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derivative at n and n + 1
This method is more accurate and stable than the explicit method.
Consider the heat equation on x E [0, 1] with initial condition u(x, 0) = sin(rcx) and boundary conditions u(0, t) = u(1, t) = 0. Using the explicit method:
un+1 = un + A un - 2un + un
Parameters: i i i-1i i+1
Ax = 0.1, At = 0.005, A = 0.5.
Solution proceeds iteratively until t = 0.1. Challenges in Numerical Solutions
• Stability: Explicit methods require small time steps to remain stable.
• Accuracy: High accuracy demands fine grids, which increase computational cost.
• Boundary Conditions: Handling complex boundaries can be challenging.
Conclusion: The finite difference method provides an efficient approach for solving parabolic PDEs like the heat equation. The choice between explicit, implicit, and Crank-Nicholson methods depends on the trade-offs between stability, accuracy, and computational cost. References
1. Smith, G. D. (1985). Numerical Solution of Partial Differential Equations.
2. Morton, K. W., Mayers, D. F. (2005). Numerical Solution of Partial Differential Equations.
3. Thomas, J. W. (1995). Numerical Partial Differential Equations: Finite Difference Methods.
© Hudayberenova A., Yazdurdyyev M., 2024
УДК 551.594.25
Khuchunaev B.M.
Doctor of Physical and Mathematical Sciences, Head of Laboratory, Federal State Budgetary Institution «High-Mountain Geophysical Institute»,
Russian Federation, Nalchik Budaev A.Kh., Junior Researcher, Federal State Budgetary Institution «High-Mountain Geophysical Institute»,
Russian Federation, Nalchik Daov I.S., Junior Researcher, Federal State Budgetary Institution «High-Mountain Geophysical Institute»,
Russian Federation, Nalchik
THE EFFECT OF ELECTRIC CHARGE ON THE ICE-FORMING PROPERTIES OF ALUMINUM OXIDE NANOSTRUCTURES
Abstract
The paper presents the research methodology and the results of laboratory experiments to study the
effect of electric charge on aluminum oxide nanostructures on their ice-forming properties. It is known that nanoscale aluminum oxide particles, depending on their shape, can amplify the electric field thousands of times near their surface, which can lead to a significant increase in the temperature threshold for the formation of an ice phase on reagent particles.
Keywords:
cloud chamber, temperature, electric field strength, electric charge, specific yield, reagent, nanotube clusters, aluminum oxide.
Introduction. To date, when actively influencing cloud processes, Agi iodide aerosol is most often used. To improve the quality of the reagent or develop a new effective composition for use in thunderclouds, it is necessary to consider the electrical phenomena accompanying thermodynamically irreversible phase transitions. In such phase transitions, the surface of the reagents is electrified [1]. With a high degree of crystallization process in the «water-ice» boundary layer, an effect similar to the Zavoysky effect occurs: there is an abnormally high resonant absorption of radiofrequency field energy, accompanied by a high electric field intensity [2].
It is known that the development of natural water aerosols, clouds and mists is accompanied by the formation of electric charges in them. As follows from observations and theory, the temperature range of -10 4 -15 °C approximately corresponds to typical cloud crystallization temperatures [2]. On the other hand, [3] discovered during experimental flights that the formation of powerful electric fields in clouds is directly related to their crystallization.
In 1960, the mechanism of charge separation in clouds was discovered. It is associated with the explosion of crystallizing water droplets [4]. For the formation of a 102 С charge cloud in a cubic kilometer, it is enough for several drops of 1 dm3 to explode. Since a drop in an explosion, as a rule, splits into several fragments, the process of increasing the field strength in the cloud should have an avalanche-like character.
The method of conducting experiments. To conduct research, a set of equipment has been created, which includes a large cloud chamber, flat capacitor plates, a high-voltage rectifier, a device for sublimating the reagent, an ultrasonic steam generator, a thermoanemometer, a video camera, scales, and an optical microscope. In laboratory experiments, the method of charging reagent particles in the electric field of a flat capacitor was used [5].
A certain amount of the prototype is weighed on an electronic scale and loaded onto a graphite substrate of a reagent sublimation device. To study the effect of electric charge on the specific yield of ice-forming nuclei, thermostatically controlled substrates are installed at the bottom of a large cloud chamber. An artificial cloud environment is created in the camera. High voltage is applied to the capacitor plates from a high-voltage rectifier and high-temperature sublimation is carried out. During the distillation, continuous video recording and measurement of the particle flow rate are carried out. With the appearance of the first ice crystals in the field of view, the substrates are opened. After the ice crystals settle, the substrates are removed from the chamber and examined under an optical microscope.
