On the dispersion of a solute in oscillating flow of a non-Newtonian fluid in a channel

The paper presents an exact analysis of the dispersion of an immiscible solute in a non-Newtonian fluid (known as an incompressible second-order fluid which shows viscoelastic behaviour) flowing slowly in a parallel plate channel in the presence of a periodic pressure gradient. Using a generalized dispersion model which is valid for all times after the solute injection, the diffusion coefficients K _i (τ)(i=1,2,3,…) are obtained as functions of time τ in the case when the initial solute distribution is in the form of a slug of finite extent. The analysis leads to the novel result that K ₂(τ) (which is a measure of the longitudinal dispersion coefficient of the solute) has a steady part S in addition to a fluctuating part D ₂(τ) due to the pulsatility of the flow. It is found that S decreases with increase in the viscoelastic parameter M for given values of the amplitude λ and frequency ω of the pressure pulsation. On the other hand, it is found that at a fixed instant τ, the amplitude of D ₂(τ) increases with increase in M for given values of λ and ω. Further it is shown that at a given instant τ, the amplitude of D ₂(τ) decreases with increase in ω for given λ and M and the profile for D ₂(τ) becomes progressively flatter with increase in ω. Finally the axial distribution of the average concentration θ _m of the solute over the channel cross-section is determined at different instants after the solute injection for several values of M, λ and ω. The present study is likely to have important bearing on the problem of dispersion of tracers in blood flow through arteries.     

Time periodic electroosmotic flow of the generalized Maxwell fluids between two micro-parallel plates

Analytical solutions are presented using method of separation of variables for the time periodic EOF flow of linear viscoelastic fluids between micro-parallel plates. The linear viscoelastic fluids used here are described by the general Maxwell model. The solution involves analytically solving the linearized Poisson–Boltzmann equation, together with the Cauchy momentum equation and the general Maxwell constitutive equation. By numerical computations, the influences of the electrokinetic width K denoting the characteristic scale of half channel width to Debye length, the periodic EOF electric oscillating Reynolds number Re and normalized relaxation time λ₁ω on velocity profiles and volumetric flow rates are presented. Results show that for prescribed electrokinetic width K, lower oscillating Reynolds number Re and shorter relaxation time λ₁ω reduces the plug-like EOF velocity profile of Newtonian fluids. For given Reynolds number Re and electrokinetic width K, longer relaxation time λ₁ω leads to rapid oscillating EOF velocity profiles with increased amplitude. With the increase of the K, the velocity variations are restricted to a very narrow region close to the EDL for small relaxation time. However, with the increase of the relaxation time, the elasticity of the fluid becomes conspicuous and the velocity variations can be expanded to the whole flow field. As far as volume flow rates are concerned, for given electrodynamic width K, larger oscillating Reynolds number Re results in a smaller volume flow rates. For prescribed oscillating Reynolds number Re, with the changes of relaxation time λ₁ω, volume flow rates will produce some peaks no matter how the electrodynamic width K varies. Moreover, the time periodic evolution of the velocity profiles provides a detail insight of the flow characteristic of this flow configuration.     

Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations

We consider a family of linearly elastic shells with thickness 2?, clamped along their entire lateral face, all having the same middle surfaceS=φ(?ω) ?R ³, whereω ?R ² is a bounded and connected open set with a Lipschitz-continuous boundaryγ, andφ ∈l ³ ( $\overline \omega$ ;R ³). We make an essential geometrical assumption on the middle surfaceS, which is satisfied ifγ andφ are smooth enough andS is “uniformly elliptic”, in the sense that the two principal radii of curvature are either both>0 at all points ofS, or both<0 at all points ofS. We show that, if the applied body force density isO(1) with respect to?, the fieldtu(?)=(u _i(?)), whereu _i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, one “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges inH ¹(Ω)×H ¹(Ω)×L ²(Ω) as?→0 to a limitu, which is independent of the transverse variable. Furthermore, the averageξ=1/2ε _?1 ¹ u dx ₃, which belongs to the space $$V_M (\omega ) = H_0^1 (\omega ) \times H_0^1 (\omega ) \times L^2 (\omega ),$$ satisfies the (scaled) two-dimensional equations of a “membrane shell” viz., $$\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta )\gamma _{\alpha \beta } (\eta ) \sqrt \alpha dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\}\eta _i \sqrt a dy$$ for allη=(η _i) εV _M(ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\gamma _{\alpha \beta } (\eta ) = \frac{1}{2}\left( {\partial _{\alpha \eta \beta } + \partial _{\beta \eta \alpha } } \right) - \Gamma _{\alpha \beta }^\sigma \eta _\sigma - b_{\alpha \beta \eta 3} $$ are the components of the linearized change of metric tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_{\alpha \beta }$ are the components of the curvature tensor ofS, andf ⁱ are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “membrane shell” are therefore justified.     

Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations

We consider as in Part I a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and?∈l ³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is any portion of?ω withlength γ ₀>0. We make an essential geometrical assumption on the middle surfaceS and on the setγ ₀, which states that the space of inextensional displacements $$\begin{gathered} V_F (\omega ) = \{ \eta = (\eta _i ) \in H^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \hfill \\ \eta _i = \partial _v \eta _3 = 0 on \gamma _0 ,\gamma _{\alpha \beta } (\eta ) = 0 in \omega \} , \hfill \\ \end{gathered}$$ where $\gamma _{\alpha \beta }$ (η) are the components of the linearized change is metric tensor ofS, contains non-zero functions. This assumption is satisfied in particular ifS is a portion of cylinder and?(γ ₀) is contained in a generatrix ofS. We show that, if the applied body force density isO(? ²) with respect to?, the fieldu(?)=(u _i (?)), whereu _i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, once “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges as?→0 inH ¹(Ω) to a limitu, which is independent of the transverse variable. Furthermore, the averageζ=1/2ts _?1 ¹ u dx ₃, which belongs to the spaceV _F (ω), satisfies the (scaled) two-dimensional equations of a “flexural shell”, viz., $$\frac{1}{3}\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta )\rho _{\alpha \beta } (\eta )\sqrt {a } dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\} \eta _i \sqrt {a } dy$$ for allη=(η _i ) ∈V _F (ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\begin{gathered} \rho _{\alpha \beta } (\eta ) = \partial _{\alpha \beta } \eta _3 - \Gamma _{\alpha \beta }^\sigma \partial _\sigma \eta _3 + b_\beta ^\sigma \left( {\partial _\alpha \eta _\sigma - \Gamma _{\alpha \sigma }^\tau \eta _\tau } \right) \hfill \\ + b_\alpha ^\sigma \left( {\partial _\beta \eta _\sigma - \Gamma _{\beta \sigma }^\tau \eta _\tau } \right) + b_\alpha ^\sigma {\text{|}}_\beta \eta _\sigma - c_{\alpha \beta } \eta _3 \hfill \\ \end{gathered} $$ are the components of the linearized change of curvature tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_\alpha ^\beta$ are the mixed components of the curvature tensor ofS, andf ⁱ are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “flexural shell” are therefore justified.     

Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations

We consider as in Parts I and II a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and? ∈ ?³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is a portion of?ω withlength γ ₀>0. For all?>0, let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $u_i^\varepsilon g^{i,\varepsilon }$ of the points of the shell, obtained by solving the three-dimensional problem; let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $\zeta _i^\varepsilon$ a ⁱ of the points of the middle surfaceS, obtained by solving the two-dimensional model ofW.T. Koiter, which consists in finding $$\zeta ^\varepsilon = \left( {\zeta _i^\varepsilon } \right) \in V_K (\omega ) = \left\{ {\eta = (\eta _\iota ) \in {\rm H}^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \eta _i = \partial _v \eta _3 = 0 on \gamma _0 } \right\}$$ such that $$\begin{gathered} \varepsilon \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta ^\varepsilon )\gamma _{\alpha \beta } (\eta )\sqrt a dy + \frac{{\varepsilon ^3 }}{3} \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta ^\varepsilon )\rho _{\alpha \beta } (\eta )\sqrt a dy \hfill \\ = \mathop \smallint \limits_\omega p^{i,\varepsilon } \eta _i \sqrt a dy for all \eta = (\eta _i ) \in V_K (\omega ), \hfill \\ \end{gathered}$$ where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor ofS, $\gamma _{\alpha \beta }$ (η) and $\rho _{\alpha \beta }$ (η) are the components of the linearized change of metric and change of curvature tensors ofS, and $p^{i,\varepsilon }$ are the components of the resultant of the applied forces. Under the same assumptions as in Part I, we show that the fields $\frac{1}{{2_\varepsilon }}\smallint _{ - \varepsilon }^\varepsilon u_i^\varepsilon g^{i,\varepsilon } dx_3^\varepsilon$ and $\zeta _i^\varepsilon$ a ⁱ , both defined on the surfaceS, have the same principal part as? → 0, inH ¹ (ω) for the tangential components, and inL ²(ω) for the normal component; under the same assumptions as in Part II, we show that the same fields again have the same principal part as? → 0, inH ¹ (ω) for all their components. For “membrane” and “flexural” shells, the two-dimensional model ofW.T. Koiter is therefore justified.     

Kinetic modeling of particles in stratified flow – Evaluation of dispersion tensors in inhomogeneous turbulence

The continuum equations for a dilute particle distribution in inhomogeneous turbulence are tested against results from a Langevin particle tracking simulation. Reeks’ version of the kinetic theory is used to generate the mass, momentum and kinetic stress equations for the particle distribution. The particle tracking data are used to directly evaluate the dispersion tensors λ and μ which serve as closure relations for the continuum equations. These exact forms are compared to approximate, local forms. Even for low Stokes numbers (corresponding to low particle inertia defined by τ/τ_p ? 1), the tensor λ is strongly affected by the inhomogeneity and depends on turbulence parameters in the volume corresponding to the particle path dispersion over the particle Lagrangian integral timescale τ. In contrast, the locally homogeneous form of the velocity dispersion tensor μ is a sufficient approximation, since it depends on the dispersion volume over the much smaller particle relaxation time τ_p. It is demonstrated that the body force due to the dispersion vector γ cannot be neglected. In the limit of passive tracers (zero stopping distance), γ is equal to the gradient of λ, if the physical setting is such that we can invoke constant tracer density in this limit.     

Low-frequency conductivity in the average-atom approximation

The quantum-mechanical average-atom model is reviewed and applied to determine scattering phase shifts, mean-free paths, and relaxation times in warm-dense plasmas. Static conductivities σ are based on an average-atom version of the Ziman formula. Applying linear response to the average-atom model leads to an average-atom version of the Kubo–Greenwood formula for the frequency-dependent conductivity σ(ω). The free–free contribution to σ(ω) is found to diverge as 1/ω² at low frequencies; however, considering effects of multiple scattering leads to a modified version of σ(ω) that is finite and reduces to the Ziman formula at ω = 0. The resulting average-atom version of the Kubo–Greenwood formula satisfies the conductivity sum rule. The dielectric function ε(ω) and the complex index of refraction n(ω) + iκ(ω) are inferred from σ(ω) using dispersion relations. Applications to anomalous dispersion in laser-produced plasmas are discussed.     

Global modes and superdirective acoustic radiation in low-speed axisymmetric jets

Global linear stability theory is used to study the resonances in a slowly diverging axisymmetric jet. The absolute frequency ω₀ is calculated as a function of slow axial position X , and analytic continuation into the complex X -plane allows a saddle point in ω₀ to be identified. A key element in the analysis is the approximation of ω₀(X) by rational functions which identifies a well-defined saddle point and leading-order global frequency. A preferred-mode Strouhal number, S_D=0.44 , is calculated which compares well with existing experimental values. The global frequency has negative imaginary part and the jet is interpreted as being marginally globally stable, so that forcing in the vicinity of the resonance frequency produces a large response above background, rather like that of a slightly damped linear oscillator. The axial shape of the global-mode amplitude is Gaussian and yields a superdirective acoustic field.     

