Natural convection over a vertical flat plate due to absorption of thermal radiation

Heat transfer by simultaneous free convection and radiation in a participating fluid has received some attention during the past few years. However most of the previous work has been focussed on gases. The present work investigates the problem of combined radiation and natural convection in liquids. Analysis are given for an optically thick cold fluid layer adjacent to a non-emitting and non-reflecting radiation-transmitting plate. The external surface of the plate is subjected to heat loss to surroundings. The governing differential equations are transformed to a dimensionless form where the solution becomes dependent on the following parameters: the plate absorpitivity,α _p; the dimensionless distance along the plate,ζ; the fluid Prandtl number,Pr; and dimensionless heat loss coefficient to surrounding,N _c. A local non-similar technique is adopted to obtain solutions atPr=6.5 and at a wide range ofα _p,ζ, andN _c. The results showed that both velocity and temperature are non-similar and they are greatly affected by the value ofα _p whenζ is small. At large values of f the effect ofα _p diminishes and for a plate without heat loss the velocity becomes similar, i.e. independent of C The heat loss from the external surface of the plate causes the maximum temperature of the fluid to depart far from the plate. The results also showed that for plates without heat loss the local heat transfer coefficient from the plate depends on the local Grashof number to the power 0.185.     

Multiple-zone sliding contact with friction on an anisotropic thermoelastic half-space

The paper studies a class of multiple-zone sliding contact problems. This class is general enough to include frictional and thermal effects, and anisotropic response of the indented material. In particular, a rigid die (indenter) slides with Coulomb friction and at constant speed over the surface of a deformable and conducting body in the form of a 2D half-space. The body is assumed to behave as a thermoelastic transversely isotropic material. Thermoelasticity of the Green–Lindsay type is assumed to govern. The solution method is based on integral transforms and singular integral equations. First, an exact transform solution for the auxiliary problem of multiple-zone (integer n > 1) surface tractions is obtained. Then, an asymptotic form for this auxiliary problem is extracted. This form can be inverted analytically, and the result applied to sliding contacts with multiple zones. For illustration, detailed calculations are provided for the case of two (n = 2) contact zones. The solution yields the contact zone width and location in terms of sliding speed, friction, die profile, and also the force exerted. Calculations for the hexagonal material zinc illustrate effects of speed, friction and line of action of the die force on relative contact zone size, location of maximal values for the temperature and the compressive stress, and the maximum temperature for a given maximum stress. Finally, from our general results, a single contact zone solution follows as a simple limit.     

Painlevé paradox during oblique impact with friction

In analyses using non-smooth dynamics, oblique impact of rough bodies in an unsymmetrical configuration can result in self-locking or “jam” at the sliding contact if the coefficient of friction is sufficiently large; this has been termed, Painlevé’s paradox. In the range of configurations and coefficients of friction where Painlevé’s paradox occurs, analyses based on rigid body dynamics give results indicating that either there are multiple solutions or the solution is nonexistent. This conundrum has been resolved by considering that the contact has small normal and tangential compliance which is representative of deformability in a local region around the contact point. An analysis using a hybrid model which includes local compliance of the contact region has calculated the time-dependent changes in relative motion of colliding bodies for a range of incident angles of obliquity, tan^?1[?V₁(0)/V₃(0)] where V₁(0)and V₃(0) are the incident tangential and normal relative velocities at the contact point, respectively. The paradox is shown to result from a negative relative acceleration of the contact points during an initial period of sliding – a negative acceleration that is inconsistent with the assumption of rigid-body contact.     

