Showing papers in "Acta Mechanica in 1999"

The effect of the slip boundary condition on the flow of fluids in a channel

Abstract: The assumption that a liquid adheres to a solid boundary (“no-slip” boundary condition) is one of the central tenets of the Navier-Stokes theory. However, there are situations wherein this assumption does not hold. In this paper we investigate the consequences of slip at the wall on the flow of a linearly viscous fluid in a channel. Usually, the slip is assumed to depend on the shear stress at the wall. However, a number of experiments suggests that the slip velocity also depends on the normal stress. Thus, we investigate the flow of a linearly viscous fluid when the slip depends on both the shear stress and the normal stress. In regions where the slip velocity depends strongly on the normal stress, the flow field in a channel is not fully developed and rectilinear flow is not possible. Also, it is shown that, in general, traditional methods such as the Mooney method cannot be used for calculating the slip velocity.

Existence and uniqueness of the integrable-exactly hypoelastic equation $$\mathop \tau \limits^ \circ = \lambda (trD){\rm I} + 2\mu D$$ and its significance to finite inelasticity

Abstract: In Eulerian rate type finite inelasticity models postulating the additive decomposition of the stretchingD, such as finite deformation elastoplasticity models, the simple rate equation indicated in the above title is widely used to characterize the elastic response withD replaced by its “elastic” part. In 1984 Simo and Pister (Compt. Meth. Appl. Mech. Engng.46, 201–215) proved that none of such rate equations with several commonly-known stress rates is exactly integrable to deliver an elastic relation, and thus any of them is incompatible with the notion of elasticity. Such incompatibility implies that Eulerian rate type inelasticity theory based on any commonly-known stress rate is self-inconsistent, and thus it is hardly surprising that some aberrant, spurious phenomena such as the so-called shear oscillatory response etc., may be resulted in. Then arises the questions: Whether or not is there a stress rate\(\mathop {\tau ^* }\limits^\bigcirc\)? The answer for these questions is crucial to achieving rational, self-consistent Eulerian rate type formulations of finite inelasticity models. It seems that there has been no complete, natural and convincing treatment for the foregoing questions until now. It is the main goal of this article to prove the fact: among all possible (infinitely many) objective corotational stress rates and other well-known objective stress rates\(\mathop {\tau ^* }\limits^\bigcirc\), there is one and only one such that the hypoelastic equation of grade zero with this stress rate is exactly integrable to define a hyperelastic relation, and this stress rate is just the newly discoveredlogarithmic stress rate by these authors and others. This result, which provides a complete answer for the aforementioned questions, indicates that in Eulerian rate type formulations of inelasticity models, the logarithmic stress rate is the only choice in the sense of compatibility of the hypoelastic equation of grade zero that is used to represent the elastic response with the notion of elasticity.

Combined heat and mass transfer in natural convection flow from a vertical wavy surface

Abstract: In the present paper, effects of combined buoyancy forces from mass and thermal diffusion by natural convection flow from a vertical wavy surface have been investigated using the implicit finite difference method. Here we have focused our attention on the evolution of the surface shear stress,f″(0), rate of heat transfer,g′(0), and surface concentration gradient,h′(0) with effect of different values of the governing parameters, such as the Schmidt number Sc ranging from 7 to 1500 which are appropriate for different species concentration in water (Pr=7.0), the amplitude of the waviness of the surface ranging from 0.0 to 0.4 and the buoyancy parameter,w, ranging from 0.0 to 1.

Coupled heat and mass transfer by free convection over a truncated cone in porous media: VWT/VWC or VHF/VMF

Abstract: The heat and mass transfer characteristics of natural convection about a truncated cone embedded in a saturated porous medium subjected to the coupled effects of thermal and mass diffusion is numerically analyzed. The surface is maintained at variable wall temperature/concentration (VWT/VWC) or variable heat/mass flux (VHF/VMF). The transformed governing equations are solved by Keller box method. Numerical data for the dimensionless temperature profiles, the dimensionless concentration profiles, the local Nusselt number and the local Sherwood number are presented for wide range of dimensionless distance ξ, the Lewis number Le, the exponent λ, and buoyancy ratioN (orN*). In general, it has been found that when the buoyancy ratio is increasing both the local Nusselt number and the local Sherwood number increase. Increasing the value of λ and ξ increases the local surface heat and mass transfer rates. The local Nusselt (Sherwood) number increases (decreases) with decreasing the Lewis number. Furthermore, it is shown that the local Nusselt number and the local Sherwood number of the truncated cone approach those of inclined plate (full cone) for the case of ξ=0 (ξ→∞).

