context of multiplicative anisotropic plasticity is discussed. In particular, evolving anisotropy combined with a Hill type of yield criterion is considered. It is shown that a sufficient, but not necessary, criterion for a non-oscillatory stress is ellipticity of the first Piola–Kirchhoff stress. Loss of ellipticity corresponds to a critical value h_cr of the generalized plastic modulus. Similarly, the absence of limit points on the stress–strain relation motivates an alternative criterion in terms of a critical value h_sh  h_cr. Finally, this criterion is demonstrated analytically as well as numerically for an important class of models with evolving anisotropy of the saturation type.     

A simple shear cell for the direct visualization of step-stress deformation in soft materials

We introduce a custom-built stress-controlled shear cell coupled to a confocal microscope for direct visualization of constant-stress shear deformation in soft materials. The torque generator is a cylindrical Taylor–Couette system with a Newtonian fluid between a rotating inner bob and a free-to-move outer cup. A spindle/cone assembly is coaxially coupled to the cup and transfers the torque exerted by the fluid to the sample of interest in a cone-and-plate geometry. We demonstrate the performance of our device in both steady-state and transient experiments with different viscoelastic materials. Our apparatus can conduct unidirectional constant-stress experiments as accurately as most commercial rheometers, with the capability to directly visualize the flow field using tracer particles. Further, our step-stress experiments on viscoelastic materials are devoid of creep ringing, which is an advantageous aspect of our torque generation mechanism. We believe that the device presented here could serve as a powerful and cost-effective tool to investigate the microstructural determinants of nonlinear rheology in complex fluids.     

Fatigue crack growth under anti-plane shear mode deformation

In this paper, the theory of the steady growth of fatigue crack in an infinite medium under the periodic anti-plane remote shear loading has been examined. The criterion of accumulative plastic work for material failure associated with the slip displacement in the fracture process zone of Dugdale type ahead of the crack tip is employed in the analysis. The effect of the locked dislocation in the fracture process zone is considered. Under the assumption that the speed of fatigue crack propagation remains uniform through the fracture process zone, the steady speed of fatigue crack can be expressed as a function of the range of the applied shearing stress and the maximum shearing stress. The effect of the crack size on the fatigue crack speed is discussed. The effect of the finite width of specimen on the speed of fatigue crack speed is investigated. The differences between the present work and the previous studies on fatigue crack speed are discussed.     

Stress-strain behavior of segmented polyurethaneureas under pure shear deformation

Elastic properties of segmented polyurethaneureas (SPUs) under pure shear deformation were investigated. Data were analyzed by using strain energy density function (W). The values of the derivatives of W with respect to the invariants of the strain tensor (I _i ; i = 1, 2, 3 ) at zero strain limit were also estimated theoretically and were compared with those estimated by experiment. The limiting value of the derivative with respect to I ₁ (W/I ₁) was shown theoretically to be 5 G/8, while the derivatives with respect to I ₂ and I ₃ (W/I ₂ and W/I ₃ were respectively-G/8 and -3G/8. The theoretical prediction could explain the asymptotic behavior of the derivatives of SPUs as well as isoprene rubber (IR) reported by Kawabata et al. at small I ₁ limit.     

Mean-field hydrodynamics brownian dynamics simulations of stabilized colloidal liquids under shear

We extend a recently proposed mean-field hydrodynamics (MFH) discrete element simulation technique to consider the effects of a shear velocity profile on a model colloidal liquid containing monodisperse spherical particles in the non-newtonian shear thinning regime. The MFH method adapts Ermak's free draining brownian dynamics algorithm to include a local density approximation for the friction coefficient, semiempirically parametrized to reproduce the experimentally determined short-time diffusion coefficient. We have also generalized further the previous treatment to allow for a friction coefficient that is dependent on local density anisotropy. The behaviour of Ermak's equations of motion and also the “isotropic” and “anisotropic” MFH schemes with shear flow are compared. We show that, at equilibrium, the MFH approaches generate the same static averages as Ermak's method, and give good agreement with the Percus-Yevick prediction for hard-sphere structure factors using an r⁻³⁶ soft-sphere interaction. However, under shear, the three equations of motion give quite different rheological behaviour. The MFH methods produce higher viscosities, although the structures remain similar (e.g. all give a “string” phase) but at different shear rates. Variation in the specific details of the MFH equations of motion can promote or delay the development of long-range order with Péclet number.     

Effect of interfacial modifier on single drop deformation and breakup in step increasing shear flow

This work deals with in situ visualisation of deformation and breakup of a copolymer modified single Newtonian drop immersed in a Newtonian homogenous matrix. The experiments were carried out on a model system made of poly-isobutylene as the suspending fluid and two poly-dimethylsiloxanes with different molecular weights as the drop phase with viscosity ratios 0.036 and 1.13, below and above but close to unity. Three weight concentrations 0.5%, 2% and 10% of the block copolymer laying below, close to and above the critical concentration of the total drop surface coverage were examined. Single drop deformation experiments were carried out in a home-designed Couette quartz cell connected to a home-modified Paar Physica Rheometer. The variation in the length-to-diameter ratio (L/D) versus shear rate and capillary number was measured both in steady and in transient regimes till breakup. The results indicated a weaker resistance of copolymer modified drops against hydrodynamic stresses at both viscosity ratios as compared to the clean drop. However, the drop deformation was found to be complex and depends on the copolymer concentration and the viscosity ratio.     

Small-diameter parallel plate rheometry: a simple technique for measuring rheological properties of glass-forming liquids in shear

The rheological characterization of glass-forming liquids is challenging due to their extreme temperature dependence and high stiffness at low temperatures. This study focuses on the special precautions that need to be taken to accommodate high sample stiffness and torsional instrument compliance in shear rheological experiments. The measurement errors due to the instrument compliance can be avoided by employing small-diameter parallel plate (SDPP) rheometry in combination of numerical instrument compliance corrections. Measurements of that type demonstrate that accurate and reliable rheological data can be obtained by SDPP rheometry despite unusually small diameter-to-gap (d/h) ratios. Specimen preparation for SDPP requires special attention, but then experiments show excellent repeatability. Advantages and some current applications of SDPP rheometry are briefly reviewed. SDPP rheometry is seen as a simple and versatile way to measure rheological properties of glass-forming liquids especially near their glass transition temperature.     

An efficient Galerkin meshfree formulation for shear deformable beam under finite deformation

This paper proposes a geometrically nonlinear total Lagrangian Galerkin meshfree formulation based on the stabilized conforming nodal integration for efficient analysis of shear deformable beam. The present nonlinear analysis encompasses the fully geometric nonlinearities due to large deflection, large deformation as well as finite rotation. The incremental equilibrium equation is obtained by the consistent linearization of the nonlinear variational equation. The Lagrangian meshfree shape function is utilized to discretize the variational equation. Subsequently to resolve the shear and membrane locking issues and accelerate the computation, the method of stabilized conforming nodal integration is systematically implemented through the Lagrangian gradient smoothing operation. Numerical results reveal that the present formulation is very effective.