Perturbation solutions for highly frictional granular media

2005 ◽  
Vol 461 (2053) ◽  
pp. 21-42 ◽  
Author(s):  
Ngamta Thamwattana ◽  
James M. Hill
Keyword(s):  
Internal Friction ◽  
Granular Media ◽  
Numerical Solutions ◽  
Solution Procedure ◽  
Perturbation Series ◽  
Regular Perturbation ◽  
Approximate Analytical Solutions ◽  
Leading Term ◽  
Flow Through ◽  
Perturbation Solutions

In this paper, we deal with the materials possessing angles of internal friction ϕ for which 1 − sin ϕ is close to zero, and we use the solution for sin ϕ = 1 as the leading term in a regular perturbation series, where the correction terms are of order 1 − sin ϕ . In this way we obtain approximate analytical solutions which can be used to describe the behaviour of real granular materials. The solution procedure is illustrated with reference to quasi–static flow through wedge–shaped and conical hoppers. For these two problems, the obtained perturbation solutions are shown to be graphically indistinguishable from the numerical solutions for high angles of internal friction, and for moderately high angles of internal friction the perturbation solutions still provide excellent approximations.

Parallel Plate Flow of a Third-Grade Fluid and a Newtonian Fluid With Variable Viscosity

2016 ◽  
Vol 71 (7) ◽  
pp. 595-606
Author(s):  
Volkan Yıldız ◽  
Mehmet Pakdemirli ◽  
Yiğit Aksoy
Keyword(s):  
Newtonian Fluid ◽  
Parallel Plate ◽  
Numerical Solutions ◽  
Variable Viscosity ◽  
Third Grade ◽  
Third Grade Fluid ◽  
Regular Perturbation ◽  
Approximate Analytical Solutions ◽  
Grade Fluid ◽  
Perturbation Solutions

AbstractSteady-state parallel plate flow of a third-grade fluid and a Newtonian fluid with temperature-dependent viscosity is considered. Approximate analytical solutions are constructed using the newly developed perturbation-iteration algorithms. Two different perturbation-iteration algorithms are used. The velocity and temperature profiles obtained by the iteration algorithms are contrasted with the numerical solutions as well as with the regular perturbation solutions. It is found that the perturbation-iteration solutions converge better to the numerical solutions than the regular perturbation solutions, in particular when the validity criteria of the regular perturbation solution are not satisfied. The new analytical approach produces promising results in solving complex fluid problems.

Torque on Eccentric Spheres Flowing in Tubes

1982 ◽  
Vol 49 (2) ◽  
pp. 279-283 ◽  
Author(s):  
H. To¨zeren
Keyword(s):  
Axisymmetric Problem ◽  
Stokes Equations ◽  
Cylindrical Tube ◽  
Perturbation Series ◽  
Linear Boundary ◽  
Regular Perturbation ◽  
Order Problem ◽  
First Order ◽  
Leading Term ◽  
Circular Cylindrical Tube

The steady flow of an eccentric sphere in a circular cylindrical tube filled with viscous fluid is considered as a regular perturbation of the axisymmetric problem. A sequence of boundary value problems are formulated involving Stokes equations and some linear boundary conditions. Solution of the first-order problem yields the leading term in the perturbation series of the torque on the sphere.

Analysis of Dynamic Systems With Periodically Varying Parameters via Chebyshev Polynomials

1991 ◽  
Author(s):  
S. C. Sinha ◽  
Der-Ho Wu ◽  
Vikas Juneja ◽  
Paul Joseph
Keyword(s):  
Dynamic Systems ◽  
Chebyshev Polynomials ◽  
Numerical Solutions ◽  
Algebraic Equations ◽  
Suggested Approach ◽  
Varying Parameters ◽  
Approximate Analytical Solutions ◽  
Linear Algebraic Equations ◽  
Mathieu’S Equation ◽  
Runge Kutta

Abstract In this paper a general method for the analysis of multidimensional second-order dynamic systems with periodically varying parameters is presented. The state vector and the periodic matrices appearing in the equations are expanded in Chebyshev polynomials over the principal period and the original differential problem is reduced to a set of linear algebraic equations. The technique is suitable for constructing either numerical or approximate analytical solutions. As an illustrative example, approximate analytical expressions for the Floquet characteristic exponents of Mathieu’s equation are obtained. Stability charts are drawn to compare the results the proposed method with those obtained by Runge-Kutta and perturbation methods. Numerical solutions for the flap-lag motion of a three blade helicopter rotor are constructed in the next example. The numerical accuracy and efficiency of the proposed technique is compared with standard numerical codes based on Runge-Kutta, Adams-Moulton and Gear algorithms. The results obtained in the both examples indicate that the suggested approach extremely accurate and is by far the most efficient one.

