On the Approximate Solution of Viscous-Flow Problems

1963 ◽  
Vol 30 (2) ◽  
pp. 263-268 ◽  
Author(s):  
J. A. Schetz
Keyword(s):  
Approximate Solution ◽  
Exact Solutions ◽  
Viscous Flow ◽  
Closed Form ◽  
New Method ◽  
Two Dimensional ◽  
General Technique ◽  
Closed Form Solutions ◽  
Flow Problems

The need for a general technique for the approximate solution of viscous-flow problems is discussed. Existing methods are considered and a new method is presented which results in simple closed-form solutions. The accuracy of the method is demonstrated by comparisons with the results of known exact solutions, and finally the general technique is employed to determine a new solution for the fully expanded two-dimensional laminar nozzle problem.

Approximate Analysis of Penetration of a Cylindrical Roller Into a Thin Elastic Coating

1990 ◽  
Vol 112 (3) ◽  
pp. 460-468 ◽  
Author(s):  
Tsung-Ju Gwo ◽  
Thomas J. Lardner
Keyword(s):  
Approximate Solution ◽  
Closed Form ◽  
Approximate Analytical Solution ◽  
Experimental Studies ◽  
Preliminary Design ◽  
Two Dimensional ◽  
Rigid Substrate ◽  
Closed Form Solutions ◽  
Finite Element Calculations ◽  
Cylindrical Roller

An approximate analytical solution to the problem of two-dimensional indentation of a frictionless cylinder into a thin elastic coating bonded to a rigid substrate has been obtained using the approach introduced by Matthewson for axisymmetric indentation. We show by comparing the results of the approximate solution to the exact solutions and to finite element calculations that the approximate solution is accurate for a/h> 2. The advantage of this approach is that the results are expressed in closed form and the accuracy of the approximate solution improves with increasing values of a/h. For a/h>2, for a given load, the theory overestimates the value of a/h compared to the exact solution by less than 10 percent. In many experimental studies and in preliminary design, it is convenient to have closed-form solutions exhibiting the dependence of the parameters.

Analytical Modeling of Mechanical Properties and Behavior of Nanotube-Based Nanocomposites

2013 ◽  
Author(s):  
Davood Askari ◽  
Mehrdad N. Ghasemi-Nejhad ◽  
Alexander L. Kalamkarov
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Axial Strain ◽  
Stress Distributions ◽  
Closed Form Solutions ◽  
Loading Case ◽  
Analytical Formulae ◽  
Plain Strain ◽  
And Behavior ◽  
Radial Loading

The objective of this paper is to introduce analytical closed form solutions for the prediction of effective axial and transverse Young’s modulus and Poisson ratios of a matrix-filled nanotube (i.e., a representative element of nanotube reinforced nanocomposites) as well as its mechanical behavior (i.e., displacements, strains and stress distributions) when it is subjected to externally applied uniform axial and radial loads. In this work, both the nanotube and its filler material are considered to be generally cylindrical orthotopic. For the derivation of exact solutions for radial loading case, no plain strain condition is assumed and effects of axial strain is taken into consideration to obtain a more precise set of solutions. Analytical formulae are developed based on the principles of linear elasticity and continuum mechanics and then exact solutions are obtained for displacements, strains and stress distributions within the domain of each individual constituent. To validate and verify the accuracy of the closed form solutions obtained from the analytical approach, a 3-D model of a matrix-filled nanotube is generated and solved for displacements, strains and stresses, numerically, using a finite element method. Excellent agreements were achieved between the results obtained from the analytical and numerical methods.

A New Approach to Thin Aerofoil Theory

1951 ◽  
Vol 3 (3) ◽  
pp. 193-210 ◽  
Author(s):  
M.J. Lighthill
Keyword(s):  
Three Dimensional ◽  
Leading Edge ◽  
Approximate Solutions ◽  
Radius Of Curvature ◽  
Two Dimensional ◽  
General Technique ◽  
New Approach ◽  
Flow Problems ◽  
Physical Problems ◽  
First Approximation

SummaryThe general technique for rendering approximate solutions to physical problems uniformly valid is here applied to the simplest form of the problem of correcting the theory of thin wings near a rounded leading edge. The flow investigated is two-dimensional, irrotational and incompressible, and therefore the results do not materially add to our already extensive knowledge of this subject, but the method, which is here satisfactorily checked against this knowledge, shows promise of extension to three-dimensional, and compressible, flow problems.The conclusion, in the problem studied here, is that the velocity field obtained by a straightforward expansion in powers of the disturbances, up to and including either the first or the second power, with coefficients functions of co-ordinates such that the leading edge is at the origin and the aerofoil chord is one of the axes, may be rendered a valid first approximation near the leading edge, as well as a valid first or second approximation away from it, if the whole field is shifted downstream parallel to the chord for a distance of half the leading edge radius of curvature ρL. It follows that the fluid speed on the aerofoil surface, as given on such a straightforward second approximation as a function of distance x along the chord, similarly is rendered uniformly valid (see equation (52)) if the part singular like x-1 is subtracted and the remainder is multiplied by .

