Error Function Profile Research Articles

A numerical modeling approach to estimate the piezoresistance of diffused resistors with experimental validation

Silicon piezoresistive sensors have been widely used for many applications in recent decades. Monocrystalline silicon resistors are realized using an ion-implantation or a thermal diffusion process with a Gaussian or complementary error function profile. However, over the years, most researchers have neglected the doping concentration profile of the piezoresistor in the modeling stages resulting in erroneous responses that are far removed from experimental results of fabricated resistive sensors. In the present work, we propose a simulation approach to accurately estimate the piezoresistance of thermally diffused resistors with a non-uniform doping profile. We have modeled the diffused resistor as a parallel combination of several small slices, each having a unique piezoresistive coefficient. Three different slicing strategies were investigated to evaluate the impact of the piezoresistive coefficients, the electrical resistivity of the resistor slices, and the stress profile across the thickness of the resistor embedded in an accelerometer device. The cumulative impact of these parameters on the sensor’s overall sensitivity is evaluated. Further, we have also studied the influence of the accelerometer’s flexure thickness on the sensor’s sensitivity. It is observed from the simulation results that one of the slicing strategies with more slices at the surface of the resistor results in less than 1% error compared to the experimental results of an accelerometer device with a 60 µm flexure thickness.

Growth laws and self-similarity in the confined mixing zone of unstratified and strongly stably stratified isokinetic mixing-layers past a splitter plate

The velocity U and the mixing scalar field Θ of turbulent wake flows developing downstream of a slender splitter plate were studied experimentally for unstratified and strongly stably stratified conditions in a square channel with water as fluid. The experimental program comprises five different Reynolds numbers spanning Re0=10 000–60 000 for u0=u1=u2 (isokinetic conditions) between the two planar, initially separated streams and five relative density stratifications from Δρ=0 to 10%, since mixing studies for these stratification strengths are rather limited. This full-channel Reynolds number uses the hydraulic diameter dh=50 mm of the mixing section as the characteristic length scale. These experiments challenge the corresponding numerical calculations based on Reynolds-averaged Navier–Stokes approaches relying on the Boussinesq approximations of the first and second kind. The development of the mixing scalar field Θ was measured by two fundamentally different approaches: laser induced fluorescence (LIF) in the downstream x–y mid-plane and wire-mesh sensors (WMSs) in discrete lateral y–z planes. This allows for a direct comparison of both sensing techniques at locations where both planes intersect. For the unstratified conditions, it is shown that the downstream developing wake velocity deficit decay agrees well with the theoretical approach. The concentration fields based on LIF and WMS profiles of the mean and fluctuating (root mean square, RMS) data confirm self-similarity by introducing a mixing scalar Θ for both techniques. In the approximate limit, it is shown that the mean concentration field can be well described by an error function and the corresponding RMS data by a Gaussian profile in self-similar coordinates. By using the momentum thickness θm at the splitter plate as the normalization parameter, the downstream developing width of the mixing zone δ(x) becomes independent for Re0≥30 000. The mean and the RMS values of the concentration field Θ for the stably stratified experiments show self-similarity in the core of the concentration field but also a departure from the error function and Gaussian profile in the outer parts. Introducing a local gradient Richardson number Rig, it is shown that both the growth and growth-suppression of the mixing zone are well described by the Miles–Howard criterion for all the stratified experiments considered, even though this criterion was originally developed for shear layers.

Design and Analysis for Hypoid Gears with Ease-Off Flank Modification

An approach of ease-off flank modification for hypoid gears was proposed to improve the meshing performance of automobile drive axle. Firstly, a conjugate pinion matching with gear globally was developed based on gear meshing theory. Secondly, a modified pinion was represented by a sum of two vector functions determining the conjugate pinion and the normal ease-off deviations expressed by both predesigned transmission error function and tooth profile modification curves to change the initial contact clearance of the tooth. Thirdly, the best ease-off deviations were determined by optimizing the minimum amplitude of loaded transmission error (ALTE) based on tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA). Finally, the results show that effective contact ratios (εe) are established by clearances both teeth space and of contact elliptical, and greatly affect ALTE. The εe is a variable value with increasing loads for the tooth with modification. ALTE decreases with increasing εe. After εe reaches the maximum, ALTE increases with increasing loads. The mismatch of the best ease-off tooth is minimal, which contributes to effective reduction in ALTE, thus significantly improving drive performance.