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

Dynamic Characteristics of a Hydrostatic Gas Bearing Driven by Oscillating Exhaust Pressure

1984 ◽  
Vol 106 (4) ◽  
pp. 477-483 ◽  
Author(s):  
C. B. Watkins ◽  
H. D. Branch ◽  
I. E. Eronini
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Source Model ◽  
Regular Perturbation ◽  
Response Characteristics ◽  
Finite Difference Approximations ◽  
Bode Plots ◽  
Gas Bearing

Vibration of a statically loaded, inherently compensated hydrostatic journal bearing due to oscillating exhaust pressure is investigated. Both angular and radial vibration modes are analyzed. The time-dependent Reynolds equation governing the pressure distribution between the oscillating journal and sleeve is solved together with the journal equation of motion to obtain the response characteristics of the bearing. The Reynolds equation and the equation of motion are simplified by applying regular perturbation theory for small displacements. The numerical solutions of the perturbation equations are obtained by discretizing the pressure field using finite-difference approximations with a discrete, nonuniform line-source model which excludes effects due to feeding hole volume. An iterative scheme is used to simultaneously satisfy the equations of motion for the journal. The results presented include Bode plots of bearing-oscillation gain and phase for a particular bearing configuration for various combinations of parameters over a range of frequencies, including the resonant frequency.

Laminar flow in symmetrical channels with slightly curved walls II. An asymptotic series for the stream function

1963 ◽  
Vol 272 (1350) ◽  
pp. 406-428 ◽  
Keyword(s):  
Laminar Flow ◽  
Asymptotic Series ◽  
Radius Of Curvature ◽  
Two Dimensional ◽  
The Third ◽  
Higher Order Terms ◽  
Curvature Parameter ◽  
Leading Term ◽  
Flow Through ◽  
Curved Walls

A class of two-dimensional channels, with walls whose radius of curvature is uniformly large relative to local channel width, is described, and the velocity field of laminar flow through these channels is obtained as a power series in the small curvature parameter. The leading term is the Jeffery-Hamel solution considered in part I, and it is shown here how the higher-order terms are found. Terms of the third approximation have been computed. The theory is applied to two examples, for one of which experimental results are available and confirm the theoretical values with fair accuracy.

scholarly journals Perturbation Solutions For Magnetohydrodynamics (Mhd) Flow of in a Non-Newtonian Fluid Between Concentric Cylinders

2019 ◽  
Vol 24 (1) ◽  
pp. 199-211
Author(s):  
M. Yürüsoy ◽  
Ö.F. Güler
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Variable Viscosity ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Mhd Flow ◽  
Model Space ◽  
Viscosity Model ◽  
Concentric Cylinders ◽  
Approximate Analytical Solutions

Abstract The steady-state magnetohydrodynamics (MHD) flow of a third-grade fluid with a variable viscosity parameter between concentric cylinders (annular pipe) with heat transfer is examined. The temperature of annular pipes is assumed to be higher than the temperature of the fluid. Three types of viscosity models were used, i.e., the constant viscosity model, space dependent viscosity model and the Reynolds viscosity model which is dependent on temperature in an exponential manner. Approximate analytical solutions are presented by using the perturbation technique. The variation of velocity and temperature profile in the fluid is analytically calculated. In addition, equations of motion are solved numerically. The numerical solutions obtained are compared with analytical solutions. Thus, the validity intervals of the analytical solutions are determined.

scholarly journals Combined numerical and experimental study of microstructure and permeability in porous granular media

2020 ◽  
Author(s):  
Philipp Eichheimer ◽  
Marcel Thielmann ◽  
Wakana Fujita ◽  
Gregor J. Golabek ◽  
Michihiko Nakamura ◽  
...  
Keyword(s):  
Fluid Flow ◽  
Numerical Simulations ◽  
Granular Media ◽  
Large Scale ◽  
Earth Science ◽  
Numerical Models ◽  
Effective Porosity ◽  
Flow Properties ◽  
Wide Range ◽  
Flow Through

Abstract. Fluid flow on different scales is of interest for several Earth science disciplines like petrophysics, hydrogeology and volcanology. To parameterize fluid flow in large-scale numerical simulations (e.g. groundwater and volcanic systems), flow properties on the microscale need to be considered. For this purpose experimental and numerical investigations of flow through porous media over a wide range of porosities are necessary. In the present study we sinter glass bead media with various porosities. The microstructure, namely effective porosity and effective specific surface, is investigated using image processing. We determine flow properties like hydraulic tortuosity and permeability using both experimental measurements and numerical simulations. By fitting microstructural and flow properties to porosity, we obtain a modified Kozeny-Carman equation for isotropic low-porosity media, that can be used to simulate permeability in large-scale numerical models. To verify the modified Kozeny-Carman equation we compare it to the computed and measured permeability values.

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