Convection in a horizontally heated sphere

2001 ◽  
Vol 438 ◽  
pp. 1-10 ◽  
Author(s):  
G. D. McBAIN
Keyword(s):  
Asymptotic Expansion ◽  
Body Force ◽  
Stokes Problem ◽  
Creeping Flow ◽  
Zeroth Order ◽  
First Order ◽  
Smooth Body ◽  
Spherical Fluid ◽  
Rayleigh Numbers ◽  
Spanwise Flow

Natural convection in horizontally heated spherical fluid-filled cavities is considered in the low Grashof number limit. The equations governing the asymptotic expansion are derived for all orders. At each order a Stokes problem must be solved for the momentum correction. The general solution of the Stokes problem in a sphere with arbitrary smooth body force is derived and used to obtain the zeroth-order (creeping) flow and the first-order corrections due to inertia and buoyancy. The solutions illustrate the two mechanisms adduced by Mallinson & de Vahl Davis (1973, 1977) for spanwise flow in horizontally heated cuboids. Further, the analytical derivations and expressions clarify these mechanisms and the conditions under which they do not operate. The inertia and buoyancy effects vanish with the Grashof and Rayleigh numbers, respectively, and both vanish if the flow is purely vertical, as in a very tall and narrow cuboid.

scholarly journals Perturbation Solution for One-Dimensional Flow to a Constant-Pressure Boundary in a Stress-Sensitive Reservoir

2021 ◽  
Author(s):  
Amarjot Singh Bhullar ◽  
Gospel Ezekiel Stewart ◽  
Robert W. Zimmerman
Keyword(s):  
Approximate Solution ◽  
Pore Pressure ◽  
Constant Pressure ◽  
Initial Permeability ◽  
Order Perturbation ◽  
Zeroth Order ◽  
One Dimensional ◽  
First Order ◽  
Outflow Boundary ◽  
Perturbation Solutions

Abstract Most analyses of fluid flow in porous media are conducted under the assumption that the permeability is constant. In some “stress-sensitive” rock formations, however, the variation of permeability with pore fluid pressure is sufficiently large that it needs to be accounted for in the analysis. Accounting for the variation of permeability with pore pressure renders the pressure diffusion equation nonlinear and not amenable to exact analytical solutions. In this paper, the regular perturbation approach is used to develop an approximate solution to the problem of flow to a linear constant-pressure boundary, in a formation whose permeability varies exponentially with pore pressure. The perturbation parameter αD is defined to be the natural logarithm of the ratio of the initial permeability to the permeability at the outflow boundary. The zeroth-order and first-order perturbation solutions are computed, from which the flux at the outflow boundary is found. An effective permeability is then determined such that, when inserted into the analytical solution for the mathematically linear problem, it yields a flux that is exact to at least first order in αD. When compared to numerical solutions of the problem, the result has 5% accuracy out to values of αD of about 2—a much larger range of accuracy than is usually achieved in similar problems. Finally, an explanation is given of why the change of variables proposed by Kikani and Pedrosa, which leads to highly accurate zeroth-order perturbation solutions in radial flow problems, does not yield an accurate result for one-dimensional flow. Article Highlights Approximate solution for flow to a constant-pressure boundary in a porous medium whose permeability varies exponentially with pressure. The predicted flowrate is accurate to within 5% for a wide range of permeability variations. If permeability at boundary is 30% less than initial permeability, flowrate will be 10% less than predicted by constant-permeability model.

