The mathematical modelling of the nonlinear heat transport in thin plate

The approximations of the nonlinear heat transport problem are based on the finite volume (FM) and averaging (AM) methods [1,2]. This procedures allows reduce the nonlinear 2‐D problem for partial differential equation (PDE) to a initial‐value problem for a system of 2 nonlinear ordinary differential equations(ODE) of first order in the time t or to a initial‐value problem for one nonlinear ODE of first order with two nonlinear algebraic equations. Netiesinio šilumos pernešimo plonose plokštelėse matematinis modeliaivimas Santrauka Darbe sprendžiamas netiesinis šilumos pernešimo uždavinys. Uždavinio specifika yra ta, kad šilumos pernešimas vyksta labai plonose plokštelese, todel nagrinejamas vienmatis apibendrintas modelis. Si redukcija atliekama baigtiniu tūriu ir vidurkinimo metodais. Gautoji dvieju netiesiniu paprastuju diferencialiniu lygčiu sistema yra sprendžiama skaitiškai. Išnagrinetas ir alternatyvus varijantas, kai po redukcijos gaunama viena netiesine paprastoji diferencialine lygtis bei dvi papildomos algebrines lygtys. First Published Online: 14 Oct 201

The vortex formation in a horizontal finite cylinder by alternating electric current

We have investigated and calculated the distribution of electromagnetic fields and forces induced by a three‐ phase axially‐symmetric system of electric current with six electrodes in a cylinder of a finite length [1, 3]. In this paper the alternating current is fed to each of nine discrete circular conductors‐electrodes, which are placed on the internal wall of the cylinder. This new mathematical model describes a real device [7], which transforms electric energy into heat. The viscous incompressible flow of weakly electroconductive liquid‐electrolyte is obtained by the finite‐difference method, using the monotonous vector finite difference schemes. The average axially‐symmetric motion of electrolyte and vortex distribution in a cylinder has been considered. The dependence of electromagnetic forces and velocity distribution at the inlet of the cylinder is investigated in the case of: • the vortex formation from the Lorentz force inside the cylinder by the electrode; • the vortex‐breakdown from the swirl velocity at the inlet of the cylinder. Straipsnyje yra nagrinejamas skysto elektrolito judejimas baigtiniame cilindre, veikiamas kintančio elektromagnetinio lauko. Panaudojant savo anksčiau sukurta metoda suvidurkintu laike elektromagnetiniu jegu reikšmiu gavimui, autoriai formuluoja stacionaru Navier‐Stokso uždavini klampiam nespudžiam elektrolito tekejimui aprašyti. Sukurto matematinio modelio skaitiniam sprendimui yra pasiūloma originali monotonine baigtiniu skirtumu schema. Yra pateikiami skaitiniu eksperimentu rezultatai su 9 žiediniais laidininkais‐elektrolitais, tolygiai paskirstytais vidiniame cilindro paviršiuje. Autoriai analizuoja gaunamus cilindre srautus priklausomai nuo elektromagnetiniu jegu pasiskirstymo (skirtingos elektrodu pajungimo schemos) ir sukurinio greičio cilindro iejime. Labiausiai autorius domina salygos, kada cilindro viduje susidaro sūkuriai. First Published Online: 14 Oct 201

A model of intensive oil burnout from glass fabric

This paper continues the previous investigations [1]‐[3] for process of intensive oil burnout in glass fabric. Aliejaus išdeginimo iš stiklo gaminių matematinis modelis Santrauka Darbe toliau sistemingai nagrinejami stiklo gaminimo procesu matematiniai modeliai. Tiriama ivairiu faktoriu (stiklo mases judejimo greičio, metalo plokšteliu savybiu, gaminio storio) itaka stiklo gaminio kokybei. Kadangi aliejaus išdeginimas vyksta labai aukštoje temperatūroje ir labai trumpai, tai matematiniai modeliai leidžia detaliai ištirti procesa bei parinkti optimalias parametru reikšmes. Matematini modeli sudaro nestacionari lygtis, aprašanti difuzijos, konvekcijos ir chemines reakcijos procesus. Vidurkinimo metodu uždavinys suvedamas i vienmati modeli. Chemines reakcijos greitis modeliuojamas panaudojant Aremijaus modeli. Pateikiami skaičiavimo eksperimento rezultatai. First Published Online: 14 Oct 201

Inverse heat transport problems for coefficients in two‐layer domains and methods for their solution

