Linear and non linear rheology of a wormlike micellar system in presence of sodium tosylate

In this experimental work, we investigate the influence of an organic counterion, sodium tosylate, on the rheological properties of an aqueous solution of CTAB at the concentration of 0.05M. With this system we can clearly see shear thickening for small salt concentrations C _s and only shear thinning behavior at higher C _s characterized by a linear evolution of η=f(γ) in a log-log representation. In these evolutions it is only in a very small domain of concentrations of the salt (near C _s =0.035M) that we can observe a nearly constant plateau of the shear stress against shear rate. The values of σ₀ (characterizing the stress plateau), G ₀ (the plateau modulus) and τ_R (the relaxation time) obtained by dynamical rheological measurements, allow to compare experimental results obtained to predicted values of the theory of Cates corresponding to the occurrence of shear induced banding structures. Received: 22 July 1997 Accepted: 3 February 1998     

Asymptotic Variational Wave Equations

We investigate the equation (u _t +(f(u)) _x ) _x =f ^{′ ′}(u) (u _x )²/2 where f(u) is a given smooth function. Typically f(u)=u ²/2 or u ³/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u _tt − c(u) (c(u)u _x ) _x =0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data.     

Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation

We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u _tt − c(u)(c(u)u _x ) _x  = 0. We allow for initial data u| _{t = 0} and u _t | _t=0 that contain measures. We assume that 0 < k^-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (u_t²+c(u)²u_x²)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples.     

Heat transfer in nucleate pool boiling of aqueous SDS and triton X-100 solutions

Variation in degree of surface wettability is presented through the application of Cooper’s correlative approach (h ∝ M ^−0.5 q _w ^″0.67) for computing enhancement (ϕ) in nucleate pool boiling of aqueous solutions of SDS and Triton X-100 and its presentation with Marangoni parameter (χ) that represents the dynamic convection effects due to surface tension gradients. Dynamic spreading coefficient defined as σ _dyn N _a , which relates spreading and wetting characteristics with the active nucleation site density on the heated surface and bubble evolution process, represents cavity filling and activation process and eliminates the concentration dependence of nucleate pool boiling heat transfer in boiling of aqueous surfactant solutions. Using the dynamic spreading coefficient (σ _dyn N _a  = 0.09q _w ^″0.71), correlation predictions within ±15% for both SDS and Triton X-100 solutions for low heat flux boiling condition (q _w^″ ≤ 100 kW/m²) characterised primarily by isolated bubble regime are presented.     

Heat transfer in pool boiling of ammonia/water mixture

The nonazeotropic binary mixtures such as, methanol/water, ethanol/water and ammonia/water, have variable boiling and dew points, depending on the combination of substance and those mass fraction. It is expected to have a higher performance as a result of decreasing the thermodynamically irreversible loss, when there is a relevant mass fraction. Therefore, ammonia/water mixture is expected to use as working fluid in small temperature difference power generation cycles and absorption refrigeration cycles. However, few experiments were carried out for measuring heat transfer coefficient for ammonia/water mixture in the world. An experimental study has been carried out to measure boiling heat transfer coefficient of an ammonia/water mixture on a horizontal heated surface at low pressure of 0.2, 0.4 and 0.7 MPa and at low mass fraction of 0 < C < 0.27 and at high pressure 0.7, 1.0 and 1.5 MPa and at mass fraction of 0.5 < C < 1.0 and at heat flux under critical heat flux the heat transfer coefficient are compared with existing correlations prediction and a revised correlation can be proposed to predict them well.     

Isolated Singularities of the 1D Complex Viscous Burgers Equation

The Cauchy problem for the 1D real-valued viscous Burgers equation u _t +uu _x  = u _xx is globally well posed (Hopf in Commun Pure Appl Math 3:201–230, 1950). For complex-valued solutions finite time blow-up is possible from smooth compactly supported initial data, see Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008). It is also proved in Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008) that the singularities for the complex-valued solutions are isolated if they are not present in the initial data. In this paper we study the singularities in more detail. In particular, we classify the possible blow-up rates and blow-up profiles. It turns out that all singularities are of type II and that the blow-up profiles are regular steady state solutions of the equation.     