Small Reynolds Number Electro-Hydrodynamic Flow Around Drops and the Resulting Deformation

1979 ◽  
Vol 46 (3) ◽  
pp. 510-512 ◽  
Author(s):  
M. B. Stewart ◽  
F. A. Morrison
Keyword(s):  
Reynolds Number ◽  
Flow Field ◽  
Order Approximation ◽  
Creeping Flow ◽  
Small Reynolds Number ◽  
Zeroth Order ◽  
Regular Perturbation ◽  
Low Reynolds Number Flow ◽  
Reynolds Number Flow ◽  
Number Flow

Low Reynolds number flow in and about a droplet is generated by an electric field. Because the creeping flow solution is a uniformly valid zeroth-order approximation, a regular perturbation in Reynolds number is used to account for the effects of convective acceleration. The flow field and resulting deformation are predicted.

scholarly journals Heat transfer in rapidly rotating convection with heterogeneous thermal boundary conditions

2017 ◽  
Vol 828 ◽  
pp. 601-629 ◽  
Author(s):  
Jon E. Mound ◽  
Christopher J. Davies
Keyword(s):  
Heat Flux ◽  
Outer Boundary ◽  
Extreme Case ◽  
Creeping Flow ◽  
Supercritical Regime ◽  
Rotating Convection ◽  
Thermal Boundary Conditions ◽  
Core Convection ◽  
Rayleigh Numbers ◽  
Lateral Variations

Convection in the metallic cores of terrestrial planets is likely to be subjected to lateral variations in heat flux through the outer boundary imposed by creeping flow in the overlying silicate mantles. Boundary anomalies can significantly influence global diagnostics of core convection when the Rayleigh number, $Ra$, is weakly supercritical; however, little is known about the strongly supercritical regime appropriate for planets. We perform numerical simulations of rapidly rotating convection in a spherical shell geometry and impose two patterns of boundary heat flow heterogeneity: a hemispherical $Y_{1}^{1}$ spherical harmonic pattern; and one derived from seismic tomography of the Earth’s lower mantle. We consider Ekman numbers $10^{-4}\leqslant E\leqslant 10^{-6}$, flux-based Rayleigh numbers up to ${\sim}800$ times critical, and a Prandtl number of unity. The amplitude of the lateral variation in heat flux is characterised by $q_{L}^{\ast }=0$, 2.3, 5.0, the peak-to-peak amplitude of the outer boundary heat flux divided by its mean. We find that the Nusselt number, $Nu$, can be increased by up to ${\sim}25\,\%$ relative to the equivalent homogeneous case due to boundary-induced correlations between the radial velocity and temperature anomalies near the top of the shell. The $Nu$ enhancement tends to become greater as the amplitude and length scale of the boundary heterogeneity are increased and as the system becomes more supercritical. This $Ra$ dependence can steepen the $Nu\propto Ra^{\unicode[STIX]{x1D6FE}}$ scaling in the rotationally dominated regime, with $\unicode[STIX]{x1D6FE}$ for our most extreme case approximately 20 % greater than the equivalent homogeneous scaling. Therefore, it may be important to consider boundary heterogeneity when extrapolating numerical results to planetary conditions.

scholarly journals ANISOTROPIC NULL STRING COSMOLOGIES

1998 ◽  
Vol 13 (39) ◽  
pp. 3169-3177 ◽  
Author(s):  
IOANNIS GIANNAKIS ◽  
K. KLEIDIS ◽  
A. KUIROUKIDIS ◽  
D. PAPADOPOULOS
Keyword(s):  
Equations Of Motion ◽  
String Tension ◽  
Perturbative Expansion ◽  
Speed Of Light ◽  
Zeroth Order ◽  
First Order ◽  
Cosmological Background ◽  
String Propagation ◽  
Analytical Expressions ◽  
Null String

We study string propagation in an anisotropic, cosmological background. We solve the equations of motion and the constraints by performing a perturbative expansion of the string coordinates in powers if c2 — the worldsheet speed of light. To zeroth order the string is approximated by a tensionless string (since c is proportional to the string tension T). We obtain exact, analytical expressions for the zeroth- and first-order solutions and we discuss some cosmological implications.

