Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr ²(t)>, of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G ^*(ω), in the Fourier frequency domain, and the stress relaxation modulus, G _r (t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of <Δr ²(t)> known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing <Δr ²(t)> as a local power law. If the logarithmic slope of <Δr ²(t)> can be accurately determined, these estimates generally perform well at the frequency extremes. Received: 8 September 2000/Accepted: 9 March 2000     

On the sensitivity of interconversion between relaxation and creep

The interconversion equation of linear viscoelasticity defines implicitly the interrelations between the relaxation and creep functions G(t) and J(t). It is widely utilised in rheology to estimate J(t) from measurements of G(t) and conversely. Because different molecular details can be recovered from G(t) and J(t), it is necessary to work with both. This leads naturally to the need to identify whether it is better to first measure G(t) and then determine J(t) or conversely. This requires an understanding of the stability (sensitivity) of the recovery of J(t) from G(t) compared with that of G(t) from J(t). Although algorithms are available that work adequately in both directions, numerical experimentation strongly suggests that the recovery of J(t) from G(t) measurements is the more stable. An elementary theoretical rationale has been given recently by Anderssen et al. (ANZIAM J 48:C346–C363, 2007) for single exponential models of G(t) and J(t). It explicitly exploits the simple algebra of such functions. In this paper, corresponding bounds are derived that hold for arbitrary sums of exponentials. The paper concludes with a discussion, from a practical rheological perspective, about the implications and implementations of the results.     

Compliance of actin filament networks measured by particle-tracking microrheology and diffusing wave spectroscopy

We monitor the time-dependent shear compliance of a solution of semi-flexible polymers, using diffusing wave spectroscopy (DWS) and video-enhanced single-particle-tracking (SPT) microrheology. These two techniques use the small thermally excited motion of probing microspheres to interrogate the local properties of polymer solutions. The solutions consist of networks of actin filaments which are long semi-flexible polymers. We establish a relationship between the mean square displacement (MSD) of microspheres imbedded in the solution and the time-dependent creep compliance of the solution, <Δr ²(t)>=(k _B T/πa)J(t). Here, J(t) is the creep compliance, <Δr ²(t)> is the mean-square displacement, and a is the radius of the microsphere chosen to be larger than the mesh size of the polymer network. DWS allows us to measure mean square displacements with microsecond temporal resolution and Ångström spatial resolution. At short times, the mean square displacement of a 0.96μm diameter sphere in a concentrated actin solution displays sub-diffusion. <Δr ²(t)>∝t ^{, with a characteristic exponent =0.78±0.05, which reflects the finite rigidity of actin. At long times, the MSD reaches a plateau, with a magnitude that decreases with concentration. The creep compliance is shown to be a weak function of polymer concentration and scales as J _p ∝c –1.2±0.3. This exponent is correctly described by a recent model describing the viscoelasticity of semi-flexible polymer solutions. The DWS and video-enhanced SPT measurements of the compliance plateau agree quantitatively with compliance measured independently using classical mechanical rheometry for a viscous oil and for a solution of flexible polymers. This paper extends the use of DWS and single-particle-tracking to probe the local mechanical properties of polymer networks, shows for the first time the proportionality between mean square displacement and local creep compliance, and therefore presents a new, direct way to extract the viscoelastic properties of polymer systems and complex fluids.}     

Rheology of SiO2/(acrylic polymer/epoxy) suspensions. II. Nonlinear stress relaxation

Large deformation, nonlinear stress relaxation modulus G(t, γ) was examined for the SiO₂ suspensions in a blend of acrylic polymer (AP) and epoxy (EP) with various SiO₂ volume fractions (?) at various temperatures (T). The AP/EP contained 70 vol.% of EP. At ??≤?30 vol.%, the SiO₂/(AP/EP) suspensions behaved as a viscoelastic liquid, and the time-strain separability, G(t, γ)?=?G(t)h(γ), was applicable at long time. The h(γ) of the suspensions was more strongly dependent on γ than that of the matrix (AP/EP). At ??=?35 vol.% and T?=?100°C, and ??≥?40 vol.%, the time-strain separability was not applicable. The suspensions exhibited a critical gel behavior at ??=?35 vol.% and T?=?100°C characterized with a power law relationship between G(t) and t; G(t)?∝?t ^???n . The relaxation exponent n was estimated to be about 0.45, which was in good agreement with the result of linear dynamic viscoelasticity reported previously. G(t, γ) also could be approximately expressed by the relation $G(t,\gamma) \propto t^{-n^{\prime}}$ at ??=?40 vol.%. The exponent n ^′ increased with increasing γ. This nonlinear stress relaxation behavior is attributable to strain-induced disruption of the network structure formed by the SiO₂ particles therein.     

