Unsteady mixed-convection boundary layer flow along a symmetric wedge with variable surface temperature

Laminar two-dimensional unsteady mixed-convection boundary-layer flow of a viscous incompressible fluid past a sharp wedge has been studied. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations is reduced to local non-similarity boundary layer equations, which are solved analytically for small time. Perturbation solutions are also obtained for small and large dimensionless time, τ. Solutions of the governing equations for all time are obtained employing the implicit finite difference method. Here we have focused our attention on the evolution of skin-friction coefficient (C_f) and local Nusselt number (Nu) (heat transfer rate), fluid velocity and fluid temperature with the effects of different governing parameters such as different time, τ, the exponent, m (=0.2, 0.4, 0.6, 0.8, 1.0), mixed convection parameter, λ (= 0.0, 0.5, 1.0) for fluids having Prandtl number, Pr = 0.1, 0.7, and 7.0.     

Moving wedge and flat plate in a micropolar fluid

Similarity solutions for a moving wedge and flat plate in a micropolar fluid may be obtained when the fluid and boundary velocities are proportional to the same power-law of the downstream coordinate. The governing partial differential equations are transformed to the ordinary differential equations using similarity variables, and then solve numerically using a finite-difference scheme known as the Keller-box method. Numerical results are given for the dimensionless velocity and microrotation profiles, as well as the skin friction coefficient for several values of the Falkner–Skan power-law parameter (m), the ratio of the boundary velocity to the free stream velocity parameter (λ) and the material parameter (K). Important features of these flow characteristics are plotted and discussed. It is found that multiple solutions exist when the boundary is moving in the opposite direction to the free stream, and the micropolar fluids display a drag reduction compared to Newtonian fluids.     

Dominant stress region for crack initiation at interface edge of microdot on a substrate

In order to examine the mechanics of crack initiation at the free interface edge of a microcomponent on a substrate, delamination tests are carried out for two specimen shapes of Cr microdots on a SiO₂ substrate. The microdots of the first specimen are shaped like the frustum of a round cone. The Cr microdots are successfully delaminated from the SiO₂ substrate in a brittle manner and the critical load is measured by atomic force microscopy (AFM) with a lateral loading apparatus. Stress analysis reveals that a singular stress field exists near the interface edge and the strength for the crack initiation is governed by the intensified normal stress field. The critical stress intensity parameter is evaluated as K_σC ≈ 0.24 MPa m^0.39. Similar delamination tests are conducted for microdots shaped like the frustum of an oval cone. The stress distributions at the crack initiation of this specimen shape show a higher normal stress than the first specimen shape in the region near the interface edge of about x < 40 nm, while it is lower in the region of about x > 50 nm (x: distance from the edge). This suggests a limitation of conventional fracture mechanics: namely, the crack initiation in these specimens is not uniquely governed by the intensity of the singular field. It is found that the delamination crack is initiated when the averaged stress σ_ya in the region of 90-130 nm reaches 190-270 MPa, regardless of the specimen shape. This indicates that the dominant stress region of crack initiation is roughly estimated as 90-130 nm and the criterion is given in terms of the averaged stress in the region.     

Non-orthogonal stagnation-point flow of a micropolar fluid

This paper considers the problem of steady two-dimensional flow of a micropolar fluid impinging obliquely on a flat plate. The flow under consideration is a generalization of the classical modified Hiemenz flow for a micropolar fluid which occurs in the boundary layer near an orthogonal stagnation point. A coordinate decomposition transforms the full governing equations into a primary equation describing the modified Hiemenz flow for a micropolar fluid and an equation for the tangential flow coupled to the primary solution. The solution to the boundary-value problem is governed by two non-dimensional parameters: the material parameter K and the ratio of the microrotation to skin friction parameter n. The obtained ordinary differential equations are solved numerically for some values of the governing parameters. The primary consequence of the free stream obliqueness is the shift of the stagnation point toward the incoming flow.     