Diffusivity, solubility and thermodynamic modelling of diffusion growth of Ga3+-doped LiTaO3 thin film for integrated optics

A thermodynamic study was performed on diffusion growth of Ga3+-doped LiTaO3(LT) thin film for integrated optics. Some Ga3+-doped LT thin films were grown on LT surface by in-diffusion of homogeneously coated Ga2O3 film at the temperature range of (1273 to 1473)K. After growth, the refractive indices at Ga3+-doped and un-doped surface parts were measured by prism coupling technique and Li composition there was evaluated from the measured refractive indices. The results show that Ga3+ dopant has small effect on the LT index. Li2O out-diffusion is not measurable. The Ga3+ profile in the grown thin film was analysed by secondary ion mass spectrometry. It is found that the grown Ga3+ ions follow a complementary error function profile. A thermodynamic model for Ga3+ diffusion growth is suggested and verified experimentally. From the measured Ga3+ profiles, some thermodynamic parameters were obtained. These include diffusivity, diffusion constant, chemical activation energy, solubility, solubility constant and enthalpy of solution. These parameters are crucial to design and growth of a desired Ga3+-doped LT thin film for integrated optics. Comparison shows that Ga3+ grows in LT two orders faster than Ti4+ because of its smaller radius. Ga3+ diffusivity/solubility in LT is one order/1.5-fold lower than in LiNbO3 because Ta has a larger atomic mass and hence a lower mobility than Nb.

An evaluation of the assumptions of the flamelet model for diesel combustion modeling

Flamelet-based models are widely used to model turbulence-chemistry interaction in Reynolds-Averaged Navier–Stokes (RANS) simulations and Large Eddy Simulations (LES). In this study, one-dimensional laminar and two-dimensional Direct Numerical Simulations (DNS) of reacting mixing layers are used to assess the validity of flamelet models in turbulent diesel jets. The pressure and initial temperature are chosen to be similar to those found in diesel engine combustion chambers at the end of compression. The validity of the flamelet model is first evaluated by comparing the flame development predicted by DNS with that predicted by solving the unsteady flamelet equations. It is shown that there are significant differences in the temperature profiles during the flame development following autoignition. This difference in the unsteady flame development can be a source of error in predicting ignition and flame development. The difference is attributed to the fact that in flamelet models, the dependence of scalar dissipation rate, χ, on the mixture fraction, Z, is assumed to have a constant functional relationship, usually an error function profile, which remains unchanged during the flame development process. The DNS results show that the chemical reactions cause significant increase in the local scalar dissipation rate near the flame. This increase leads to higher diffusion rates for the active radicals near the flame and thus faster flame development. It was shown that solving the unsteady flamelet equations using the scalar dissipation rate profiles obtained from one-dimensional direct simulations lead to less than 5% error in the temperature profiles during the flame development. Since this approach is not computationally feasible for practical applications, an ad hoc modification of the error function profile employed in generating flamelet libraries is proposed for the scalar dissipation rate to account for the chemical reactions and the results show an improved performance compared to the traditional flamelet approach.

Wall-crossing made smooth

In $D=4,N=2$ theories on $R^{3,1}$, the index receives contributions not only from single-particle BPS states, counted by the BPS indices, but also from multi-particle states made of BPS constituents. In a recent work [arXiv:1406.2360], a general formula expressing the index in terms of the BPS indices was proposed, which is smooth across walls of marginal stability and reproduces the expected single-particle contributions. In this note, I analyze the two-particle contributions predicted by this formula, and show agreement with the spectral asymmetry of the continuum of scattering states in the supersymmetric quantum mechanics of two non-relativistic, mutually non-local dyons. This provides a physical justification for the error function profile used in the mathematics literature on indefinite theta series, and in the physics literature on black hole partition functions.

