Instability of channel flow of a shear-thinning White–Metzner fluid

We consider the inertialess planar channel flow of a White–Metzner (WM) fluid having a power-law viscosity with exponent n. The case n = 1 corresponds to an upper-convected Maxwell (UCM) fluid. We explore the linear stability of such a flow to perturbations of wavelength k⁻¹. We find numerically that if n < n_c ≈ 0.3 there is an instability to disturbances having wavelength comparable with the channel width. For n close to n_c, this is the only unstable disturbance. For even smaller n, several unstable modes appear, and very short waves become unstable and have the largest growth rate. If n exceeds n_c, all disturbances are linearly stable. We consider asymptotically both the long-wave limit which is stable for all n, and the short-wave limit for which waves grow or decay at a finite rate independent of k for each n.The mechanism of this elastic shear-thinning instability is discussed.     

Crystallization of alkanes under quiescent and shearing conditions

We report molecular dynamics simulation of crystallization of model alkane systems conducted under constant pressure conditions. We have studied crystallization of n-eicosane (C₂₀H₄₂) and n-hexacontane (C₆₀ H₁₂₂) under quiescent and shearing conditions. We find preshearing before subjecting the melt to quiescent crystallization enhances the crystallization of higher molecular weight hexacontane, whereas, for low molecular weight eicosane, no significant change can be detected. For both alkanes applying steady planar shear significantly speeds up the crystallization. The crystal growth rate increases with the shear rate. However, we find that the critical shear rate above which the crystallization is enhanced, is inversely proportional to the size of the chains. In all cases the Weissenberg numbers of the sheared systems are moderate. We estimate them to be in the range of 0.01–10. Our quiescent simulations for eicosane predict crystallization temperature and lattice parameters of the crystalline phase in good agreement with experimental measurements. We have compared an order parameter used in the simulations against one analogous to that used in dilatometry experiments. Using this order parameter as a measure of crystallinity we predict the crystal growth rate of n-eicosane to be a maximum at ∼300 K in good agreement with experiments. Fitting crystallization growth data to Avrami's model we have calculated Avrami growth functions and exponents for many cases. For quiescent crystallization of n-eicosane we found the Avrami exponent calculated using our order parameter for defining degree of crystallinity, agrees well with that obtained in the experiments. For C60 the crystallization process is very slow at quiescent conditions; however preshearing enhances the crystal growth.     

Axisymmetric spreading of a thin liquid drop with suction or blowing at the horizontal base

The axisymmetric spreading under gravity of a thin liquid drop on a horizontal plane with suction or blowing of fluid at the base is considered. The thickness of the liquid drop satisfies a non-linear diffusion equation with a source term. For a group invariant solution to exist the normal component of the fluid velocity at the base, v_n, must satisfy a first-order quasi-linear partial differential equation. The general form of the group invariant solution for the thickness of the liquid drop and for v_n is derived. Two particular solutions are considered. Each solution depends essentially on only one parameter which can be varied to yield a range of models. In the first solution, v_n is proportional to the thickness of the liquid drop. The base radius always increases even for suction. In the second solution, v_n is proportional to the gradient of the thickness of the liquid drop. The thickness of the liquid drop always decreases even for blowing. A special case is the solution with no spreading or contraction at the base which may have application in ink-jet printing.     

Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction

We consider quasistatic motion and stability of a single degree of freedom elastic system undergoing frictional slip. The system is represented by a block (slider) slipping at speed V and connected by a spring of stiffness k to a point at which motion is enforced at speed V₀ We adopt rate and state dependent frictional constitutive relations for the slider which describe approximately experimental results of Dieterich and Ruina over a range of slip speeds V. In the simplest relation the friction stress depends additively on a term A In V and a state variable θ; the state variable θ evolves, with a characteristic slip distance, to the value ? B In V, where the constants A, B are assumed to satisfy B > A > 0. Limited results are presented based on a similar friction law using two state variables.Linearized stability analysis predicts constant slip rate motion at V₀ to change from stable to unstable with a decrease in the spring stiffness k below a critical value k_cr. At neutral stability oscillations in slip rate are predicted. A nonlinear analysis of slip motions given here uses the Hopf bifurcation technique, direct determination of phase plane trajectories, Liapunov methods and numerical integration of the equations of motion. Small but finite amplitude limit cycles exist for one value of k, if one state variable is used. With two state variables oscillations exist for a small range of k which undergo period doubling and then lead to apparently chaotic motions as k is decreased.Perturbations from steady sliding are imposed by step changes in the imposed load point motion. Three cases are considered: (1) the load point speed V₀ is suddenly increased; (2) the load point is stopped for some time and then moved again at a constant rate; and (3) the load point displacement suddenly jumps and then stops. In all cases, for all values of k:, sufficiently large perturbations lead to instability. Primary conclusions are: (1) ‘stick-slip’ instability is possible in systems for which steady sliding is stable, and (2) physical manifestation of quasistatic oscillations is sensitive to material properties, stiffness, and the nature and magnitude of load perturbations.     

Spectral analysis of the Pekeris operator in the theory of acoustic wave propagation in shallow water

ThePekeris differential operator is defined by $$Au = - c^2 (x_n )\rho (x_n )\nabla \cdot \left( {\frac{1}{{\rho (x_n )}}\nabla u} \right),$$ wherex=(x ₁,x ₂,...x _n )∈R _n ,?=(?/?x ₁, ?/?x ₂,...?/?x _n ), and the functionsc(x _n),σ(x _n) satisfy $$c(x_n ) = \left\{ \begin{gathered} c_1 , 0 \leqq x_n< h, \hfill \\ c_2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$ and $$\rho (x_n ) = \left\{ \begin{gathered} \rho _1 , 0 \leqq x_n< h, \hfill \\ \rho _2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$ wherec ₁,c ₂,? ₁,? ₂, andh are positive constants. The operator arises in the study of acoustic wave propagation in a layer of water having sound speedc ₁ and density? ₁ which overlays a bottom having sound speedc ₂ and density? ₂. In this paper it is shown that the operatorA, acting on a class of functions u (x) which are defined for x_n≧0 and vanish for x_n=0, defines a selfadjoint operator on the Hilbert space whereR ₊ ⁿ ={x∈R ⁿ :x _n >0} anddx =dx ₁ dx ₂...dx _n denotes Lebesgue measure in R ₊ ⁿ . The spectral family ofA is constructed and the spectrum is shown to be continuous. Moreover an eigenfunction expansion for A is given in terms of a family of improper eigenfunctions. Whenc ₁≧c ₂ each eigenfunction can be interpreted as a plane wave plus a reflected wave. When c₁< c₂, additional eigen-functions arise which can be interpreted as plane waves that are trapped in the layer 0n h by total reflection at the interface x_n=h.     

On solitary waves. Part 2 A unified perturbation theory for higher-order waves

A unified perturbation theory is developed here for calculating solitary waves of all heights by series expansion of base flow variables in powers of a small base parameter to eighteenth order for the one-parameter family of solutions in exact form, with all the coefficients determined in rational numbers. Comparative studies are pursued to investigate the effects due to changes of base parameters on (i) the accuracy of the theoretically predicted wave properties and (ii) the rate of convergence of perturbation expansion. Two important results are found by comparisons between the theoretical predictions based on a set of parameters separately adopted for expansion in turn. First, the accuracy and the convergence of the perturbation expansions, appraised versus the exact solution provided by an earlier paper [1] as the standard reference, are found to depend, quite sensitively, on changes in base parameter. The resulting variations in the solution are physically displayed in various wave properties with differences found dependent on which property (e.g. the wave amplitude, speed, its profile, excess mass, momentum, and energy), on what range in value of the base, and on the rank of the order n in the expansion being addressed. Secondly, regarding convergence, the present perturbation series is found definitely asymptotic in nature, with the relative error δ(n) (the relative mean-square difference between successive orders n of wave elevations) reaching a minimum, δ _m , at a specific order, n=n _m , both depending on the base adopted, e.g. n _{m , α} =11-12 based on parameter α (wave amplitude), n _{m , β} =15 on β (amplitude-speed square ratio), and n _{m , ∈} =17 on ∈ ( wave number squared). The asymptotic range is brought to completion by the highest order of n=18 reached in this work.     