Turbulent Couette–Poiseuille flow with zero wall shear

A particular pressure-driven flow in a plane channel is considered, in which one of the walls moves with a constant speed that makes the mean shear rate and the friction at the moving wall vanish. The Reynolds number considered based on the friction velocity at the stationary wall (u_τ,S) and half the channel height (h) is Re_τ,S = 180. The resulting mean velocity increases monotonically from the stationary to the moving wall and exhibits a substantial logarithmic region. Conventional near-wall streaks are observed only near the stationary wall, whereas the turbulence in the vicinity of the shear-free moving wall is qualitatively different from typical near-wall turbulence. Large-scale-structures (LSS) dominate in the center region and their spanwise spacing increases almost linearly from about 2.3 to 4.2 channel half-heights at this Re_τ,S. The presence of LSS adds to the transport of turbulent kinetic energy from the core region towards the moving wall where the energy production is negligible. Energy is supplied to this particular flow only by the driving pressure gradient and the wall motion enhances this energy input from the mean flow. About half of the supplied mechanical energy is directly lost by viscous dissipation whereas the other half is first converted from mean-flow energy to turbulent kinetic energy and thereafter dissipated.     

Effects of large spanwise wavelength on the wake of a sinusoidal wavy cylinder

The wake of a sinusoidal wavy cylinder with a large spanwise wavelength λ/D_m (=3.79–7.57) and a constant wave amplitude a/D_m=0.152, where D_m is the mean diameter of the cylinder, is investigated using three dimensional (3D) large eddy simulation (LES) at a subcritical Reynolds number Re=3×10³, based on incoming free-stream velocity (U_∞) and D_m. Attention is paid to assimilating the effects of λ/D_m on the cylinder wake, including vortex shedding frequency, spanwise vortex formation length, streamwise velocity distribution, flow separation angle, 3D vortex structure, and turbulent kinetic energy (TKE) distribution. Based on the predominant role of λ/D_m in the near wake modification, three regimes are identified, i.e., regime I at λ/D_m<6.0, regime II at λ/D_m≈6.0 and regime III at λ/D_m>6.0. A dramatic decrease in fluid forces is observed at λ/D_m=6.06, about 16% and 93% reduction in time-averaged drag and fluctuating lift, respectively, compared to those of a smooth cylinder. We identified, for the first time, an optimum λ/D_m (=6.06) for the wavy cylinder with relatively large λ/D_m (>3.5) in the subcritical flow regime. The underlying mechanisms of force reduction are discussed, including the flow characteristics at the three λ/D_m regimes. A comparison is also made between the results of λ/D_m effects on the near wakes of a circular and a square cylinder.     

A minimization problem related to the estimation of a lower bound for the temperature in a free boundary value problem related to the combustion of a solid

This paper investigates the least time τ_* of the first zero of the bounded solution to an initial boundary value problem for the heat equation. The heat equation is considered in the domain $$\left\{ {(x,t)| - \infty< x< s(t),0< t \leqslant T} \right\}$$ . The initial conditionu(x, 0)=φ(x) and the boundary conditionu _x (s(t),t)=?R are specified. Let τ=τ(φ,R, s) denote the first zero ofu onx=s(t), that is,u(s(τ), τ)=0. Let τ_*=min τ, where the minimum is taken over a class of functionss=s(t). The existence of τ_* is demonstrated, and a generalization of the problem is discussed.     

