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.     

Transient laminar forced convection to power-law fluids inside ducts resulting from a sudden change in wall temperature

The transient laminar forced convection to power-law fluids in thermally developing, hydrodynamically developed flow inside parallel-plate ducts and circular tubes resulting from a sudden change in wall temperature is studied. The generalized integral transform and the Laplace transform techniques are employed to develop approximate analytic solutions. The local Nusselt number and average fluid temperature are presented over the range of the dimensionless axial coordinate Z varying from 10^?4 to 10^?1 for several dimensionless times. Three different values of the power-law index are considered in the study includedn=1/3,n=1 andn=3 corresponding to, respectively, the pseudoplastic, Newtonian and dilatant fluids.     

Squeezing flow of elastic liquids

The theoretical analysis is made of the relation between applied force and plate separation for squeezing flows of viscoelastic liquids between closely-spaced parallel disks. The lubrication approximation and the quasi-steady-state assumption are employed in the development. Elastic effects are incorporated through inclusion of normal stresses. Solutions are presented for liquids with power-law viscometric functions, and a numerical procedure is used for fluids having viscometric functions of arbitrary form. For fast and slow squeezing, calculated values of t_{$\text{1}\text{2}$}, the time required to squeeze out half the fluid, are found to agree with the constant force data of Leider [1,2].     

Some remarks on “Drag coefficients of a slowly moving sphere in Non-Newtonian fluids”

Viscoelastic solutions were ejected vertically downwards into air and various Newtonian fluids. The measured swell increased significantly when ejected into a liquid rather than air. The observed increase is considered a result of both bouyancy and drag forces on the solution. The following dimensions expression relating the ratio of the swell diameter in liquid and air D_L/D_A to the elastic shear compliance of the ejected solution J_e was experimentally observed.(D_L/D_A)⁶-1=30(Δ?/?_s)^{?$\text{1}\text{2}$}([g²η²_N?_s]$\text{1}\text{3}$J_e)^{$\text{3}\text{5}$}, where Δ? is the density difference between the extruded and Newtonian fluid, ?_s is the solution density, g is the gravitational constant, and η_N is the Newtonian fluid viscosity. Thus with this expression a simple extrudate swell technique exists to estimate the elastic shear compliance of a viscoelastic solution.     

Fractional vapour content of a liquid pool through which vapour is bubbled

Data from a large number of Russian, American and German sources are examined and found to be correlated in general by

$\text{α}\text{1?α)}\text{1}\text{2}\text{= K[F}{}_{\mathrm{D}}\text{P}{}^{\mathrm{m}}\text{]}{}^{\mathrm{n}}$

where α is voidage or fractional vapour content, K is a constant, F_D is a Froude number and P is a physical properties group. However, the exponent m is found to vary from 0 to 0.3 and the exponent n from $\text{2}\text{3}$ to 0.79, depending upon the sources of the data. The most probable value for n is $\text{2}\text{3}$ but a firm choice cannot be made for m, which is either 0.16 or 0.3. The different values of m depend chiefly upon the method of measurement of the voidage.     

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.     

The end correction for power-law fluids in the capillary rheometer

The finite element scheme developed by Nickell, Tanner and Caswell is used to compute the entry and exit losses for creeping flow of power-law fluids in a capillary rheometer. The predicted entry losses for a Newtonian fluid agree well with available experimental and theoretical results. The entry losses for inelastic power-law fluids increased with decreasing flow behaviour index and show an increasing deviation from available upper bound results as the flow behaviour index in the power-law decreases.The exit losses are found to be finite for inelastic power-law fluids and increase as the flow behaviour index decreases. The predicted die swell for Newtonian fluids agrees well with the available experimental data while the influence of shear thinning is to reduce the die swell.The end correction which is the sum of the entry and exit losses relative to twice the viscometric wall shear stress varies from 0.834 for n = 1 to 2.917 for n = 1/6. This figure reaches a very high value as n tends to zero. The experimental variation in the Couette correction factor in capillary rheometry is explained in terms of the shear thinning characteristics of the fluid. It is concluded that the exit flow is not viscometric, contrary to a common assumption.     