Effect of a hydrotrope on the viscoelastic properties of polymer-like micellar solutions

Low-viscosity micellar aqueous solutions of cetyltrimethylammonium bromide (CTAB) undergo a major change in the presence of the hydrotrope, potassium 1-phenylmethylsulfate (KPhMS), producing a highly viscoelastic entanglement network of polymer-like micelles. The system studied here shows typical shear banding flow behavior, which tends to disappear with increasing the hydrotrope-to-surfactant concentration ratio (C _H / C _S). The linear rheological response was analyzed with the model of Granek–Cates, whereas the nonlinear behavior was reproduced with the Bautista–Manero–Puig (BMP) model. Both models introduce a kinetic equation to account for the breaking and reformation of the micelles, and they predict the linear and nonlinear rheological data very well. This paper was presented at Annual European Rheology Conference (AERC) held in Hersonisos, Crete, Greece, April 27–29, 2006.     

Momentum and energy balances for dispersed two-phase flow

Summary The equations of motion and the mechanical energy balances for two-phase flow systems are derived by integration over a volume containing a large number of elements of the dispersed phase.List of symbols A, A boundary of volumes V, V - dA, dA surface element of A, A - A _s boundary of particles in V - dA _s surface element of A _s - F force per unit volume of the system - g –gz=gravity vector - g acceleration by gravity - I unit tensor - p pressure - Q dissipation in the continuous phase - Q _s dissipation in the dispersed phase - R compression work in the continuous phase - R _s compression work in the dispersed phase - t time - u velocity of continuous phase - u _s velocity of dispersed phase - u magnitude of u - u _s magnitude of u _s - V volume in the two-phase system - V part of V occupied by the continuous phase - W work done by F - z vertical coordinate - local volume fraction of the dispersed phase - pI–=stress tensor of the continuous phase - _s turbulent particle stress tensor - density of the continuous phase - _s density of the dispersed phase - shearing-stress tensor of the continuous phase - _s turbulent particle shearing-stress tensor - nabla operator - u, u _s velocity gradient tensor - substantial derivative (Shell Internationale Research Maatschappij N.V.)(Bataafse Internationale Petroleum Maatschappij N.V.)     

Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form u _t = Δu + f(t, u) on \mathbbR^N{\mathbb{R}^N} . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.     

Double diffusive natural convection in a partially heated enclosure with Soret and Dufour effects

The effect of double-diffusive natural convection of water in a partially heated enclosure with Soret and Dufour coefficients around the density maximum is studied numerically. The right vertical wall has constant temperature θ_c, while left vertical wall is partially heated θ_h, with θ_h > θ_c. The concentration in right wall is maintained higher than left wall (C_c < C_h) for case I, and concentration is lower in right wall than left wall (C_h > C_c) for case II. The remaining left vertical wall and the two horizontal walls are considered adiabatic. Water is considered as the working fluid. The governing equations are solved by control volume method using SIMPLE algorithm with QUICK scheme. The effect of the various parameters (thermal Rayleigh number, center of the heating location, density inversion parameter, Buoyancy ratio number, Schmidt number, and Soret and Dufour coefficients) on the flow pattern and heat and mass transfer has been depicted. Comprehensive Nusselt and Sherwood numbers data are presented as functions of the governing parameters mentioned above.     

Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent

In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l₁ u = m₁ u³+buv²,     x ? W,-Dv +l₂ v = m₂ v³+bvu²,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.     

A 3D LIF system for turbulent buoyant jet flows

A Laser-Induced Fluorescence (LIF) system for mapping the three dimensional tracer concentration field in turbulent flows is described. The system is particularly suited to studies of single or multiple buoyant jets discharged into unstratified and stratified flowing environments for conditions typical of wastewater discharges into surface water bodies. A laser beam is scanned through the flow and LIF images are obtained in parallel planes with a high-speed synchronized CCD camera. Refractive index matching is used to minimize refractive index variations due to local density gradients. An application to vertical round buoyant jets discharging into unstratified and stratified cross flows is presented. The three-dimensional system can obtain vastly more data than is possible with probe-based techniques and can yield far more insight into the flow and mixing processes.Abbreviations a Combined attenuation coefficient, cm^-1 - a_w Attenuation coefficient for fresh water, cm^-1 - B Buoyancy flux of buoyant jet, cm⁴/s³ - c Tracer concentration, g/l - C _salt, C _eth Salt and ethanol concentrations, g/l - d Port diameter, cm - g Acceleration due to gravity, cm/s² - Modified acceleration due to gravity at source, cm/s² - Q Volume flux of buoyant jet, cm³/s - I Image gray scale level, DN - l _a,l _Q , l _M , l _s, l _t Buoyant jet length scales, cm (Eq. 8) - L _y Distance from camera to image plane, cm - M Momentum flux of buoyant jet, cm⁴/s² - M _y Image scale factor - N Buoyancy frequency, s^-1 - u _a Ambient flow velocity, cm/s - u _j Jet exit velocity, cm/s - P Laser power, W - S ₀ Lowest dilution on the vertical center-plane through the nozzle - S _m Minimum dilution: lowest dilution in a vertical plane perpendicular to the flow - x, y, z Coordinates, cm - z _m Maximum rise height of buoyant jets, cm - z _e Equilibrium rise height of buoyant jets, cm - Image calibration constant - ₀ Effluent density, g/cm³ - _a Ambient density, g/cm³     

