scholarly journals A new approach to the classical Stokes flow problem: Part II Series solutions and higher-order applications

1997 ◽  
Vol 78 (2) ◽  
pp. 233-254 ◽  
Author(s):  
H. Ta§eli ◽  
R. Eid
Keyword(s):  
Stokes Flow ◽  
Flow Problem ◽  
Higher Order ◽  
Series Solutions ◽  
New Approach

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

2021 ◽  
Vol 477 (2255) ◽  
Author(s):  
Shou-Fu Tian ◽  
Mei-Juan Xu ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Vector Fields ◽  
Higher Order ◽  
Beam Equation ◽  
Series Solutions ◽  
Local Conservation ◽  
Power Series Method ◽  
Potential System ◽  
Power Series Solutions ◽  
The Difference

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.

On Asymptotic Approximations to a Class of Regular Perturbation Problems

1977 ◽  
Vol 99 (3) ◽  
pp. 369-375 ◽  
Author(s):  
D. A. MacDonald
Keyword(s):  
Algebraic Manipulation ◽  
Higher Order ◽  
Standard Theory ◽  
Asymptotic Approximations ◽  
Rounding Error ◽  
Common Occurrence ◽  
Regular Perturbation ◽  
New Approach ◽  
Exact Agreement ◽  
Perturbation Problems

A new approach to a class of regular perturbation problems of common occurrence in lubrication theory is presented. The approach is not dependent on extensive algebraic manipulation and predicts results which, apart from numerical rounding error, are in exact agreement with standard theory. Three illustrative examples are studied in detail and it is demonstrated that asymptotic approximations obtained through use of the approach can be of a substantially higher order than approximations at present in the literature.

Slow steady rotation of axially symmetric bodies in a viscous fluid

1961 ◽  
Vol 10 (1) ◽  
pp. 17-24 ◽  
Author(s):  
R. P. Kanwal
Keyword(s):  
Angular Velocity ◽  
Flow Field ◽  
Viscous Fluid ◽  
Stokes Flow ◽  
Flow Problem ◽  
Analytic Solutions ◽  
The Body ◽  
Axially Symmetric ◽  
Steady Rotation ◽  
Axis Of Symmetry

The Stokes flow problem is considered here for the case in which an axially symmetric body is uniformly rotating about its axis of symmetry. Analytic solutions are presented for the heretofore unsolved cases of a spindle, a torus, a lens, and various special configurations of a lens. Formulas are derived for the angular velocity of the flow field and for the couple experienced by the body in each case.

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