Experimental study of the induced flow birefringence of a suspension of rigid particles at the transition from Couette flow to Taylor vortex flow

The flow birefringence induced in solutions of rigid particles is studied experimentally in the region of the axisymmetrical Taylor vortex flow which arises once the velocity gradient G in the annular gap of a conventional Couette cell reaches a critical value G _c .The measurements are performed for several values of G > G _c and for 10 radial observation points in the annular gap. Solutions of two types of rigid particles are investigated: the first is a suspension of flattened clay particles like bentonite, while the second contains rod-like particles of tobacco mosaic virus (TMV). The variations of the birefringence intensity n and of the extinction angle measured in the domain of the axisymmetrical flow show a different behavior according to the shape of the particle in solution. This fact is confirmed theoretically with a good agreement for the measurements performed with solutions of flat particles.     

Mode III effects on interface delamination

For crack growth along an interface between dissimilar materials the effect of combined modes I, II and III at the crack-tip is investigated. First, in order to highlight situations where crack growth is affected by a mode III contribution, examples of material configurations are discussed where mode III has an effect. Subsequently, the focus is on crack growth along an interface between an elastic-plastic solid and an elastic substrate. The analyses are carried out for conditions of small-scale yielding, with the fracture process at the interface represented by a cohesive zone model. Due to the mismatch of elastic properties across the interface the corresponding elastic solution has an oscillating stress singularity, and this solution is applied as boundary conditions on the outer edge of the region analyzed. For several combinations of modes I, II and III crack growth resistance curves are calculated numerically in order to determine the steady-state fracture toughness. For given values of K_I and K_II the minimum fracture toughness corresponds to K_III=0 in most of the range analyzed, but there is a range where the minimum occurs for a nonzero value of K_III.     

On shear correction factors in the non-linear theory of elastic shells

Theoretical values of two correction factors α_s = 5/6 and α_t = 7/10 are established for the respective transverse shear stress resultants and stress couples within the general, dynamically and kinematically exact, six-field theory of elastic shells. These values do not depend on the shell material symmetry, geometry of the base surface, the shell thickness, or any kind of kinematic and/or dynamic constraints. The analysis is based on the complementary energy density following from the transverse shear stresses acting only on the shell cross section. The appropriate quadratic and cubic distributions of the stresses across the thickness allow one to derive the consistent constitutive equations for the transverse shear stress resultants and stress couples with α_s and α_t as the respective correction factors. Four numerical examples of highly non-linear shell structures illustrate the influence of different values of α_s and α_t on the results. In particular, some influence of α_t is noticed on the placement of bifurcation points. In dynamic problem of flight of three intersecting plates analysed with Newmark-type temporal algorithm, the value of α_t influences the moment at which the relative error of total energy of the system begins to grow indefinitely leading to the solution failure.     

A Property of Equations of Rigid/Plastic Material Obeying a Voce–Type Hardening Law

Using a simple example, the rotation of a rigid cone in rigid/plastic hardening material, the paper shows a qualitative difference between the solutions for two groups of hardening laws. The first group includes hardening laws with no saturation stress. In this case the solution under sticking conditions exists at any rotation angle of the cone up to infinity. The second group includes hardening laws with a saturation stress. For such laws the solution exists up to a finite value of the rotation angle. Once this angle has been reached, the solution breaks down. At the beginning of the process the behavior of the solution for both groups of hardening materials is similar. However, at the final stage the behavior of rigid/plastic hardening materials of the second group is similar to the behavior of rigid perfectly plastic materials. A specific hardening law with a saturation stress is applied to illustrate the general solution and the restrictions imposed by this law, and a priori specified interfacial law (sticking) on existence of the solution. This revised version was published online in July 2006 with corrections to the Cover Date.     

A note on the contact problem in an ultrasonic travelling wave motor

In this paper we discuss the non-linear contact problem between the stator and the rotor of an ultrasonic travelling were motor. For a first simplified mathematical model the problem is formulated for a linear motor in which the stator is modelled as a Bernoulli-Euler beam and the slider (rotor) is assumed to be rigid. A thin layer of visco-elastic material is assumed to exist between stator and slider. Expressions are obtained for the contact pressure between the two parts. The frictional forces both in the sticking and in the sliding zone can then be easily obtained assuming Coulomb friction.     