Chebyshev finite difference approach to modeling the thermoviscosity effect in a power-law liquid film on an unsteady stretching surface

The effect of thermoviscosity (temperature-dependent viscosity) on the heat transfer in a power-law liquid film over an unsteady stretching sheet is investigated. Similarity analysis is used to transform the governing equations for mass, momentum and energy into a system of ordinary differential equations, which contain a thermoviscosity parameter θ_r, unsteadiness parameter S, generalized Prandtl number Pr and power-law index n. The film thickness, the temperature distributions, the local heat transfer rate, and the local skin-friction coefficient were obtained using the Chebyshev finite difference method (ChFD). The results show that thermoviscosity significantly increases the film thickness and the local heat transfer rate while decreasing the local skin-friction coefficient as θ_r → 1. It is found that liquids with a higher power-law index will have a larger film thickness and a higher free-surface temperature, which indicate a lower local heat transfer rate, ?θ′(0).     

A modified Lin equation for the energy balance in isotropic turbulence

At sufficiently large Reynolds numbers, turbulence is expected to exhibit scale-invariance in an intermediate (“inertial”) range of wavenumbers, as shown by power law behavior of the energy spectrum and also by a constant rate of energy transfer through wavenumber. However, there is an apparent contradiction between the definition of the energy flux (i.e., the integral of the transfer spectrum) and the observed behavior of the transfer spectrum itself. This is because the transfer spectrum T(k) is invariably found to have a zero-crossing at a single point (at k = k_*), implying that the corresponding energy flux cannot have an extended plateau but must instead have a maximum value at k = k_*. This behavior was formulated as a paradox and resolved by the introduction of filtered/partitioned transfer spectra, which exploited the symmetries of the triadic interactions (J. Phys. A: Math. Theor., 2008). In this paper we consider the more general implications of that procedure for the spectral energy balance equation, also known as the Lin equation. It is argued that the resulting modified Lin equations (and their corresponding Navier–Stokes equations) offer a new starting point for both numerical and theoretical methods, which may lead to a better understanding of the underlying energy transfer processes in turbulence. In particular the filtered partitioned transfer spectra could provide a basis for a hybrid approach to the statistical closure problem, with the different spectra being tackled using different methods.     

Bifurcation phenomena of generalized newtonian fluids in curved rectangular ducts

The bifurcation phenomenon in flow through a curved rectangular duct is investigated in this study. The non-linear equations of motion governing the steady, fully developed laminar flow of an incompressible generalized Newtonian fluid have been solved numerically. Extensive results have been generated in an effort to map the regions of multiple solution in the parameter space of Dean number, Dn, aspect ratio, γ, power-law index, n, and radius of curvature, r. For a Newtonian fluid (n = 1), at a fixed curvature (r = 100), the transition between a symmetric 2-cell and a symmetric 4-cell solution appears to follow a tilted cusp. The extent of the stable, symmetric 2-cell solution surface is critically influenced by the length scale γ. In the non-Newtonian case, at a fixed aspect ratio (γ = 1) and a fixed curvature (r = 100), the flow transition follows that of a fold catastrophe. The influence of the curvature is reasonably well accounted for in Dn. The bifurcation set determined in the Dn-γ space remains qualitatively the same at any value of n or r. These parameters merely shift and/or stretch the equilibrium surface determined by Dn and γ.     