Transfer matrix solutions for three-dimensional consolidation of a multi-layered soil with compressible constituents

In this paper, transfer matrix solutions for three-dimensional consolidation of a multi-layered soil considering the compressibility of pore fluid are presented. The derivation of the solutions starts with the fundamental differential equations of Biot’s three-dimensional consolidation theory, takes into account the compressibility of pore fluid in the Cartesian coordinate system, and introduces the extended displacement functions. The relationship of displacements, stresses, excess pore water pressure, and flux between the ground surface (z = 0) and an arbitrary depth z is established for Biot’s three-dimensional consolidation problem of a finite soil layer with compressible pore fluid by taking the Laplace transform with respect to t and the double Fourier transform with respect to x and y, respectively. Based on this relationship of the transfer matrix, the continuity between layers, and the boundary conditions, the solutions for Biot’s three-dimensional consolidation problem of a multi-layered soil with compressible constituents in a Laplace-Fourier transform domain is obtained. The final solutions in the physical domain are obtained by inverting the Laplace-Fourier transforms. Numerical analysis is carried out by using a corresponding program based on the solutions developed in this study. This analysis demonstrates that the compressibility of pore fluid has a remarkable effect on the process of consolidation.     

Flow and heat transfer of a micropolar fluid past a continuously moving porous plate

The boundary layer flow and heat transfer of a micropolar fluid past a semi-infinite porous plate moving continuously is studied. Similarity solutions for the velocity, the temperature, and the microrotation equations are derived. These are shown graphically. The numerical values of the skin-friction coefficient C_f and the rate of heat transfer [?θ'(0)] are entered in tables for different values of suction and injection parameters. The effects of suction and injection, K (the coupling parameter) and G (the microrotation parameter) are discussed.     

Low-frequency scattering from perfectly conducting spheroidal bodies in a conductive medium with magnetic dipole excitation

Inductive electromagnetic means that are currently employed in the exploration of the Earth’s subsurface and embedded voluminous bodies often call for an intensive use, primary at the modeling stage and later on at the inversion stage, of analytically demanding tools of field calculation. Under the aim of modeling implementation, this contribution is concerned with some interesting aspects of the low-frequency interaction of arbitrarily orientated (i.e. three-dimensional) time-harmonic magnetic dipoles, with 3-D perfectly conducting spheroidal bodies embedded in an otherwise homogeneous conductive medium. For many practical applications involving buried obstacles such as Earth’s subsurface electromagnetic probing at low-frequency or any other physical cases (e.g. geoelectromagnetics), non-axisymmetric spheroidal geometry approximates sufficiently such kind of metallic shapes. On the other hand, our analytical approach deals with prolate spheroids, since the corresponding results for the oblate spheroidal geometry can be readily obtained through a simple transformation. The particular physical model concerns a solid impenetrable (metallic) body under a magnetic dipole excitation, where the scattering boundary value problem is attacked via rigorous low-frequency expansions for the incident, scattered and total electric and magnetic fields in terms of positive integral powers of (ik), that is (ik)ⁿ for n ? 0, where k stands for the complex wavenumber of the exterior medium. The purpose of the modeling is to contribute to a simple yet versatile tool to infer information on an unknown body from measurements of the three-component electric and magnetic fields nearby. Our goal is to obtain the most important terms of the low-frequency expansions of the electromagnetic fields, that is the static (for n = 0) and the dynamic (n = 1, 2, 3) terms. In particular, for n = 1 there are no incident fields and thus no scattered ones, while for n = 0 the Rayleigh electromagnetic expression is easily obtained in terms of infinite series. Emphasis is given on the calculation of the next two non-trivial terms (at n = 2 and at n = 3) of the aforementioned fields. Consequently, those are found in closed form from exact solutions of coupled (at n = 2, to the one at n = 0) or uncoupled (at n = 3) Laplace equations and they are given in compact fashion, as infinite series expansions for n = 2 or finite forms for n = 3. Nevertheless, the difficulty of the Poisson’s equation that has to be solved for n = 2 is presented, whereas our analytical approach demands the use of the well-known cut-off method in order to obtain an analytical closed solution. Finally, this research adds useful reference results to the already ample library of scattering by simple shapes using analytical methods.     

