Smith Equation Research Articles

Levinson–Smith Dissipative Equations and Geometry of GENERIC Formalism and Contact Hamiltonian Mechanics

We apply Jacobi’s Last Multiplier theory to construct the non-standard Lagrangian and Hamiltonian structures for the Levinson–Smith equations satisfying the Chiellini integrability condition. Then after a brief exposition of the contact geometry, we explore its connection with the non-standard Hamiltonian structures. We present the formulation of the Levinson–Smith equation in terms of General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) method and also study the gradient-type flow. We give a geometric formulation of GENERIC and apply this to general Levinson–Smith equations.

A turbulence model: Model, analysis, and algorithm

The standard eddy viscosity model is commonly used to simulate turbulence, but there exist difficulties in simulating the backscatter of the non-statistical equilibrium turbulence. In this work, we construct a novel modified eddy viscosity model based on a simplified Cebeci–Smith equation, which is applied to the non-equilibrium turbulence. Moreover, we analyze some properties of the model and propose a fully discrete scheme, which is proven to be stable and convergent. Finally, some numerical examples are presented to test the theoretical prediction of the proposed scheme, and intermittent backscatter is captured and observed by simulating the established model.

The Darboux Polynomials and Integrability of Polynomial Levinson–Smith Differential Equations

We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.

The existence of a stable limit cycle in the Liénard–Levinson–Smith (LLS) equation beyond the LLS theorem

We have examined a class of Liénard–Levinson–Smith (LLS) oscillators having a stable limit cycle which demonstrates the fact that the LLS theorem can only be applied under an extended condition. The problem has been partly raised in a recent communication by Saha et al. (2020) (last paragraph of Sec 4.2.2). For the existence criteria we have provided a perturbative approach through Krylov–Bogolyubov averaging method to address this problem using the concept of energy consumption per cycle. We have also established the idea and its validity through proper numerical demonstration by considering a generalized model system. Such issues have potential utility in nonlinear vibration control.

Lie Algebra Classification, Conservation Laws, and Invariant Solutions for a Generalization of the Levinson–Smith Equation

We obtain the optimal system’s generating operators associated with a generalized Levinson–Smith equation; this one is related to the Liénard equation which is important for physical, mathematical, and engineering points of view. The underlying equation has applications in mechanics and nonlinear dynamics as well. This equation has been widely studied in the qualitative scheme. Here, we treat the equation by using the Lie group method, and we obtain certain operators; using those operators, we characterized all invariants solutions associated with the generalized equation of Levinson Smith considered in this paper. Finally, we classify the Lie algebra associated with the given equation.

Bubble tree convergence of conformally cross product preserving maps

We study a class of weakly conformal $3$-harmonic maps, called associative Smith maps, from $3$-manifolds into $7$-manifolds that parametrize associative $3$-folds in Riemannian $7$-manifolds equipped with $\mathrm{G}_2$-structures. Associative Smith maps are solutions of a conformally invariant nonlinear first order PDE system, called the Smith equation, that may be viewed as a $\mathrm{G}_2$-analogue of the Cauchy-Riemann system for $J$-holomorphic curves. In this paper, we show that associative Smith maps enjoy many of the same analytic properties as $J$-holomorphic curves in symplectic geometry. In particular, we prove: (i) an interior regularity theorem, (ii) a removable singularity result, (iii) an energy gap result, and (iv) a mean-value inequality. While our approach is informed by the holomorphic curve case, a number of nontrivial extensions are involved, primarily due to the degeneracy of the Smith equation. At the heart of above results is an $\varepsilon$-regularity theorem that gives quantitative $C^{1,\beta}$-regularity of $W^{1,3}$ associative Smith maps under a smallness assumption on the $3$-energy. The proof combines previous work on weakly $3$-harmonic maps and the observation that the associative Smith equation demonstrates a certain "compensation phenomenon" that shows up in many other geometric PDEs. Combining these analytical properties and the conformal invariance of the Smith equation, we explain how sequences of associative Smith maps with bounded $3$-energy may be conformally rescaled to yield bubble trees of such maps. When the $\mathrm{G}_2$-structure is closed, we prove that both the $3$-energy and the homotopy are preserved in the bubble tree limit. This result may be regarded as an associative analogue of Gromov's Compactness Theorem in symplectic geometry.

Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

Soil physicochemical properties to evaluate soil degradation under different land use types in a high rainfall tropical region: A case study from South Sulawesi, Indonesia

Intensive cropping in the tropical region always becomes one of important driving forces of soil degradation. The primary aim of this study is to analyze the states and the dynamics of soil physicochemical properties to evaluate soil degradation in the tropical region a high rainfall on agricultural areas in South Sulawesi. A number of soil characteristics were analyzed for physical and chemical properties, and clay minerals with X-ray diffractometer. The degree of soil degradation is determined using Wischmeier and Smith equation. This study reveals that mean annual precipitation in 1979-2016 ranged from 1853.15 to 2981.30 mm/year. For land used for paddy field, palm oil, cacao and coffee plantation, the texture dominated with silt loam-clay loam, cation exchange capacity was 18.63-26.32 cmol+ kg-1, 0.98-2.91% of C-organic, 32-55% of base saturation, 0.1-3.5 cm h-1 of permeability, soil clay minerals were montmorillonite-kaolinite-halloysite, and the index erodibility was 0.3-0.5. Land used for mixed plants and shrubs, the texture dominated with silt loam-sandy clay loam, cation exchange capacity was 18.63-27.12 cmol+ kg-1, 1.09-2.89% of C-organic, 32-55% of base saturation, 0.2-4.9 cm/h of permeability, soil clay minerals were kaolinite-halloysite, and index erodibility was 0.1-0.3. Land use for cultivated in the high intensity of rainfall has changed the physicochemical properties of soils, but cultivated in monoculture has at some degree increased soil erodibility.

Исследование решений уравнений Левинсона – Смита

Исследование решений уравнений Левинсона – Смита

The use of archived precipitation data in the assessment of soil erosion risk in the Świętokrzyskie Province of central-southern Poland

The usefulness of archived IMGW–PIB published reports was discussed as a source of information on high total and intensity precipitation that generates the risk of soil erosion. The study area consist of theŚwiętokrzyskie Province of central-southern Poland. The data were obtained mainly from yearbooks: The Atmospheric Precipitation Yearbooks and Results of pluviographic studies and precipitation of high intensity for the period 1959–1981. The analysis is limited to the occurrence of rainfall events that produced at least 30 mm of rainwater and were classified as A3 or higher on the Chomicz scale of rainstorms and downpours. A total of 247 rainstorms and downpours were recorded at 74 weather stations in the Świętokrzyskie Province. The utilized data sets allowed he estimation of erosivity index values using a simplified Wischmeier and Smith equation. Their erosivity index reached up to 6,387.8 MJ∙mm∙ha−1∙h−1 with a median value of 455.8 MJ∙mm∙ha−1∙h−1. This maximum value of EI was recorded at the Słupia weather station in the southwestern part of the study area. However, higherosivity rainfalls most often occurred in the lower section of the Nida Valley (in the Wiślica weather station). Precipitation characterized by the greatest erosivity occurred in June and July.