Size effects in generalised continuum crystal plasticity for two-phase laminates

The solutions of a boundary value problem are explored for various classes of generalised crystal plasticity models including Cosserat, strain gradient and micromorphic crystal plasticity. The considered microstructure consists of a two-phase laminate containing a purely elastic and an elasto-plastic phase undergoing single or double slip. The local distributions of plastic slip, lattice rotation and stresses are derived when the microstructure is subjected to simple shear. The arising size effects are characterised by the overall extra back stress component resulting from the action of higher order stresses, a characteristic length l_c describing the size-dependent domain of material response, and by the corresponding scaling law lⁿ as a function of microstructural length scale, l. Explicit relations for these quantities are derived and compared for the different models. The conditions at the interface between the elastic and elasto-plastic phases are shown to play a major role in the solution. A range of material parameters is shown to exist for which the Cosserat and micromorphic approaches exhibit the same behaviour. The models display in general significantly different asymptotic regimes for small microstructural length scales. Scaling power laws with the exponent continuously ranging from 0 to −2 are obtained depending on the values of the material parameters. The unusual exponent value −2 is obtained for the strain gradient plasticity model, denoted “curl H^p” in this work. These results provide guidelines for the identification of higher order material parameters of crystal plasticity models from experimental data, such as precipitate size effects in precipitate strengthened alloys.     

Fluid forces on a very low Reynolds number airfoil and their prediction

This paper presents the measurements of mean and fluctuating forces on an NACA0012 airfoil over a large range of angle (α) of attack (0-90°) and low to small chord Reynolds numbers (Re_c), 5.3 × 10³-5.1 × 10⁴, which is of both fundamental and practical importance. The forces, measured using a load cell, display good agreement with the estimate from the LDA-measured cross-flow distributions of velocities in the wake based on the momentum conservation. The dependence of the forces on both α and Re_c is determined and discussed in detail. It has been found that the stall of an airfoil, characterized by a drop in the lift force and a jump in the drag force, occurs at Re_c ? 1.05 × 10⁴ but is absent at Re_c = 5.3 × 10³. A theoretical analysis is developed to predict and explain the observed dependence of the mean lift and drag on α.     

Progressive Frictional Delamination of an Infinite Elastic Film on a Rigid Substrate Due to In-Plane Point Loading

The paper presents a solution to a delamination problem of an infinite elastic film resting on a rigid substrate and loaded by a monotonically increasing in-plane point force. A?rigid-slip contact is assumed between the film and the substrate, leading to the development of two regions at the interface: a damaged zone with a relative slip between the materials, and a region where the interface remains intact. Both film natural and essential boundary conditions are zero on the boundary between these two interfacial zones with the shape of the boundary being a part of the solution. Problem??s self-similarity enables us to obtain an approximate distribution of interfacial traction within the delaminated zone and a shape of the zone itself. For film??s Poisson??s ratio ??=?1 the approximate solution becomes exact. It is argued that this can be treated as a special case of a rigid film sliding on a rigid substrate. The presented approach can be used to obtain approximate closed-form solutions to similar delamination problems.     

Similarity solutions in free convection boundary-layer flows adjacent to vertical permeable surfaces in porous media: II prescribed surface heat flux

The equation which governs the similarity solution for free convection boundary-layer flow along a vertical permeable surface with prescribed surface heating and mass transfer rate is discussed. The solution is seen to depend on two non-dimensional parameters;m, the power-law exponent, and γ, the mass transfer parameter. It is shown that solutions exist for allm>?1 for γ>0 (fluid injection) whereas for γ<0 (fluid withdrawal), solution exist form>m ₀(γ), wherem ₀ is determined as a function of γ. Solutions for large mass transfer rates are obtained, for both γ>0 and γ<0. For γ>0 the form of the asymptotic solution for γ large is seen to depend on the value ofm. Solutions form large are derived, these are seen to be different depending on whether γ is positive or negative.     

