Some steady axisymmetric vortex flows past a sphere

2001 ◽  
Vol 433 ◽  
pp. 315-328 ◽  
Author(s):  
ALAN ELCRAT ◽  
BENGT FORNBERG ◽  
KENNETH MILLER
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Stream Function ◽  
Vortex Rings ◽  
Vortex Flows ◽  
Partial Differential ◽  
Axisymmetric Vortex ◽  
Spherical Vortex ◽  
Attached Vortex ◽  
Vortex Tubes

Steady, inviscid, axisymmetric vortex flows past a sphere are obtained numerically as solutions of a partial differential equation for the stream function. The solutions found include vortex rings, bounded vortices attached to the sphere and infinite vortex tubes. Four families of attached vortices are described: vortex wakes behind the sphere, spherically annular vortices surrounding the spherical obstacle (which can be given analytically), bands of vorticity around the sphere and symmetric pairs of vortices fore and aft of the sphere. Each attached vortex leads to a one-parameter family of vortex rings, analogous to the connection between Hill's spherical vortex and the vortex rings of Norbury.

scholarly journals The steady broadside motion of an anchor ring in an infinite viscous liquid

1931 ◽  
Vol 134 (823) ◽  
pp. 47-57 ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Viscous Liquid ◽  
Stream Function ◽  
Steady Motion ◽  
Second Order ◽  
Meridian Plane ◽  
Legendre Functions ◽  
Order Partial Differential Equation ◽  
Partial Differential

The ring is translated along its axis of revolution with constant velocity in an infinite viscous liquid. The motion of the liquid is due to the motion of the ring, each particle moving in a meridian plane to which the vector vorticity is perpendicular. The analysis is conducted in orthogonal curvilinear “ring coordinates” using vectors, and the condition of continuity leads to a stream function which is connected with the vorticity by a partial differential equation of the second order. The equation of steady motion, on ignoring the inertia terms, is a partial differential equation of the second order in which the dependent variable is the vorticity. The motion thus comes to depend on a fourth-order partial differential equation in which the dependent variable is the stream function. Two independent types of solution of this equation are obtained in trigonometrical series involving associated Legendre functions of degree half an odd integer, the solutions tending to zero at infinity. The arbitrary constants are determined from the boundary conditions of no slip at the surface of the ring. By means of the usual dyadics an expression, is obtained for the resistance to the motion. Numerical values are omitted in the absence of the necessary tables, a defect which it is hoped to remedy in the near future.

Qxygen transfer in gravity flow sewers

2000 ◽  
Vol 42 (3-4) ◽  
pp. 417-422 ◽  
Author(s):  
T.Y. Pai ◽  
C.F. Ouyang ◽  
Y.C. Liao ◽  
H.G. Leu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dissolved Oxygen ◽  
Flow Velocity ◽  
Molecular Diffusion ◽  
Eddy Diffusion ◽  
Partial Differential ◽  
Diffusion Term ◽  
Sewer Pipes ◽  
Gravity Sewer

Oxygen diffused to water in gravity sewer pipes was studied in a 21 m long, 0.15 m diameter model sewer. At first, the sodium sulfide was added into the clean water to deoxygenate, then the pump was started to recirculate the water and the deoxygenated water was reaerated. The dissolved oxygen microelectrode was installed to measure the dissolved oxygen concentrations varied with flow velocity, time and depth. The dissolved oxygen concentration profiles were constructed and observed. The partial differential equation diffusion model that considered Fick's law including the molecular diffusion term and eddy diffusion term were derived. The analytic solution of the partial differential equation was used to determine the diffusivities by the method of nonlinear regression. The diffusivity values for the oxygen transfer was found to be a function of molecular diffusion, eddy diffusion and flow velocity.

scholarly journals The Boundary Proportion Differential Control Method of Micro-Deformable Manipulator with Compensator Based on Partial Differential Equation Dynamic Model

Micromachines ◽  
2021 ◽  
Vol 12 (7) ◽  
pp. 799
Author(s):  
Xiangli Pei ◽  
Ying Tian ◽  
Minglu Zhang ◽  
Ruizhuo Shi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dynamic Model ◽  
Control Method ◽  
System Modeling ◽  
Working Environment ◽  
Partial Differential ◽  
External Interference ◽  
Differential Control ◽  
Differential Control Method

It is challenging to accurately judge the actual end position of the manipulator—regarded as a rigid body—due to the influence of micro-deformation. Its precise and efficient control is a crucial problem. To solve the problem, the Hamilton principle was used to establish the partial differential equation (PDE) dynamic model of the manipulator system based on the infinite dimension of the working environment interference and the manipulator space. Hence, it resolves the common overflow instability problem in the micro-deformable manipulator system modeling. Furthermore, an infinite-dimensional radial basis function neural network compensator suitable for the dynamic model was proposed to compensate for boundary and uncertain external interference. Based on this compensation method, a distributed boundary proportional differential control method was designed to improve control accuracy and speed. The effectiveness of the proposed model and method was verified by theoretical analysis, numerical simulation, and experimental verification. The results show that the proposed method can effectively improve the response speed while ensuring accuracy.

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