Diffusion Properties of Tm3+ in Congruent LiNbO3 Crystal

Diffusion properties of Tm3+ in congruent LiNbO3 crystal have been investigated, together with other two related issues, i.e., Tm3+‐doping contribution to refractive index of LiNbO3 substrate and Li out‐diffusion. Four X‐cut and four Z‐cut congruent LiNbO3 substrates locally coated with 15–31 nm‐thick Tm‐metal films were annealed in surrounding air under different temperatures of 1030°C–1130°C for different durations of 20–70 h. After anneal, refractive index at Tm3+‐doped and Tm3+‐free parts of crystal surface was measured at the wavelengths of 1311 and 1553 nm and surface Li2O contents were evaluated from measured refractive index. The results show that Tm3+ doping has a weak effect on substrate index and a small contribution to index increment in waveguide layer in comparison with Ti4+‐ or Zn2+ doping. The Li2O content at the Tm3+‐doped surface equals that at the Tm3+‐free surface. The Li out‐diffusion depends mainly on the diffusion temperature. Below 1100°C, the Li out‐diffusion is not measurable. At 1130°C, a 30‐h diffusion procedure may cause 0.2–0.3 mol% slight loss of Li2O content. Secondary ion mass spectrometry was used to study the Tm3+ diffusion properties. The results show that the diffused Tm3+ ions in all samples follow a complementary error function profile. From measured Tm3+ profiles, characteristic diffusion parameters such as diffusivity, diffusion constant, activation energy, solubility, solubility constant, and heat of solution were obtained and discussed in comparison with the case of Er3+ diffusion. In comparison with Er3+ diffusion, the Tm3+ diffusion shows similar anisotropy and temperature dependence of solubility. In the aspect of diffusivity, under lower temperature the Tm3+ has a lower diffusivity than the Er3+, and their diffusivity difference reduces with the increased temperature and becomes null at 1130°C.

Using electron spectroscopy to verify the model of Ga implanted during focused ion beam circuit editing

The authors use electron spectroscopy for chemical analysis to study the implantation of 50 keV Ga+ ions into electronic grade silicon, silicon oxide, silicon nitride, and silicon oxynitride typical of focused ion beam circuit editing work. The Ga+ dose naturally splits into low dose and high dose regimes with a transitional, median regime between the two. In the low dose regime (<1016 cm−2), the amount of milled material is small (∼10 nm) relative to typical layer thicknesses of circuits. A Gaussian implanting model with an implanting yield of ∼0.6 ± 0.1, range of 43 ± 9 nm, and straggle of 15 ± 5 nm describe the data for all studied materials. The Ga atoms bind to the oxygen and nitrogen anions for oxides and nitrides, and silicon for elemental silicon. In the median regime (1016–1017 cm−2), Ga accumulation and substrate defect formation alter a simple Gaussian model for Ga implantation although a steady state condition has not yet been reached. The high dose regime (>1017 cm−2) entails a steady state condition where the implanted Ga assumes an error-function profile comprising a surface concentration as high as 50 at. % and a penetration depth of ∼60 nm. In the high dose regime, Ga implanted into nitride and oxide displays metallic valence (Ga0) on the surface, gradually transitioning to nitride and oxide valence (Ga3+) with depth.

Locally Er-Doped High-Solubility LiNbO3 Crystal Prepared by Li-Poor Vapor Transport Equilibration and Er Codiffusion

We demonstrate the feasibility of preparing locally Er-doped high-solubility LiNbO3 (LN) crystal for the use of integrated optics by simultaneous work of Er in-diffusion and Li-poor vapor transport equilibration (VTE) treatment. Four X-cut and four Z-cut congruent LN substrates locally coated with 30–44 nm thick Er-metal films underwent a Li-poor VTE treatment of 1030/240°, 1050/180°, 1070/180°, and 1050/180°C/h, respectively. After the VTE procedure, the Li2O content in the crystal, determined from the measured birefringence, decreases from original 48.5±0.1 mol% to (48.0–48.2)±0.1 mol% with an approximately homogeneous profile and shows less effect of the cut of plate. Secondary ion mass spectrometry was used to study the Er diffusion properties. The results show that all of the Er diffusion reservoirs were not exhausted. The diffused Er ions in all of the samples studied follow the complementary error function profile. From the measured Er profiles, characteristic diffusion parameters such as diffusion depth, diffusivity, diffusion constant, activation energy, and solubility were determined and discussed in comparison with the case of the traditional diffusion in the argon or O2 atmosphere. It is found that the diffusion in the Li-poor atmosphere shows an anisotropy similar to that in the traditional argon or O2 atmosphere. Under the same temperature, the Er diffusivity in the Li-poor atmosphere is considerably larger than in the O2. From the logarithmic plot of diffusivity versus 1/T and the Arrhenius law, the diffusion constant and activation energy are determined to be (1.2±0.2) × 107/(2.2±0.3) × 104μm2/h and (2.32±0.02)/(1.51±0.02) eV for the diffusion in the X-/Z-cut plate. The solubility in the X-/Z-cut plate is evaluated to be (1.42±0.24)/(1.14±0.15), (2.18±0.27)/(2.01±0.19), (2.62±0.26)/(2.68±0.19), and (2.54±0.20)/(2.30±0.16) × 1020 ions/cm3 for the VTE temperature of 1030°, 1050°, 1070°, and 1100°C, respectively. As expected, it depends on both the VTE temperature and the Li2O content in crystal. Under the same VTE temperature, the solubilities in the X- and Z-cut plates can be regarded as same within the error range. Under the same temperature, the solubility in the Li-poor atmosphere case is considerably higher than that in either the argon or the O2 atmosphere case.

