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.     

Large-scale edge waves generated by a moving atmospheric pressure

Long waves generated by a moving atmospheric pressure distribution, associated with a storm, in coastal region are investigated numerically. For simplicity the moving atmospheric pressure is assumed to be moving only in the alongshore direction and the beach slope is assumed to be a constant in the on-offshore direction. By solving the linear shallow water equations we obtain numerical solutions for a wide range of physical parameters, including storm size (2a), storm speed (U), and beach slope (α). Based on the numerical results, it is determined that edge wave packets are generated if the storm speed is equal to or greater than the critical velocity, U_cr, which is defined as the phase speed of the fundamental edge wave mode whose wavelength is scaled by the width of the storm size. The length and the location of the positively moving edge wave packet is roughly Ut/2 ≤ y ≤ Ut, where y is in the alongshore direction and t is the time. Once the edge wave packet is generated, the wavelength is the same as that of the fundamental edge wave mode corresponding to the storm speed and is independent of the storm size, which can, however, affect the wave amplitude. When the storm speed is less than the critical velocity, the primary surface signature is a depression directly correlated to the atmospheric pressure distribution.     

Moving boundary in a non-Newtonian fluid

Moving boundary value problem in non-Newtonian fluid is considered. Exact analytical solution for the flow of second-grade fluid for a rigid moving plate oscillating in its own plane, is obtained. The Doppler effect has been observed due to the motion of the plate. The shearing stress on the plate is also calculated. It is concluded that the solutions for stationary porous boundaries can be obtained from the solutions of moving rigid boundaries.     

Thermal waves induced by a rapidly moving line source

For the cases involving a fast moving heat source or extremely short pulses emitted by lasers or short time after the start of transients, the classical theory of heat conduction breaks down since the wave nature of heat transport dominates. In this study, the temperature field due to a fast moving line source was determined analytically using the wave concept. The results are given for different values of thermal Mach number (M=V/C). For M>1 the heat affected zone is confined in a wedge shape region behind the source. The wedge half angle is equal to sin^?1 (1/M). It was confirmed that the difference between the results of diffusion and wave models depends on the corresponding time scale and the relaxation time.     

The unsteady heat transfer due to a heat source in an MHD stretching sheet flow

The time evolution in the temperature field resulting from the sudden introduction of a heat source into the already fully established steady MHD flow of an electrically conducting fluid past a linearly stretching isothermal surface is considered. The problem is shown to be fully described by two dimensionless parameters, a modified magnetic field strength ?? and a heat source strength Q. Numerical solutions of the initial-value problem show that there is a critical value Q _c of the parameter Q, dependent on ??, such that, for Q<Q _c , the solution approaches a steady state at large times and, for Q>Q _c , the solutions grows exponentially large as time increases. This growth rate is determined through an eigenvalue problem which also determines the critical value Q _c . The limits of Q _c for both small and large values of ?? are discussed.     

Dynamic fracture of a kane-mindlin plate

We study dynamic crack problems for an elastic plate by using Kane-Mindlin's kinematic assumptions. The general solutions of the Laplace transformed displacements and stresses are first derived. Path independent integrals for stationary cracks subjected to transient loads and steadily growing cracks are deduced. For a stationary crack in a very thin plate subjected to impact loads, the crack tip dynamic stress intensity factor (DSIF), K₁(t), is related to the far field plane stress one, K₁⁰(t), by where ν is Poisson's ratio. For a crack steadily growing with speed V, the crack tip DSIF, K₁(V), is given by where K₁⁰(V) is the plane stress DSIF and A(V) and B(V) are known functions of V. These results are applied to compute the DSIF for a semi-infinite stationary crack in an unbounded plate subjected to impact pressure on the crack faces. The results of DSIF for a finite crack in an infinite plate under uniform impact pressure on the crack surfaces show that for each plate thickness, the maximum DSIF is higher than that for the plane stress case.     