Rheology of polydisperse polymers: relationship between intermolecular interactions and molecular weight distribution

An expression of the relaxation function of linear polydisperse polymers is proposed in terms of intermolecular couplings of reptative chains. The relaxation times of each molecular weight are assumed to be shifted according to a tube renewal mechanism accounting for the diffusion of the surrounding chains. The subsequent shift is applied to the relaxation function of each molecular weight obtained from an analytical expression of the complex compliance J ^*(). Therefore the complex shear modulus G ^*() is derived from the overall relaxation function using the probability density accounting for the molecular weight distribution and four species-dependent parameters: a front factor A for zero-shear viscosity, plateau modulus G _N ⁰ , activation energy E and characteristic temperature T . All the main features of the theology of polydisperse polymers are described by the proposed model.     

Crack repair using an elastic filler

The effect of repairing a crack in an elastic body using an elastic filler is examined in terms of the stress intensity levels generated at the crack tip. The effect of the filler is to change the stress field singularity from order 1/r^1/2 to 1/r^(1-λ) where r is the distance from the crack tip, and λ is the solution to a simple transcendental equation. The singularity power (1-λ) varies from (the unfilled crack limit) to 1 (the fully repaired crack), depending primarily on the scaled shear modulus ratio γ_r defined by G₂/G₁=γ_rε, where 2πε is the (small) crack angle, and the indices (1, 2) refer to base and filler material properties, respectively. The fully repaired limit is effectively reached for γ_r≈10, so that fillers with surprisingly small shear modulus ratios can be effectively used to repair cracks. This fits in with observations in the mining industry, where materials with G₂/G₁ of the order of 10^-3 have been found to be effective for stabilizing the walls of tunnels. The results are also relevant for the repair of cracks in thin elastic sheets.     

Finite element calculation of elastodynamic stress field around a notch tip via contour integrals

Direct computation of the mixed-mode dynamic asymptotic stress field around a notch tip is difficult because the mode I and mode II stresses are in general governed by different orders of singularity. In this paper, we propose a pair of elastodynamic contour integrals J_kR(t). The integrals are shown to be path-independent in a modified sense and so they can be accurately evaluated with finite element solutions. Also, by defining a pair of generalized stress intensity factors (SIFs) K_I,β(t) and K_II,β(t), the relationship between J_kR(t) and the SIF’s is derived and expressed as functions of the notch angle β. Once the J_kR(t)-integrals are accurately computed, the generalized SIF’s and, consequently, the asymptotic mixed-mode stress field can then be properly determined. No particular singular elements are required in the calculation. The proposed numerical scheme can be used to investigate the dynamic amplifying effect in the near-tip stress field.     

An alternative approach to the linear theory of viscoelasticity and some characteristic effects being distinctive of the type of material

In the introduction some postulates on which the linear theory of viscoelasticity is based are recalled, and the postulate of passivity is substituted by a stronger postulate called detailed passivity.Next, a symmetric formulation of this theory is presented which is founded in a well-balanced way on the limiting properties of elasticity and viscosity. This leads to the introduction of the basic functions of creep compliance J ⁺(t) and stressing viscosity ⁺(t) associated to one another, whereas the basic functions retardation fluidity ⁺(t) and relaxation modulus G ⁺(t) emerge as their time derivatives. Correspondingly, four complex basic functions are defined as their Carson transforms.In addition to the proper retardation and relaxation terms, these basic functions contain the non-disappearing constants of either instantaneous compliance J ₀ or instantaneous viscosity ₀ and also of either ultimate fluidity or ultimate modulus G . Therefrom ensues a classification of linear viscoelastic materials into four types: instantaneous elasticity or viscosity is allowed to combine with ultimate viscosity or elasticity. The latter alternative, signifying fluidlike or solidlike materials, leads, of course, to a quite different behavior in many situations; however, remarkable distinctive features are associated to the first one as well.A few respective examples are outlined: 1) propagation of shear waves in a half-space with periodic and step-shaped excitation, 2) dissipation of work in a torsional vibration damper, and 3) shear flow between two parallel porous plates with injection and suction.Finally, materials with viscous initial behavior are defended against the notion that they be of no or almost no real significance.Delivered as a Plenary Lecture at the Fourth European Rheology Conference, Seville (Spain), 4–9 September 1994. The herein only outlined topics are taken from a recently pulished monograph (Geisekus, 1994) in which complete derivations of the results and more detailed discussions are given.Dedicated to Professor K. Walters on the occasion of his 60th birthday.     