Rainfall Characteristics at Makurdi, North–Central Nigeria II

Rainfall characteristics were evaluated in Makurdi in the Guinea Savanna agroclimatic belt of Nigeria. Pluviograhic rainfall data were collected for the period 1985 to 1987. The mean Annual rainfall was 1140mm.The highest rainfall amount per storm event was 71.7mm while the highest six- minute rainfall intensity was 240mmh-1. The mean monthly Kinetic energy (E) using the Wischmeier and Smith (1978) equation ranged from 2.0 to 84.1MJha-1 whereas the value ranged from 2.5 to 128.0MJha-1 using Kowal and Kassam (1976) equation (designated Ek).Conventional rainfall erosivity used for evaluation were the EI30, K.E > 25 and AIm. The ranges of monthly erosivity based on these indices were 24 to 406 MJ.mmha-1h-1, 0.7 to 9.0MJha-1 and 270 to 4280 mm2h-1 respectively. Following Obi and Salako (1995) additional indices, namely, E(A), EkI30, EkIm ,E(A)I30 and EkAIm were evaluated. Mean monthly erosivity values ranges based on these indices were 18 to 471 MJ.mmha-1h-1, 31 to 595 MJ.mmha-1h-1, 76 to 1594 MJ.mmha-1h-1, 218 to 26992 MJ.mmha-1h-1 and 527 to 67293 MJ.mmha-1h-1 respectively. The findings enable better understanding of the rainfall effects on soil erosion in the region. The generally high erosivity values are pointers to the compelling need for soil protective covers in particular and the integration of other serious conservation measures as key strategies for sustainable production in the agro-ecological zone.Keywords: Rainfall, Eosivity, USLE

Coexistence of Near-Field and Far-Field Sources: the Angular Resolution Limit

Passive source localization is a well known inverse problem in which we convert the observed measurements into information about the direction of arrivals. In this paper we focus on the optimal resolution of such problem. More precisely, we propose in this contribution to derive and analyze the Angular Resolution Limit (ARL) for the scenario of mixed Near-Field (NF) and Far-Field (FF) Sources. This scenario is relevant to some realistic situations. We base our analysis on the Smith's equation which involves the Cramér-Rao Bound (CRB). This equation provides the theoretical ARL which is independent of a specific estimator. Our methodology is the following: first, we derive a closed-form expression of the CRB for the considered problem. Using these expressions, we can rewrite the Smith's equation as a 4-th order polynomial by assuming a small separation of the sources. Finally, we derive in closed-form the analytic ARL under or not the assumption of low noise variance. The obtained expression is compact and can provide useful qualitative informations on the behavior of the ARL.

Determination of the moisture sorption behavior of osmotically dehydrated mackerel fillets by means of binary and ternary solutions

In this study, the moisture sorption isotherm of osmotically dehydrated mackerel fillets (Scomber japonicus) was experimentally determined. The fillets were osmotically dehydrated with solutions of salt (NaCl) (120 and 180 g per liter of solution) or in combination with sugar (350 to 700 g per liter of solution). The sorption isotherms were determined using the static gravimetric methodology with six salts for the water activity range of 0.33-0.98 at 5 degrees C and 25 degrees C. All the sorption curves were found to be type III. Temperature and the final tissue salt content had significant (p < 0.05) effects on the sorption isotherms. A regression program was used to fit the Halsey, Oswin and Smith moisture sorption isotherm models. Oswin equation gave the best fit for the whole range of water activity and temperatures. The Smith equation only presented valuable results for the mackerel fillets samples with the higher salt content.

Effect of eutectic particles on the grain size control and the superplasticity of aluminium alloys

Abstract Aluminium alloys containing eutectic particles of the Al–Ni, Al–Mg–Si, Al–Ni–Ce and Al–Cu–Ce systems are investigated. The particles which control grain growth and stimulate grain nucleation are studied. The Zener–Smith law about dependence between grain size and particle parameters is confirmed and experimental coefficients are found. Experimental coefficients of the Zener–Smith equation obtained in this study depend on the particle size and differ from theoretical coefficients proposed by Zener and Smith. Some alloys with grain size about 3 μm demonstrate very good superplasticity indicators, namely: the strain rate sensitivity index m = 0.5–0.6 and the elongation over 400% at constant strain rate 5 × 10 −3 s −1 .

