An iterative method for solution of a boundary value problem in non-newtonian fluid flow

The boundary value problem

arises in boundary layer equations for the steady flow of a power-law fluid over an impermeable, semi-infinite flat plane. The parameter μ is equal to $\text{1}\text{n}$ where n is the exponent of the strain rate in the expression for the shear stress. We develop and prove the convergence of an iterative method for the solution of the given boundary value broblem for dilatant fluids (0 < μ <1). The iterative method can be easily implemented computationally. An added feature of our technique is that it accurately yields y(0), an important parameter which is related to the drag at the plate. The iterative method works well computationally not only for 0 < μ < 1 but for the range 1 < μ < 4 (pseudoplastic fluids with 1 > n > $\text{1}\text{4}$), as well.     

The theory of a rigid circular disc ground anchor buried in an elastic soil either with adhesion or without adhesion

This paper presents a classical elastostatic analysis of the following situation. A rigid circular disc of radius a is buried in an elastic soil at a depth h below a stress-free surface. The disc is subject to a normal force T resulting in uniform normal displacement of the disc of amount α. Two problems are solved. In the first, the elastic soil is assumed to adhere to the underside of the disc and a solution is obtained by perturbation methods for $\text{a}\text{h}\text{< 0·97}$. For the second, the material on the underside of the disc is assumed to have broken away; here, an exact solution is found for the limiting case $\text{a}\text{h}\text{→ 0}$. The analysis is pertinent to the recently innovated civil engineering technique which utilizes ground anchors to support the retaining walls of excavations.     

Mass and heat transfer by natural convection in a vertical cavity

This paper reports a fundamental study of laminar natural convection in a rectangular enclosure with heat and mass transfer from the side, when the bouyancy effect is due to density variations caused by either temperature or concentration variations. In the first part of the study scale analysis is used to determine the scales of the flow, temperature and concentration fields in boundary layer flow for all values of Prandtl and Lewis numbers. In particular, scale analysis shows that in the extreme case where the flow is driven by bouyancy due to temperature variations, the ratio of mass transfer rate divided by heat transfer rate scales as $\text{Le}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ only if (Pr > 1, Le < 1) or (Pr < 1, Sc < 1), and as $\text{Le}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$ if (Pr > 1, Le > 1) or (Pr < 1, Sc > 1). In the second part of the study, the boundary layer scales derived in the first part are used to determine the heat and mass transport characteristics of a vertical slot filled with fluid. Criteria for the existence of distinct thermal and concentration boundary layers in the slot are determined. Numerical solutions for the flow and concentration fields in a slot without distinct thermal boundary layers are reported. These solutions support further the method of scale analysis employed in the first part of the study     

An experimental study of choked foam flows in a convergent-divergent' nozzle

Choked flow of a foam in a convergent-divergent nozzle has been investigated. The foam consisted of air and a solution of a surface active agent in water. The upstream gas-liquid volume ratio $\text{δ}{}_0$ was in the range 0.053–1.57. The experimental results are in very good agreement with a homogeneous frictionless nozzle flow theory, assuming isothermal behaviour of the gas and no relative motion between the phases, for throat gas-liquid volume ratios $\text{δ}{}_1$ as high as 0.8; for ratios in the range $\text{0.8 < δ}{}_{\mathrm{t}}\text{< 2.98}$ the agreement, while only approximate, is still quite close. Departures from the homogeneous theory are explained in terms of (a) the failure of the assumption of the isothermal behaviour and (b) the existence of relative velocity between the phases. The latter effect predominates at low values of $\text{δ}{}_1$ but at large values, it appears that both contribute to errors in the predictions.     

Two new methods for the approximate determination of the initial coefficient of the first normal stress difference

Two methods for determining the initial coefficient of the first normal stress difference are presented. They are based on the evaluation of the steady viscosity function η($\text{γ}\text{.}$) and the viscosity function η⁺($\text{γ}\text{.}$, t) at the start-up of a flow with a very small rate of deformation $\text{γ}\text{.}$ < $\text{γ}\text{.}$₀. For the functions η($\text{γ}\text{.}$) and η⁺($\text{γ}\text{.}$), equations are given which can be used for a simple evaluation of the integral relationships obtaiend for ψ₁₀. The values for ψ₁₀ calculated by the two methods are compared with values obtained by the well-known methods via measurement of the ψ₁($\text{γ}\text{.}$) or η″(ω)/ω functions and extrapolation to zero). Both methods give values which are in satisfactory agreement with the experimental values.     

