The constitutive equation for a dilute emulsion

1970 ◽  
Vol 44 (1) ◽  
pp. 65-78 ◽  
Author(s):  
N. A. Frankel ◽  
Andreas Acrivos
Keyword(s):  
Constitutive Equation ◽  
Numerical Solutions ◽  
Extensional Flow ◽  
Small Deformation ◽  
Order Theory ◽  
Droplet Surface ◽  
Time Dependent ◽  
Deformation Analysis ◽  
First Order Theory ◽  
Freely Suspended

A constitutive equation for dilute emulsions is developed by considering the deformations, assumed infinitesimal, of a small droplet freely suspended in a time-dependent shearing flow. This equation is non-linear in the kinematic variables and gives rise to ‘fluid memory’ effects attributable to the droplet surface dynamics. Furthermore, it has the same form as the corresponding expression for a dilute suspension of Hookean elastic spheres (Goddard & Miller 1967), and reduces to a relation previously proposed by Schowalter, Chaffey & Brenner (1968) when time-dependent effects become small.Numerical solutions are also presented for the case of a small bubble in a steady extensional flow for the purpose of estimating the range of validity of the small deformation analysis. It is shown that, unlike the drag of a bubble which, in creeping motion, is known to be relatively insensitive to its exact shape, the macroscopic stress field in an emulsion is not well described by the present analysis unless the shapes of the deformed bubbles agree closely with those given by the first-order theory. Thus, the present rheological equation should prove of value in a qualitative rather than a quantitative sense.

A numerical study of the deformation and burst of a viscous drop in an extensional flow

1978 ◽  
Vol 89 (1) ◽  
pp. 191-200 ◽  
Author(s):  
J. M. Rallison ◽  
A. Acrivos
Keyword(s):  
Numerical Study ◽  
Extensional Flow ◽  
Small Deformation ◽  
Surface Velocity ◽  
Deformation Analysis ◽  
Slender Body Theory ◽  
Spherical Drop ◽  
Viscous Drop ◽  
Freely Suspended ◽  
Body Theory

We study the deformation and conditions for breakup of a liquid drop of viscosity λμ freely suspended in another liquid of viscosity μ with which it is immiscible and which is being sheared. The problem at zero Reynolds number is formulated exactly as an integral equation for the unknown surface velocity, which is shown to reduce to a particularly simple form when Δ = 1. This equation is then solved numerically, for the case in which the impressed shear is a radially symmetric extensional flow, by an improved version of the technique used, for Δ = 0, by Youngren & Acrivos (1976) so that we model the time-dependent distortion of an initially spherical drop. It is shown that, for a given Δ, a steady shape is attained only if the dimensionless group Ω ≡4πGμa/γ lies below a critical value Ωc(Δ), where G refers to the strength of the shear field, a is the radius of the initial spherical drop and γ is the interfacial tension. On the other hand, when Ω > Ωc the drop extends indefinitely along its long axis. The numerical results for Δ = 0·3, 0·5, 1, 2, 10 and 100 are in good agreement with the predictions of the small deformation analysis by Taylor (1932) and Barthès-Biesel & Acrivos (1973) and, at the smaller Δ, with those of slender-body theory (Taylor 1964; Acrivos & Lo 1978).

Dimensional changes during shear without normal tractions (the Poynting effect) in nonlinear viscoelastic fiber-reinforced solids

2019 ◽  
Vol 25 (3) ◽  
pp. 582-596
Author(s):  
Alan Wineman
Keyword(s):  
Constitutive Equation ◽  
Numerical Solutions ◽  
Elastic Fiber ◽  
Time Dependent ◽  
Nonlinear Material ◽  
Fiber Reinforced ◽  
Dimensional Changes ◽  
Fiber Reinforced Materials ◽  
The Matrix ◽  
Poynting Effect