On Temporal Stability Analysis for Hydromagnetic Flow in a Channel Filled with a Saturated Porous Medium

In this paper, a linear stability analysis is presented to trace the time evolution of an infinitesimal, two-dimensional disturbance imposed on the base flow of an electrically conducting fluid in a channel filled with a saturated porous medium under the influence of a transversely imposed magnetic field. An eigenvalue problem is obtained and solved numerically using the Chebyshev collocation spectral method. The critical Reynolds number Re _c, the critical wave number α _c and the critical wave speed c _c are obtained for a wide range of the porous medium shape factor parameter S and Hartmann number H. It is found that an increase in the magnetic field intensity and a decrease in porous medium permeability have a stabilizing effect on the fluid flow.     

Dissipative interface waves and the transient response of a three-dimensional sliding interface with Coulomb friction

We investigate the linearized response of two elastic half-spaces sliding past one another with constant Coulomb friction to small three-dimensional perturbations. Starting with the assumption that friction always opposes slip velocity, we derive a set of linearized boundary conditions relating perturbations of shear traction to slip velocity. Friction introduces an effective viscosity transverse to the direction of the original sliding, but offers no additional resistance to slip aligned with the original sliding direction. The amplitude of transverse slip depends on a nondimensional parameter η=c_sτ₀/μv₀, where τ₀ is the initial shear stress, 2v₀ is the initial slip velocity, μ is the shear modulus, and c_s is the shear wave speed. As η→0, the transverse shear traction becomes negligible, and we find an azimuthally symmetric Rayleigh wave trapped along the interface. As η→∞, the inplane and antiplane wavesystems frictionally couple into an interface wave with a velocity that is directionally dependent, increasing from the Rayleigh speed in the direction of initial sliding up to the shear wave speed in the transverse direction. Except in these frictional limits and the specialization to two-dimensional inplane geometry, the interface waves are dissipative. In addition to forward and backward propagating interface waves, we find that for η>1, a third solution to the dispersion relation appears, corresponding to a damped standing wave mode. For large-amplitude perturbations, the interface becomes isotropically dissipative. The behavior resembles the frictionless response in the extremely strong perturbation limit, except that the waves are damped. We extend the linearized analysis by presenting analytical solutions for the transient response of the medium to both line and point sources on the interface. The resulting self-similar slip pulses consist of the interface waves and head waves, and help explain the transmission of forces across fracture surfaces. Furthermore, we suggest that the η→∞ limit describes the sliding interface behind the crack edge for shear fracture problems in which the absolute level of sliding friction is much larger than any interfacial stress changes.     

Interaction of thermal contact resistance and frictional heating in thermoelastic instability

Thermoelastic contact problems can posess non-unique and/or unstable steady-state solutions if there is frictional heating or if there is a pressure-dependent thermal contact resistance at the interface. These two effects have been extensively studied in isolation, but their possible interaction has never been investigated. In this paper, we consider an idealized problem in which a thermoelastic rod slides against a rigid plane with both frictional heating and a contact resistance. For sufficiently low sliding speeds, the results are qualitatively similar to those with no sliding. In particular, there is always an odd number of steady-state solutions; if the steady-state is unique it is stable and if it is non-unique, stable and unstable solutions alternate, with the outlying solutions being stable. However, we identify a sliding speed V₀ above which the number of steady states is always even (including zero, implying possible non-existence of a steady-state) and again stable and unstable states alternate. A parallel numerical study shows that for V>V₀ there are some initial conditions from which the contact pressure grows without limit in time, whereas for V<V₀ the system will always tend to one of the stable steady states.     

Non-linear modulation of SH waves in a two-layered plate and formation of surface SH waves

Non-linear modulation of shear horizontal (SH) waves in a two-layered elastic plate of uniform thickness is considered. Both layers are assumed to be homogeneous, isotropic and incompressible elastic and having different mechanical properties. The problem is investigated by a perturbation method and in the analysis it is assumed that between the linear shear velocities of the top layer, c₁, and the bottom layer, c₂, the inequality c₁<c₂ is valid. In the layered structure then an SH wave exists if the wave velocity c of the wave satisfies either the condition c₁<c?c₂ or the one c₁<c₂?c. Here the problem is examined under the former condition and it is shown that the non-linear modulation of SH waves is governed by a non-linear Schrödinger equation. In this case the formation of surface SH (Love) waves is also revealed if the top layer is thinner when compared with the bottom layer. Then the stability condition is discussed and the existence of bright (envelope) and dark solitons are manifested.     