Irreducible structure, symmetry and average of Eshelby's tensor fields in isotropic elasticity

The strain field ?(x) in an infinitely large, homogenous, and isotropic elastic medium induced by a uniform eigenstrain ?⁰ in a domain ω depends linearly upon . It has been a long-standing conjecture that the Eshelby's tensor field S^ω(x) is uniform inside ω if and only if ω is ellipsoidally shaped. Because of the minor index symmetry , S^ω might have a maximum of 36 or nine independent components in three or two dimensions, respectively. In this paper, using the irreducible decomposition of S^ω, we show that the isotropic part S of S^ω vanishes outside ω and is uniform inside ω with the same value as the Eshelby's tensor S⁰ for 3D spherical or 2D circular domains. We further show that the anisotropic part A^ω=S^ω-S of S^ω is characterized by a second- and a fourth-order deviatoric tensors and therefore have at maximum 14 or four independent components as characteristics of ω's geometry. Remarkably, the above irreducible structure of S^ω is independent of ω's geometry (e.g., shape, orientation, connectedness, convexity, boundary smoothness, etc.). Interesting consequences have implication for a number of recently findings that, for example, both the values of S^ω at the center of a 2D C_n(n?3,n≠4)-symmetric or 3D icosahedral ω and the average value of S^ω over such a ω are equal to S⁰.     

Capillary breakup extensional rheometry (CaBER) on semi-dilute and concentrated polyethyleneoxide (PEO) solutions

Semi-dilute ( $c^\ast < c < c_{\rm e}$ ) as well as concentrated, entangled (c?>?c _e) solutions of PEO yield uniformly thinning, cylindrical filaments in capillary breakup extensional rheometry (CaBER) experiments. Up to c?≈ c _e thinning can be characterized by a single elongational relaxation time λ _E. Comparison with the longest shear relaxation time, λ _S reveals that λ _E/λ _S decreases with increasing concentration or molecular weight according to (c[η])^???4/3. This is attributed to the large deformation the solutions experience during filament thinning. A factorable integral model including a single relaxation time and a Soskey or Wagner damping function accounting for the large deformation in CaBER experiments is used to calculate λ _E/λ _S and provides good agreement with experimental results. Irrespective of concentration or molecular weight a beads-on-a-string structure occurs prior to filament breakup at a diameter ratio D/D ₀?≈ 0.01. This instability is supposed to be closely related to a flow-induced phase separation.     

The near wake of sinusoidal wavy cylinders: Three-dimensional POD analyses

Three-dimensional (3D) proper orthogonal decomposition (POD) analyses are conducted to investigate the near wake of sinusoidal wavy cylinders. For a wave amplitude a/D_m = 0.152, three typical spanwise wavelengths (λ_z) of the wavy cylinder are taken into account, i.e., λ_z/D_m = 1.89, 3.79 and 6.06, where D_m is the mean diameter of the wavy cylinder, among which λ_z/D_m = 1.89 and 6.06 are the optimum wavelengths corresponding to the largest reduction/suppression of fluid forces acting on the wavy cylinder. Time- and space-resolved three-component velocities of the near wake flow, obtained from large eddy simulation (LES) at a subcritical Reynolds number Re = 3 × 10³, are used in the 3D POD analyses. Comparison is made among the wavy cylinders of the three λ_z/D_m values as well as between them and a smooth cylinder, in terms of POD modes, mode energy, mode coefficients, as well as reconstructed flow structures by lower modes. For the optimum λ_z/D_m = 1.89 and 6.06, energy associated with the first two POD modes is significantly reduced compared with that for λ_z/D_m = 3.79 and the smooth cylinder. Distinct characteristics are observed on the lower POD modes for the wavy cylinders. It is found that the first two POD modes for λ_z/D_m = 1.89 and 6.06 are linked to large-scale streamwise vortices that are additionally introduced into the near wake due to the wavy geometry. Meanwhile, POD mode 3 suggests that the wavy cylinder with the larger optimum λ_z/D_m (= 6.06) generates dominant hairpin-like and spanwise coherent structures (CSs) shedding from the saddle at a different frequency from those shedding from the node. Evolutionary development of these CSs is discussed based on reconstructed flows.     

