Cumulative damage in creep

The effect of two step stress variation and intermittent loading on the creep behaviour of commercially pure aluminium and stainless steel has been investigated. In the two step stress variation, the first stress σ₁ was applied for a given time t₁ and the stress level was switched over to σ₂. The resultant creep rate ?_s2⁺ and the failure time t₂ have been observed. Under the intermittent loading programme, the stress cycle was applied in the order σ₁-zero-σ₁ and the average creep rate on each reloading has been observed. The experimental data appear to give a cumulative damage rule in the form

$\text{t}{}_1\text{t}{}_{\mathrm{r1}}\text{+}\text{t}{}_2\text{t}{}_{\mathrm{r2}}\text{e}\text{?}{}_{\mathrm{s2}}{}^{\mathrm{+}}\text{e}\text{?}{}_{\mathrm{s2}}\text{=1}$

where t_r is the creep rupture time corresponding to a given stress and ?_s2 is the creep rate under the second stress in normal creep.     

Effect of loading history on stress corrosion cracking in high strength steel

The effect of preloading on crack nucleation time was examined with compact tension specimens having various notch radius in 0.1N-H²SO⁴ aqueous solution for 200°C tempered AISI 4340 steel. Crack nucleation time t_n increases by preloading for a given apparent stress intensity factor K_p2. The curve K_?2 vs. t_n deviates upward from the curve for the non preloading case. A linear relationship between the crack nucleation time and parameter is seen in semi-log diagram, where is taken as the value at t_n=α due to preloading. The apparent threshold stress intensity factor increases with K_?2 which is the apparent stress intensity factor of preloading. A detached crack is nucleated at some distance from the notch root and extends in a form of circle. This distance increases with increasing K_?2. The effect of load reduction during crack growth was examined. When the K-value was reduced from K₁ to K₂, an incubation time was observed before the crack started growing under the K₂-value. The incubation time t_m tends to increase with increasing ΔK = K₁-K₂. The threshold stress intensity factor was also found to increase for high load reduction.In order to explain these experimental results, a new dislocation model is proposed on the basis of stress induced diffusion of hydrogen in high stress region ahead of the notch root or a crack. This model suggests that the change in the crack nucleation time and the increase of the incubation time due to preloading or load reduction are caused by reducing the hydrostatic pressure and by spreading the hydrogen saturated region which requires more time for the hydrogen accumulation due to preloading or load reduction. The theory predicts the experimentally observed relations between and t_n and between log t_in and ΔK.     

Nucleation mechanism of stress corrosion cracking from notches

Crack nucleation mechanism of hydrogen assisted cracking at notched cracks in aqueous solutions is investigated, using the compact type specimens with various notch radius in low-tempered 4340 steel. A detached crack initiates at some distance ahead of the notch root. The crack nucleation at the notched root is determined by the electrical potential method. When the crack initiates, the voltage difference starts to increase. The crack nucleation site is examined by SEM. The time for crack nucleation increases with the notch root radius, ρ, and decreases with the apparent stress intensity factor K_ρ. A linear relationship between the crack nucleation time, t_n, and the parameter ${}^2\text{K}{}_{\mathrm{\rho }}\text{/(πρ)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{-(2K}{}_{\mathrm{\rho }}\text{/(πρ)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}_{\mathrm{th}}\text{}}$ is seen in semi-log diagram, where $\text{(2K}{}_{\mathrm{\rho }}\text{/(πρ)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}_{\mathrm{th}}$ is almost equal to the yield shear strength.In order to explain these experimental results, a new model of micromechanics is proposed on the basis of stress induced diffusion of hydrogen in the high stress region ahead of the notch root. This model suggests that the detached crack initiates at the elasto-plastic boundary where the hydrogen concentration is from 2 to 5 times higher than that of the notch root surface. The theory agrees with experiments with respect to $\text{{2K}{}_{\mathrm{\rho }}\text{/(πρ)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{-(2K}{}_{\mathrm{\rho }}\text{/(πρ)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}_{\mathrm{th}}\text{}}$ vs t_n and t_n vs ρ.The empirical equation holds under constant t_n, K_ρ = K_o(ρ/ρ_eff)^m where K₀ is the stress intensity factor with ρ ≈ 0 under the present environment, ρeff is the effective notch radius and m is constant. The value of m is 0.25 for the crack nucleation time (t_n)_th corresponding to the threshold stress intensity factor (K_ρ)_th, 0.5 for t_n < (t_n)_th and 0 for ρ ≦ ρ_eff. The above equation agrees with the theoretical equation proposed by Tanaka and Mura for any t_n and ρ_eff.     