Phenomenological study of parabolic and spherical indentation of elastic-ideally plastic material

A phenomenological study of parabolic and spherical indentation of elastic ideally plastic materials was carried out by using precise results of finite elements calculations. The study shows that no “pseudo-Hertzian” regime occurs during spherical indentation. As soon as the yield stress of the indented material is exceeded, a deviation from the, purely elastic Hertzian contact behaviour is found. Two elastic–plastic regimes and two plastic regimes are observed for materials of very large Young modulus to Yield stress ratio, E/σ_y. The first elastic–plastic regime corresponds to a strong evolution of the indented plastic zone. The first plastic regime corresponds to the commonly called “fully plastic regime”, in which the average indentation pressure is constant and equal to about three times the yield stress of the indented material. In this regime, the contact depth to penetration depth ratio tends toward a constant value, i.e. h_c/h = 1.47. h_c/h is only constant for very low values of yield strain (σ_y/E lower than 5 × 10^?6) when aE¹/Rσ_y is higher than 10,000. The second plastic regime corresponds to a decrease in the average indentation pressure and to a steeper increase in the pile-up. For materials with very large E/σ_y ratio, the second plastic regime appears when the value of the non-dimensional contact radius a/R is lower than 0.01. In the case of spherical and parabolic indentation, results show that the first plastic regime exists only for elastic-ideally plastic materials having an E/σ_y ratio higher than approximately 2.000.     

Thrust enhancement due to flexible trailing-edge of plunging foils

Drag reduction for hydrofoils is studied through thrust generation on foils plunging at low Strouhal numbers in order to simulate the action of the ocean waves. Force, deformation and flow field measurements are presented for a partially flexible plunging foil in water tunnel experiments. The foil is predominantly rigid with a short flexible trailing-edge plate of length: L=0.1c, 0.2c, or 0.3c. Using flexible plates, whose natural structural frequency is much higher than the frequency of the plunge oscillations, increases thrust compared to the rigid case. Flexibility is generally more effective for larger lengths of the flexible plate and smaller plunge amplitudes. The maximum observed is therefore for the largest length and smallest amplitude studied: L=0.3c and a=0.1c and equates to 28% more thrust than the rigid case. Optima are observed in the non-dimensional rigidity (λ) versus flap angle amplitude (δ, which is a measure of the relative deformation) parameter space. These occur at λ≈2 and δ≈7–13° for a wide range of flexible plate length and plunge amplitude. Whilst a satisfactory explanation of why there is an optimal flap amplitude remains unavailable, the case of optimal flap angle amplitude results in increased trailing-edge vortex circulation, giving a stronger reverse Kármán vortex street and thus a stronger time-averaged jet.     

On the shearing zone around rotating vanes in plastic liquids: theory and experiment

The fracture zone (or shearing surface) around a four-bladed vane rotating in a Bingham liquid has a diameterD_c which is significantly larger than the vane diameter. In typical plastic liquidsD_c/D ≈ 1.00–1.05. We have measuredD_c and the rheology of some automotive greases. We have also photographed the fractural surface in a transparent Bingham liquid and have found it to be approximately cylindrical.Computer simulations of a four-bladed vane rotating in a Bingham liquid have produced a value ofD_c/D = 1.025 which agrees well with experimental data on two viscoelastic automotive greases. We have not been able to obtain agreement between experiment and theory on the dependence ofD_c/D on τ_y/η_p.The experimental and theoretical data presented here support the use of the vane for yield stress measurements if the diameter correction is applied. The present experimental data suggests the vane diameter has little or no effect onD_c/D.     