Generation of dynamic biochemical signals with a tube mixer: effect of dispersion in an oscillatory flow

The generation of controlled dynamic biochemical signals has many applications in the life sciences. This paper presents an analysis of the dispersion of an oscillatory biochemical signal in an incompressible viscous oscillatory flow in a mixing tube. By using the method of Gill and Sankarasubramanian, the dispersion coefficients $K_i(\tau)\;(i=1, 2,\ldots)The generation of controlled dynamic biochemical signals has many applications in the life sciences. This paper presents an analysis of the dispersion of an oscillatory biochemical signal in an incompressible viscous oscillatory flow in a mixing tube. By using the method of Gill and Sankarasubramanian, the dispersion coefficients K_i(t)  (i=1, 2,?)K_i(\tau)\;(i=1, 2,\ldots) are determined as the functions of dimensionless time τ. With the assumptions of quasi-steady flow and steady flow, the dispersion coefficients K_i(t)  (i=1, 2)K_i(\tau)\;(i=1, 2) are simplified. The effects of the frequencies of the oscillatory flow and the oscillatory biochemical signal, and the length of the mixing tube on the average solute concentrations, θ _m , over the tube cross-section at the outlet of the mixing tube are analyzed by numerical simulations with and without the assumptions of the quasi-steady and steady flows. It is concluded that for the dispersion in an oscillatory flow, an excellent accuracy can be achieved by using quasi-steady flow assumption while the steady flow assumption would lead to inaccurate results. However, if the frequency of oscillatory flow is sufficiently high, the steady flow assumption can be used to further simplify the calculation while still maintaining sufficient accuracy. These results are of practical importance in producing dynamic biochemical signals as the stimuli of biological cells by a tube mixer.     

Some new classes of inverse coefficient problems in non-linear mechanics and computational material science

Three classes of inverse coefficient problems arising in engineering mechanics and computational material science are considered. Mathematical models of all considered problems are proposed within the J₂-deformation theory of plasticity. The first class is related to the determination of unknown elastoplastic properties of a beam from a limited number of torsional experiments. The inverse problem here consists of identifying the unknown coefficient g(ξ²) (plasticity function) in the non-linear differential equation of torsional creep −(g(|∇u|²)u_x1)_x1−(g(|∇u|²)u_x2)_x2=2?, x∈Ω⊂R², from the torque (or torsional rigidity) T(?), given experimentally. The second class of inverse problems is related to the identification of elastoplastic properties of a 3D body from spherical indentation tests. In this case one needs to determine unknown Lame coefficients in the system of PDEs of non-linear elasticity, from the measured spherical indentation loading curve P=P(α), obtained during the quasi-static indentation test. In the third model an inverse problem of identifying the unknown coefficient g(ξ²(u)) in the non-linear bending equation is analyzed. The boundary measured data here is assumed to be the deflections w_i[τ_k]?w(λ_i;τ_k), measured during the quasi-static bending process, given by the parameter τ_k, , at some points , of a plate. An existence of weak solutions of all direct problems are derived in appropriate Sobolev spaces, by using monotone potential operator theory. Then monotone iteration schemes for all the linearized direct problems are proposed. Strong convergence of solutions of the linearized problems, as well as rates of convergence is proved. Based on obtained continuity property of the direct problem solution with respect to coefficients, and compactness of the set of admissible coefficients, an existence of quasi-solutions of all considered inverse problems is proved. Some numerical results, useful from the points of view of engineering mechanics and computational material science, are demonstrated.     