Theorems of zaremba type for thermoelastic solid

It was shown in [2] that any solution of the equation of coupled dynamic thermoelasticity (a) $\text{[(?}{}^2\text{??}{}_{\mathrm{t}}{}^2\text{)(?}{}^2\text{??}{}_{\mathrm{t}}\text{)?∈?}{}_{\mathrm{t}}\text{?}{}^2\text{]phi = 0, (?}{}_{\mathrm{t}}\text{= ?/?t)}$, subject to homogeneous initial conditions admits the representation $\text{phi = phi}{}_1\text{+ phi}{}_2$, where (b) $\text{(?}{}^2\text{??}{}_{\mathrm{t}}{}^2\text{?∈?}{}_{\mathrm{t}}\text{?∈?∈K 1)phi}{}_1\text{= 0}$, and (c) $\text{?}{}^2\text{??}{}_{\mathrm{t}}\text{+∈+teK (s)phi}{}_2\text{= 0}$. Here $\text{K = K(t, epsi)}$ is a given function and ¹ denotes the operation of convolution with respect to time $\text{t}$. In the present paper three uniqueness theorems associated with (a), (b) and (c) are given, and a domain of influence theorem for (b) is established. Next, these theorems are used to show that there exist such couplings between the external mechanical and thermal fields applied to the boundary of a thermoelastic solid that inside the body the temperature is either of a wave type or of a diffusive type, but not of both types.     

Fracture of thin sections containing surface cracks

Surface-cracked specimens of several thicknesses of 7075-T651 and 7075-T6 aluminum were tested in uniaxial tension. For thicknesses t less than 0.25 in., the gross fracture stress σ_f of 7075-T651 Al was empirically related to flaw size by the following expression:

$\text{δ}{}_{\mathrm{f}}\text{σ}{}_{\mathrm{ult}}\text{= 1 + S(}\text{a}\text{φ}{}^2\text{.t}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$

where σ_ult is the ultimate strength, a the crack depth, φ a function of crack shape, and S a proportionality constant equal to ?1.7 in.^{$\text{?1}\text{2}$}. For 0.25-in. thick 7075-T651 aluminum, σ_f was found to obey this relationship only when $\text{a}\text{φ}{}^2$ is less than 0.065 in.; for larger flaws, such that $\text{0.065 <}\text{a}\text{φ}{}^2\text{< 0.11, σ}{}_{\mathrm{f}}$ is better predicted by Irwin's surface-crack equation with an apparent K_IC value of $\text{32.2}\text{ksi-in.}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.Fracture data for thin sections of 2014-T6 and 2014-T651 Al tested at ?423°F are analyzed in terms of the empirical relationship above and are found to be in good agreement. For these alloys, S has a value of ?2.6 in.^{$\text{?1}\text{2}$}.Applicability of the empirical relationship and Irwin's surface-crack analysis to the fracture of thin sections is discussed in terms of crack size, section thickness, and plastic zone size.     

Models for the effect of chemisorbed gases on the electrical resistance of metal films

Three hypotheses are considered for the proportionate change of resistance $\text{ΔR}\text{R}{}_1$ when a continuous metal film of resistance R₁ adsorbs gas: (a) a loss of metallic properties at the surface; (b) a change in the electron scattering parameter at the surface; and (c) a change in the electron concentration throughout the film. Two models of the film structure are used to compare the predictions of these hypothesis with our experimental data for the systems $\text{O}{}_2\text{Er}$, $\text{CO}\text{Er}$, $\text{CO}\text{Ti}$ and $\text{O}{}_2\text{Ti}$. The flat plate model accounts for neither R₁ nor $\text{ΔR}\text{R}{}_1$. A simple version of the columnar model accounts for $\text{ΔR}\text{R}{}_1$ by either (a) or (b) for the first three systems and by (c) for the fourth, but it does not explain all the properties of the clean films. The linear approximation given by Mola and Heras for the columnar model of Mayadas and Shatzkes accounts satisfactorily for R₁ and $\text{ΔR}\text{R}{}_1$ for $\text{O}{}_2\text{Er}$, $\text{CO}\text{Er}$ and $\text{CO}\text{Ti}$ by increases in the electron scattering coefficient at the grain boundaries. It does not account for $\text{ΔR}\text{R}{}_1$ with $\text{O}{}_2\text{Ti}$ because that system incorporates gas into the metal lattice.     