A Slender-Ship Theory of Wave Resistance

1983 ◽  
Vol 27 (01) ◽  
pp. 13-33
Author(s):  
Francis Noblesse
Keyword(s):  
Froude Number ◽  
Wave Resistance ◽  
Successive Approximations ◽  
Zeroth Order ◽  
Remarkable Effect ◽  
Ship Wave ◽  
First Order ◽  
Wigley Hull ◽  
Low Froude Number ◽  
The Waves

A new slender-ship theory of wave resistance is presented. Specifically, a sequence of explicit slender-ship wave-resistance approximations is obtained. These approximations are associated with successive approximations in a slender-ship iterative procedure for solving a new (nonlinear integro-differential) equation for the velocity potential of the flow caused by the ship. The zeroth, first, and second-order slender-ship approximations are given explicitly and examined in some detail. The zeroth-order slender-ship wave-resistance approximation, r(0) is obtained by simply taking the (disturbance) potential, ϕ, as the trivial zeroth-order slender-ship approximation ϕ(0) = 0 in the expression for the Kochin free-wave amplitude function; the classical wave-resistance formulas of Michell [1]2 and Hogner [2] correspond to particular cases of this simple approximation. The low-speed wave-resistance formulas proposed by Guevel [3], Baba [4], Maruo [5], and Kayo [6] are essentially equivalent (for most practical purposes) to the first-order slender-ship low-Froude-number approximation, rlF(1), which is a particular case of the first-order slender-ship approximation r(1): specifically, the first-order slender-ship wave-resistance approximation r(1) is obtained by approximating the potential ϕ in the expression for the Kochin function by the first-order slender-ship potential ϕ1 whereas the low-Froude-number approximation rlF(1) is associated with the zero-Froude-number limit ϕ0(1) of the potentialϕ(1). A major difference between the first-order slender-ship potential ϕ(1) and its zero-Froude-number limit ϕ0(1) resides in the waves that are included in the potential ϕ(1) but are ignored in the zero-Froude-number potential ϕ0(1). Results of calculations by C. Y. Chen for the Wigley hull show that the waves in the potential ϕ(1) have a remarkable effect upon the wave resistance, in particular causing a large phase shift of the wave-resistance curve toward higher values of the Froude number. As a result, the first-order slender-ship wave-resistance approximation in significantly better agreement with experimental data than the low-Froude-number approximation rlF(1) and the approximations r(0) and rM.

The Effect of Blade Count on Body Force Model Performance for Axial Fans

2020 ◽  
Author(s):  
Palak Saini ◽  
Jeff Defoe
Keyword(s):  
Body Force ◽  
Computational Cost ◽  
Model Performance ◽  
Aircraft Engine ◽  
The Body ◽  
Force Model ◽  
Cooling Flows ◽  
Cooling Fans ◽  
Axial Fans ◽  
Spanwise Flow

Abstract Body force models enable inexpensive numerical simulations of turbomachinery. The approach replaces the blades with sources of momentum/energy. Such models capture a “smeared out” version of the blades’ effect on the flow, reducing computational cost. The body force model used in this paper has been widely used in aircraft engine applications. Its implementation for low speed, low solidity (few blades) turbomachines, such as automotive cooling fans, enables predictions of cooling flows and component temperatures without calibrated fan curves. Automotive cooling fans tend to have less than 10 blades, which is approximately 50% of blade counts for modern jet engine fans. The effect this has on the body force model predictions is unknown and the objective of this paper is to quantify how varying blade count affects the accuracy of the predictions for both uniform and non-uniform inflow. The key findings are that reductions in blade metal blockage combined with spanwise flow redistribution drives the body force model to more accurately predict work coefficient as the blade count decreases, and that reducing the number of blades is found to have negligible impacts on upstream influence and distortion transfer in non-uniform inflow until extremely low blade counts (such as 2) are applied.