Self-similar collapse of 2D and 3D vortex filament models

In this article, a very simple toy model for a candidate blow-up solution of the Euler equation by Boratav and Pelz (vortex dodecapole) is investigated. In this model, vortex tubes are replaced with straight vortex filaments of infinitesimal thickness, and the entire motion is monitored by tracing the motion of a representative point on one vortex filament. It is demonstrated that this model permits a self-similar collapse solution which provides the time dependence of the length scale as (t _c ? t)^1/2, (t < t _c), where the collapse time t _c depends on the initial configuration. From the conservation of circulation, this time dependence implies that vorticity ω scales as (t _c ? t) ^?1, which agrees with the one observed in the direct numerical (pseudo spectral) simulations of the vortex dodecapole. Finally, possible modification of the model is considered.     

Magnetic field and magneto elastic stress in an infinite plate containing an elliptical hole with an edge crack under uniform electric current

Two-dimensional solutions of the electric current, magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack under uniform electric current. Using a rational mapping function, the each solution is obtained as a closed form. The linear constitutive equation is used for the magnetic field and the stress analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate which raises a plane stress state for a thin plate and the deformation of the plate thickness. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, electric current, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that the stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H_x, H_y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H_x, H_y is the same as the pure shear stress state. Solving the present magneto elastic stress problem, dislocation and rotation terms appear, which makes the present problem complicate. Solutions of the magneto elastic stress are nonlinear for the direction of electric current. Stresses in the direction of the plate thickness are caused and the solution is also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length and the electric current direction.     

Optimizing strength and toughness of nanofiber-reinforced CMCs

Nanofibers used in current ceramic matrix composites (CMCs) are usually wavy and of finite length. Here, a shear-lag model for predicting the tensile strength and work of fracture of a composite containing a “single matrix crack”, as a function of all the material parameters including constant plus Coulomb interfacial friction, is presented for a CMC containing wavy, finite-length nanofibers having a statistical distribution of strengths. Literature results are recovered for straight infinite fibers, with strength and toughness depending on a characteristic strength σ_c and a characteristic length δ_c. For nanofibers of finite length L, radius of curvature R, and with Coulomb friction coefficient μ, the strength and toughness are found to depend only on σ_c, δ_c, and two new dimensionless parameters μδ_c/R and L/δ_c. Parametric results for a typical carbon nanotube CMC show an optimal region of morphology (L and R) that maximizes composite strength and toughness, exceeding the properties of composites with straight (R=∞) and/or long (L=∞) reinforcements. Therefore, while factors such as the nanofiber strength distribution and the nanofiber–matrix interface sliding resistance may not be easily controlled, the tuning, via processing, of purely geometrical properties of the nanofibers (L and R) alone can aid in maximizing composite properties. These results have important application in hybrid CMCs such as “fuzzy fiber” CMCs, where micron-scale fibers are covered with a forest of nanofibers such that the nanofiber array can bridge longitudinal cracks and thus improve delamination resistance.     

Instabilities in Dynamic Anti-plane Sliding of an Elastic Layer on a Dissimilar Elastic Half-Space

The stability of dynamic anti-plane sliding at an interface between an elastic layer and an elastic half-space with dissimilar elastic properties is studied. Friction at the interface is assumed to follow a rate- and state-dependent law, with a positive instantaneous dependence on slip velocity and a rate weakening behavior in the steady state. The perturbations at the interface are of the form exp(ikx ₁+pt), where k is the wavenumber, x ₁ is the coordinate along the interface, p is the time response to the perturbation and t is time. A key feature of the problem is that interfacial waves both in freely slipping contact as well as in bonded contact exist for the problem. Attention is focused on the role of the interfacial waves on slip stability. Instabilities are plotted in the $\operatorname{Re} (pL/V_{o})$ versus $\operatorname{Im} (p/|k|c_{s})$ plane, where L is a length scale in the friction law, V _o is the unperturbed slip velocity and c _s is the shear wave speed of the layer. Stability of both rapid and slow slip is studied. The results show one mechanism by which instabilities occur is the destabilization by friction of the interfacial waves in freely slipping contact/bonded contact. This occurs even in slow sliding, thus confirming that the quasi-static approximation is not valid for slow sliding. The effect of material properties and layer thickness on the stability results is studied.