Structure and evolution of the airflow generated by a slender body entry near a tube

When a slender body moving forward in open air enters into a confined region, two important unsteady aerodynamic phenomena are generated. An exiting flow is created with a direction opposite to the body movement and inside the confined region, a compression wave is formed. Generation mechanism of compression wave have been extensively studied but so far, no detailed investigation of the exiting flow has ever been reported. The experimental study presented in this paper was undertaken to gain insight into the structure and the evolution of the exit-flow. Experiments were conducted with an axisymmetric apparatus and the explored range of the moving body speed was 5–50 m/s. The study focused on the influence of the body speed and the body nose geometry on the flow. It was shown that the air ejected from the tube entrance generates an annulus jet accompanied by a vortex ring. The vortex development was clarified using laser sheet visualizations associated with unsteady pressure and velocity measurements at the tube entrance. It is constituted by four phases, the pre-vortex phase, the vortex development phase, the vortex convection phase and the vortex breakdown phase. The duration of each of these steps was found to be independent of both the studied parameters in a non-dimensional time scale. Furthermore, neither the body speed nor the nose geometry induced significant changes on the vortex ring evolution, except for extreme conditions (low body speed, V_M.B.<15 m/s, and/or very long nose geometry, L_nose/D_M.B.>6). The evolution of the vortex ring was compared to that of ‘classical’ vortex ring generated at a tube exit by a piston motion with large non-dimensional stroke length. Main similarities and differences were discussed in the paper. In particular, the formation number of vortex ring observed in our experiments was found to be significantly smaller.     

The stability of the modified plane Poiseuille flow in the presence of a transverse magnetic field

The stability against small disturbances of the pressure-driven plane laminar motion of an electrically conducting fluid under a transverse magnetic field is investigated. Assuming that the outer regions adjacent to the fluid layer are electrically non-conducting and not ferromagnetic, the appropriate boundary conditions on the magnetic field perturbations are presented. The Chebyshev collocation method is adopted to obtain the eigenvalue equation, which is then solved numerically. The critical Reynolds number R_c, the critical wave number α_c, and the critical wave speed c_c are obtained for wide ranges of the magnetic Prandtl number P_m and the Hartmann number M. It is found that except for the case when P_m is sufficiently small, the magnetic field has both stabilizing and destabilizing effects on the fluid flow, and that for a fixed value of M the fluid flow becomes more unstable as P_m increases.     

Constant moving crack in a magnetoelectroelastic material under anti-plane shear loading

Analytical solutions for an anti-plane Griffith moving crack inside an infinite magnetoelectroelastic medium under the conditions of permeable crack faces are formulated using integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic material. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities. The moving speed of the crack have influence on the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF), while the dynamic stress intensity factor (DSIF) does not depend on the velocity of the moving crack. When the crack is moving at very lower or very higher speeds, the crack will propagate along its original plane; while in the range of M_c1 < M < M_c2, the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.     

Generation of Transient Waves by Impulsive Disturbances in an Inviscid Fluid with an Ice-Cover

The dynamic responses of an ice-covered fluid to impulsive disturbances are analytically investigated for two- and three-dimensional cases. The initially quiescent fluid of infinite depth is assumed to be inviscid, incompressible and homogenous. The thin ice-cover is modelled as a homogenous elastic plate with negligible inertia. Four types of impulsive concentrated disturbances are considered, namely an instantaneous mass source immersed in the fluid, an instantaneously dynamic load on the plate, an initial impulse on the surface of the fluid, and an initial displacement of the ice plate. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The solutions in integral form for the vertical deflexions at the ice-water interface are obtained by means of a joint Laplace-Fourier transform. The asymptotic representations of the wave motions for large time with a fixed distance-to-time ratio are derived by making use of the method of stationary phase. It is found that there exists a minimal group velocity and the wave system observed depends on the moving speed of the observer. For an observer moving with the speed larger than the minimal group velocity, there exist two trains of waves, namely the long gravity waves and the short flexural waves, the latter riding on the former. Moreover, the deflexions of the ice-plate for an observer moving with a speed near the minimal group velocity are expressed in terms of the Airy functions. The effects of the presence of an ice-cover on the resultant wave amplitudes, the wavelengths and periods are discussed in detail. The explicit expressions for the free-surface gravity waves can readily be recovered by the present results as the thickness of ice-plate tends to zero.     