Internal waves in a compressible two-layer model atmosphere: Hamiltonian description

Slow flows of an ideal compressible fluid (gas) in the gravity field in the presence of two isentropic layers are considered, with a small difference of specific entropy between them. Assuming irrotational flows in each layer [that is ${\bf v}_{1,2}=\nabla\phi_{1,2}$], and neglecting acoustic degrees of freedom by means of the conditions ${div}(\bar\rho(z)\nabla\phi_{1,2})\approx0$, where $\bar\rho(z)$ is a mean equilibrium density, we derive equations of motion for the interface in terms of the boundary shape $z=\eta(x,y,t)$ and the difference of the two boundary values of the velocity potentials: $\psi(x,y,t)=\psi_1-\psi_2$. A Hamiltonian structure of the obtained equations is proved, which is determined by the Lagrangian of the form ${\cal L}=\int \bar\rho(\eta)\eta_t\psi \,dx dy -{\cal H}\{\eta,\psi\}$. The idealized system under consideration is the most simple theoretical model for studying internal waves in a sharply stratified atmosphere, where the decrease of equilibrium gas density with the altitude due to compressibility is essentially taken into account. For planar flows, a generalization is made to the case when in each layer there is a constant potential vorticity. Investigated in more details is the system with a model density profile $\bar\rho(z)\propto \exp(-2\alpha z)$, for which the Hamiltonian ${\cal H}\{\eta,\psi\}$ can be expressed explicitly. A long-wave regime is considered, and an approximate weakly nonlinear equation of the form $u_t+auu_x-b[-\hat\partial_x^2+\alpha^2]^{1/2}u_x=0$ (known as Smith's equation) is derived for evolution of a unidirectional wave.

INTEGRATED SYSTEM FOR AIR POLLUTION AROUND REFINERIES

A mathematical model for integrated air pollution modelling around refineries is built and named as Computerized System for Integrated Air Pollution Modelling Around Oil Refineries (CSIAPMAOR).The model based on Gaussian equation to estimate concentration of pollutants (SO2, NO2, CO, Particulates) that are emitted from a continuous air pollution elevated source. The model is designed by using Visual Basic as a main core of the system and linked with auxiliary models such as ArcMap (GIS), Surfer software, Ms-Excel and Ms-Access. The model has flexibility to select either rural, urban or stability (Smith equation) wind speed profile. It also has the option of using three types of dispersion coefficients equations for rural condition (PGT-Briggs-Martin equation) and one equation for urban condition (Briggs equation).The model has many options to display results as concentrations versus center line-downwind distance or as Three-Dimensional (3D) map. The model can compute maximum concentration with the contribution of each stack to the overall maximum concentration. Moreover the model has the ability to perform a sensitivity analysis for the effect of the most important parameters according to the Gaussian equation. AL-Doura Oil Refinery was taken as a case study using the available observed data of two sites1 and 2 for periods 15th -21st and 23rd -29th August 1997 in order to check the performance potential of the model. Results showed that Briggs equation for dispersion coefficients with rural wind speed profile has the best degree of agreement with the observed values of 0.86, 0.90 for SO2; 0.69, 0.80 for NO2; 0.73, 0.79 for CO; 0.63, 0.60 for particulates at site 1 and 2 respectively. It is found, that for AL-Doura Oil Refinery stacks number 6, 2, 7, and 3 have a large contribution on the overall maximum concentration. The model demonstrates the influence of atmospheric stability, wind speed, emission rate, exit velocity, physical height, exit temperature and rural-urban area in reducing the concentrations of pollutants. Sensitivity analysis shows that the concentrations are sensitive to stability class in comparison with other input parameters.

Determination of kinetic and equilibrium parameters of the batch adsorption of Cr(VI) onto waste acorn of Quercus ithaburensis