Rheology of suspensions of spherical particles in a newtonian and a non-newtonian fluid

Properties of suspensions of spherical glass beads (25–38 μm dia.) in a Newtonian fluid and a non-Newtonian (NBS Fluid 40) fluid were measured at volume fractions, φ, of 0%, 10%, 20% and 30%. Measurements were made using a modified and computerized Weissenberg Rheogoniometer. Properties measured included steady shear viscosity, η($\text{γ}\text{.}$), first normal stress difference, N₁($\text{γ}\text{.}$), linear viscoelastic properties, η′(ω) and G′(ω), shear stress relaxation, σ^? ($\text{γ}\text{.}$, t), and growth, σ⁺($\text{γ}\text{.}$, t) and normal stress relaxation, N₁^?($\text{γ}\text{.}$, t).For a the Newtonian fluid, increasing φ causes both η and η′ to increase, with η′ showing a slight frequency dependence. Both N₁ and G′ are zero and stress relaxation and growth occur essentially instantaneously. For the NBS fluid, both η and η′ increse with φ at all $\text{γ}\text{.}$ and ω, respectively, the increase being greater as $\text{γ}\text{.}$ and ω approach zero. N₁ and G′ are less affected by the presence of the particles than η and η′ with the effect on G′ being more pronounced than on N₁. For fixed $\text{γ}\text{.}$, stress relaxation and growth exhibit greater non-linear effects as φ is increased. A model for predicting a priori the linear viscoelastic properties for suspensions was found to yeild reasonable estimates up to φ = 20%.     

Nonlinear rheological behavior of a concentrated spherical silica suspension

Linear and nonlinear viscoelastic properties were examined for a 50 wt% suspension of spherical silica particles (with radius of 40 nm) in a viscous medium, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The effective volume fraction of the particles evaluated from zero-shear viscosities of the suspension and medium was 0.53. At a quiescent state the particles had a liquid-like, isotropic spatial distribution in the medium. Dynamic moduli G^* obtained for small oscillatory strain (in the linear viscoelastic regime) exhibited a relaxation process that reflected the equilibrium Brownian motion of those particles. In the stress relaxation experiments, the linear relaxation modulus G(t) was obtained for small step strain (0.2) while the nonlinear relaxation modulus G(t, ) characterizing strong stress damping behavior was obtained for large (>0.2). G(t, ) obeyed the time-strain separability at long time scales, and the damping function h() (–G(t, )/G(t)) was determined. Steady flow measurements revealed shear-thinning of the steady state viscosity () for small shear rates (< ^–1; = linear viscoelastic relaxation time) and shear-thickening for larger (>^–1). Corresponding changes were observed also for the viscosity growth and decay functions on start up and cessation of flow, + (t, ) and _– (t, ). In the shear-thinning regime, the and dependence of ₊(t,) and _–(t,) as well as the dependence of () were well described by a BKZ-type constitutive equation using the G(t) and h() data. On the other hand, this equation completely failed in describing the behavior in the shear-thickening regime. These applicabilities of the BKZ equation were utilized to discuss the shearthinning and shear-thickening mechanisms in relation to shear effects on the structure (spatial distribution) and motion of the suspended particles.Dedicated to the memory of Prof. Dale S. Parson     

Linearized Stability for Semiflows Generated by a Class of Neutral Equations,with Applications to State-Dependent Delays

We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form x′(t) = g(? x _t , x _t ). The state space is a closed subset in a manifold of C ²-functions. Applications include equations with state-dependent delay, as for example x′(t) = a x′(t + d(x(t))) + f (x(t + r(x(t)))) with \({a\in\mathbb{R}, d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\).     

Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales

Our aim is to establish some sufficient conditions for the oscillation of the second-order quasilinear neutral functional dynamic equation

Some remarks on “Drag coefficients of a slowly moving sphere in Non-Newtonian fluids”

Viscoelastic solutions were ejected vertically downwards into air and various Newtonian fluids. The measured swell increased significantly when ejected into a liquid rather than air. The observed increase is considered a result of both bouyancy and drag forces on the solution. The following dimensions expression relating the ratio of the swell diameter in liquid and air D_L/D_A to the elastic shear compliance of the ejected solution J_e was experimentally observed.(D_L/D_A)⁶-1=30(Δ?/?_s)^{?$\text{1}\text{2}$}([g²η²_N?_s]$\text{1}\text{3}$J_e)^{$\text{3}\text{5}$}, where Δ? is the density difference between the extruded and Newtonian fluid, ?_s is the solution density, g is the gravitational constant, and η_N is the Newtonian fluid viscosity. Thus with this expression a simple extrudate swell technique exists to estimate the elastic shear compliance of a viscoelastic solution.     

On the particle inertia-free collision with a partially contaminated spherical bubble

The collision between a contaminated spherical bubble and fine particles in suspension is considered for r_p/r_b ? 1 (r_p being the radius of the particles in suspension and r_b the radius of the bubble). The collision probability or efficiency is defined as the number of particles colliding the bubble surface to the number of particles initially present in the volume swept out by the bubble. In this note we show that the collision probability can be expressed as P_c(r_p/r_b,Re) = g(r_p/r_b)f(Re) for both mobile and immobile interfaces. For partially contaminated bubbles a linear or quadratic dependency in r_p/r_b is found depending on the level of contamination and the value of r_p/r_b. These behaviors are given by the flux of particles near the surface which is controlled by the tangential velocity for mobile interfaces and by the velocity gradient for immobile interfaces. The threshold value (r_p/r_b)_th between the r_p/r_b and (r_p/r_b)² evolution is shown to vary as sin^n(Re)(θ_clean/n(Re))sin(3θ_clean/4), θ_clean being the angle describing the front clean part of the bubble and n(Re) varying from n = 2 to n = 1 from small to large Reynolds number.     

Orientational anisotropy for Rouse eigenmodes during creep and recovery process

The Rouse model is a well established model for nonentangled polymer chains and its dynamic behavior under step strain has been fully analyzed in the literature. However, to the knowledge of the authors, no analysis has been made for the orientational anisotropy for the Rouse eigenmodes during the creep and creep recovery processes. For completeness of the analysis of the Rouse model, this anisotropy is calculated from the Rouse equation of motion. The calculation is simple and straightforward, but the result is intriguing in a sense that respective Rouse eigenmodes do not exhibit the single Voigt-type retardation. Instead, each Rouse eigenmode has a distribution in the retardation time. This behavior, reflecting the interplay among the Rouse eigenmodes of different orders under the constant stress condition, is quite different from the behavior under rate-controlled flow (where each eigenmode exhibits retardation/relaxation associated with a single characteristic time).List of abbreviations and symbols a Average segment size at equilibrium - A_p(t) Normalized orientational anisotropy for the p-th Rouse eigenmode defined by Eq. (14) - p-th Fourier component of the Brownian force (=x, y) - F_B(n,t) Brownian force acting on n-th segment at time t - G(t) Relaxation modulus - J(t) Creep compliance - J_R(t) Recoverable creep compliance - k_B Boltzmann constant - N Segment number per Rouse chain - Q_j(t) Orientational anisotropy of chain sections defined by Eq. (21) - r(n,t) Position of n-th segment of the chain at time t - S(n,t) Shear orientation function (S(n,t)=a^–2<u_x(n,t)u_y(n,t)>) - T Absolute temperature - u(n,t) Tangential vector of n-th segment at time t (u = r/n) - V(r(n,t)) Flow velocity of the frictional medium at the position r(n,t) - X_p(t), Y_p(t), and Z_p(t) x-, y-, and z-components of the amplitudes of p-th Rouse eigenmode at time t - Strain rate being uniform throughout the system - Segmental friction coefficient - ₀ Zero-shear viscosity - _p Numerical coefficients determined from Eq. (25) - Gaussian spring constant ( = 3k_BT/a²) - Number of Rouse chains per unit volume - (t) Shear stress of the system at time t - _steady Shear stress in the steadily flowing state - _R Longest viscoelastic relaxation time of the Rouse chain     