The specific yield of ice-forming nuclei is determined by the formula:
— «jr
N = SJ—,
where Sch is the area of the reagent application chamber, mm2; Sfr - the area of the frame in the field
of view of the microscope, mm2; mr - the mass of the introduced reagent, g; nfr - the number of crystals in the frame, m-2.
Results of the study. The connection of the charge on aluminum oxide clusters with the electric field strength is investigated. It is found that when a negative potential is applied to one of the plates of a flat capacitor
mr
and the other plate is grounded, the particles acquire a positive charge. The flow of reagent particles is deflected onto a plate with a negative potential (Figure 1, a). When a positive potential is applied to one of the plates and the other plate is grounded, the particles acquire a negative charge. The flow of charged particles is deflected onto a plate with a positive potential (Figure 1, b).
Fig. 1 - Deviation of the reagent particle flow in an electric field (a - polarity «0», «-»; b - polarity «0», «+»)
Table 1 and Figure 2 show the dependence of the specific charge on aluminum oxide particles on the electric field strength.
Each point on the graph represents the average value of the experiments for each value of the electric field strength. Therefore, the table shows the average value of Q=q/m.
Table 1
Dependence of the specific charge on aluminum oxide particles on the electric field strength
Electric field strength, V/cm Specific charge, x10-5, C/kg
Negatively charged particles Positively charged particles
750 6,86 3,36
1500 1,67 2,52
2250 2,01 1,61
3000 0,59 0,58
Electric field strength, V/cm
Fig. 2 - Dependence of the specific charge on aluminum oxide particles on the electric field strength
With a decrease in the electric field strength, the values of the specific electric charge on negatively and positively charged aluminum oxide particles increase. At the same time, the absolute values of the specific charge on negatively charged particles are 2 times higher than the values of the specific charge on positively charged particles. This is due to the peculiarities of the formation of positive and negative charges on the reagent particles.
Additionally, experiments conducted to determine the specific yield of ice-forming nuclei showed that in the temperature range of -10...-12 °C, negatively charged particles give a specific yield of ice-forming nuclei of the order of 1012 particles per 1 gram of reagent, which is 2 times higher than positively charged ones. Conclusion
A complex of laboratory equipment and a method of conducting experiments for the study of aluminum oxide nanostructures have been developed. It is found that the electric charge on the particles affects their specific yield of ice-forming nuclei.
With a decrease in the electric field strength, the values of the specific electric charge on negatively and positively charged aluminum oxide particles increase.
The specific yield of ice-forming nuclei showed that in the temperature range of -10...-12 °C, negatively charged particles give a specific yield of ice-forming nuclei of the order of 1012 particles per 1 gram of reagent. References
1. Kachurin L.G., Bekryaev V.I. Investigation of the electrification process of crystallizing water. - Reports of the Academy of Sciences USSR, 1960. vol. 130, №1, pp. 57-60.
2. Kachurin L.G. Physical bases of influence on atmospheric processes: An experiment. atmospheric physics: [Study for universities on spec. «Meteorology»] / L.G. Kachurin. - L.: Hydrometeoizdat. 1990. - 462 p.
3. Imyanitov I.M., Chuvaev A.P. On the question of the main processes leading to electrification in thunderclouds. Proceedings of the MGO, 1957. issue 67 (129).
4. Mason. B.J., Maybank J. The fragmentation and electrification of freezing water drops. Quarterly Journal of the Royal Meteorological Society, 1960, 86(371), 176-185.
5. Khuchunaev B.M., Gekkieva S.O., Budaev A.Kh. Laboratory studies of the effect of electric field strength on the specific charge on reagent particles formed during the sublimation of pyrotechnic compositions / «Science. Innovation. Technologies», 2021 - №4. pp. 209-226.
© Khuchunaev B.M., Budaev A.Kh., Daov I.S., 2024
УДК 53
Owezov B.,
student. Pudakov B.,
teacher.
Oguzhan Egineering and Technology University of Turkmenistan.
Ashgabat, Turkmenistan.
APPLICATIONS OF MATHEMATICAL METHODS FOR ANALYZING BIOLOGICAL DATA
Abstract
The increasing complexity of biological data has made mathematical methods indispensable for