Spatial Instability Of Two‐Phase Mixing Layers

AbstractThe initial stage of liquid atomization by a fast gas stream is considered by means of spatial stability analysis of two–phase shear layers. We solve the viscous linear stability problem for parallel flow modelled by error–function profiles. The unstable modes known from temporal theory are recovered in the convectively unstable case for fluid properties close to the air/water system. The most unstable wavelength depends only weakly on the absolute velocity relative to the laboratory system. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Intrinsic and extrinsic diffusion of phosphorus, arsenic, and antimony in germanium

Diffusion experiments of phosphorus (P), arsenic (As), and antimony (Sb) in high purity germanium (Ge) were performed at temperatures between 600 and 920 °C. Secondary ion mass spectrometry and spreading resistance profiling were applied to determine the concentration profiles of the chemically and electrically active dopants. Intrinsic and extrinsic doping conditions result in a complementary error function and box-shaped diffusion profiles, respectively. These profiles demonstrate enhanced dopant diffusion under extrinsic doping. Accurate modeling of dopant diffusion is achieved on the basis of the vacancy mechanism taking into account singly negatively charged dopant-vacancy pairs and doubly negatively charged vacancies. The activation enthalpy and pre-exponential factor for dopant diffusion under intrinsic condition were determined to 2.85 eV and 9.1 cm2 s−1 for P, 2.71 eV and 32 cm2 s−1 for As, and 2.55 eV and 16.7 cm2 s−1 for Sb. With increasing atomic size of the dopants the activation enthalpy decreases. This is attributed to differences in the binding energy of the dopant-vacancy pairs.

Aggregation, Adsorption, and Surface Properties of Multiply End-Functionalized Polystyrenes

The properties of polystyrene blends containing deuteriopolystyrene, multiply end-functionalized with C8F17 fluorocarbon groups, are strikingly analogous to those of surfactants in solution. These materials, denoted FxdPSy, where x is the number of fluorocarbon groups and y is the molecular weight of the dPS chain in kg/mol, were blended with unfunctionalized polystyrene, hPS. Nuclear reaction analysis experiments show that FxdPSy polymers adsorb spontaneously to solution and blend surfaces, resulting in a reduction in surface energy inferred from contact angle analysis. Aggregation of functionalized polymers in the bulk was found to be sensitive to FxdPSy structure and closely related to surface properties. At low concentrations, the functionalized polymers are freely dispersed in the hPS matrix, and in this range, the surface excess concentration grows sharply with increasing bulk concentration. At higher concentrations, surface excess concentrations and contact angles reach a plateau, small-angle neutron scattering data indicate small micellar aggregates of six to seven F2dPS10 polymer chains and much larger aggregates of F4dPS10. Whereas F2dPS10 aggregates are miscible with the hPS matrix, F4dPS10 forms a separate phase of multilamellar vesicles. Using neutron reflectometry (NR), we found that the extent of the adsorbed layer was approximately half the lamellar spacing of the multilamellar vesicles. NR data were fitted using an error function profile to describe the concentration profile of the adsorbed layer, and reasonable agreement was found with concentration profiles predicted by the SCFT model. The thermodynamic sticking energy of the fluorocarbon-functionalized polymer chains to the blend surface increases from 5.3kBT for x = 2 to 6.6kBT for x = 4 but appears to be somewhat dependent upon the blend concentration.

