Numerical Solutions of the Basic Equations

2008 ◽  
pp. 587-626
1971 ◽  
Vol 10 (5) ◽  
pp. 566-570 ◽  
Author(s):  
Masayuki Yoshida ◽  
Ken Kimura

1987 ◽  
Vol 109 (3) ◽  
pp. 635-641 ◽  
Author(s):  
S. Akagi ◽  
K. Uchida

Fluid motion and heat transfer of a high-viscosity fluid contained in a two-dimensional rectangular ship’s tank subjected to oscillating motion are investigated by a finite difference technique. The study is motivated by the thermal design of the heating system of oil tanks on a tanker which is moving in a wavy sea. The bottom of the tank is heated and its side walls are cooled. The motion of the tank is assumed to be a simple harmonic rolling motion. The isotherms and flow velocity vectors are determined by numerical solutions of the basic equations describing the convection flows in a tank with oscillating motion. The heat transfer rates to the tank walls are predicted. The influence of the frequency of the oscillating motion on the heat transfer rate is examined.


1968 ◽  
Vol 183 (1) ◽  
pp. 631-645 ◽  
Author(s):  
E. J. Hahn ◽  
C. F. Kettleborough

The authors have shown previously by solving numerically the relevant mathematical equations describing lubricant flow, heat transfer and thermal distortion of bearing components in a slider bearing of infinite width, that, for the particular operating and boundary conditions assumed, thermal distortion rather than variation of the lubricant properties is the cause of hydrodynamic lubrication of initially parallel, radially grooved, thrust bearings. In this paper, numerical solutions are presented to show the effect of relaxing these conditions. The beneficial effect of thermal distortion is proved to hold generally and the importance of proper mixing of the lubricant at the inlet is illustrated. Comparison of numerical solutions with existing experimental observations shows only a qualitative agreement. A realistic three-dimensional mathematical model of a slider bearing of finite width is derived from basic equations, and the solution of the resulting equation, allowing for variable lubricant properties and thermal distortion of the bearing components, is developed and discussed.


2017 ◽  
Vol 371 ◽  
pp. 49-58 ◽  
Author(s):  
Katarzyna Tkacz-Śmiech ◽  
Bartek Wierzba ◽  
Bogdan Bożek ◽  
M. Danielewski

Low-temperature nitriding of austenitic stainless steels or chromium containing alloys can produce expanded austenite, known as S-phase, with combined improvement in wear and corrosion resistance. In the paper a critical review of various models for nitrogen diffusion during nitriding is presented. A special attention is paid to the expanded austenite growth. A new model based on bi-velocity method and including stresses is presented. Basic equations and boundary conditions are discussed. Composition dependent nitrogen diffusion coefficient is assumed. Numerical solutions are obtained for the growth of the S-phase layer in steel. The results are compared with previous experiment and calculations.


1982 ◽  
Vol 21 (Part 1, No. 3) ◽  
pp. 446-450 ◽  
Author(s):  
Hajime Kitagawa ◽  
Kimio Hashimoto ◽  
Masayuki Yoshida

2020 ◽  
Vol 496 (1) ◽  
pp. 434-441
Author(s):  
Hanifeh Ghanbarnejad ◽  
Maryam Ghasemnezhad

ABSTRACT Numerical simulations of self-gravitating accretion discs have shown that the evolution of such systems depends strongly on the rate at which it cools. In this work, we study the vertical structure of the self-gravitating accretion discs and also investigate the effect of the cooling rate on the latitudinal structure of such accretion discs. In the spherical coordinates, we write the hydrodynamics equations and simplify the basic equations based on the assumptions of axisymmetric and steady state. We use the self-similar method for solving the equations in the radial direction and we find proper boundary conditions. We find inflow–outflow solutions by considering the meridional component of the velocity field. In order to formulate the cooling term in energy equation, we introduce the new parameter β as a free constant that is the cooling time-scale in units of the dynamical time-scale. Our numerical solutions show that the thickness of the disc decreases with smaller β (or increasing the cooling term in energy equation) and it makes the disc colder and outflows form in the regions with lower latitude. So by increasing the cooling rate in the disc, the regions which belong to inflow decrease.


Author(s):  
D.E. Jesson ◽  
S. J. Pennycook

It is well known that conventional atomic resolution electron microscopy is a coherent imaging process best interpreted in reciprocal space using contrast transfer function theory. This is because the equivalent real space interpretation involving a convolution between the exit face wave function and the instrumental response is difficult to visualize. Furthermore, the crystal wave function is not simply related to the projected crystal potential, except under a very restrictive set of experimental conditions, making image simulation an essential part of image interpretation. In this paper we present a different conceptual approach to the atomic imaging of crystals based on incoherent imaging theory. Using a real-space analysis of electron scattering to a high-angle annular detector, it is shown how the STEM imaging process can be partitioned into components parallel and perpendicular to the relevant low index zone-axis.It has become customary to describe STEM imaging using the analytical treatment developed by Cowley. However, the convenient assumption of a phase object (which neglects the curvature of the Ewald sphere) fails rapidly for large scattering angles, even in very thin crystals. Thus, to avoid unpredictive numerical solutions, it would seem more appropriate to apply pseudo-kinematic theory to the treatment of the weak high angle signal. Diffraction to medium order zero-layer reflections is most important compared with thermal diffuse scattering in very thin crystals (<5nm). The electron wave function ψ(R,z) at a depth z and transverse coordinate R due to a phase aberrated surface probe function P(R-RO) located at RO is then well described by the channeling approximation;


1961 ◽  
Vol 38 (3) ◽  
pp. 469-472 ◽  
Author(s):  
K. R. Laumas ◽  
J. F. Tait ◽  
S. A. S. Tait

ABSTRACT Reconsideration of the question of the validity of the calculations of the secretion rates from the specificity activity of a urinary metabolite after the single injection of a radioactive hormone has led us to conclude that the basic equations used in a previous theoretical treatment are not generally applicable to the nonisotopic steady state if the radioactive steroid and hormone are introduced into the same compartment. If this is so, in a two compartmental model with metabolism occurring in both pools, it is now shown that the calculation (S = R — τ) is rigorously valid if certain precautions are taken. This is in contrast to the previous treatment which concluded (in certain special circumstances) that the calculation might not be correct. However, if the hormone is secreted in both compartments and the radioactive steroid is injected into only one, then the calculation (S = R — τ) may not be correct in certain circumstances as was previously concluded (Laumas et al. 1961).


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