Heat transfer in oblique stagnation-point flow of an incompressible viscous fluid towards a stretching surface

Steady two-dimensional oblique stagnation-point flow of an incompressible viscous fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that the flow has a boundary layer structure for values of a/c (> 1), where ax+2by and cx are the x-component of the free stream velocity and the stretching velocity of the plate respectively, x being the distance from the stagnation-point. On the other hand when a/c < 1, the flow has an inverted boundary layer structure. It is also observed that the velocity at a point increases with increase in the free stream shear. For a fixed value of a/c, the streamlines becomes more and more oblique towards the left of the stagnation-point with increase in b/c where b > 0. On the other hand the streamlines become increasingly oblique to the right of the stagnation-point with increase in |b/c| when b < 0. For a fixed value of the Prandtl number Pr, temperature at a point decreases with increase in a/c. Further for a given value of a/c, the surface heat flux increases with increase in Pr.     

Structuration de la convection mixte en milieu poreux confiné latéralement et chauffé par le bas : effets d''inertiePattern formation of mixed convection in a porous medium confined laterally and heated from below: effect of inertia

We consider the onset of convection in a porous medium heated from below and subjected to a horizontal mean flow. The effect of porous inertia is studied, and the transverse aspect ratio a of the medium is taken into accout. We find that the dominant modes are longitudinal rolls (L.R) if a is an integer or transverse traveling rolls (T.R) if a is below a_c with a_c<1. When a is not an integer with a>a_c, the setting on patterns are oscillatory three-dimensional structures (3D) for a>1 or T.R for a_c<a<1 provided that the Reynolds number remains below a critical value Re_K^*. We show that these structures are replaced by L.R if Re_K>Re_K^*. To cite this article: A. Delache et al., C. R. Mecanique 330 (2002) 885–891.     

Marangoni instabilities in small circular containers under microgravity

Circular containers of various aspect ratios a with flat free upper liquid surfaces were heated from below under microgravity to generate the Marangoni instability (MI). We realized “liquid lateral sidewalls” for the containers to come near to the “slippery sidewalls” introduced by Rosenblat et al. (J Fluid Mech 120:91–122, 1982a) and Echebarría et al. (Physica D 99:487–502, 1997), henceforth referred to as RHD and EKP, respectively. The flow structure was visualized by aluminium flakes and recorded on videotape. The MI was clearly observed in all containers above a critical Marangoni number Ma ^c which depends on a. In the first microgravity experiment in a container with a=7.5, we found significant convective heat transport and reported a Nusselt number Nu=1.8 for Ma=4×Ma ^c . In a second microgravity experiment with containers with a=0.5, 0.75, 1.0, 1.5, 2.0, 4.0 and 5.0, various flow structures (azimuthal and radial wave numbers) were observed, depending on a and Ma. The observed scenario compares qualitatively well with the stability curves calculated by RHD and EKP. Frequent switching between modes (2,1) and (1,1) was observed in the container with a=2 at supercritical Ma that is exactly the case for which this behaviour was predicted by EKP for reduced gravity.     

On Asymptotics of Solutions of Nonlinear Differential Equations of the Second Order

We study the problem of asymptotics of unbounded solutions of differential equations of the form y″ = α₀ p(t)ϕ(y), where α₀ ∈ {−1, 1}, p: [a, ω[→]0, +∞[, −∞ < a < ω ≤ +∞, is a continuous function, and ϕ: [y ₀, +∞[→]0, +∞[ is a twice continuously differentiable function close to a power function in a certain sense.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 18–28, January–March, 2005.     

On thermodynamic potentials in linear thermoelasticity

The four thermodynamic potentials, the internal energy u=u(ε_ij,s), the Helmholtz free energy f=f(ε_ij,T), the Gibbs energy g=g(σ_ij,T) and the enthalpy h=h(σ_ij,s) are derived, independently of each other, by using the Duhamel–Neumann extension of Hooke's law and an assumed linear dependence of the specific heat on temperature. A systematic procedure is then presented to express all thermodynamic potentials in terms of four possible pairs of independent state variables. This procedure circumvents a tedious transition from one potential to another, based on the formal change of variables, and inversions of the stress–strain and entropy–temperature relations. The general results are applied to uniaxial loading paths under isothermal, adiabatic, constant stress, and constant strain conditions. An interplay of adiabatic and isothermal elastic constants in the expressions for exchanged heat along certain thermodynamic paths is indicated.     

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.     

Finest Morse Decompositions for Semigroup Actions on Fiber Bundles

Let S{\mathcal{S}} be a semigroup acting on a topological space M. We study finest Morse decompositions for the action of S{\mathcal{S}} on M. This concept depends on a family of subsets of S{\mathcal{S}} . For certain semigroups and families it recovers the concept of Morse decomposition for flows and semiflows. This paper also studies the behaviour of Morse decompositions for semigroup actions on principal bundles and their associated bundles. The emphasis is put on the study of those decompositions considering their projections onto the base space and their intersections with the fibers.     