Permeability evolution induced by the precipitation of an advected solute: Effect of the microscopic and the macroscopic scales competition on the clogging mechanism

A model is proposed for coupling the one-dimensional transport of solute with surface precipitation kinetics which induces the clogging of an initially homogeneous porous medium. The aim is to focus the non-linear feedback effect between the transport and the chemical reaction through the permeability of the medium. A Lagrangian formulation, used to solve the coupled differential equations, gives semi-analytical expressions of the hydrodynamic quantities. A detailed analysis reveals that the competition between the microscopic and macroscopic scales controls the clogging mechanism, which differs depends on whether short or long times are considered. In order to illustrate this analysis, more quantitative results were obtained in the case of a second and zeroth order kinetic. It was necessary to circumvent the semi-analytic character of the solutions problem by successive approximation. A comparison with results obtained by simulations displays a good agreement during the most part of the clogging time.Nomenclature a(x, t) Capillary tube radius (L) - A _(aq) Chemical species in the aqueous phase - A _n(s) Chemical species of the solid phase - C(x, t) Aqueous concentration in a capillary tube ([mole/L³] in the case of a permanent injection - [mole/L³/L] in the case of an instantaneous injection) - C(x, t) C(x, t)/C ₀ Dimensionless aqueous concentration in a capillary tube - C ₀ Aqueous concentration imposed at the inlet and also initial concentration in an elementary volume of fluid (mole/L³) - C _i(t) Concentration in a fluid element i (mole/L³) - C(_R) (t, C_o) Aqueous concentration in a stirred reactor (mole/L³) - d_ij (t) Length belonging to the volume, inside a fluid element i, which interacts with a precipitate element j (L) - dM _ij(t) Mass exchange between a fluid element i and a precipitate element j (M) - dN ₀ Number of molecules in an elementary volume of fluid injected at the inlet of a capillary tube during dt ₀ - dN(x, t, C₀) Number of molecules in an elementary volume of fluid - dt ₀ Time injection of an elementary volume of fluid (T) - D(x, t) Dispersion coefficient (L²/T) - Da(t, x) Damköhler number - D _m Molecular diffusion coefficient (L²/T) - F(x, t) Advective flux (mole/L²/T) - k ₁ Kinetic constant of dissolution (mole/L³/T) - k ₂ Kinetic constant of precipitation ([mole/L³]^{1 - n} /T) - k ₂ Kinetic constant of precipitation in the case of a zeroth order kinetics (mole/L³/T) - K(x, t) Permeability in a capillary tube (L²) - K(x, t) K (x, t)/K₀ Dimensionless permeability - k ₀ Permeability of a capillary tube at t = 0 (L²) - L Length of a capillary tube (L) - m Molecular weight of the reactive species (M/mole) - n Stochiometry of the chemical reaction and kinetic order of the precipitation reaction - P(x, t) Precipitate concentration in a capillary tube (mole/L³) - P _j(t) Concentration in a precipitate element j (mole/L³) - P(_r) (t, C_o) Precipitate concentration in a stirred reactor (mole/L³) - Pr(x, t) Local pressure in a capillary tube - (M/T²/L³) Pr(x, t) Pr(x, t)/Pr(x, 0) Dimensionless local pressure in a capillary tube - Q(t) Flow rate (L³/T) - Q(t) Q(t)U ₀/S₀ Dimensionless flow rate - R(x, t) Chemical flux between the aqueous and the solid phase in a capillary tube (mole/L³/T) - R _i(t) Chemical flux between an aqueous element i and the solid phase (mole/L³/T) - R _(R)[t, C₀] Chemical flux between the aqueous and the solid phase in a stirred reactor (mole/L³/T) - S(x, t) Cross sectional area of a capillary tube accessible to the aqueous phase (L²) - S(x, t) S(x, t)/S₀ Dimensionless cross-sectional area - S ₀ Cross-sectional area of a capillary tube at t = 0 (L²) - tlim(x) Time at which the precipitation front concentration vanishes in the case of zeroth order kinetics (T) - t _max Time of maximum propagation of the precipitation front in the case of zeroth order kinetics (T) - tmin(x) Time at which the precipitation front arrives at x (T) - t _p L/U ₀. Time necessary for an elementary volume of fluid, moving with the velocity U ₀, to reach the oulet of the medium - t _U max Time of maximum value of the velocity field in the case of zeroth order kinetics (T) - t ₀ Time at which an elementary volume of fluid has left the inlet of a capillary tube (T) - t _0m (x, t) Time at which the last elementary volume of fluid has left the inlet of a capillary tube to reach x at a time lower or equal to t (T) - U(x, t) Fluid velocity (L/T) - U(x, t) U(x, t)/U ₀. Dimensionless fluid velocity - U _j(x, t) Fluid velocity defined from the precipitate element j (L/T) - U _l (t₀, t) Lagrangian fluid velocity (L/T) - U _l (t ₀, t) U _l (x, t)/U ₀. Dimensionless lagrangian fluid velocity - U ₀ Velocity of the fluid at t = 0 (L/T) - V _ij(t) Volume, inside a fluid element i, which interacts with a precipitate element j (L³) - x _i(t) Front position of the fluid element i (L) - x _j Front position of the precipitate element j (L) - X _front(t) Position of the precipitation front (L) - x _lim(t) Position of the precipitation front when the value of its concentration is zero (L) - xmax Position of the maximum propagation of the precipitation front in the case of zeroth order kinetics and for high value of C ₀ (L) - Xmin (t) Position of the precipitation front (L) - x _inf* ^supmax Position of the maximum propagation of the precipitation front in the case of zeroth order kinetics and for small value of C_o (L) Greek Symbols t Time step used during the numerical computation (T) - Pro Imposed pressure drop (M/L/T²) - Injection time of reactive species (T) - Density of the precipitate (M/L³) - Dynamic viscosity (M/L/T) - <Ri(t)> _infi ^supj Mean chemical flux between a precipitate element j and all the fluid elements i susceptible to interact with the precipitate element j (mole/L³/T)     