Comparative study of forced convection of a power-law fluid in a channel with a built-in square cylinder

A detailed comparison between the lattice Boltzmann method and the finite element method is presented for an incompressible steady laminar flow and heat transfer of a power-law fluid past a square cylinder between two parallel plates. Computations are performed for three different blockage ratios (ratios of the square side length to the channel width) and different values of the power-law index n covering both pseudo-plastic fluids (n < 1) and dilatant fluids (n > 1). The methodology is validated against the exact solution. The local and averaged Nusselt numbers are also presented. The results show that the relatively simple lattice Boltzmann method is a good alternative to the finite element method for analyzing non-Newtonian fluids.     

Plastic flow and dislocation structure at small strains in OFHC copper deformed in tension,torsion and combined tension-torsion

OFHC copper specimens of 39 μm grain size were deformed to small strains (up to 8%) in tension, torsion and combined tension-torsion at 300 K and the resulting dislocation structures, distributions and densities were determined using transmission electron microscopy. Employing the von Mises yield criterion and the plastic-work hypothesis good agreement was obtained for the three testing conditions for (i) equivalent stress $\text{}\text{?}$gs vs equivalent strain $\text{}\text{?}$g3^p curves, (ii) the dislocation structure, distribution and density ρ as a function of $\text{}\text{?}$g3^p, and (iii) $\text{}\text{?}$gs as a function of $\text{ρ}\text{1}\text{2}$. Furthermore, upon comparing the $\text{}\text{?}$gs vs $\text{ρ}\text{1}\text{2}$ curve for polycrystalline copper with the τ_RSS vs $\text{ρ}\text{1}\text{2}$ curve for single crystals, an average Taylor factor M= (σ/τ_RSS) of approximately 3.2 was obtained, which is in good accord with that predicted theoretically for FCC metals. Almost equally good correlations for the stressstrain curves and for the dislocation density were obtained on the basis of maximum shear stress τ_max and maximum shear strain γ^p_max as on the basis of $\text{}\text{?}$gs and $\text{}\text{?}$g3^P. Therefore, the present results do not permit a positive decision on the question whether the dislocation density correlates better with $\text{}\text{?}$gs and $\text{}\text{?}$g3^P or with τ_max and γ^P_max.A single test in which the direction of straining in torsion was reversed yielded a density and distribution of dislocations (and a corresponding value of $\text{}\text{?}$gs) equivalent to those that developed at a smaller strain in unidirectional straining.     

Rheology of suspensions of spherical particles in a newtonian and a non-newtonian fluid

Properties of suspensions of spherical glass beads (25–38 μm dia.) in a Newtonian fluid and a non-Newtonian (NBS Fluid 40) fluid were measured at volume fractions, φ, of 0%, 10%, 20% and 30%. Measurements were made using a modified and computerized Weissenberg Rheogoniometer. Properties measured included steady shear viscosity, η($\text{γ}\text{.}$), first normal stress difference, N₁($\text{γ}\text{.}$), linear viscoelastic properties, η′(ω) and G′(ω), shear stress relaxation, σ^? ($\text{γ}\text{.}$, t), and growth, σ⁺($\text{γ}\text{.}$, t) and normal stress relaxation, N₁^?($\text{γ}\text{.}$, t).For a the Newtonian fluid, increasing φ causes both η and η′ to increase, with η′ showing a slight frequency dependence. Both N₁ and G′ are zero and stress relaxation and growth occur essentially instantaneously. For the NBS fluid, both η and η′ increse with φ at all $\text{γ}\text{.}$ and ω, respectively, the increase being greater as $\text{γ}\text{.}$ and ω approach zero. N₁ and G′ are less affected by the presence of the particles than η and η′ with the effect on G′ being more pronounced than on N₁. For fixed $\text{γ}\text{.}$, stress relaxation and growth exhibit greater non-linear effects as φ is increased. A model for predicting a priori the linear viscoelastic properties for suspensions was found to yeild reasonable estimates up to φ = 20%.     