An influence function for the intensity factor in tensile fracture

A semi-infinite plane-strain tensile crack occupies the region z = 0, ? ∞ < x < l(t) of an elastic medium. The faces of the crack are loaded symmetrically with an arbitrary time dependent load ($\text{σ}{}_{\mathrm{zz}}\text{= ?(x, t)}$. We derive a closed form expression for a kernel function K(l; t₁; x, t) such that the stress-intensity factor k(L; t₁) at the point (l (t₁), 0, t₁) is given by

$\text{k}\text{(l;t}{}_1\text{)= ∫}\text{K}\text{(l;t}{}_1\text{x, t)?(x, t)}\text{d}\text{x}\text{d}\text{t}$

the domain of integration being a suitable characteristic triangle in the x, t plane. Because of the special form of K we are able to recover Freund's 1973 result that, given f(x, t), k(i;t_l) depends upon/only through l(t_l) and l'(t_l), the latter through a multiplicative factor k(l').     

Une modelisation du comportement magnetique d'un compose de l'iridium pentavalent: LaLi12Ir12O3

A new interpretation is proposed for the magnetic properties of perovskite-type iridium (+V) oxide $\text{LaLi}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{Ir}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{O}{}_3$. In its unusual +5 oxidation state iridium has a t⁴_2ge⁰_g configuration. The magnetic susceptibility has been calculated assuming cubic symmetry of the crystal field and a Coulomb repulsion of the same order of magnitude than spin-orbit coupling. Fitting of the experimental data leads to a single spin-orbit constant $\text{ζ ? 3470}\text{cm}{}^{\mathrm{?1}}$ close to that of previously investigated Ir(+V) compounds.     

Fracture toughness characterization of several aluminum alloys in semi-brittle fracture

A method has recently been developed for determining a nonlinear fracture toughness parameter defined by the relation $\text{G}\text{?}{}_{\mathrm{c}}\text{=}\text{C}\text{?}\text{G}{}_{\mathrm{c}}$ where G_c is the critical elastic strain energy rate as defined by Irwin. The $\text{C}\text{?}$ term is a function of the nonlinearity of the load-displacement test record and has been evaluated using the three parameter Ramberg-Osgood approach, although other curve fitting techniques could be applied as well. The method is quite straightforward and is applicable to plane stress, plane strain and mixed mode testing although only plane stress conditions are considered in this paper. For the case of a linear load-displacement record $\text{C}\text{?}\text{→ 1}$ and $\text{G}\text{?}{}_{\mathrm{c}}$ reduces to the linear elastic result.The toughness parameter $\text{G}\text{?}{}_{\mathrm{c}}$ has been evaluated for a number of high strength aluminum alloys and compared with published G_c values for these materials. The tests were conducted on center-cracked sheets of 2014-T6, 2024-T81, 7075-T6 and 7475-T61 aluminum alloys under conditions of varying specimen geometry and displacement gage length. It was found that the values of $\text{G}\text{?}{}_{\mathrm{c}}$ obtained from displacement readings with a gage length of 2 in. generally agreed with published values of $\text{G}{}_{\mathrm{c}}\text{=}\text{K}{}_{\mathrm{c}}{}^2\text{E}$. The $\text{G}\text{?}{}_{\mathrm{c}}$ values were found to vary inversely with gage length and a/w ratios. The variation in values for $\text{G}\text{?}{}_{\mathrm{c}}$ is of the same order of magnitude as the scatter in published values for G_c. However, $\text{G}\text{?}{}_{\mathrm{c}}$ appears to be less sensitive than G_c to changes in a/w.     

