scholarly journals Adiabatic Compression of Ideal Collisionless Gas in One-Dimensional Space

2020 ◽  
pp. 6-12
Author(s):  
Д. А. Быковских ◽  
В. А. Галкин
Keyword(s):  
Analytical Solution ◽  
Exact Solutions ◽  
Particle Velocity ◽  
Software Verification ◽  
Distribution Density ◽  
Dimensional Space ◽  
Adiabatic Compression ◽  
One Dimensional ◽  
Over Time

В статье рассматривается задача об адиабатическом сжатии бесстолкновительного газа с подвижной и неподвижной границами в одномерном пространстве. Для этой задачи получен класс точных решений. Идея нахождения класса точных решений заключается в определении плотности распределения молекул в пространстве координат и скоростей с течением времени. Поскольку пространство скоростей дискретное, то для вычисления макроскопических величин необходимо суммировать плотность распределения частиц по скоростям. Представлены результаты сравнения численного исследования методом Монте-Карло и аналитического решения задачи при различном числе частиц и скоростях движения стенки. Выполнена оценка погрешностей результатов. Полученный класс аналитического решения можно использовать для верификации комплексов программ. The study focuses on adiabatic compression of collisionless gas with moving and fixedboundaries in onedimensionalspace. A class of exact solutions is found. The key concept for findingthese exact solutions is the determination of the molecule distribution density in the coordinate and velocityspaces over time. Since the velocity space is discrete, the particle velocity distribution density is integratedover the velocities to obtain the macroscopic gas properties. The analytical solution and numerical MonteCarlo solution results are compared for different particle numbers and boundary velocities, and the errorsare performed. The class of exact solutions can be used for software verification.

The Method of Rotated Solutions: A Highly Efficient Procedure for Code Verification

2020 ◽  
Author(s):  
Marc Horner
Keyword(s):  
Analytical Solution ◽  
Analytical Solutions ◽  
Source Code ◽  
Numerical Algorithms ◽  
Coordinate Transformations ◽  
Code Verification ◽  
Efficient Procedure ◽  
One Dimensional ◽  
Engineering Problems

Abstract Code verification provides mathematical evidence that the source code of a scientific computing software platform is free of bugs and that the numerical algorithms are consistent. The most stringent form of code verification requires the user to demonstrate agreement between the formal and observed orders of accuracy. The observed order is based on a determination of the discretization error, and therefore requires the existence of an analytical solution. One drawback of analytical solutions based on traditional engineering problems is that most derivatives are identically zero, which limits their scope during code verification. The Method of Rotated Solutions is introduced herein as a methodology that utilizes coordinate transformations to generate additional non-zero derivatives in the numerical and analytical solutions. These transformations extend the utility of even the simplest one-dimensional solutions to be able to perform more thorough evaluations. This paper outlines the rotated solutions methodology and provides an example that demonstrates and confirms the utility of this new technique.

scholarly journals Analytical solution of one-dimensional Pennes’ bioheat equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1084-1092
Author(s):  
Hongyun Wang ◽  
Wesley A. Burgei ◽  
Hong Zhou
Keyword(s):  
Analytical Solution ◽  
Radiofrequency Energy ◽  
Bioheat Equation ◽  
Surface Cooling ◽  
One Dimensional ◽  
The Fourier Transform ◽  
The One ◽  
Parameter Values ◽  
Mathematical Derivation ◽  
Effect Of Surface

Abstract Pennes’ bioheat equation is the most widely used thermal model for studying heat transfer in biological systems exposed to radiofrequency energy. In their article, “Effect of Surface Cooling and Blood Flow on the Microwave Heating of Tissue,” Foster et al. published an analytical solution to the one-dimensional (1-D) problem, obtained using the Fourier transform. However, their article did not offer any details of the derivation. In this work, we revisit the 1-D problem and provide a comprehensive mathematical derivation of an analytical solution. Our result corrects an error in Foster’s solution which might be a typo in their article. Unlike Foster et al., we integrate the partial differential equation directly. The expression of solution has several apparent singularities for certain parameter values where the physical problem is not expected to be singular. We show that all these singularities are removable, and we derive alternative non-singular formulas. Finally, we extend our analysis to write out an analytical solution of the 1-D bioheat equation for the case of multiple electromagnetic heating pulses.