When a rectangular block of a nonlinear material is subjected to a simple shearing deformation, specific normal tractions are required to ensure that the distances between the faces of the block, i.e. its dimensions, do not change. This work investigates the time-dependent dimensional changes during shear in the absence of these normal tractions (the Poynting effect) that occur in a block composed of an incompressible nonlinearly viscoelastic fiber-reinforced solid. The material is modeled using the Pipkin–Rogers nonlinear single integral constitutive equation for viscoelasticity. This constitutive equation is used because (1) it exhibits the essential features of nonlinear viscoelasticity; (2) it is straightforward to include the material symmetry restrictions due to the reinforcing fibers. A system of nonlinear Volterra integral equations is formulated for the dimensional changes in the block. Numerical solutions are presented for the case when the standard reinforcing model for nonlinearly elastic fiber-reinforced materials is incorporated in the Pipkin–Rogers constitutive framework. The results illustrate how the time-dependent dimensional changes depend on the fiber orientation and the viscoelastic properties of the fibers relative to those of the matrix.

scholarly journals CASIMIR FRICTION FORCE FOR MOVING HARMONIC OSCILLATORS

2012 ◽  
Vol 14 ◽  
pp. 141-154 ◽  
Author(s):  
JOHAN S. HØYE ◽  
IVER BREVIK
Keyword(s):  
Friction Force ◽  
Constant Velocity ◽  
Order Effect ◽  
Order Theory ◽  
Harmonic Oscillators ◽  
Time Dependent ◽  
Zero Temperature ◽  
Dielectric Particles ◽  
First Order Theory ◽  
Statistical Mechanical

Casimir friction is analyzed for a pair of dielectric particles in relative motion. We first adopt a microscopic model for harmonically oscillating particles at finite temperature T moving non-relativistically with constant velocity. We use a statistical-mechanical description where time-dependent correlations are involved. This description is physical and direct, and, in spite of its simplicity, is able to elucidate the essentials of the problem. This treatment elaborates upon, and extends, an earlier theory of ours back in 1992. The energy change ΔE turns out to be finite in general, corresponding to a finite friction force. In the limit of zero temperature the formalism yields, however, ΔE → 0, this being due to our assumption about constant velocity, meaning slowly varying coupling. For couplings varying more rapidly, there will also be a finite friction force at T = 0. As second part of our work, we consider the friction problem using time-dependent perturbation theory. The dissipation, basically a second order effect, is obtainable with the use of first order theory, the reason being the absence of cross terms due to uncorrelated phases of eigenstates. The third part of the present paper is to demonstrate explicitly the equivalence of our results with those recently obtained by Barton (2010); this being not a trivial task since the formal results are seemingly quite different from each other.

scholarly journals CASIMIR FRICTION FORCE FOR MOVING HARMONIC OSCILLATORS

2012 ◽  
Vol 27 (15) ◽  
pp. 1260011 ◽  
Author(s):  
JOHAN S. HØYE ◽  
IVER BREVIK
Keyword(s):  
Friction Force ◽  
Constant Velocity ◽  
Order Effect ◽  
Order Theory ◽  
Harmonic Oscillators ◽  
Time Dependent ◽  
Zero Temperature ◽  
Dielectric Particles ◽  
First Order Theory ◽  
Statistical Mechanical

Casimir friction is analyzed for a pair of dielectric particles in relative motion. We first adopt a microscopic model for harmonically oscillating particles at finite temperature T moving nonrelativistically with constant velocity. We use a statistical-mechanical description where time-dependent correlations are involved. This description is physical and direct, and, in spite of its simplicity, is able to elucidate the essentials of the problem. This treatment elaborates upon, and extends, an earlier theory of ours back in 1992. The energy change ΔE turns out to be finite in general, corresponding to a finite friction force. In the limit of zero temperature the formalism yields, however, ΔE → 0, this being due to our assumption about constant velocity, meaning slowly varying coupling. For couplings varying more rapidly, there will also be a finite friction force at T = 0. As second part of our work, we consider the friction problem using time-dependent perturbation theory. The dissipation, basically a second order effect, is obtainable with the use of first order theory, the reason being the absence of cross terms due to uncorrelated phases of eigenstates. The third part of the present paper is to demonstrate explicitly the equivalence of our results with those recently obtained by Barton (2010); this being not a trivial task since the formal results are seemingly quite different from each other.