Percolation Approach to the Anisotropy Factor in an Unsaturated Porous Medium

The anisotropy factor is defined as the ratio of the effective (macroscopic) conductivities parallel to the bedding plane and perpendicular to it. The anisotropy factor A(p,a) is a function of both the saturation degree, p, in the void space of the disordered medium and the anisotropy parameter, a, that characterizes the ratio of the local conductivities parallel and normal to the bedding plane. There are two opposite behaviors of the anisotropy factor as a function of the saturation degree described in literature. One presents the anisotropy factor as a curve with a maximum and the other as a curve with a minimum. The main result of calculating the conductivities of a uniaxial percolation anisotropic model is that at the saturation threshold value, p_c, A(p_c, a) = 1, wherefrom it increases at a >1 (or decreases at a < 1) with saturation. An extension of the computed results below the threshold is also proposed.     

An effective interaction potential model for the shape memory alloy AuCd

The unusual properties of shape memory alloys (SMAs) result from a lattice level martensitic transformation (MT) corresponding to an instability of the SMAs crystal structure. Currently, there exists a shortage of material models that can capture the details of lattice level MTs occurring in SMAs and that can be used for efficient computational investigations of the interaction between MTs and larger-scale features found in typical materials. These larger-scale features could include precipitates, dislocation networks, voids, and even cracks. In this article, one such model is developed for the SMA AuCd. The model is based on effective interaction potentials (EIPs). These are atomic interaction potentials that are explicit functions of temperature. In particular, the Morse pair potential is used and its adjustable coefficients are taken to be temperature dependent. An extensive exploration of the Morse pair potential is performed to identify an appropriate functional form for the temperature dependence of the potential parameters. A fitting procedure is developed for the EIPs that matches, at suitable temperatures, the stress-free equilibrium lattice parameters, instantaneous bulk moduli, cohesive energies, thermal expansion coefficients, and heat capacities of FCC Au, HCP Cd, and the B2 cubic austenite phase of the Au-47.5at%Cd alloy. The resulting model is explored using branch-following and bifurcation techniques. A hysteretic temperature-induced MT between the B2 cubic and B19 orthorhombic crystal structures is predicted. This is the behavior that is observed in the real material. In addition to reproducing the important properties mentioned above, the model predicts, to reasonable accuracy, the transformation strain tensor and captures the latent heat and thermal hysteresis to within an order of magnitude.     

The effect of contact stress intensity factors on fatigue crack propagation

Using the technique of Dimensional Analysis the phenomenon of crack closure is modelled using the concept of a contact stress intensity factor Kc. For constant amplitude loading, a simple expression, Kc_max = g(R) ΔK, is obtained without making idealized assumptions concerning crack tip behaviour. Further, by assuming that crack closure arises from the interaction of residual plasticity in the wake of the crack and crack tip compressive stresses, the function g(R) is shown to be constant for non-workhardening materials. This implies that any dependency of Kc_max on R must be attributed to the workhardening characteristic of the material. With Kc known, an “effective” stress intensity factor Ke may be calculated and incorporated into a crack growth law of the form da/dn = f(ΔKe). From analysis, it can be deduced that for a workhardening material, Kc_max will decrease as R increases and the effective stress intensity factor will increase. This means that the fatigue crack propagation rate will increase with R, in accordance with experimental observations.     

Cover Image

The cover image is based on the Research Article Pressure boundary condition in a multi-phase lattice Boltzmann method and its applications on simulations of two-phase flows by Lei Wang and Ze-Rui Peng, https://doi.org/10.1002/fld.4800 .

     

k‐Version of finite element method in 2‐D polymer flows: Oldroyd‐B constitutive model

In this paper, a new mathematical framework based on h, p, k and variational consistency (VC) of the integral forms is utilized to develop a finite element computational process of two‐dimensional polymer flows utilizing Oldroyd‐B constitutive model. Alternate forms of the choices of dependent variables in the governing differential equations (GDEs) are considered and is concluded that u, v, p, τ choice yielding strong form of the GDEs is meritorious over others. It is shown that: (a) since, the differential operator in the GDEs is non‐linear, Galerkin method and Galerkin method with weak form are variationally inconsistent (VIC). The coefficient matrices in these processes are non‐symmetric and hence may have partial or completely complex basis and thus the resulting computational processes may be spurious. (b) Since the VC of the VIC integral forms cannot be restored through any mathematically justifiable means, the computational processes in these approaches always have possibility of spurious solutions. (c) Least squares process utilizing GDEs in u, v, p, τ (strong form of the GDEs) variables (as well as others) is variationally consistent. The coefficient matrices are always symmetric and positive definite and hence always have a real basis and thus naturally yield computational processes that are free of spurious solutions. (d) The theoretical solution of the GDEs are generally of higher order global differentiability. Numerical simulations of such solutions in which higher order global differentiability characteristics of the theoretical solution are preserved, undoubtedly requires local approximations in higher order scalar product spaces . (e) LSP with local approximations in spaces provide an incomparable mathematical and computational framework in which it is possible to preserve desired characteristics of the theoretical solution in the computational process. Numerical studies are presented for fully developed flow between parallel plates and a lid driven square cavity. M1 fluid is used in all numerical studies. The range of applicability of the Oldroyd‐B model or lack of it is examined for both model problems for increasing De. A mathematical idealization of the corners where stationary wall meets the lid is presented and is shown to simulate the real physics when the local approximations are in higher order spaces and when h_d→0. For both model problems shear rate is examined in the flow domain to establish validity of the Oldroyd‐B constitutive model. Copyright © 2006 John Wiley & Sons, Ltd.     