On frame-indifference and form-invariance in constitutive theory

Abstract: The purpose of this paper is to present an alternative formulation of the principles of Euclidean frame-indifference and indifference with respect to superimposed rigid body motions, i.e., rotations and translations. This is accomplished on the basis of the action of the proper Euclidean group on constitutive relationsinduced by the action of this group on the Euclidean tensors appearing in these relations. The resulting formulation of these concepts can then be used to show that the usual concept of material frame-indifference actually consists of twoindependent concepts, i.e., Euclidean frame-indifference andform-invariance, the latter being generally overlooked as an independent concept. On this basis, one can in addition show that any two of the concepts of Euclidean frame-indifference, form-invariance, and indifference with respect to superimposed rigid body motions, automatically implies the third. As an application of this formulation, we discuss the constitutive relations for a simple (elastic) material and a kinetic gas. In this context, it follows straighforwardly that the latter satisfy Euclidean frame-indifference, as shown by Murdoch [1], but not indifference with respect to superimposed rigid body motion, as shown by Muller [2]. As such, the current formation yields immediately that these are not form-invariant, and so not material frame-indifferent.

Scattering of antiplane shear waves by a finite crack in piezoelectric laminates

Abstract: Following the dynamic theory of linear piezoelectricity, we consider the scattering of horizontally polarized shear waves by a finite crack in a composite laminate containing a piezoelectric layer. The piezoelectric layer is bonded between two half-spaces of a different elastic solid. The crack is normal to the interfaces and is placed at an equal distance away from them. Both cases of a partially broken layer and a completely broken layer are studied. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a singular integral equation. The propagation of symmetric first mode is studied numerically, and the dynamic stress intensity factor and the dynamic energy release rate are obtained for some piezoelectric laminates.

Forced convection over a continuous sheet with suction or injection moving in a flowing fluid

Abstract: The analysis of forced convection flow and heat transfer about a flat sheet with suction or injection continuously moving in a quiescent or flowing fluid has been carried out. This kind of problem finds applications in a variety of manufacturing processes such as hot rolling, extrusion of plastic sheets, continuous casting, and cooling of a metallic plate in a cooling bath. The governing differential equations are reduced to nonlinear ordinary differential equations by similarity transformations. These equations are solved numerically based on a finite difference algorithm. Representative velocity and temperature profiles within the boundary layer are presented at selected values of free stream velocity and injection parameter. The friction factor and Nusselt number are illustrated for a wide range of governing parameters. For the same injection parameter, Prandtl number, and normalized velocity difference |Uw−U∞|, higher values of the Nusselt number and friction factor result fromUw>U∞ than fromUw

Impact response of a finite crack in an orthotropic piezoelectric ceramic

Abstract: The transient dynamic stress intensity factor and dynamic energy release rate were determined for a cracked piezoelectric ceramic under normal impact in this study. A plane step pulse strikes the crack and stress wave diffraction takes place. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion technique is used to compute the values of the dynamic stress intensity factor and the dynamic energy release rate for some piezoelectric ceramics, and the results are graphed to display the electroelastic interactions.

Streamline topologies near a fixed wall using normal forms

Abstract: Streamline patterns and their bifurcations in two-dimensional incompressible viscous flow in the vicinity of a fixed wall have been investigated from a topological point of view by Bakker [11]. Bakker's work is revisited in a more general setting allowing a curvature of the fixed wall and a time dependence of the streamlines. The velocity field is expanded at a point on the wall, and the expansion coefficients are considered as bifurcation parameters. A series of nonlinear coordinate changes results in a much simplified system of differential equations for the streamlines (a normal form) encapsulating all the features of the original system. From this, a complete description of bifurcations up to codimension three close to a simple linear degeneracy is obtained. Further, the case of a non-simple degeneracy is considered. Finally the effect of the Navier-Stokes equations on the local topology is investigated.

On the rocking-uplifting motion of a rigid block in free and forced motion: influence of sliding and bouncing

Abstract: The problem of rocking response of a rigid block in free and forced motion has been studied for a number of technical reasons. Apart from the technical interests, the problem of rigid block rocking is intrinsically of interest from a theoretical point of view. In fact, the problem is highly nonlinear in nature.