Model predictions of higher-order normal alkane ignition from dilute shock-tube experiments

Shock-induced oxidation of two higher-order linear alkanes was measured using a heated shock tube facility. Experimental overlap in stoichiometric ignition delay times obtained under dilute (99 % Ar) conditions near atmospheric pressure was observed in the temperature-dependent ignition trends of n-nonane (n-C₉H₂₀) and n-undecane (n-C₁₁H₂₄). Despite the overlap, model predictions of ignition using two different detailed chemical kinetics mechanisms show discrepancies relative to both the measured data as well as to one another. The present study therefore focuses on the differences observed in the modeled, high-temperature ignition delay times of higher-order n-alkanes, which are generally regarded to have identical ignition behavior for carbon numbers above C₇. Comparisons are drawn using experimental data from the present study and from recent work by the authors relative to two existing chemical kinetics mechanisms. Time histories from the shock-tube OH* measurements are also compared to the model predictions; a double-peaked structure observed in the data shows that the time response of the detector electronics is crucial for properly capturing the first, incipient peak near time zero. Calculations using the two mechanisms were carried out at the dilution level employed in the shock-tube experiments for lean ${({\rm {\phi}} = 0.5)}$ , stoichiometric, and rich ${({\rm {\phi}} = 2.0)}$ equivalence ratios, 1230–1620 K, and for both 1.5 and 10 atm. In general, the models show differing trends relative to both measured data and to one another, indicating that agreement among chemical kinetics models for higher-order n-alkanes is not consistent. For example, under certain conditions, one mechanism predicts the ignition delay times to be virtually identical between the n-nonane and n-undecane fuels (in fact, also for all alkanes between at least C₈ and C₁₂), which is in agreement with the experiment, while the other mechanism predicts the larger fuels to ignite progressively more slowly.     

Axisymmetric stagnation point flow and mass transfer for power-law fluids II. Moderate schmidt number correction

The theory for axisymmetric stagnation point flow of power-law fluids has been extended to include the correction terms for convective diffusion at moderate Schmidt numbers. The dimensionless mass transfer rate is expressed as an asymptotic series that is valid for Re^{(1 ? n)/3(1 + n)}Sc^{?$\text{1}\text{3}$} < 1. The result can be used to predict accurate diffusion coefficients for dilute species in fluids with specified power-law characteristics.     

Post-buckling behavior of columns with non-linear constitutive equations

The post-buckling behavior of a column made of a material with cubic constitutive equation, σ = E₁? + E₂?³, is unstable for a range of values of E₂. In these cases, the imperfection sensitivity is qualitatively described using catastrophe theory. A numerical method is given to compute the post-buckling deflections.     

The effect of suction or injection on the boundary layer flows induced by continuous surfaces stretched with prescribed skin friction

The boundary layer flow and heat transfer on a stretched surface moving with prescribed skin friction is studied for permeable surface. Three major cases are studied for isothermal surface (n=0) stretched corresponding to different dimensional skin friction boundary conditions namely; skin friction at the surface scales as (x ^?1/2) at m=0, constant skin friction at m=1/3 and skin friction scales as (x) at m=1. The constants m and n are the indices of the power law velocity and temperature exponent respectively. Similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity variation. The effect of various governing parameters, such as Prandtl number Pr, suction/injection parameter f _w , m and n are studied. The results show that for isothermal surface increasing m enhances the dimensionless heat transfer coefficient for fixed f _w at the suction case and the reverse is true at the injection case. Furthermore, for fixed m, as f _w increases the dimensionless heat transfer coefficient increases. Large enhancements are observed in the heat transfer coefficient as the temperature boundary condition along the surface changes from uniform to linear where the dimensional skin friction is of order (x) at m=1. This enhancement decreases as the suction increases.     

A Concentration Phenomenon for Semilinear Elliptic Equations

For a domain ${\Omega \subset \mathbb{R}^{N}}$ we consider the equation $$-\Delta{u} + V(x)u = Q_n(x)|{u}|^{p-2}u$$ with zero Dirichlet boundary conditions and ${p\in(2, 2^*)}$ . Here ${V \geqq 0}$ and Q _n are bounded functions that are positive in a region contained in ${\Omega}$ and negative outside, and such that the sets {Q _n  > 0} shrink to a point ${x_0 \in \Omega}$ as ${n \to \infty}$ . We show that if u _n is a nontrivial solution corresponding to Q _n , then the sequence (u _n ) concentrates at x ₀ with respect to the H ¹ and certain L ^q -norms. We also show that if the sets {Q _n  > 0} shrink to two points and u _n are ground state solutions, then they concentrate at one of these points.     