On the Chebyshev collocation spectral approach to stability of fluid flow in a porous medium

In this paper, the temporal development of small disturbances in a pressure‐driven fluid flow through a channel filled with a saturated porous medium is investigated. The Brinkman flow model is employed in order to obtain the basic flow velocity distribution. Under normal mode assumption, the linearized governing equations for disturbances yield a fourth‐order eigenvalue problem, which reduces to the well‐known Orr–Sommerfeld equation in some limiting cases solved numerically by a spectral collocation technique with expansions in Chebyshev polynomials. The critical Reynolds number Re_c, the critical wave number α_c, and the critical wave speed c_c are obtained for a wide range of the porous medium shape factor parameter S. It is found that a decrease in porous medium permeability has a stabilizing effect on the fluid flow. Copyright © 2008 John Wiley & Sons, Ltd.     

Wave propagation in an inhomogeneous transversely isotropic material obeying the generalized power law model

The analytical solutions for body-wave velocity in a continuously inhomogeneous transversely isotropic material, in which Young’s moduli (E, E′), shear modulus (G′), and material density (ρ) change according to the generalized power law model, (a+b z) ^c , are set down. The remaining elastic constants of transversely isotropic media, ν, and ν′ are assumed to be constants throughout the depth. The planes of transversely isotropy are selected to be parallel to the horizontal surface. The generalized Hooke’s law, strain-displacement relationships, and equilibrium equations are integrated to constitute the governing equations. In these equations, utilizing the displacement components as fundamental variables, the solutions of three quasi-wave velocities (V _SV , V _P ,?V _SH ) are generated for the present inhomogeneous transversely isotropic materials. The proposed solutions are compared with those of Daley and Hron (Bull Seismol Soc Am 67:661–675, (1977)), and Levin (Geophysics 44:918–936, (1979)) when the inhomogeneity parameter c?=?0. The agreement between the present results and previously published ones is excellent. In addition, the parametric study results reveal that the magnitudes of wave velocity are remarkably affected by (1) the inhomogeneity parameters (a, b, c); (2) the type and degree of material anisotropy (E/E′, ν/ν′, G/G′); (3) the phase angle (θ); and (4) the depth of the medium (z). Consequently, it is imperative to consider the effects of inhomogeneity when investigating wave propagation in transversely isotropic media.     

Limiting velocity of crack propagation in dynamically fractured materials

A model of a fractal crack is considered. It is found that the limiting velocity of crack propagation is determined by the fractal dimension of the crack contour. It is shown that for commercial steels, the limiting crack velocity is in the range V _lim = (0.155–0.537)c ₁ (c ₁ is the speed of sound).     

Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I–V Relations and Critical Potentials. Part II: Numerics