Fracture mechanics approach to fatigue crack initiation from deep notches

Fracture mechanics approach is applied to fatigue crack initiation at the tips of deep, blunt notches including those with very small notch-tip radius. The theoretical relations between the stress intensity range ΔK_ρ and the notch-tip radius ρ for a fixed life for crack initiation were derived based on the models of dislocation-dipole accumulation and blocked slip-band. Those are approximated by a simpler equation: $\text{ΔK}{}_{\mathrm{\rho }}\text{ΔK}{}_{\mathrm{o}}\text{= (1 + ρ/ρ0)}\text{1}\text{2}$ where ΔK₀ and ρ0 are material constants which are related to the fatigue strength of smooth specimens Δρ0 as $\text{Δρ0 =}\text{2ΔK}{}_0\text{(πρ0)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. The results of experiments done with bluntly notched compact tension specimens of a structural low-carbon steel agree with the above relation between $\text{ΔK}{}_{\mathrm{gr}}\text{ΔK}{}_{\mathrm{o}}$ and ρ/ρ_o. The method to predict ΔK_o, ρ_o and Δρ_o from the fatigue data of cracked and smooth specimens is proposed.     

High temperature standard gibbs free energy determinations for Co-o systems by e.m.f. measurements. A statistical approach to evaluate the reliability of the current methods

The standard Gibbs free energies of the following reactions: $\text{Co +}\text{1}\text{2}\text{O}{}_2\text{? CoO}$ (1) and $\text{3CoO +}\text{1}\text{2}\text{O}{}_2\text{? Co}{}_3\text{O}{}_4$ (2) have been calculated from the e.m.f. measurements carried out using a simple compartment solid state galvanic cell, as well for the reaction: 3Co + 20₂ ? Co₃O₄ (3) The temperature-equilibrium oxygen partial pressure relationship has been evaluated. A phase diagram for Co, CoO and Co₃O₄ is proposed in the temperature range 600–900 °C. A statistical analysis on reported ΔG_f⁰ values at 1100 K for reaction (1) has been performed and the results are discussed.     

A low-energy high-brightness electron gun for inverse photoemission

We describe the design and construction of an electron gun which produces a 1 mm focused beam with a 5° full width of angular convergence. The maximum achievable beam current depends on the energy, υ₀, and is given as $\text{I}{}_{\mathrm{<mtext>max</mtext>}}\text{? 0.1 (}\text{μA}\text{V}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{) × V}{}_0{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$. The gun is used in inverse photoelectron spectroscopy (IPES), and is operated at energies as low as 5 eV. The gun employs a BaO cathode operating at 850–1100°C. The low thermal energy spread of the beam (kT ≈ 0.1 eV) and the narrow convergence angle are ideal for momentum resolved IPES. The focused beam is well suited for use with a grating spectrograph IPES detector.The output beam parameters could easily be modified, within certain fundamental constraints, to suit purposes other than IPES. The design principles and the factors controlling beam parameters are discussed in detail.     

Studies on crack growth rate under high temperature creep,fatigue and creep-fatigue interaction—II: On the role of fracture mechanics parameter aeffσg

For high temperature creep, fatigue and creep-fatigue interaction, several authors have recently attempted to express crack growth rate in terms of stress intensity factor $\text{K}{}_{\mathrm{I}}\text{= α}\text{aσ}{}_{\mathrm{g}}$, where a is the equivalent crack length as the sum of the initial notch length a₀ and the actual crack length $\text{a}{}^1$, that is, $\text{a = a}{}_0\text{+ a}{}^1$. On the other hand, it has been shown by Yokobori and Konosu that under the large scale yielding condition, the local stress distribution near the notch tip is given by the fracture mechanics parameter of $\text{aσ}{}_{\mathrm{g}}\text{?(σ}{}_{\mathrm{g}}$), where a is the cycloidal notch length, σ_g is the gross section stress and $\text{?(σ}{}_{\mathrm{g}}$) is a function of σ_g. Furthermore, when the crack growth from the initial notch is concerned, it is more reasonable to use the effective crack length a_eff taking into account of the effect of the initial notch instead of the equivalent crack length a. Thus we believe mathematical formula for the crack growth rate under high temperature creep, fatigue and creep-fatigue interaction conditions may be expressed at least in principle as function of $\text{a}{}_{\mathrm{eff}}\text{σ}{}_{\mathrm{g}}\text{, σ}{}_{\mathrm{g}}$ and temperature.In the present paper, the geometrical change of notch shape from the instant of load application was continuously observed during the tests without interruption under high temperature creep, fatigue and creep-fatigue interaction conditions. Also, the effective crack length a_eff was calculated by the finite element method for the accurate estimation of local stress distribution near the tip of the crack initiated from the initial notch root. Furthermore, experimental data on crack growth rates previously obtained are analysed in terms of the parameter of $\text{a}{}_{\mathrm{eff}}\text{σ}{}_{\mathrm{g}}$ with gross section stresses and temperatures as parameters, respectively.     