Experimental studies of the deformation of a synthetic capsule in extensional flow

1993 ◽  
Vol 250 ◽  
pp. 587-608 ◽  
Author(s):  
K. S. Chang ◽  
W. L. Olbricht
Keyword(s):  
Elastic Modulus ◽  
Large Strain ◽  
Extensional Flow ◽  
Experimental Studies ◽  
Small Deformation ◽  
Polymeric Membrane ◽  
Roll Mill ◽  
Independent Technique ◽  
Freely Suspended ◽  
Flow Experiments

Experiments are described to study the motion and deformation of a synthetic, liquid-filled capsule that is freely suspended in hyperbolic extensional flow. The capsule is a composite particle consisting of a viscous liquid drop surrounded by a thin polymeric membrane. The method used to fabricate capsules suitable for macroscopic flow experiments is described. The deformation of the capsule is measured as a function of strain rate for an extensional flow generated in a four-roll mill. The data agree well with results from small-deformation theory developed by Barthes-Biesel and co-workers, provided the deformation of the capsule is not too large. Using the theory to correlate the experimental data produces an estimate for the elastic modulus of the membrane that agrees reasonably well with the elastic modulus obtained by an independent technique. However, for sufficiently large strain rates, the membrane exhibits strain hardening and a permanent change in its structure, both of which are reflected in the shape of the capsule.

scholarly journals Nutritional and Rheological Features of Lentil Protein Isolate for Yoghurt-Like Application

Foods ◽  
2021 ◽  
Vol 10 (8) ◽  
pp. 1692
Author(s):  
Theresa Boeck ◽  
Emanuele Zannini ◽  
Aylin W. Sahin ◽  
Juergen Bez ◽  
Elke K. Arendt
Keyword(s):  
Laser Scanning ◽  
Base Material ◽  
Small Deformation ◽  
Protein Isolate ◽  
Protein Supplementation ◽  
Water Holding Capacity ◽  
Laser Scanning Microscopy ◽  
Confocal Laser ◽  
Deformation Analysis ◽  
Water Holding

The substitution of animal protein with proteins of plant origin is a viable way to decrease the negative impact caused by animal husbandry on the environment. Pulse consumption has been widely promoted as a nutritious contribution to protein supplementation. In this study, an emulsion of lentil (Lens culinaris) protein isolate is fermented with lactic acid bacteria (LAB) to manufacture a yoghurt alternative and the techno-functional properties compared to a dairy- and a soy-based product with similar protein contents. The yoghurt-like products are subjected to large and small deformation analysis, quantification of fermentable oligosaccharides, disaccharides, monosaccharides and polyols (FODMAP), water holding capacity tests, protein profile analysis and the gel structure is visualised by confocal laser scanning microscopy (CLSM). The lentil yoghurt alternative shows good water holding capacity, high firmness and consistency values in large deformation analysis, with cohesiveness and viscosity not significantly different from that of dairy yoghurt. The high gel strength and rigidity of the lentil yoghurt gels measured by small deformation analysis is well-reflected in the dense protein matrix in the CLSM graphs. FODMAP content of the lentil yoghurt is very low, making it suitable for consumption by irritable bowel syndrome (IBS) patients. Our results show that lentil protein isolate is an excellent base material for producing a plant-based yoghurt alternative.

scholarly journals A first-order theory of Ulm type

Computability ◽  
2019 ◽  
Vol 8 (3-4) ◽  
pp. 347-358
Author(s):  
Matthew Harrison-Trainor
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory
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