On resolution to Wu＇s conjecture on Cauchy function＇s exterior singularities

This is a series of studies on Wu’s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy’s function f(z) (z = x + iy) and its integral J[f(z) ] ≡(2πi) -1 C f(t)(t z) -1dt taken along the unit circle as contour C,inside which(the open domain D+) f(z) is regular but has singularities distributed in open domain Doutside C. Resolution is given to the inverse problem that the singularities of f(z) can be determined in analytical form in terms of the values f(t) of f(z) numerically prescribed on C(|t| = 1) ,as so enunciated by Wu’s conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle,for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction,and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics,engineering and technology in analogy with resolving the inverse problem presented here.     

An inversion method for identification of elastoplastic properties of a beam from torsional experiment

A new method of determining elastoplastic properties of a beam from an experimentally given value T?T(φ) of torque (or torsional rigidity), during the quasistatic process of torsion, given by the angle of twist φ∈[φ_*,φ^*], is proposed. The mathematical model leads to the inverse problem of determining the unknown coefficient g=g(ξ²), ξ?|∇u|, of the non-linear differential equation −∇(g(|∇u|²)∇u)=2φ, x∈Ω⊂R². The inversion method is based on the parametrization of the unknown coefficient, according to the discrete values of the gradient ξ?|∇u|. Within the range of J₂-deformation theory, it is shown that the considered inverse coefficient problem is an ill-conditioned one. A numerical reconstruction algorithm based on parametrization of the unknown coefficient g=g(ξ²), with optimal selection of the experimentally given data T_m?T(φ_m), is proposed as a new regularization scheme for the considered inverse problem. Numerical results with noise free and noisy data illustrate applicability and high accuracy of the proposed method.     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

A delta-function method for the n-th approximation of relaxation or retardation time spectrum from dynamic data

A function series g(x; n, m) is presented that converges in the limiting case n and m = constant to the delta-function located at x = = 1. For every finite n, there exists 2n+1(–nmn) approximations of the delta-function _(n)(x–x _n,m ). x _n,m is the argument where the function reaches its maximum. A formula for the calculation is given.The delta-function approximation is the starting point for the approximative determination of the logarithmic density function of the relaxation or retardation time spectrum. The n-th approximation of density functions based on components of the complex modulus (G*) or the complex compliance (J*) is given. It represents an easy differential operator of order n.This approach generalizes the results obtained by Schwarzl and Staverman, and Tschoegl. The symmetry properties of the approximations are explained by the symmetry properties of the function g(x; n, m). Therefore, the separate equations for each approximation given by Tschoegl can be subsumed in a single equation for G and G, and in another for J and J.