Bounds on the effective thermal conductivity of a dispersion of fully penetrable spheres

We evaluate upper and lower bounds on the effective thermal conductivity K_e of a model of two-phase composite materials in which one of the phases consists of spherical inclusions (or voids) of conductivity K₂ and volume fraction φ₂, dispersed randomly throughout a matrix phase of conductivity K₁ and volume fraction φ₁. Our evaluations compare third-order bounds of Beran and of Brown, which utilize the three-point matrix probability function of the model, with bounds of De Vera and Strieder (which apply only to the aforementioned “fully penetrable sphere” model) and of Hashin and Shtrikman. The comparisons are made over extended ranges of values of both φ₂ and $\text{α =}\text{K}{}_2\text{K}{}_1$ and reveal that the best bounds among those we have tested (generally those of Beran) are sharp enough to give quantitatively useful estimates of K_e for $\text{0.1 ≤}\text{K}{}_2\text{K}{}_1\text{≤ 10}$ over a wide range of φ₂ values. They are sharp at high φ₂ values (i.e., φ₂ = 0.9) and very sharp at low φ₂ values (e.g. φ₂ = 0.1) where they remain useful for $\text{K}{}_2\text{K}{}_1\text{≈ 100}$. They are less sharp at intermediate values (e.g. φ₂ = 0.5). As is well known, such results immediately translate into equivalent results for the electrical conductivity, dielectric constant, or magnetic permeability of composites.     

Syntheses and magnetic properties of Eu3B2O6 and Sr3B2O6

Europium orthoborate and strontium orthoborate crystallize in the rhombohedral system with two formula units in a cell of dimensions $\text{a}{}_{\mathrm{R}}\text{=6.697}\text{A}\text{?}$, α_R=85.17° for Eu₃B₂O₆, and $\text{a}{}_{\mathrm{R}}\text{=6.695}\text{A}\text{?}$, α_R=85.00° for Sr₃B₂O₆. The equivalent hexagonal lattice parameters are $\text{a}{}_{\mathrm{H}}\text{=9.069}\text{A}\text{?}$, $\text{c}{}_{\mathrm{H}}\text{=12.542}\text{A}\text{?}$, and $\text{a}{}_{\mathrm{H}}\text{=9.046}\text{A}\text{?}$, $\text{c}{}_{\mathrm{H}}\text{=12.566}\text{A}\text{?}$ respectively. Eu₃B₂O₆ appears to be ferromagnetic below 7.5K.     

Deformation and fracture processes during creep of 12Cr12Mo14V ferritic steel

The creep and fracture properties of $\text{1}\text{2}\text{Cr}\text{1}\text{2}\text{Mo}\text{1}\text{4}\text{V}$ ferritic steel have been determined over the stress range 125 to 362 MNm^?2 at 838 K using high precision, constant stress equipment. When the variation of the rupture life, t_f, and the secondary creep rate, $\text{?}\text{dot}{}_{\mathrm{<mtext>s</mtext>}}$, with stress, σ, was described as ${}_{\mathrm{<mtext>f</mtext>}}\text{α}\text{?}\text{dot}{}_{\mathrm{<mtext>s</mtext>}}\text{ασ}{}^{\mathrm{n}}$ a stress exponent of was recorded. Comparison with long term data then established that, at stresses below ～ 100 MNn^?2, n decreased gradually until values close to unity were obtained at stresses approaching those encountered in electricity generating plant. This decrease in stress exponent is shown to be attributable to a progressive loss of creep resistance associated with changes in carbide dispersion rather than to any change in the mechanisms by which deformation and failure occur.     