In various fields of science and technology it is often necessary to solve inverse problems, where from measurements of state of the system or process it is required to determine a certain typesetting of the causal characteristics. It is known that infringement of the natural causal relationships can entail incorrectness of the mathematical stating of inverse problems. Therefore the development of efficient methods for solving such problems allows one to considerably simplify experimental research and to increase the accuracy and reliability of the obtained results due to certain complication of algorithms for processing the experimental data. The problem of determination of thermal diffusivity coefficients considering other known characteristics of heat transport process is among incorrect inverse problems. These inverse problems for coefficients are quite difficult even in the case of homogeneous media. In this paper it is supposed that the heat transport equation is non‐homogeneous and an algorithm for determination of the thermal diffusivity coefficients for both the media is proposed. At the first step, the non‐homogeneous inverse problem with piecewise‐constant function of non‐homogeneity is solved. For this auxiliary inverse problem, the proposed method allows one to determine both the coefficients of thermal diffusivity and to restore the heat transport process without any additional information, i.e. the algorithm also solves the direct problem. Then the initial non‐homogeneous inverse problem with a piecewise‐continuous function of non‐homogeneity is solved. The proposed method reduces the non‐homogeneous inverse problem for coefficients to a set of two transcendent algebraic equations. Finally, the analytical solution to direct problem is obtained using Green's function. Atvirkštiniai šilumos laidumo  uždaviniai dvisluoksnėse srityse ir jų sprendimo metodai Santrauka Ivairiose mokslo ir technologijos srityse dażnai tenka spresti atvirkštinius użdavinius, kada remiantis sistemos ar proceso būsenos parametru matavimais reikia nustatyti priežastines charakteristikas. Yra żinoma, kad natūraliu prieżastiniu priklausomybiu nepaisymas gali nulemti neteisinga atvirkštinio użdavinio matematine formuluote. Todel efektyvūs tokiu użdaviniu sprendimo metodai leis żymiai supaprastinti eksperimentinius tyrimus, padidinti gaunamu rezultatu tiksluma ir patikimuma, jeigu bus pritaikyti tam tikri sudetingesni eksperimentiniu rezultatu apdorojimo būdai. Difuzijos koeficientu nustatymas naudojant kitas żinomas šilumos laidumo proceso charakteristikas priklauso nekorektišku użdaviniu kategorijai. Atvirkštiniai użdaviniai koeficientams yra sudetingi net ir homogeninese terpese. Šiame darbe daroma prielaida, kad terpe nehomogeniška, ir pasiūlytas algoritmas difuzijos koeficientu nustatymui tokiu atveju. Pirmajame etape sumażinamas nehomogeninis atvirkštinis użdavinys, laikant, kad nehomogeniškumas aprašomas dalimis pastoviomis funkcijomis. Šiam pagalbiniam atvirkštiniam użdaviniui siūlomas metodas leidżia apibreżti abu šilumos difuzijos koeficientus ir atkurti šilumos laidumo proceso eiga be papildomos informacijos, t.y., algoritmas sprendżia taip pat ir tiesiogini użdavini. Po to yra sprendżiamas atvirkštinis pradinis użdavinys esant dalimis tolydżiai nehomogeniškuma aprašančiai funkcijai. Siūlomas metodas redukuoja nehomogenini atvirkštini użdavini i dvieju transcendentiniu lygčiu sprendima. Taip pat yra gautas tiesioginio użdavinio analizinis sprendinys, taikant Gryno formule. First Published Online: 14 Oct 201

Influence of input data on the solution of the multiphase liquid flow model

In the paper, a filtration model for immiscible ‐ compressible and incompressible ‐ two‐phase liquids is considered. The 1D case of the model is analyzed in detail and it is shown that approximation of the Buckley‐Leverett function by a step‐wise linear function leads to new fictitious jumps of the solution. Skusčio tekėjimo modelio sprendinio priklausomybė nuo pradinių duomenų  Santrauka Darbe nagrinejamas dvisluoksnis filtracijos modelis spudžiuir nespudžiuskysčiuatvejais. Detaliai išnagrinetas vienmatis modelis ir parodyta, kad Buckley‐Leverett funkcijaaproksimuo‐jant dalimis tiesine funkcija, sprendinyje atsiranda fiktyvus šuolis. First Published Online: 14 Oct 201

A mathematical model for the heat treatment of glass fabric sheets

During the industrial process of glass fabric manufacture in Latvia, an oil that has earlier been added to the fabric must be removed. To accomplish this, the fabric is passed through a furnace where the oil is removed by burning. It is known that the oil removal process reduces the tensile strength of the fabric. The burning process is analysed via a simple mathematical model that allows the furnace parameters to be optimized. Using some of the well-established properties of glass, it is shown that it is the cooling rather than the heating process that adversely affects the final product. A thermoviscoelastic model is developed to predict the permanent stress in the glass fabric. Some suggestions for modifications to the process are also examined, and it is shown that, to a large extent, the strength of the fabric may be preserved without significantly adding to the cost or intricacy of the production process

Calculation of electromagnetic fields, forces and temperature in a finite cylinder

The distribution of electromagnetic fields, forces and temperatures induced by a three‐phase axially‐symmetric system of electric current in a conducting cylinder of a finite length has been calculated. An original method was used to calculate the radial and axial components of the magnetic fields and the mean values of electromagnetic forces, as well as the azimuthal component of the electric field and of the mean curl of the electromagnetic forces. For finding the source term in the temperature equation we applied an approximation of the heat transport problem based on the finite‐difference method. Such a procedure allows one to calculate the distribution of temperatures inside the cylinder depending on that of currents in the wires. Elektromagnetinių laukų, jėgų ir temperatūros skaičiavimas baigtiniame cilindre Santrauka Apskaičiuojamas elektromagnetiniu lauku, jegu ir temperatūru pasiskirstymas trifazes elektros sroves ašineje simetrineje sistemoje baigtiniame cilindre. Panaudotas originalus metodas apskaičiuojant radialine ir ašine komponentes, vidurkines elektromagnetiniu jegu reikšmes, elektrinio lauko azimutine komponente ir elektromagnetiniu jegu rotoriu. Temperatūros šaltinio nustatymui, naudojama šilumos laidumo uždavinio aproksimacija baigtiniu skirtumu metodu. Tokia procedūra leidžia skaičiuoti temperatūros pasiskirstyma cilindro viduje, priklausomai nuo sroviu jo paviršiuje. First Published Online: 14 Oct 201