Effect of surfactant concentration on saturated flow boiling in vertical narrow annular channels

Saturated flow boiling of environmentally acceptable nonionic surfactant solutions of Alkyl (8–16) was compared to that of pure water. The concentration of surfactant solutions was in the range of 100–1000 ppm. The liquid flowed in an annular gap of 2.5 and 4.4 mm between two vertical tubes. The heat was transferred from the inner heated tube to two-phase flow in the range of mass flux from 5 to 18 kg/m² s and heat flux from 40 to 200 kW/m². Boiling curves of water were found to be heat flux and channel gap size dependent but essentially mass flux independent. An addition of surfactant to the water produced a large number of bubbles of small diameter, which, at high heat fluxes, tend to cover the entire heater surface with a vapor blanket. It was found that the heat transfer increased at low values of relative surfactant concentration C/C₀, reaches a maximum close to the value of C/C₀ = 1 (where C₀ = 300 ppm is the critical micelle concentration) and decreased with further increase in the amount of additive. The dependence of the maximal values of the relative heat transfer enhancement, obtained at the value of relative concentration of C/C₀ = 1, on the boiling number Bo may be presented as single curve for both gap sizes and the whole range of considered concentrations.     

Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder

The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x₃ and find the solution (u, p) having the following form

Turbulent pipe flow of a drag-reducing rigid “rod-like” polymer solution

Fully developed turbulent pipe flow of an aqueous solution of a rigid “rod-like” polymer, scleroglucan, at concentrations of 0.005% (w/w) and 0.01% (w/w) has been investigated experimentally. Fanning friction factors were determined from pressure-drop measurements for the Newtonian solvent (water) and the polymer solutions and so levels of drag reduction for the latter. Mean axial velocity u and complete Reynolds normal stress data, i.e. u′, v′ and w′, were measured by means of a laser Doppler anemometer at three different Reynolds numbers for each fluid. The measurements indicate that the effectiveness of scleroglucan as a drag-reducing agent is only mildly dependent on Reynolds number. The turbulence structure essentially resembles that of flexible polymer solutions which also lead to low levels of drag reduction.     

A natural neighbour method based on Fraeijs de Veubeke variational principle for materially non-linear problems

Abstract The natural neighbour method can be considered as one of many variants of the meshless methods. In the present paper, a new approach based on the Fraeijs de Veubeke （FdV） functional, which is initially developed for linear elasticity, is extended to the case of geometrically linear but materially non-linear solids. The new approach provides an original treatment to two classical problems： the numerical evaluation of the integrals over the domain A and the enforcement of boundary conditions of the type ui = hi on Su. In the absence of body forces （Fi = 0）, it will be shown that the calculation of integrals of the type fA .dA can be avoided and that boundary conditions of the type ui = hi on Su can be imposed in the average sense in general and exactly if hi is linear between two contour nodes, which is obviously the case for tTi = O.     

Time Periodic Solutions to the One-Dimensional Nonlinear Wave Equation

This paper is concerned with the time periodic solutions to the one-dimensional nonlinear wave equation with either variable or constant coefficients. By adjusting the basis of L ² function space, we can circumvent the difficulties caused by η _u  = 0 and obtain the existence of a weak periodic solution, which was posed as an open problem by Baubu and Pavel in (Trans Am Math Soc 349:2035–2048, 1997). Finally, an application to the forced Sine-Gordon equation is presented to illustrate the utility of this technique.