Boundary effects on the Bénard-Marangoni instability under an electric field

This article theoretically studies the Bénard-Marangoni instability problem for a liquid layer with a free upper surface, which is heated from below by a heating coil through a solid plate in ana.c. electric field. The boundary effects of the solid plate, which include its thermal conductivity, electric conductivity and thickness, have great influences on the onset of convective instability in the liquid layer. The stability analysis in this study is based on the linear stability theory. The eigenvalue equations obtained from the analysis are solved by using the fourth order Runge-Kutta-Gill's method with the shooting technique. The results indicate that the solid plate with a higher thermal or electric conductivity and a bigger thickness tends to stabilize the system. It is also found that the critical Rayleigh numberR _c, the critical Marangoni numberM _c, and the criticala.c. Rayleigh numberE _ac become smaller as the intensity of thea.c. electric field increases.     

Dynamics of vorticity stretching and breakup of isolated viscoelastic droplets in an immiscible viscoelastic matrix

The effect of droplet elasticity on transient deformation of isolated droplets in immiscible polymer blends of equal viscosity was investigated. In terms of the deformation parameter, Def*=a*–c/a*+c where a* and c are apparent drop principal axes, it undergoes two cycles of positive oscillations before reaching a negative value, followed by one cycle of oscillation before attaining a steady state negative value. This behavior was observed when Capillary number, Ca, was varied between 3 and 9 at a fixed Weissenber number, Wi, of 0.31, and when Ca number was held fixed at 8 and Wi number was varied between 0.21 and 0.40. In another blend of relatively lower Wi number of 0.21, one cycle of oscillation in Def* was observed before reaching steady state negative values when Ca number was varied between 3 and 14. The steady state Def* varies inversely with Ca number, with a stronger dependence for the blend with higher Wi number. The magnitude of oscillation increases with increasing Ca and Wi numbers. The critical Ca was found to be 12 and 14 for the two blends studied; these values are about 30 times greater than that of Newtonian blends.     

Multiscale Creep Compliance of Epoxy Networks at Elevated Temperatures

Epoxy networks are thermoset polymers for which an important structural length scale, molecular weight between crosslinks (M _c), influences physical and mechanical properties. In the present work, creep compliance was measured for three aliphatic epoxy networks of differing M _c using both macroscale torsion and microscale depth-sensing indentation at temperatures of 25 and 55°C. Analytical relations were used to compute creep compliance (J(t)) for each approach; similar results were observed for the two techniques at 25°C, but not at 55°C. Although creep compliance measurement differed at elevated temperatures, there were clear correlations between M _c, glass transition temperature, T _g, and the observed time-dependent mechanical behavior via both techniques at 55°C, but these correlations could not be seen at 25°C. This work demonstrates the capacity of depth-sensing indentation to differentiate among epoxy networks of differing structural configurations via J(t) for small material volumes at elevated temperatures.     

Mapping the stiffness-temperature-moisture relationship of solid biomaterials at and around their glass transition

Mechanical changes in biomaterials at and around their glass transition are key factors in their functionality and/or stability. They are described in terms of a relationship betwen a relative stiffness R(T,M) defined as the ratio between a modulus or storage modulus at a temperature T, and moisture M, and its magnitude in the glassy state. The relationship, in turn, is expressed by the model R(T,M) = 1/[1+exp [{T-T _c,(M))/a(M)]} where T _c(M) is a critical temperature identifying the transition temperature range and a(M) a constant representing the relationship's slope. The proposed model correctly accounts for the downward concavity of the stiffness vs temperature relationship at the transition onset. Published data on biosolids indicate that T _c(M) can be described by a single exponential decay term, and so most probably also a(M). Incorporation of these terms into the model enables the creation of realistic three-dimensional maps of the relative stiffness-temperature-moisture relationship at and in the neighborhood of the glass transition region. In principle, the same method can also be used to describe the effect of plasticizers other than water if their influence on T _cand the steepness parameter can be formulated as an algebraic expression.