Ternary Mutual Diffusion Coefficients from Error-Function Dispersion Profiles: Aqueous Solutions of Triton X-100 Micelles + Poly(ethylene glycol)

Taylor dispersion has gained widespread popularity for measuring diffusion in liquids. The usual procedure is to inject small volumes of solution containing solute at concentration c + Δc into carrier streams of composition c. Binary mutual diffusion coefficients D are evaluated from the Gaussian distribution of the dispersed solute measured at the outlet of a long capillary tube. As a result of strong dilution of the injected solute with the carrier solution, obtaining favorable signal-to-noise ratios for the measured profiles can require unacceptably large Δc values for solutions with strongly composition-dependent diffusion coefficients or broad dispersion profiles produced by slowly diffusing solutes. For these systems, D can be reliably evaluated from error-function profiles generated by changing the solution flowing into dispersion tube from composition c − (Δc/2) to c + (Δc/2). There are no dilution factors, so Δc can be orders of magnitude smaller than the values employed in conventional pulse...

Pb-diffusion in monazite: Constraints from a high- T contact aureole setting

We report measurements of Pb diffusion in natural monazite that was subjected to a static, anhydrous, high-temperature overprint in the contact aureole of the Makhavinekh Lake Pluton (MLP), northern Labrador, Canada. The 1850 Ma monazite grains we studied were reheated to temperatures of 900 ± 25 °C during intrusion of MLP at 1322 Ma. Ion-microprobe depth-profiling reveals the outermost ∼ 0.3 μm of these grains to contain Pb/U gradients that can be modeled as a result of one-dimensional concentration-independent diffusion (i.e., an error-function profile) at 1322 Ma. These gradients are combined with temperature–time histories predicted by a numerical model of conductive heat flow in the contact aureole to calculate Pb diffusion coefficients between 3.2 × 10 − 28 and 1.3 × 10 − 27 m 2/s. This range overlaps with the down-temperature extrapolation of experimental Pb diffusion data obtained under dry conditions confirming the slow diffusivity of Pb in monazite at crustal temperatures and timescales.

Inviscid instability of an unbounded shear layer: effect of surface tension, density and velocity profile

The inviscid temporal stability analysis of an unbounded shear layer of two fluids of different density is investigated. Two background velocity profiles are considered: the piecewise-linear profile and the more realistic error-function profile. The disturbance kinetic energy is analyzed to physically understand the mechanism that causes instability. The surface-tension effect is investigated extensively. Surface tension is found to destabilize the neutrally stable waves that exist when surface tension is absent. This surface-tension-induced unstable mode is generally weaker than the dominant mode and extremely less evident when the density and/or viscosity difference increases. Short-wavelength instability is observed with a background viscosity jump at the interface. A comparison between the two velocity profiles is presented. The piecewise-linear profile does not match the more realistic results obtained with the error-function profile in the short wavelength range, especially in nonhomogeneous shear-layer flows; however, the phase-speed results are in a good agreement with those of the error-function profile.

Self-similarly developing, premixed, turbulent flames: A theoretical study

The development of a premixed turbulent flame in a statistically stationary and uniform, planar, one-dimensional flow of a homogeneous mixture is theoretically studied in the case of frozen turbulence. A balance equation for the mean combustion progress variable is proposed to generalize a number of currently available models of premixed turbulent combustion. The equation invokes two unspecified scales and three arbitrary functions to model turbulent diffusion, countergradient transport, and the mean rate of product creation, the rate being assumed to depend straightforwardly on the flame development time t. The equation supplemented with the averaged mass balance equation is analyzed by invoking a single assumption of the self-similarity of the mean flame structure. The assumption, well supported by numerous experiments, allows us to split the aforementioned partial differential balance equations into two ordinary differential equations, which separately model spatial variations of the mean combustion progress variable c¯ and time development of flame thickness Δt. The complementary error function profile of c¯, documented in numerous experiments, is predicted. The profile is independent of mixture properties and turbulence characteristics, as indicated by currently available experimental data. Closures of the countergradient transport term and of the mean rate of product creation are obtained. They substantiate the so-called Zimont equation and generalize it. They also imply that gradient diffusion dominates during the initial stage of flame development, followed by the transition to countergradient turbulent scalar flux ρu″c″¯ in a sufficiently developed flame. A criterion of the transition, obtained in the paper, shows that the transition is promoted by the density ratio. An analytical dependence of a fully developed flame thickness Δt,∞=Δt(t→∞) on the scale of countergradient transport and density ratio is found. Analytical expressions for the development of flame thickness are obtained in the simplified case of a time-independent mean rate of product creation.