Some simple flows of a Johnson-Segalman fluid

Abstract: Unlike most other fluid models, the Johnson-Segalman fluid allows for a non-monotonic relationship between the shear stress and rate of shear in a simple shear flow for certain values of the material parameter. This has been used for explaining a phenomenon such as “spurt”. Here, we study three simple flows of a Johnson-Segalman fluid with a view towards understanding its response characteristics. We find that boundary conditions can have a very interesting effect on the regularity of the solution; changing them continuously leads to solutions that change their regularity. First, we consider the flow through a circular pipe and find solutions that have discontinuous velocity profiles which have been used to explain the phenomenon of “spurt” (cf. [10], [11]). Second, we consider the flow past an infinite porous plate and show that it will not admit solutions which have discontinuous velocity gradients, the solutions being necessarity smooth. Lastly, we study Poiseuille flow in a concentric annulus with porous boundaries. While “spurt” could be explained alternatively by allowing for “stick-slip” at the wall, the Johnson-Segalman model seems particularly suited in describing the appearance of “shear-layers” (cf. [13]).

Multi-component convection-diffusion with internal heating or cooling

Abstract: The onset of convection in a fluid layer with an internal heat source is studied. In addition to the temperature field there are present two different, dissolved salt fields. Thus, this paper investigated the effect of an internal heat source on the problem of triply-diffusive convection. The effect of the boundary conditions is found to be important. For two surfaces free of tangential stress a disconnected oscillatory neutral curve can be found which has the same minimum as the stationary convection one. Thus the possibility of simultaneous initiation of convection by two different mechanisms, but with two different aspect ratios, is found. It is also found that the above effect is prsent when the lower surface is fixed while the upper surface is free of tangential stress, even if the container of the fluid is of finite horizontal extent. When both surfaces are fixed we have not observed the twin minima effect.

Investigation of the scattering of harmonic elastic shear waves by two collinear symmetric cracks using the non-local theory

Abstract: In this paper, the scattering of harmonic shear waves by two collinear symmetric cracks is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then, a set of triple integral equations is solved using a new method, namely Schmidt's method. This method is simple and convenient for solving this problem. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length.

On apparent properties of nonlinear heterogeneous bodies smaller than the representative volume

Abstract: Huet's model for overall properties of specimens smaller than the representative volume is generalized on nonlinear heterogeneous elastic materials with imperfect interfaces. A modified definition for the apparent properties of heterogeneous nonlinear elastic bodies is given. The size effect relationships are established between experimental results obtained on a big specimen and on an appropriate set of smaller specimens. Hierarchies between the apparent properties of the families of specimens of different sizes are constructed.

On the moment of a plane disk in a non-Newtonian fluid

Abstract: Exact analytic solution for the flow of non-Newtonian fluid of grade two generated by periodic oscillations of a plane disk is obtained. The velocity field and the moment of the frictional forces are calculated and the results are compared with those for Newtonian fluid. The moments caused by certain special oscillations are also discussed.

Heat and mass transfer by natural convection in a non-Darcy porous medium

Abstract: The Forchheimer free convection heat and mass transfer near a vertical surface embedded in a fluid saturated porous medium has been analyzed. A similarity solution is presented for constant wall temperature and concentration distributions with specified power function form (Ax−1/2) of mass flux parameter. The effect of Grashof number (Gr), the buoyancy ratio (N), the Lewis number (Le) and the surface mass flux (fw) on the nondimensional heat and mass transfer coefficients are presented.

A constitutive model in finite thermoviscoelasticity based on the concept of transient networks

Abstract: Constitutive equations are derived for the non-isothermal viscoelastic behavior of polymers at finite strains. The model is based on the concept of transient reversible networks with temperature-dependent rates of creation and breakage for active chains. It is demonstrated that the constitutive relations adequately predict the response of polyisobutylene in isothermal and non-isothermal static tests.

Unsteady convective diffusion in a pulsatile flow through a channel

Abstract: The paper presents an exact analysis of the streamwise dispersion of passive contaminant molecules released in an incompressible viscous fluid flowing through a channel under the influence of a periodic pressure gradient. Using the Aris-Barton method of moments which is valid for all time after the injection of the solute, the dispersion coefficients of a passive contaminant cloud are obtained separately for three different cases: steady, periodic and for comparison the combined effect of steady and periodic currents. Here it is shown how the injected material disperses due to the shear effect caused by the combined effects of flow (steady or periodic) and lateral diffusion about its mean position, and how the centre of gravity of mass moves, when the initial distribution of contaminant is uniform over the cross-section of the channel. The comparison reveals that for all cases the dispersion coefficient asymptotically reaches a stationary state after a certain time, but it changes cyclically with dispersion time even in the stationary state for the case of oscillatory flows. The analysis leads to the interesting result that the dispersion coefficient consists of a steady part and a fluctuating part due to the pulsatility of the flow.