Titanium diffusion in gold thin films

In the present study, diffusion phenomena in titanium/gold (Ti/Au) thin films occurring at temperatures ranging between 200 and 400 °C are investigated.The motivation is twofold: the first objective is to characterize Ti diffusion into Au layer as an effect of different heat-treatments. The second goal is to prove that the implementation of a thin titanium nitride (TiN) layer between Ti and Au can remarkably reduce Ti diffusion.It is observed that Ti atoms can fully diffuse through polycrystalline Au thin films (260 nm thick) already at temperatures as a low as 250 °C. Starting from secondary ion mass spectroscopy data, the overall diffusion activation energy ΔE = 0.66 eV and the corresponding pre-exponential factor D₀ = 5 × 10^− 11 cm²/s are determined. As for the grain boundary diffusivity, both the activation energy range 0.54 < ΔE_gb < 0.66 eV and the pre-exponential factor s₀D_gb0 = 1.14 × 10^− 8 cm²/s are obtained. Finally, it is observed that the insertion of a thin TiN layer (40 nm) between gold and titanium acts as an effective diffusion barrier up to 400 °C.     

Longitudinal oscillations of a sphere in a micropolar fluid

Summary A general expression for the force exerted on a sphere excuting longitudinal oscillations, with small amplitude, in an incompressible micropolar fluid is obtained. This is accomplished by using direct integral consequences of the full field and the constitutive equations written in cartesian coordinates. The results which are independent of any boundary conditions are then applied to calculate the hydrodynamic force experienced by a sphere moving with rectilinear oscillating velocityu(t)=(u _oe^it, 0, 0) in an unbounded micropolar fluid. As a special case, a general expression for the drag in a similar viscous flow is also derived.     

Boundary layer equations and stretching sheet solutions for the modified second grade fluid

A modified second grade non-Newtonian fluid model is considered. The model is a combination of power-law and second grade fluids in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows. The boundary layer equations are derived from the equations. Symmetries of the boundary layer equations are calculated using Lie Group theory. For a special power law index of m = −1, the principal Lie algebra extends. Using one of the symmetries, the partial differential system is transferred to an ordinary differential system. The ordinary differential equations are numerically integrated for the stretching sheet boundary conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions. The shear stress on the boundary is also calculated.     

Peristaltic pumping of a micropolar fluid in a tube

Summary.  Peristaltic transport of a micropolar fluid in a circular tube is studied under low Reynolds number and long wavelength approximations. The closed form solutions are obtained for velocity, microrotation components, as well as the stream function and they contain new additional parameters namely, N the coupling number and m the micropolar parameter. In the case of free pumping (pressure difference Δp=0) the difference in pumping flux is observed to be very small for Newtonian and micropolar fluids but in the case of pumping (Δp>0) the characteristics are significantly altered for different N and m. It is observed that the peristalsis in micropolar fluids works as a pump against a greater pressure rise compared with a Newtonian fluid. Streamline patterns which depict trapping phenomena are presented for different parameter ranges. The limit on the trapping of the center streamline is obtained. The effects of N and m on friction force for different Δp are discussed. Received June 20, 2002; revised October 23, 2002 Published online: April 17, 2003 The authors thank the referees for pointing out some mistakes in the governing equations and for the suggestions to improve the presentation of the paper.     

Starting solutions for some unsteady unidirectional flows of a second grade fluid

Exact solutions corresponding to the motions of a second grade fluid, due to the cosine and sine oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient are determined by means of the Fourier sine transforms. These solutions, presented as sum of the steady-state and transient solutions, satisfy both the governing equations and all associate initial and boundary conditions. In the special case when α₁ → 0, they reduce to those for a Navier-Stokes fluid.     