Composition and microstructure of arc-sprayed 13% Cr steel coatings

A series of 13% Cr steel wear-resistant coatings have been produced by arc spraying using a range of spraying conditions. The resultant coatings have been examined with optical and scanning microscopes. It has been established that the range of particle sizes varies across the spray beam particularly if the gun-substrate distance is short. At the centre of the beam where the particles are larger, it has been shown that the mean particle are S_A is related to the spraying distance d by a relationship of the form.

where n is an exponent. It is suggested that the gun emits large particles which disintegrate in flight. The use of a small aperture in an arc-restricting air cap produced a general reduction in particle size across the spray beam at all gun-substrate distances.X-ray analysis demonstrated that the combined iron and chromium contents of the deposits were reduced when comparison was made with the composition of the original wire. The longer spraying distances (i.e.d > 50 mm) and use of the finer air caps also reduced the metallic content of the coating. Spot analysis on spray particles revealed a central metallic region with a small residual oxygen content and an outer region of varying thickness (depending on spraying conditions) which contains a significant fraction of oxide.     

Stability of the perovskite phase LaBO3 (B = V,Cr, Mn,Fe, Co,Ni) in reducing atmosphere I. Experimental results

The chemical stability of perovskites LaBO₃ where B = V, Cr, Mn, Fe, Co, Ni was studied by thermogravimetry at 1000°C in gas mixtures of $\text{CO}{}_2\text{H}{}_2$, $\text{O}{}_2\text{CO}{}_2$ and $\text{O}{}_2\text{Ar}$ at 1 bar.The stability limits of the perovskite phases expressed in terms of -log Po₂★ (Po₂★ = critical oxygen partial pressure in bar) were for LaCrO₃ and LaVO₃ (greater than 21.1), LaFeO₃ (16.95), LaMnO₃ (15.05), LaCoO₃ (7.0) and for LaNiO₃ (～0.6). The changes in standard enthalpy ΔH° and entropy ΔS° of the following reactions were obtained.$\text{LaVO}{}_4\text{=}\text{LaVO}{}_3\text{+}\text{1}\text{20}{}_2\text{δ=328}\text{kJ}\text{mol}\text{°=135}\text{J}\text{mol·deg.}$,$\text{LaMnO}{}_3\text{=}\text{1}\text{2}\text{La}{}_2\text{O}{}_3\text{+}\text{MnO}$     

Analysis of a turbulent boundary layer subjected to a strong adverse pressure gradient

We define two non-dimensional parameters $\text{Λ =}\text{τ}{}_{\mathrm{w}}\text{p}{}_{\mathrm{x}}\text{δ}$ and $\text{R}{}_{\mathrm{p}}\text{=}\text{U}{}_{\mathrm{p}}\text{δ}\text{ν}$ where τ_w is the wall stress, p_x(?0) is the pressure gradient to which the turbulent boundary layer (of thickness δ) is subjected, ν is the kinematic viscosity, $\text{U}{}_{\mathrm{p}}\text{= (}\text{νp}{}_{\mathrm{x}}\text{p}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$ is a characteristic velocity and p is the density. The limit corresponding to the strong adverse pressure gradient is formulated as Λ → 0, R_p → ∞, ΛR_p finite. Using appropriate inner and outer asympcotic expansions, both above a wall layer possibly scaling with τ_w and ν, it is found by an application of Millikan's argument that there is an inertial sublayer where the streamwise velocity distribution obeys a half-power law, whose slope depends on Λ, and intercept on ΛR_p. Indeed comparison with available experimental data shows the inner law to be well represented by $\text{u}\text{Up}\text{= (3.5 + 19Λ)(}\text{yU}{}_{\mathrm{p}}\text{ν}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ 2.5ΛR}{}_{\mathrm{p}}$. The outer flow obeys a generalized defect law; use of constant eddy viscosity closure yields results in good agreement with experiment.