A frictional Cosserat model for the flow of granular materials through a vertical channel

Abstract: A rigid-plastic Cosserat model has been used to study dense, fully developed flow of granular materials through a vertical channel. Frictional models based on the classical continuum do not predict the occurrence of shear layers, in contrast to experimental observations. This feature has been attributed to the absence of a material length scale in their constitutive equations. The present model incorporates such a material length scale by treating the granular material as a Cosserat continuum. Thus, localized couple stresses exist, and the stress tensor is asymmetric. The velocity profiles predicted by the model are in close agreement with available experimental data. The predicted dependence of the shear layer thickness on the width of the channel is in reasonable agreement with data. In the limit of small e (ratio of the particle diameter to the half-width of the channel), the model predicts that the shear layer thickness scaled by the particle diameter grows as e -1/3 .

Closed form solutions for partially debonded circular inclusion in piezoelectric materials

Abstract: This paper deals with the problem of a partially debonded piezoelectric circular inclusion in a piezoelectric matrix. This boundary value problem is reduced to two Riemann-Hilbert problems through the use of the analytical continuation theory.Closed form solutions are obtained by considering the behavior of the complex field potentials at origin and infinity. The formulae for the electro-elastic field intensity factors of the interfacial crack are derivedexplicitly. Several particular cases are provided to show the effect of the crack angle, the mechanical and electrical properties and the loads on the electroelastic field singularities.

A micromechanical method for predicting the precipitation hardening response of particle strengthened alloys hardened by ordered precipitates

Abstract: A micromechanical method was developed for predicting the precipitation hardening response of particle strengthened alloys hardened by ordered precipitates based on the microstructure, composition, and heat treatment, and utilizing a minimum number of experimental tests to evaluate the microstructural constants of the overall model. The overall approach was based on incorporating the dislocation particle interaction mechanics, particle growth and coarsening theory, thermodynamics, and particle strengthening mechanisms applicable to precipitation hardened alloys as part of the overall micromechanical method. The method/model evaluates, from a minimum number of experimental tensile tests, microstructural constants necessary in determining the precipitation srengthening response of a particle strengthened alloy. The materials that were used as vehicles to demonstrate and evaluate the model were precipitation hardenable aluminium-lithium-zirconium and nickel-aluminum alloys. Utilizing these demonstration alloys, the method used a total of four tensile tests to evaluate the necessary microstructural constants and thus predict the variation in strength as a function of aging time, aging temperature, and composition, for the underaged, the peak-aged, and the overaged conditions. Predictions of the precipitation strengthening response were made incorporating the Wagner particle distribution model to evaluate the size distributions of particles in the microstructures. The predicted variation of strength with aging practice and composition using the Wagner distribution model compared well with the corresponding experimental yield strength results.

Nonuniqueness of transonic flows

Abstract: The objective of this paper is to present nonunique (numerical) solutions of potential, Euler and Navier-Stokes equations for steady transonic flows over the same airfoil at the same Mach number. It seems, therefore, that the nonuniqueness is associated with the common inherent nonlinearity of the different models.

Boundary and inertia effects on convection in porous media with throughflow

Abstract: The effects of a nonuniform temperature gradient and inertia arising due to throughflow on the onset of convection in a porous layer for different types of boundaries are investigated. Closed form solutions are obtained for the boundaries which are insulating to temperature perturbations, and for the conducting boundaries solutions are obtained using Galerkin technique. It is found that when the two boundaries are of the same type, the effect of throughflow is to stabilize the system irrespective of its direction. However, when the lower and upper boundaries are of different types, a small amount of throughflow in one particular direction destabilizes the system depending upon the values of the Prandtl number and the porous parameter. The standard results available are obtained as limiting cases.

The effect of a uniform vertical magnetic field on the onset of oscillatory marangoni convection in a horizontal layer of conducting fluid

Abstract: In this paper we use classical linear stability theory to undertake a detailed investigation of the effect of a uniform vertical magnetic field on the onset of oscillatory thermocapillary-driven Marangoni convection in a horizontal layer of quiescent, electrically conducting fluid heated from below. For simplicity we restrict our attention to the simplest case of a fluid layer with a non-deformable free surface and perfectly electrically conducting boundaries in which the onset of convection is always steady in the absence of the magnetic field. The present numerical calculations show that the presence of the magnetic field can cause the preferred mode of instability to be oscillatory rather than steady convection. Nevertheless, in all the cases investigated the effect of the magnetic field is always to stabilise the layer relative to the case of no magnetic field.