Effects of viscosities on the transient current in homeotropic nematic liquid crystal cells

Effects of the Leslie viscosity coefficients α_i (i = 1, 2, …, 5) of nematic liquid crystals (NLCs) with negative dielectric anisotropy (Δε < 0) on the electric-field-induced director reorientation in homeotropic NLC cells have been studied from the analysis of the transient current induced by step voltage application. The transient current in a homeotropic NLC cell with Δε < 0 was well reproduced by computer simulation, based on the theory of NLCs in which the flow effects and the free-slip boundary condition are taken into account. It is found that the response time of vertical alignment NLC displays is dominantly governed by α₂ and α₄ + α₅ of NLCs with Δε < 0.     

Dependence of Seebeck coefficient on a load resistance and energy conversion efficiency in a thermoelectric composite

The thermo-emf ΔV and current ΔI generated by imposing the alternating temperature gradients (ATG) at a period of T and the steady temperature gradient (STG) on a thermoelectric (TE) composite were measured as a function of t, where t is the lapsed time and T was varied from 60 to or ∞ s. The STG and ATG were produced by imposing steadily and alternatively a source voltage V in the range from 1.0 to 4.0 V on two Peltier modules sandwiching a composite. ΔT, ΔV, ΔI and V_P oscillate at a period T and their waveforms vary significantly with a change of T, where ΔV and V_P are the voltage drops in a load resistance R_L and in resistance R_P of two modules. The resultant Seebeck coefficient |α| = |ΔV|/ΔT of a composite under the STG was found to be expressed as |α| = |α₀|(1 − R_comp/R_T), where R_T is the total resistance of a circuit for measuring the output signals and R_comp is the resistance of a composite. The effective generating power ΔW_eff has a local maximum at T = 960 s for the p-type composite and at T = 480 s for the n-type one. The maximum energy conversion efficiency η of the p- and n-type composites under the ATG produced by imposing a voltage of 4.0 V at an optimum period were 0.22 and 0.23% at ΔT_eff = 50 K, respectively, which are 42 and 43% higher than those at ΔT = 42 K under the STG. These maximum η for a TE composite sandwiched between two Peltier modules, were found to be expressed theoretically in terms of R_P, R_T, R_L, α_P and α, where α_P and α are the resultant Seebeck coefficients of Peltier modules and a TE composite.     

Mechanics of layered anisotropic poroelastic media with applications to effective stress for fluid permeability

The mechanics of vertically layered porous media has some similarities to and some differences from the more typical layered analysis for purely elastic media. Assuming welded solid contact at the solid-solid interfaces implies the usual continuity conditions, which are continuity of the vertical (layering direction) stress components and the horizontal strain components. These conditions are valid for both elastic and poroelastic media. Differences arise through the conditions for the pore pressure and the increment of fluid content in the context of fluid-saturated porous media. The two distinct conditions most often considered between any pair of contiguous layers are: (1) an undrained fluid condition at the interface, meaning that the increment of fluid content is zero (i.e., δζ = 0), or (2) fluid pressure continuity at the interface, implying that the change in fluid pressure is zero across the interface (i.e., δp_f = 0). Depending on the types of measurements being made on the system and the pertinent boundary conditions for these measurements, either (or neither) of these two conditions might be directly pertinent. But these conditions are sufficient nevertheless to be used as thought experiments to determine the expected values of all the poroelastic coefficients. For quasi-static mechanical changes over long time periods, we expect drained conditions to hold, so the pressure must then be continuous. For high-frequency wave propagation, the pore-fluid typically acts as if it were undrained (or very nearly so), with vanishing of the fluid increment at the boundaries being appropriate. Poroelastic analysis of both these end-member cases is discussed, and the general equations for a variety of applications to heterogeneous porous media are developed. In particular, effective stress for the fluid permeability of such poroelastic systems is considered; fluid permeabilities characteristic of granular media or tubular pore shapes are treated in some detail, as are permeabilities of some of the simpler types of fractured materials.     