Heat transfer to non-Newtonian flows over a cylinder in cross flow

Over a range of 102<Re^*<5800, 6.5<Pr^*<79, and 0.6<n<1, circumferential wall temperatures for water and aqueous polymer (purely viscous) solution flows over a smooth cylinder were measured experimentally. The cylinder was heated by passing direct electric current through it. Aqueous solutions of Carbopol 934 and EZ1 were used as power-law non-Newtonian fluids. The peripherally averaged heat transfer coefficient for purely viscous non-Newtonian fluids, at any fixed flow rate, decreases with increasing polymer concentration. A new correlation is proposed for predicting the peripherally averaged Nusselt number for power-law fluid flows over a heated cylinder in cross flow.     

Creep transition in bending of rectangular plates

Transitional stresses of a rectangular plate bent into the form of a circular cylinder have been derived in closed form. The effect of compressibility is presented graphically. The result indicate that for n (measure index)>1, the circumferential stress is maximum at the inner surface for an incompressible material and not for the compressible material, while for $\text{n=}\text{1}\text{N}$,(N ? 1), it is found to be maximum at some inner point and not at the inner surface. The neutral surface alters with compressibility of the material and n.     

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).     

Homogenized model of reaction–diffusion in a porous mediumUn modèle homogénéisé de réaction–diffusion dans un milieu poreux

We study the initial boundary value problem for the reaction–diffusion equation,

$\text{?}{}_{\mathrm{t}}\text{u}{}^{\mathrm{\epsilon }}\text{??·(a}{}^{\mathrm{\epsilon }}\text{?u}{}^{\mathrm{\epsilon }}\text{)+g(u}{}^{\mathrm{\epsilon }}\text{)=h}{}^{\mathrm{\epsilon }}$

in a bounded domain $\text{Ω}$ with periodic microstructure $\text{F}{}^{\mathrm{(\epsilon )}}\text{∪}\text{M}{}^{\mathrm{(\epsilon )}}$, where a^ε(x) is of order 1 in $\text{F}{}^{\mathrm{(\epsilon )}}$ and κ(ε) in $\text{M}{}^{\mathrm{(\epsilon )}}$ with κ(ε)→0 as ε→0. Combining the method of two-scale convergence and the variational homogenization we obtain effective models which depend on the parameter θ=lim_ε→0κ(ε)/ε². In the case of strictly positive finite θ the effective problem is nonlocal in time that corresponds to the memory effect. To cite this article: L. Pankratov et al., C. R. Mecanique 331 (2003).     

Diffusion and wave behaviour in linear Voigt modelDiffusion et comportement ondulatoire dans le modèle linéaire de Voigt

A boundary value problem $\text{P}{}_{\mathrm{\epsilon }}$ related to a third order parabolic equation with a small parameter ε is analized. This equation models the one-dimensional evolution of many dissipative media as viscoelastic fluids or solids, viscous gases, superconducting materials, incompressible and electrically conducting fluids. Moreover, the third order parabolic operator regularizes various nonlinear second order wave equations. In this paper, the hyperbolic and parabolic behaviour of the solution of $\text{P}{}_{\mathrm{\epsilon }}$ is estimated by means of slow timeτ=εt and fast timeθ=t/ε. As consequence, a rigorous asymptotic approximation for the solution of $\text{P}{}_{\mathrm{\epsilon }}$ is established. To cite this article: M. De Angelis, P. Renno, C. R. Mecanique 330 (2002) 21–26     

Stokes paradox for power-law flow around a cylinder

A simple analysis for power-law fluids shows that the Stokes paradox for creeping flow around a cylinder is removed for shear-thinning (n < 1) but not for shear-thickening (n 1) fluids. An approximate drag value is found for n < 1 and is compared with computed results.     

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.