Two-phase flow in a vertical pipe and the phenomenon of choking: Homogeneous diffusion model—I. Homogeneous flow models

The paper examines the topological structure of all possible solutions which can exist in flows through adiabatic constant-area ducts for which the homogeneous diffusion model has been assumed. The conservation equations are one-dimensional with the single space variable $\text{z.}$ but gravity effects are included. The conservation equations are coupled with three equations of state: a pure substance, a perfect gas with constant specific heats, and a homogeneous two-phase system in thermodynamic equilibrium. The preferred state variables are pressure $\text{P.}$ enthalpy $\text{h.}$ and mass flux $\text{G}{}^2$.The three conservation equations are first-order but nonlinear. They induce a family of solutions which are interpreted as curves in a four-dimensional phase space conceived as a union of three-dimensional spaces ($\text{P, h, G}{}^2\text{, z}$) with $\text{G}{}^2\text{=}\text{const}$ treated as a parameter. It is shown that all points in these spaces are regular, so that no singular solutions need to be considered. The existence and uniqueness theorem leads to the conclusion that through every point in phase space there passes one and only one solution-curve.The set of differential equations, treated as a system of algebraic equations of each point of the phase space, determines the components of a rate-of-change vector which are obtained explicitly by Cramer's rule. This vector is tangent to the solution curve. Each solution curve turns downward in $\text{z}$ at some specific elevation $\text{z}{}^1$, and this determines the condition for choking. Choking occurs always when the exit flow velocity at $\text{L = z}{}^1$ is equal to the local velocity of propagation of small plane disturbances of sufficiently large wavelength, that is when the flow rate $\text{G}$ becomes equal to a specified, critical flow rate, $\text{G}{}^1$. (The possible dependence of the sonic velocity on frequency in a real flow is ignored, because it has not been allowed for in the equations of the model under study.) A criterion, analogous to the Mach number, which indicates the presence or absence of choking in a cross section is the ratio $\text{K = G/G}{}^7$ of the mass-flow rate $\text{G}$ to the local critical mass flow rate. $\text{G}{}^7$, $\text{K = 1}$ denoting choking. The critical parameters depend only on the thermodynamic properties of the fluid and are independent of the gravitational acceleration and shearing stress at the wall.The topological characteristics of the solutions allow us to study all flow patterns which can, and which cannot, occur in a pipe of given length $\text{L}$ into which fluid is discharged through a rounded entrance from a stagnation reservoir and whose back-pressure is slowly lowered. The set of flow patterns is analogous to that which occurs with a perfect gas, except that the characteristic numerical values are different. They must be obtained by numerical integration and the influence of gravity must be allowed for.The preceding conclusions are valid for all assumptions concerning the shearing stress at the wall which make if dependent on the state parameters only, but not on their derivatives with respect to $\text{z}$. However, the study is limited to upward flows for which the shearing stress at the wall and the gravitational acceleration are codirectional.     

The effect of inextensibility on elastic surface waves

It is shown that when the complications associated with material anisotropy are absent a simple exact analysis can be given of the effect of unidirectional inextensibility on the propagation of surface waves in a semi-infinite elastic body. Provided that the direction of inextensibility e is not orthogonal to either m or m Λ n (m being the outward unit normal to the traction-free boundary of the body and n the wave normal), a unique surface wave exists with displacement everywhere orthogonal to e. The surface-wave solution is assembled from inhomogeneous plane waves in the usual manner, but a novel feature is the presence of a degenerate wave producing no displacement yet perturbing sinusoidally the tension in the inextensible fibres. When the aforementioned provisos are not met the surface wave either degenerates continuously into a shear wave (when $\text{(}\text{m}\text{Λ}\text{n}\text{)·}\text{e}\text{= 0,}\text{m}\text{·}\text{e}\text{≠ 0)}$, ceases to exist (when $\text{m}\text{·}\text{e}\text{= 0,}\text{n}\text{·}\text{e}\text{≠ 0)}$, or merges smoothly into a Rayleigh wave (when $\text{(}\text{e}\text{=±}\text{m}\text{Λ}\text{n}$, the inextensibility constraint then being inoperative).     

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.     

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.     

Small amplitude pressure wave in subsonic and supersonic bubble flows in convergent-divergent nozzles

This paper makes a theoretical analysis of the propagation phenomena of the small amplitude pressure wave in the subsonic and supersonic bubble flow with a velocity slip between bubble and liquid in the convergent-divergent nozzle. From an analysis of the time-mean flow, the nondimensional parameter $\text{m = {u}\text{2}\text{G}\text{·α(1 ? α)ρ}\text{l}\text{β(2 ? 1/S)/P·[αβS + (1 ? α)βS}{}^2\text{+ α(1 ? α)]}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ corresponds to Mach number is gasdynamics where u_G is the gas velocity, α: the void fraction, ρ_L: the liquid density, P: the pressure, S: the velocity ratio of the gas and liquid flows and β: the proportional constant for the virtual mass. From a theoretical analysis of the small disturbance field, it is clarified that the parameter m also plays an essential and important role as Mach number, although the propagation performance of the disturbance is very complicated compared with that in gasdynamics. It is also shown that the pressure waves are divided into four groups depending on the velocity ratio S. Two of them are rather realistic, but the other two are required of a further investigation in future.     