Effects of suction-injection-combination (SIC) on the onset of Rayleigh-Benard magnetoconvection in a micropolar fluid

The effects of suction-injection-combination (SIC) and magnetic field on the linear stability analysis of Rayleigh-Benard convection in a horizontal layer of an Boussinesq micropolar fluid is studied using a Rayleigh-Ritz techinque. The eigenvalues are obtained for free-free, rigid-free and rigid-rigid velocity boundary combinations with isothermal and adiabatic temperature conditions on the spin-vanishing boundaries. The eigenvalues are also obtained for lower rigid isothermal and upper free adiabatic boundaries with vanishing spin. The influence of various micropolar fluid parameters on the onset of convection has been analysed. It is found that the effect of Prandtl number on the stability of the system is dependent on the SIC being pro-gravity or anti-gravity. A similar Pe-sensitivity is found in respect of the critical wave number. It is observed that the micropolar fluid layer heated from below is more stable compared to the classical fluid layer.     

Bi-velocity hydrodynamics: Single-component fluids

Acceptance of the Navier-Stokes-Fourier (NSF) equations as the fundamental equations of single-component continuum fluid mechanics for liquids and gases is noted to be inseparably linked to Euler’s implicit, but unproved, hypothesis that but a single-velocity field is required to characterize the four physically different, context-specific, velocities appearing in the mass, momentum, and energy equations. To test Euler’s hypothesis, velocity is added to the usual list of quantities requiring constitutive formulation - namely the heat flux q and viscous stress T - in order to effect closure of the mass, momentum, and energy equations. Establishment of this enlarged set of constitutive relations is effected by using conventional linear irreversible thermodynamics (LIT) principles governing the behavior of simple fluid continua, importantly including satisfaction of Onsager reciprocity as a fundamental continuum requirement. The resulting analysis shows that, in general, two velocities rather than one are required and, concomitantly, that additional driving forces must be added to each of the standard constitutive equations for the Fourier’s-law heat flux q = −k∇T and the Newton’s-law viscous stress (wherein the “mass velocity” v_m is the context-specific velocity appearing in the continuity equation ∂ρ/∂t + ∇ · (ρv_m) = 0). For the particular case of dilute gaseous continua explicit expressions are established for the phenomenological coefficients appearing in these additional constitutive contributions. Determination of these coefficients is effected using data derived from the Chapman-Enskog-Burnett constitutive expressions for q and T, the latter obtained by solving the Boltzmann equation at small Knudsen numbers, including so-called rarefied-gas contributions. These coefficients are found to be nonzero, confirming the conclusion, inter alia, that two velocities are constitutively required to quantify hydrodynamic behavior for gases and, by inference, for liquids too. Collectively, these velocity, heat flux, and stress constitutive findings collectively negate the current belief that the NSF equations fully describe the physics of viscous fluid continua. Rather, they do so only in limiting cases where the additional constitutive terms than we have found necessary for completeness are asymptotically small.     

Effect of Pb(Fe1/2Nb1/2)O3 modification on dielectric and piezoelectric properties of Pb(Mg1/3Nb2/3)O3-PbZr0.52Ti0.48O3 ceramics