Generalized Couette Flow of a Third-Grade Fluid with Slip: The Exact Solutions

2010 ◽  
Vol 65 (12) ◽  
pp. 1071-1076 ◽  
Author(s):  
Rahmat Ellahi ◽  
Tasawar Hayat ◽  
Fazal Mahmood Mahomed
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Closed Form ◽  
Analytical Solutions ◽  
Present Note ◽  
Nonlinear Problems ◽  
Third Grade ◽  
Third Grade Fluid ◽  
Flow Problems ◽  
Grade Fluid

The present note investigates the influence of slip on the generalized Couette flows of a third-grade fluid. Two flow problems are considered. The resulting equations and the boundary conditions are nonlinear. Analytical solutions of the governing nonlinear problems are found in closed form.

Some Exact Solutions of Magneto-Hydrodynamic Viscous Flow Problems

1962 ◽  
Vol 41 (1-4) ◽  
pp. 116-131
Author(s):  
H. P. Greenspan ◽  
L. A. Peletier
Keyword(s):  
Exact Solutions ◽  
Viscous Flow ◽  
Flow Problems

Lie symmetry reductions, abound exact solutions and localized wave structures of solitons for a (2 + 1)-dimensional Bogoyavlenskii equation

2021 ◽  
pp. 2150252
Author(s):  
Sachin Kumar ◽  
Monika Niwas
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Evolution Equations ◽  
Dynamical Behavior ◽  
Lie Symmetry ◽  
Nonlinear Evolution Equations ◽  
Mathematical Methods ◽  
Closed Form Solutions ◽  
Symmetry Reductions ◽  
Engineering Sciences

By applying the two efficient mathematical methods particularly with regard to the classical Lie symmetry approach and generalized exponential rational function method, numerous exact solutions are constructed for a (2 + 1)-dimensional Bogoyavlenskii equation, which describes the interaction of Riemann wave propagation along the spatial axes. Moreover, we obtain the infinitesimals, all the possible vector fields, optimal system, and Lie symmetry reductions. The governing Bogoyavlenskii equation is converted into various nonlinear ordinary differential equations through two stages of Lie symmetry reductions. Accordingly, abundant exact closed-form solutions are obtained explicitly in terms of independent arbitrary functions, rational functions, trigonometric functions, and hyperbolic functions with arbitrary free parameters. The dynamical behavior of the resulting soliton solutions is presented through 3D-plots via numerical simulation. Eventually, single solitons, multi-solitons with oscillations, kink wave with breather-type solitons, and single lump-type solitons are obtained. The proposed mathematical techniques are effective, trustworthy, and reliable mathematical tools to work out new exact closed-form solutions of various types of nonlinear evolution equations in mathematical physics and engineering sciences.

Closed-Form Exact Solution to H∞ Optimization of Dynamic Vibration Absorbers (Application to Different Transfer Functions and Damping Systems)

2003 ◽  
Vol 125 (3) ◽  
pp. 398-405 ◽  
Author(s):  
Toshihiko Asami ◽  
Osamu Nishihara
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Optimization Problem ◽  
Vibration Suppression ◽  
Transfer Functions ◽  
New Method ◽  
Algebraic Solution ◽  
Vibration Absorbers ◽  
Dynamic Vibration ◽  
Dynamic Vibration Absorbers

H ∞ optimization of the dynamic vibration absorbers is a classical optimization problem, and has been already solved more than 50 years ago. It is a well-known solution, but we know this solution is only an approximate one. Recently, one of the authors has proposed a new method for attaining the H∞ optimization of the absorber in linear systems. The new method enables us to obtain the exact algebraic solution of the H∞ optimization problem of the absorber. In this paper, we first apply this method to the design optimization of a viscous damped (Voigt type) absorber and a hysteretic damped absorber attached to undamped primary systems. For each absorber, six different transfer functions are taken here as performance indices to vibration suppression or isolation. As a result, we found the closed-form exact solutions to all transfer functions. The solutions obtained here are then compared with those of the approximate ones. Finally, we present the closed-form exact solutions to the hysteretic damped absorber attached to damped primary systems.

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