Thermal boundary layer flow of a micropolar fluid past a wedge with constant wall temperature

Abstract: The steady laminar flow of micropolar fluids past a wedge has been examined with constant surface temperature. The similarity variables found by Falkner and Skan are employed to reduce the streamwise-dependence in the coupled nonlinear boundary layer equation. Numerical solutions are presented for the heat transfer characteristics with Pr=1 using the fourth-order Runge-Kutta method, and their dependence on the material parameters is discussed. The distributions of dimensionless temperature and Nusselt number across the boundary layer are compared with the corresponding flow problems for a Newtonian fluid over wedges. Numerical results show that for a constant wedge angle with a given Prandtl number Pr=1, the effect of increasing values ofK results in an increasing thermal boundary thickness for a micropolar fluid, as compared with a Newtonian fluid. For the case of the constant material parameterK, however, the heat transfer rate for a micropolar fluid is lower than that of a Newtonian fluid.

Flow past an accelerated horizontal plate in a rotating fluid

Abstract: A semi-infinite mass of an incompressible viscous fluid bounded by an infinite flat plate is initially rotating with uniform angular velocity Ω about an axis normal to the plate. An analysis is presented for the subsequent flow when the plate started impulsively from rest relative to the rotating fluid moves with uniform acceleration in its own plane. It is found that when Ω≠0, the velocity profiles for varying times are nonsimilar in contrast to the velocity profiles which are similar in the absence of rotation (Ω=0). At a given instant, the velocity component along the direction of motion of the plate decreases with an increase in rotation but the transverse velocity component (induced by the Coriolis force) increases with increasing rotation. Due to the gradual thinning of the boundary layer with rotation, both the skin-friction components along and transverse to the direction of motion of the plate increase with increasing rotation. A study of the asymptotic behavior of the velocity field for large time reveals a novel feature of the flow; it develops inertial oscillations with frequency 2Ω, which grow with time. This behavior has not been reported in the absence of rotation.

Thermoelastic plane waves in a rotating isotropic medium

Abstract: We discuss the theory of thermoelastic wave propagation in a rotating isotropic material. In any given direction there are four waves. In general, all of these waves are attenuated, and none of them is purely dilatational or transverse. Some earlier published results are found to be false.

Wave structure in Stokes' second problem for a dipolar fluid with nonclassical heat conduction

Abstract: The classical heat conduction law of Fourier associates an infinite speed of propagation to a thermal disturbance in a material body. Such behavior is a violation of the causality principle. In recent years, several modifications of Fourier's heat law have been proposed. In this work, a modified form of Fourier's heat law, based on the Maxwell-Cattaneo-Fox (MCF) model, is used to analyze the heat conduction effects in Stokes' second problem for a dipolar fluid. The structure of the waves and the influence of the dipolar constants on the velocity field is investigated. These results are then compared to the viscous fluid case. In addition, the displacement thickness and skin friction at the plate are determined.

Wave propagation modeling in human long bones

Abstract: The dynamic behavior of a dry long bone that has been modeled as a piezoelectric hollow cylinder of crystal class 6 is investigated. The solution for the wave propagation problem is expressed in terms of a potential function which satisfies an eighth-order partial differential equation, whose solutions lead to the derivation of the explicit solution of the wave equation. The mechanical boundary conditions correspond to those of stress free lateral surfaces, while the electrical boundary conditions correspond to those of short circuit. The satisfaction of the boundary conditions leads to the dispersion relation which is solved numerically. The frequencies obtained are presented as functions of various parameters and they compare well with other researchers' theoretical results.

Uniform transpiration effect on coupled heat and mass transfer in mixed convection about inclined surfaces in porous media: the entire regime

Abstract: A boundary layer analysis is used to investigate the effect of uniform transpiration velocity on the heat and mass transfer characteristics of mixed convection about inclined surfaces in saturated porous media under the coupled effects of thermal and mass diffusion. The surfaces are maintained at variable wall temperature (VWT) and variable wall concentration (VWC). Nonsimilar governing equations are obtained by using a suitable transformation and solved by Keller box method. Numerical results are presented for the local Nusselt number as well as the local Sherwood number. The local Nusselt number and the local Sherwood number increase (decrease) due to the effect of suction (blowing). Increasing the buoyancy ratio N increases the local Nusselt number and the local Sherwood number. It is apparent that the Lewis number has a pronounced effect on the local Sherwood number than it does on the local Nusselt number. Furthermore, increasing the Lewis number decreases (increases) the local heat (mass) transfer rate.