10 mol% Pb(Fe_1/2Nb_1/2)O₃ (PFN) modified Pb(Mg_1/3Nb_2/3)O₃-PbZr_0.52Ti_0.48O₃ (PMN-PZT) relaxor ferroelectric ceramics with compositions of (0.9 − x)PMN-0.1PFN-xPZT (x = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9) were prepared. X-ray diffraction investigations indicated that as-prepared ceramics were of pure perovskite phase and the sample with composition of x = 0.8 was close to morphotropic phase boundary (MPB) between rhombohedral and tetragonal phase. Dielectric properties of the as-prepared ceramics were measured, and the Curie temperature (T_c) increased sharply with increasing PZT content and could be higher than 300 °C around morphotropic phase boundary (MPB) area. At 1 kHz, the sample with composition of x = 0.1 had the largest room temperature dielectric constant ?_r = 3519 and maximum dielectric constant ?_m = 20,475 at T_m, while the sample with composition of x = 0.3 possessed the maximum dielectric relaxor factor of γ = 1.94. The largest d₃₃ = 318 pC/N could be obtained from as-prepared ceramics at x = 0.9. The maximum remnant polarization (P_r = 28.3 μC/cm²) was obtained from as-prepared ceramics at x = 0.4.     

Characterization of a fracture specimen for crack growth in epoxy due to thermal fatigue

Finite element simulations are carried out to characterize a new fracture specimen, consisting of an outer circular epoxy ring bonded to an inner circular invar plate for accelerated thermal fatigue testing. Radial cracks are introduced in the epoxy ring. The growth of these radial cracks is correlated to the applied energy release rate G. We studied the dependence of G on the crack length, the specimen geometry and the elastic modulus. For short cracks, G is obtained in closed form. Analysis is carried out to determine the critical thermal buckling load the specimen can withstand. Experimental results show that the fatigue crack growth rate per thermal cycle da/dN is given by da/dN = 0.51(ΔG)^0.38 for cycling between 4 and 100 °C but by da/dN = 0.25(ΔG)^0.24 for cycling between 20 and 85 °C, where ΔG is the difference of the energy release rate between the highest and lowest temperatures during a thermal cycle. More severe thermal cycles produce considerably larger fatigue crack growth rates than less severe ones at the same ΔG. This result also implies that isothermal fatigue tests will probably be inadequate to predict thermal fatigue crack growth in epoxies.     

Thermal boundary layer of a micropolar fluid jet impinging normally on a flat plate

Summary The heat transfer is studied in the boundary layer formed on a flat plate by the impingement of an incompressible micropolar fluid jet. The thermal boundary layer equations are obtained after writing the governing equations for the steady two-dimensional flow of an incompressible micropolar fluid in cartesian co-ordinate system. The solution for the energy equation inside the boundary layer is obtained as a polynomial in terms of the distance from the stagnation point. The temperature of the plate and the temperature outside the boundary layer are assumed to be constant. The temperature distribution and the dimensionless heat transfer coefficient are presented graphically for various values of the material parameters which arise due to the micropolar property of the fluid. These results have been compared with the corresponding results for a Newtonian fluid.

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Thermische Grenzschicht eines mikropolaren Flüssigkeitsstrahles, der senkrecht auf eine ebene Platte auftrifft

Zusammenfassung Der Wärmeübergang in der Grenzschicht einer ebenen Platte, zufolge des Auftreffens eines inkompressiblen, mikropolaren Flüssigkeitsstrahles wird untersucht. Die thermischen Grenzschichtgleichungen werden aus den Grundgleichungen für die stationäre zweidimensionale Strömung einer inkompressiblen, mikropolaren Flüssigkeit in kartesischen Koordinaten erhalten. Die Lösung der Energiegleichung innerhalb der Grenzschicht wird als Polynom in Termen des Abstandes zum Staupunkt angegeben. Die Temperatur der Platte, sowie die Temperatur außerhalb der Grenzschicht werden als konstant vorausgesetzt. Die Temperaturverteilung und der dimensionslose Wärmeübergangskoeffizient sind graphisch für verschiedene Werte der Materialparameter, zufolge der mikropolaren Eigenschaften der Flüssigkeit, dargestellt. Diese Ergebnisse wurden mit den entsprechenden Ergebnissen für Newtonsche Flüssigkeiten verglichen.

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With 6 Figures