On the use of stretched-exponential functions for both linear viscoelastic creep and stress relaxation

The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G _∞)/(G ₀-G _∞) and the retardation function r(t) = (J _∞+t/η-J(t))/(J _∞-J ₀) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (?(t/τ)^β), can r(t) be represented as exp (?(t/λ)^µ) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, η is the viscosity (η^?1 is finite for a fluid and zero for a solid), G _∞ is the equilibrium modulus G _e for a solid or zero for a fluid, J _∞ is 1/G _e for a solid or the steady-state recoverable compliance for a fluid, G ₀= 1/J ₀ is the instantaneous modulus, and t is the time. It is concluded that g(t) and r(t) cannot both exactly by stretched-exponential functions for a given material. Nevertheless, it is found that both g(t) and r(t) can be approximately represented by stretched-exponential functions for the special case of a fluid with exponents β=µ in the range 0.5 to 0.6, with the correspondence being very close with β=µ=0.5 and λ=2τ. Otherwise, the functions g(t) and r(t) differ, with the deviation being marked for solids. The possible application of a stretched-exponential to represent r(t) for a critical gel is discussed.     

Experimental study on the capillary thinning of entangled polymer solutions

The transient elongation behavior of entangled polymer and wormlike micelles (WLM) solutions has been investigated using capillary breakup extensional rheometry (CaBER). The transient force ratio X = 0.713 reveals the existence of an intermediate Newtonian thinning region for polystyrene and WLM solutions prior to the viscoelastic thinning. The exponential decay of X(t) in the first period of thinning defines an elongational relaxation time λ _x which is equal to elongational relaxation time λ _e obtained from exponential diameter decay D(t) indicating that the initial stress decay is controlled by the same molecular relaxation process as the strain hardening observed in the terminal regime of filament thinning. Deviations in true and apparent elongational viscosity are discussed in terms of X(t). A minimum Trouton ratio is observed which decreases exponentially with increasing polymer concentration leveling off at Tr_min = 3 for the solutions exhibiting intermediate Newtonian thinning and Tr_min ≈ 10 otherwise. The relaxation time ratio λ _e/ λ _s, where λ _s is the terminal shear relaxation time, decreases exponentially with increasing polymer concentration and the data for all investigated solutions collapse onto a master curve irrespective of polymer molecular weight or solvent viscosity when plotted versus the reduced concentration c[ η], with [ η] being the intrinsic viscosity. This confirms the strong effect of the nonlinear deformation in CaBER experiments on entangled polymer solutions as suggested earlier. On the other hand, λ _e ≈λ _s is found for all WLM solutions clearly indicating that these nonlinear deformations do not affect the capillary thinning process of these living polymer systems.     

Trigonometric simplification of a class of conservative nonlinear oscillators

This paper deals with a class of conservative nonlinear oscillators of the form $\ddot x(t)+f(x(t))=0$ , where f(x) is analytic. A transformation of time from t to a new time coordinate τ is defined such that periodic solutions can be expressed in the form x(τ) = A ₀+A ₁ cos 2τ. We refer to this process of trigonometric simplification as trigonometrification. Application is given to the stability of nonlinear normal modes (NNMs) in two-degree-of-freedom systems.     

Anomalies associated with energy release parameters for cracks in piezoelectric materials

For a central crack in a piezoelectric plate, the mode-I stress intensity factor (K_I), electric displacement intensity factor (K_D), energy release rates (G, G_M) and energy density factor (S) are obtained from the finite element results. For the impermeable crack, the numerical results of K_I and K_D are coupled; this error is contrary to the uncoupled analytical solutions. The error has little effect on the total energy release rate G and energy density factor S, but in some cases, large errors in the mechanical energy release rate G_M are observed. G is global while SED is local. Also G is negative which defies physics where energy cannot be created while crack attempts to extend as implied by G. Computations should be made for the J-integral and also show that J becomes negative. What this shows is that the global fracture energy criterion is not suitable to address the local release of energy because it includes the overall energy which are irrelevant to fracture initiation being a local behavior. In addition, the case study shows that the energy density theory is the better fracture criterion for the piezoelectric material. According to the results of S, it retards the crack growth when the external electric field and piezoelectric poling are on opposite directions. This conclusion agrees with analytical and experimental evidence in the past references.