The efficiency of waste acorn of Quercus ithaburensis (WAQI) as an adsorbent for removing Cr(VI) ions from synthetic wastewater has been studied. Batch adsorption experiments were carried out as a function of pH, adsorbent mass, Cr(VI) ion concentration, agitation rate and temperatures. Maximum metal sorption was found to occur at initial pH 2.0. The adsorption capacity of WAQI was found to be 31.48 mg g −1 for initial Cr(VI) concentration of 400 mg L −1 at 25 °C. Batch adsorption models, based on the assumption of the pseudo first-order, pseudo second-order mechanism and Elovich equation were applied to examine the kinetics of the adsorption. The results showed that kinetic data were followed fitting the pseudo second-order model than the pseudo first-order and Elovich equation. Thermodynamic parameters such as Δ H°, Δ S° and Δ G° were calculated. The adsorption process was found to be endothermic and spontaneous. The results were analyzed by the Langmuir, Freundlich, Dubinin–Radushkevich (D–R), Temkin, Frumkin, Harkins–Jura, and Smith equation using linearized correlation coefficient at different temperature. The characteristic parameters for each isotherm have been determined. Models and the isotherm constant were evaluated depending on temperature. Langmuir and Freundlich equation is found to best represent the equilibrium data for Cr(VI) ions–waste acorn systems.

Uniformity of Strawberry Yield and Incidence of Botrytis Fruit Rot in an Annual Production System.

A plant yield and disease incidence uniformity trial was conducted to provide information concerning the efficiency and precision of field trials used to evaluate Botrytis fruit rot control methods on strawberry. Fruit yield and Botrytis fruit rot incidence were recorded for individual strawberry plants of cultivars Sweet Charlie and Camarosa grown in an annual production system over two growing seasons. A nested analysis of variance model was used to measure plot edge effects and to obtain variance components to describe the relationship between plot size and plot variance. Mean seasonal yield for Sweet Charlie was 599 g/plant and for Camarosa 972 g/plant. Mean seasonal fruit rot incidence was 10.3% for Sweet Charlie and 3.0% for Camarosa. Plants growing on the edge of plots next to aisles had higher yields (637 versus 577 g/plant for Sweet Charlie and 1047 versus 923 g/plant for Camarosa), but there was no edge effect for disease incidence. Smith's equation was used to determine the relationship between plot size and plot variance within mulched beds. Smith's index was relatively high for yield (0.92 and 0.95) and Botrytis fruit rot incidence (0.91 and 0.69) for Sweet Charlie and Camarosa, respectively. This suggests a relatively uniform distribution of seasonal yield and disease incidence among plants within beds. Using plot variances estimated from Smith's equation, the power of hypothesis tests to discriminate hypothetical treatment effects of different magnitudes was examined for each cultivar. Power analysis suggests that treatment effects ranging from 11.3 to 21.3% of the mean seasonal yield observed for Sweet Charlie and 8.6 to 16.5% of the mean seasonal yield observed for Camarosa can be detected with a power of 0.80 using four replicates of each treatment group and plot sizes ranging from 32 to eight plants/plot. For Botrytis fruit rot incidence, a power of 0.80 is achieved with treatment differences ranging from 24.9 to 44.1% of mean seasonal Botrytis fruit rot incidence for Sweet Charlie and from 44.1 to 64.9% of the mean observed for Camarosa using plot sizes ranging from 32 to 8 plants/plot. Before planting, the crown diameter and leaf number of transplants were recorded. There was no consistent significant correlation between any of these growth traits and seasonal yields or Botrytis fruit rot incidence. However, there was a consistent positive correlation for both cultivars between yield and crown diameter and also yield and leaf number during the months of December and January in the early part of the season (0.1513 ≤ r ≤ 0.314 for crown diameter and 0.106 ≤ r ≤ 0.264 for leaf number).

Valence force field theory, elastic, acoustic and thermodynamic properties and a potential function of TeO 2 glass