We consider a one-dimensional steady-state Poisson–Nernst–Planck type model for ionic flow through membrane channels. Improving the classical Poisson–Nernst–Planck models where ion species are treated as point charges, this model includes ionic interaction due to finite sizes of ion species modeled by hard sphere potential from the Density Functional Theory. The resulting problem is a singularly perturbed boundary value problem of an integro-differential system. We examine the problem and investigate the ion size effect on the current–voltage (I–V) relations numerically, focusing on the case where two oppositely charged ion species are involved and only the hard sphere components of the excess chemical potentials are included. Two numerical tasks are conducted. The first one is a numerical approach of solving the boundary value problem and obtaining I–V curves. This is accomplished through a numerical implementation of the analytical strategy introduced by Ji and Liu in [Poisson–Nernst–Planck systems for ion flow with density functional theory for hard-sphere potential: I–V relations and critical potentials. Part I: Analysis, J. Dyn. Differ. Equ. (to appear)]. The second task is to numerically detect two critical potential values V _c and V ^c .The existence of these two critical values is first realized for a relatively simple setting and analytical approximations of V _c and V ^c are obtained in the above mentioned reference. We propose an algorithm for numerical detection of V _c and V ^c without using any analytical formulas but based on the defining properties and numerical I–V curves directly. For the setting in the above mentioned reference, our numerical values for V _c and V ^c agree well with the analytical predictions. For a setting including a nonzero permanent charge in which case no analytic formula for the I–V relation is available now, our algorithms can still be applied to find V _c and V ^c numerically.     

On two distinct types of drag-reducing fluids,diameter scaling,and turbulent profiles

Two distinct scaling procedures were found to predict the diameter effect for different types of drag-reducing fluids. The first one, which correlates the relative drag reduction (DR) with flow bulk velocity (V), appears applicable to fluids that comply with the 3-layers velocity profile model. This model has been applied to many polymer solutions; but the drag reduction versus V scaling procedure was successfully tested here for some surfactant solutions as well. This feature, together with our temperature profile measurements, suggest that these surfactant solutions may also show this type of 3-layers velocity profiles (3L-type fluids).The second scaling procedure is based on a correlation of τ_w versus V, which is found to be applicable to some surfactant solutions but appears to be applicable to some polymer solutions as well. The distinction between the two procedures is therefore not simply one between polymer and surfactants. It was also seen that the τ_w versus V correlation applies to fluids which show a stronger diameter effect than those scaling with the other procedure. Moreover, for fluids that scale according to the τ_w versus V procedure, the drag-reducing effects extend throughout the whole pipe cross section even at conditions close to the onset of drag reduction, in contrast to the behavior of 3L fluids. This was shown by our measurements of temperature profiles which exhibit a fan-type pattern for the τ_w versus V fluids (F-type), unlike the 3-layers profile for the fluids well correlated by drag reduction versus V. Finally, mechanically-degraded polymer solutions appeared to behave in a manner intermediate between the 3L and F fluids.Furthermore, we also showed that a given fluid in a given pipe may transition from a Type A drag reduction at low Reynolds number to a Type B at high Reynolds number, the two types apparently being more representative of different levels of fluid/flow interactions than of fundamentally different phenomena of drag reduction. After transition to the non-asymptotic Type B regime, our results suggest that, without degradation, the friction becomes independent of pipe diameter and that the drag reduction level becomes also approximately independent of the Reynolds number, in a strong analogy to Newtonian flow.     

Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I–V Relations and Critical Potentials. Part I: Analysis

In this work, we analyze a one-dimensional steady-state Poisson–Nernst–Planck type model for ionic flow through a membrane channel including ionic interactions modeled from the Density Functional Theory in a simple setting: Two oppositely charged ion species are involved with electroneutrality boundary conditions and with zero permanent charge, and only the hard sphere component of the excess (beyond the ideal) electrochemical potential is included. The model can be viewed as a singularly perturbed integro-differential system with a parameter resulting from a dimensionless scaling of the problem as the singular parameter. Our analysis is a combination of geometric singular perturbation theory and functional analysis. The existence of a solution of the model problem for small ion sizes is established and, treating the sizes as small parameters, we also derive an approximation of the I–V (current–voltage) relation. For this relatively simple situation, it is found that the ion size effect on the I–V relation can go either way—enhance or reduce the current. More precisely, there is a critical potential value V _c so that, if V > V _c , then the ion size enhances the current; if V < V _c , it reduces the current. There is another critical potential value V ^c so that, if V > V ^c , the current is increasing with respect to λ =? r ₂/r ₁ where r ₁ and r ₂ are, respectively, the radii of the positively and negatively charged ions; if V < V ^c , the current is decreasing in λ. To our knowledge, the existence of these two critical values for the potential was not previously identified.