Behavior of granular materials under cyclic shear

The design and development of a parallel plate shear cell for the study of large-scale shear flows in granular materials is presented. The parallel plate geometry allows for shear studies without the effects of curvature found in the more common Couette experiments. A system of independently movable slats creates a well with side walls that deform in response to the motions of grains within the pack. This allows for true parallel plate shear with minimal interference from the containing geometry. The motions of the side walls also allow for a direct measurement of the velocity profile across the granular pack. Results are presented for applying this system to the study of transients in granular shear and for shear-induced crystallization. Initial shear profiles are found to vary from packing to packing, ranging from a linear profile across the entire system to an exponential decay with a width of approximately six bead diameters. As the system is sheared, the velocity profile becomes much sharper, resembling an exponential decay with a width of roughly three bead diameters. Further shearing produces velocity profiles which can no longer be fit to an exponential decay, but are better represented as a Gaussian decay or error function profile. Cyclic shear is found to produce large-scale ordering of the granular pack, which has a profound impact on the shear profile. There exist periods of time in which there is slipping between layers as well as periods of time in which the layered particles lock together resulting in very little relative motion.

Viscous versus inviscid instability of two-phase mixing layers with continuous velocity profile

We consider the temporal instability of parallel two-phase mixing layers. The viscous case is examined using a composite error-function velocity profile. The inviscid case is considered for the broken-line velocity profile, where the thickness of the boundary layer in each fluid next to the interface is chosen to match the viscous error-function profile at the interface and far away from it. Viscosity modifies the inviscid stability properties quantitatively, but we can also discern an additional unstable mode exclusively related to viscous shear. In the absence of interfacial tension, this mode dominates at large wavenumbers when the Reynolds number is sufficiently high. The various viscous modes cannot generally be attributed to either one of the phases due to mode mixing or exchange. For parameters resembling those of atomization experiments and applications, the most unstable wavelength and growthrate in the viscous case can exceed the inviscid values significantly. The viscous stability analysis also provides better agreement with recent experimental results for air and water than inviscid stability calculations.

Instability of two-phase mixing layers: analysis of exact and approximate base flows from boundary layer theory

The utility of composite error-function velocity profiles for the modeling of gas–liquid shear layers is examined by comparing temporal stability results from such approximate error-function profiles with results from the exact velocity profiles based on the two-fluid boundary layer equations. The stability analysis is restricted to two-dimensional perturbations. The two-fluid boundary layer equations are solved numerically using a shooting method in each fluid layer. The composite error-function profile is constructed by matching displacement thicknesses with the exact solution. With given fluid properties, the displacement thicknesses depend on the asymptotic velocity ratio of the liquid and gas stream in the laboratory frame. For different sets of fluid properties, the maximum growth rates of the Kelvin–Helmholtz instability show good numerical agreement between the exact and the approximate velocity profiles, especially when the asymptotic velocities of the gas and liquid phase are close.

Real-Gas Effects on Mean Flow and Temporal Stability of Binary-Species Mixing Layers

Real-gas effects on the mean flow and inviscid stability of temporal mixing layers are examined for supercritical heptane/nitrogen and oxygen/hydrogen mixtures. The analysis is based on the compressible Navier–Stokes equations for conservation of mass, momentum, total energy, and species mass, with heat and species-mass fluxes derived from fluctuation-dissipation theory and incorporating Soret and Dufour effects. An approximate form of the equations is used to obtain a system of similarity equations for the streamwise velocity, the temperature, and the mass fraction. The similarity profiles show important real-gas nonideal-mixture effects, particularly for the temperature, in departing from the incompressible error-function similarity solution. Realistic Schmidt and Prandtl numbers were found to be important to the similarity profiles. A linear, inviscid stability analysis is then performed using the similarity profile, as well as analytical error-function profiles, as its basic flow. The stability analysis shows that the similarity profile has larger growth rates at a given wavelength and a shorter more unstable wavelength than the error-function profiles and than an incompressible flow. The similarity profile also has a larger range of unstable wavelengths than the error-function profiles.