Improved thin-layer chromatographic determination of phospholipids in gastric aspirate from newborns, for assessment of lung maturity.

1988 ◽  
Vol 34 (4) ◽  
pp. 736-738 ◽  
Author(s):  
D Serrano de la Cruz ◽  
E Santillana ◽  
A Mingo ◽  
G Fuenmayor ◽  
A Pantoja ◽  
...  
Keyword(s):  
Thin Layer ◽  
Fetal Lung ◽  
Chromatographic Determination ◽  
Solvent Mixture ◽  
Gastric Aspirate ◽  
One Dimensional ◽  
Acetic Acid Water ◽  
Chromatographic Resolution ◽  
Lung Maturity

Abstract This one-dimensional thin-layer chromatographic method is used for assay of phospholipids in the gastric aspirate of newborns. The solvent mixture (chloroform/hexane/methanol/glacial acetic acid/water, 12/7/4/3/0.3 by vol) completely resolves lecithin, sphingomyelin, phosphatidylinositol, phosphatidylserine, phosphatidylethanolamine, and phosphatidylglycerol. The method is simple, precise, inexpensive, and rapid (chromatographic development takes less than 25 min) and gives high chromatographic resolution. We used this method to determine the lecithin/sphingomyelin densitometric ratio (L/S ratio) and the phosphatidylglycerol percentage in 200 samples of gastric aspirate and found an L/S ratio of 2.5 to be a satisfactory cutoff value for distinguishing fetal lung maturity and immaturity. We confirmed that the presence of phosphatidylglycerol excluded the possibility of respiratory distress.

Formalism for the determination of intermediate stress gradients using X-ray diffraction

2009 ◽  
Vol 42 (2) ◽  
pp. 192-197 ◽  
Author(s):  
Thomas Gnäupel-Herold
Keyword(s):  
Lattice Strain ◽  
Narrow Line ◽  
X Ray Diffraction ◽  
Beam Spot ◽  
Strain Measurements ◽  
One Dimensional ◽  
X Ray ◽  
Stress Gradients ◽  
Shaped Beam

A method is outlined that allows the determination of one-dimensional stress gradients at length scales greater than 0.2 mm. By using standard four-circle X-ray diffractometer equipment and simple aperture components, length resolutions down to 0.05 mm in one direction can be achieved through constant orientation of a narrow, line-shaped beam spot. Angle calculations are given for the adjustment of goniometer angles, and for the effective azimuth and tilt of the scattering vector for general angle settings in a four-circle goniometer. The latter is necessary for the computation of stresses from lattice strain measurements.

Building Vertical Walls by Deposition of Molten Metal Droplets

2005 ◽  
Author(s):  
M. Fang ◽  
S. Chandra ◽  
C. B. Park
Keyword(s):  
Surface Layer ◽  
Analytical Solution ◽  
Substrate Temperature ◽  
Layer By Layer ◽  
Droplet Temperature ◽  
Droplet Generator ◽  
One Dimensional ◽  
Metallurgical Bonding ◽  
Experimental Parameters ◽  
Vertical Walls

Experiments were conducted to determine conditions under which good metallurgical bonding was achieved in vertical walls composed of multiple layers of droplets that were fabricated by depositing tin droplets layer by layer. Molten tin droplets (0.75 mm diameter) were deposited using a pneumatic droplet generator on an aluminum substrate. The primary parameters varied in experiments were those found to most affect bonding between droplets on different layers: droplet temperature (varied from 250°C to 325°C) and substrate temperature (varied from 100°C to 190°C). Considering the cooling rate of droplet is much faster than the deposition rate previous deposition layer cooled down too much that impinging droplets could only remelt a thin surface layer after impact. Assuming that remelting between impacting droplets and the previous deposition layer is a one-dimensional Stefan problem with phase change an analytical solution can be found and applied to predict the minimum droplet temperature and substrate temperature required for local remelting. It was experimentally confirmed that good bonding at the interface of two adjacent layers could be achieved when the experimental parameters were such that the model predicted remelting.

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