Synthesis, Properties and Application in Optical Memory of a Photochromic Diarylethene Bearing Pyrimidine and Pyridine Ring

2012 ◽  
Vol 490-495 ◽  
pp. 3733-3737
Author(s):  
Shu Hong Jing ◽  
Shou Zhi Pu ◽  
Shi Qiang Cui
Keyword(s):  
Uv Light ◽  
Optical Memory ◽  
Order Reaction ◽  
Pmma Film ◽  
Zeroth Order ◽  
First Order ◽  
Synthesis Properties ◽  
Fluorescence Switching ◽  
Photochromic Reaction ◽  
Hexane Solution

A new photochromic diarylethene compound, 1-(2,4-dimethoxy-5-pyrimidine)-2-[2-methyl-5-(3-pyridine)-3-thienyl]perfluorocyclopentene(1a), was synthesized, and its photochromic reactivity, fluorescent and electrochemical property were also investigated. Diarylethene 1a changed the color from colorless to pink upon irradiation with UV light, in which absorption maxima were observed at 520 and 519 nm in hexane and PMMA film, respectively. The the photochromic reaction kinetics indicated that the cyclization processes of 1 belong to the zeroth order reaction and the cycloreversion process belong to the first order reaction. This new photochromic system also exhibited remarkable fluorescence switching in hexane solution and this new photochromic system also exhibited remarkable optical storage character.

Stability Increase of Aerodynamically Unstable Rotors Using Intentional Mistuning

2006 ◽  
Author(s):  
Carlos Martel ◽  
Roque Corral ◽  
Jose´ Miguel Llorens
Keyword(s):  
Asymptotic Expansion ◽  
Small Parameter ◽  
Structural Dynamics ◽  
Low Pressure ◽  
Vibration Characteristics ◽  
Stability Properties ◽  
First Order ◽  
Low Pressure Turbine ◽  
Turbine Rotors ◽  
The Stability

A new simple asymptotic mistuning model (AMM), which constitutes an extension of the well known Fundamental Mistuning Model for groups of modes belonging to a modal family exhibiting a large variation of the tuned vibration characteristics, is used to analyze the effect of mistuning on the stability properties of aerodynamically unstable rotors. The model assumes that both, the aerodynamics and the structural dynamics of the assembly are linear, and retains the first order terms of a fully consistent asymptotic expansion of the tuned system where the small parameter is the blade mistuning. The simplicity of the model allows the optimization of the blade mistuning pattern to achieve maximum rotor stability. The results of the application of this technique to realistic welded-in-pair and interlock low-pressure-turbine rotors are also presented.

scholarly journals Study on the Effect of Fractional Derivative on the Hyperspectral Data of Soil Organic Matter Content in Arid Region

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Cheng-Biao Fu ◽  
Hei-Gang Xiong ◽  
An-Hong Tian
Keyword(s):  
Organic Matter ◽  
Soil Organic Matter ◽  
Fractional Derivative ◽  
Arid Region ◽  
Arid Zone ◽  
Organic Matter Content ◽  
Matter Content ◽  
Second Order ◽  
Zeroth Order ◽  
First Order

Discussion on the application of fractional derivative algorithm in monitoring organic matter content in field soil is scarce. This study is aimed at improving the accuracy of soil organic matter (SOM) content estimation in arid region, and the undesirable model precision caused by the missing information associated with the larger discrepancy between conventional integer-order, i.e., first order and second order, derivative, and raw spectral data. We utilized fractional derivative (of zeroth order to second order in 0.2-order interval) processing on the field spectral reflectance (R) of the salinized soil sample from Fukang, Xinjiang, and its square root-transformed (R), log-transformed (lgR), inverse-transformed (1/R), and inverse log-transformed (1/lgR) values. The correlation coefficient of each fractional derivative of transformed value with SOM content was calculated. The simulation showed the derivative reflectance value approximates zero. When increasing from zeroth order to first order, the derivative curve gradually aligns to the first-order curve, and the destination alignment was also seen while increasing from first order to second order. The significance test of 0.05 showed initial increase and later decay of bands in the five spectral transformations as the order increases. For specific bands, the derivative algorithm clearly justifies the correlation between soil spectra and organic matter content, and all of the absolute highest correlation coefficient values were obtained at fractional orders. When compared with integer-order derivative, fractional derivative is significantly better in improving correlation, showing overall superiority. The result supports the application of fractional derivative in the hyperspectral remote monitor of SOM in arid zone, which may in turn realize the timely and accurate SOM monitor in arid zone, and provides the basis for ecological restoration.

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