A note on an improvement of heat transfer of tube banks

An experimental investigation has been conducted for exploring a possibility to improve the heat transfer of tube banks of in-line arrangement, in which the first cylinder was roughened with pyramids. Measured were the heat transfer characteristics of the first cylinder for several cylinder spacings. It is found that there exists the critical Reynolds numberRe _c , beyond which the heat transfer rate increases drastically by about 30 to 50% as compared with that for the smooth cylinder, though the increasing rate is small for the case of very narrow spacing such asC _y /d×C _x /d =1.2×1.2. In the region ofRe>Re _c , the separation point shifts downstream to θ=120° to 130° from the forward stagnation point, and it results in the decrease of the form drag.     

Effect of a Uniform Horizontal Throughflow on the Darcy Free Convection over a Permeable Vertical Plate with Volumetric Heat Generation

This article deals with the steady Darcy free convection adjacent to a heated or cooled permeable vertical flat plate of constant temperature, which is embedded in a fluid-saturated porous medium of uniform ambient temperature T _∞. There is a uniform horizontal throughflow of the fluid and a volumetric heat generation q′′′ takes place, which is considered to be a power-law function of the local temperature difference T ? T _∞, i.e., q′′′ ~ (T ? T _∞) ⁿ . To be specific, two cases of this type of volumetric heat generation are considered in the analysis in some detail, namely, the linear and the quadratic cases, n = 1 and n = 2, respectively.     

The Failure of Rank-One Connections

This article is concerned with interface problems for Lipschitz mappings f ₊:? ⁿ ₊→? ⁿ and f _?:? ⁿ _?→? ⁿ in the half spaces, which agree on the common boundary ? ^{n ? 1}=?? ⁿ ₊=?? ⁿ _?. These naturally occur in mathematical models for material microstructures and crystals. The task is to determine the relationship between the sets of values of the differentials Df ₊ and Df _?. For some time it has been thought that the polyconvex hulls [Df ₊] ^pc and [Df _?] ^pc satisfy Hadamard's jump condition or are at least rank-one connected. Our examples here refute this idea.The estimates of the Jacobians we obtain in the course of solving the so-called Monge-Ampère inequalities seem also to be of independent interest. As an application, we construct uniformly elliptic systems of first order partial differential equations in the same homotopy class as the familiar Cauchy-Riemann equations, for which the unique continuation property fails.     

Rich n-heptane and diesel combustion in porous media

Rich n-heptane and diesel flames in two-layer porous media are experimentally investigated in the context of syngas production. The stable operating points of n-heptane reforming have been determined and the mole fractions of H₂, CO, CO₂ and light hydrocarbons have been measured in the exhaust gas at an equivalence ratio of 2 for different thermal input values. The reformer performance has been assessed also from the point of view of the heat losses and the mixture homogeneity. The pre-vapouriser produces an approximately uniform vapour–air mixture upstream of the flame front. The range of flow rates for stable flames decreased with increasing equivalence ratio. Heat losses were about 10% of the thermal input at high firing rates. A 77.2% of the equilibrium H₂ was achieved at a flame speed of 0.82 m/s. The same reactor with a different porous matrix for the reforming stage demonstrates diesel reforming to syngas with a conversion efficiency of 77.3% for a flame speed of 0.65 m/s.     

Qualitative Convergence in the Discrete Approximation of the Euler Problem∗

ABSTRACT

There are two mathematically rigorous ways to derive Euler's differential equation of the elastica. The first is to start from integral rules and use variational principles, whereas the second is to regard the continuous rod as a limit of a discrete sequence of elastically connected rigid elements when the length of the elements decreases to zero. Discrete models of the Euler buckling problem are investigated. The global number s of solutions of the boundary-value problem is expressed as a function of the number of elements in the discrete model, s = s(n), at constant loading P. The functions s (n) are described by the characteristic parameters n ₁ and n ₂, and graphs of n ₁(P) and n ₂(P) are plotted. Observations related to these diagrams reveal interesting features in the behavior of the discrete model, from the point of view of both theory and application.