A short survey has been made on the extensive work that is being done on the pressure derivatives of the second order elastic constants (SOEC) to ascertain various properties of substances. Hence an attempt has been made to correlate the pressure derivatives to some properties of the substances. Thus some equations have been derived to correlate the Grüneisen parameter which is evaluated from Schofield's equations and Bhatia-Singh's (BS) parameters. They have been used to compute the longitudinal ( γ g L) and transverse ( γ g T) Grüneisen constants. γ g L calculated by different methods agree well with experiment. γ g T obtained from BS parameters gives rather higher value while Schofield's equations give results in agreement with experiment. The DeLaunay–Nath–Smith (DNS) equation has been used to derive a relation to compute γ g el (elastic). A method has been extended to calculate the third order elastic constants (TOEC) and it is found to give excellent values of TOECs in agreement with experiment. The absorption band position of TeO 2 has been predicted to occur at 276 cm −1. The phonon dispersion curves have been calculated through BS equations for TeO 2. Several other properties of TeO 2 have been computed such as thermal Grüneisen parameter γ g th, its pressure derivatives ( γ g th)′≡(d γ g th/d P), the pressure variation of bulk modulus C 1≡(d K T /d P) T and its pressure derivatives that is (d C 1/d P) T which is in turn related to ( γ g th)′, the heat capacity at constant volume C V, and the second Grüneisen constant Q. In some cases we calculated these quantities by different methods and the agreement between them is good. Besides we evaluated δ T AG the Anderson Grüneisen parameter. Another important aspect of the present investigations is the formulation of the potential function (PF) of TeO 2 from which we calculated SOECs and these are found to be in excellent agreement with experiment. All other properties mentioned already have also been calculated through the use of the newly formulated PF and the calculated values obtained through various other equations are in good agreement with those obtained from PF. According to valence force field (VFF) all atomic forces can be resolved into bond bending β and bond stretching α forces. It is shown that TeO 2 does not satisfy Martins unity rule. Hence it is concluded that there is an effective dynamic charge on Te in TeO 2. Using the experimental elastic constants the bond bending force β and bond stretching force α and also their pressure derivatives have been evaluated. In addition the reststrauhlen optic frequency ω has been calculated. A self consistent check has been made by evaluating C 44 through the calculated values of α and β.

Valence force field theory, elastic, acoustic and thermodynamic properties and a potential function of TeO2 glass

A short survey has been made on the extensive work that is being done on the pressure derivatives of the second order elastic constants (SOEC) to ascertain various properties of substances. Hence an attempt has been made to correlate the pressure derivatives to some properties of the substances. Thus some equations have been derived to correlate the Grüneisen parameter which is evaluated from Schofield's equations and Bhatia-Singh's (BS) parameters. They have been used to compute the longitudinal (γgL) and transverse (γgT) Grüneisen constants. γgL calculated by different methods agree well with experiment. γgT obtained from BS parameters gives rather higher value while Schofield's equations give results in agreement with experiment. The DeLaunay–Nath–Smith (DNS) equation has been used to derive a relation to compute γgel (elastic). A method has been extended to calculate the third order elastic constants (TOEC) and it is found to give excellent values of TOECs in agreement with experiment. The absorption band position of TeO2 has been predicted to occur at 276 cm−1. The phonon dispersion curves have been calculated through BS equations for TeO2. Several other properties of TeO2 have been computed such as thermal Grüneisen parameter γgth, its pressure derivatives (γgth)′≡(dγgth/dP), the pressure variation of bulk modulus C1≡(dKT/dP)T and its pressure derivatives that is (dC1/dP)T which is in turn related to (γgth)′, the heat capacity at constant volume CV, and the second Grüneisen constant Q. In some cases we calculated these quantities by different methods and the agreement between them is good. Besides we evaluated δTAG the Anderson Grüneisen parameter. Another important aspect of the present investigations is the formulation of the potential function (PF) of TeO2 from which we calculated SOECs and these are found to be in excellent agreement with experiment. All other properties mentioned already have also been calculated through the use of the newly formulated PF and the calculated values obtained through various other equations are in good agreement with those obtained from PF. According to valence force field (VFF) all atomic forces can be resolved into bond bending β and bond stretching α forces. It is shown that TeO2 does not satisfy Martins unity rule. Hence it is concluded that there is an effective dynamic charge on Te in TeO2. Using the experimental elastic constants the bond bending force β and bond stretching force α and also their pressure derivatives have been evaluated. In addition the reststrauhlen optic frequency ω has been calculated. A self consistent check has been made by evaluating C44 through the calculated values of α and β.