Size effect of cylindrical specimens with fatigue cracks

The rate at which energy is accumulated within a unit volume of material in fatigue is assumed to depend not only on load-time history but also on the specimen size and geometry in addition to material type. A threshold level for the hysteresis strain energy density function accumulated in the material is used for predicting macrocrack growth. This is accomplished by application of the incremental theory of plasticity for each increment of crack growth. The accumulated hysteresis strain energy density factor ΔS to crack growth increment Δa ratio is found to be constant for fixed specimen size and loading, i.e., $\text{ΔS}\text{Δa}\text{=}\text{const}$. Results are obtained for the cylindrical bar specimens with a penny-shaped defect at the center subjected to a constant amplitude and frequency loading. The resistance curves in the ΔS versus Δa plot are parallel lines as specimen size is altered. This information provides a rational means for predicting the influence of specimen size on fatigue lifetime.The results are also compared with those found for geometrically similar plate specimens with line cracks. Cylinder bar specimens of the same material are found to sustain more load cycles prior to catastrophic failure.     

Exact solution of the non-linear differential equation RR̈ + 32R2 − AR−4 + B = 0

The non-linear equation $\text{R}\text{R}\text{?}\text{+}\text{3}\text{2}\text{R}{}^2\text{- AR}{}^{\mathrm{?4}}\text{+ B = 0}$ is shown to represent simply periodic motion with a minimum at R₁ and a maximum at R₁R₀ or a maximum at R₁ and a minimum at R₁R₀^?1. R₀ is a function of the ratio $\text{A}\text{B}$ and is greater than 1 for $\text{A}\text{B}$ > 1 and less than 1 for $\text{A}\text{B}$ > 1. The period of the motion satisfies the simple relation T(R₀^?1) = R₀^?1T(R₀). The exact solution to the above equation is represented in terms of elliptic integrals of the first and second kinds and a simple algebraic function.     

Scattering of a plane longitudinal wave by an elastic quarter space

We study the two-dimensional problem of the scattering of a plane longitudinal wave incident in a homogeneous, isotropic, linearly elastic quarter space. The complex-valued amplitudes of the Rayleigh waves propagating on the free surfaces are plotted versus Poisson's ratio. Also plotted are the farfield scattering patterns for Poisson's ratio $\text{μ=}\text{1}\text{4}$ and $\text{1}\text{3}$.     

Internal gravity waves: parametric instability and deep ocean mixing

The Boussinesq approximation provides a convenient framework to describe the dynamics of stably-stratified fluids. A fundamental motion in these fluids consists of internal gravity waves, whatever the strength of the stratification. These waves may be unstable through parametric instability, which results in turbulence and mixing. After a brief review of the main properties of internal gravity waves, we show how the parametric instability of a monochromatic internal gravity wave organizes itself in space and time, using energetics arguments and a simple kinematic model. We provide an example, in the deep ocean, where such instability is likely to occur, as estimates of mixing from in situ measurements suggest. We eventually discuss the fundamental role of internal gravity wave mixing in the maintenance of the abyssal thermal stratification. To cite this article: C. Staquet, C. R. Mecanique 335 (2007).     

Two phase flow through vertical capillaries—existence of a stratified flow pattern

Two phase downflows were investigated in vertical capillaries with internal diameters in the range 0.5–7.1 mm. Flow pattern regime maps were developed for water-air, distillate-air and water-distillate systems. Over the range of flowrates studied, stratified flow was found to occur in the smaller diameter ($\text{<3}\text{mm}$) capillaries where interfacial tension forces dominate. A modified Hagen-Poiseuille model, which incorporated surface tension effects and which employed the concept of an effective hydraulic diameter, was developed for each of the phases. These equations were used to predict pressure drop and holdup for laminar flow of the two phases in the capillaries. Comparison of pressure drop predictions and measurements gave an RMS error of 14% for the water-distillate system and 8% for the water-air system.     

Investigation of droplet deposition from a turbulent gas stream

Turbulent deposition of particles from two-phase flow onto the smooth wall of a tube has been studied theoretically and experimentally. A model is proposed for the deposition motion of large particles based on turbulent diffusion in the core followed by a free flight towards the wall. The theory shows that within the Stokes regime, the dimensionless deposition velocity $\text{k-}{}_{\mathrm{<mtext>d</mtext>}}\text{/}\text{u}$* depends on Re and $\text{τ}{}^{\mathrm{+}}$ only, where u* is the friction velocity, Re is the tube Reynolds number and $\text{τ}{}^{\mathrm{+}}$ is the dimensionless particle relaxation time. Deposition data are obtained for air-water droplet flow through a 12.7-mm i.d. acrylic tubing at $\text{Re}\text{= 52,500}$ and 94,600. The proposed theory satisfactorily describes the existing deposition data as well as present measurements, covering a wide range of Re and $\text{τ}{}^{\mathrm{+}}$.