Perturbation Solutions

1990 ◽  
pp. 458-537
Author(s):  
J. Kevorkian
Keyword(s):  
Perturbation Solutions

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.

First-order perturbation solutions of embedded strained wires

2006 ◽  
Vol 100 (12) ◽  
pp. 123506 ◽  
Author(s):  
C.-H. Chiu ◽  
Hangyao Wang
Keyword(s):  
Order Perturbation ◽  
First Order ◽  
First Order Perturbation ◽  
Perturbation Solutions

scholarly journals Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

2018 ◽  
Vol 14 (1) ◽  
Author(s):  
Feras K. Alfosail ◽  
Amal Z. Hajjaj ◽  
Mohammad I. Younis
Keyword(s):  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Hopf Bifurcations ◽  
Chaotic Motions ◽  
Different Types ◽  
Quadratic Nonlinearities ◽  
Multiple Scales Perturbation Method ◽  
Good Agreement ◽  
Perturbation Solutions

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

Axial flow in a two-dimensional microchannel induced by a travelling temperature wave imposed at the bottom wall

2018 ◽  
Vol 848 ◽  
pp. 1040-1072 ◽  
Author(s):  
Chenguang Zhang ◽  
Harris Wong ◽  
Krishnaswamy Nandakumar
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Axial Flow ◽  
Temperature Wave ◽  
Bottom Wall ◽  
Moving Frame ◽  
Global Maximum ◽  
System Of Equations ◽  
Two Dimensional ◽  
Perturbation Solutions

Fluid flow in microchannels has wide industrial and scientific applications. Hence, it is important to explore different driving mechanisms. In this paper, we study the net transport or fluid pumping in a two-dimensional channel induced by a travelling temperature wave applied at the bottom wall. The Navier–Stokes equations with the Boussinesq approximation and the convection–diffusion heat equation are made dimensionless by the height of the channel and a velocity scale obtained by a dominant balance between buoyancy and viscous resistance in the momentum equation. The system of equations is transformed to an axial coordinate that moves with the travelling temperature wave, and we seek steady solutions in this moving frame. Four dimensionless numbers emerge from the governing equations and boundary conditions: the Reynolds number $Re$, a Reynolds number $Rc$ based on the wave speed, the Prandtl number $Pr$ and the dimensionless wavenumber $K$. The system of equations is solved by a finite-volume method and by a perturbation method in the limit $Re\rightarrow 0$. Surprisingly, the leading and first-order perturbation solutions agree well with the computed axial flow for $Re\leqslant 10^{3}$. Thus, the analytic perturbation solutions are used to study systematically the effects of $Re$, $Rc$, $Pr$ and $K$ on the dimensionless induced axial flow $Q$. We find that $Q$ varies linearly with $Re$, and $Q/Re$ versus any of the three remaining dimensionless groups always exhibits a maximum. The global maximum of $Q/Re$ in the parameter space is subsequently determined for the first time. This induced axial flow is driven solely by the Reynolds stress.

Oscillatory Third-Order Perturbation Solutions for Sums of Interacting Long-Crested Stokes Waves on Deep Water

1993 ◽  
Vol 37 (04) ◽  
pp. 354-383
Author(s):  
Willard J. Pierson
Keyword(s):  
Free Surface ◽  
Deep Water ◽  
Perturbation Expansion ◽  
The United States ◽  
Order Perturbation ◽  
Naval Academy ◽  
Third Order ◽  
First Order ◽  
Stokes Waves ◽  
Perturbation Solutions

Oscillatory third-order perturbation solutions for sums of interacting long-crested Stokes waves on deep water are obtained. A third-order perturbation expansion of the nonlinear free boundary value problem, defined by the coupled Bernoulli equation and kinematic boundary condition evaluated at the free surface, is solved by replacing the exponential term in the potential function by its series expansion and substituting the equation for the free surface into it. There are second-order changes in the frequencies of the first-order terms at third order. The waves have a Stokes-like form when they are high. The phase speeds are a function of the amplitudes and wave numbers of all of the first-order terms. The solutions are illustrated. A preliminary experiment at the United States Naval Academy is described. Some applications to sea keeping are bow submergence and slamming, capsizing in following seas and bending moments.

scholarly journals Entropy Analysis of an MHD Synthetic Cilia Assisted Transport in a Microchannel Enclosure with Velocity and Thermal Slippage Effects

Coatings ◽  
2020 ◽  
Vol 10 (4) ◽  
pp. 414
Author(s):  
Sufian Munawar ◽  
Najma Saleem
Keyword(s):  
Flow Direction ◽  
Pressure Rise ◽  
Velocity Slip ◽  
Fluid Temperature ◽  
Thermal Slip ◽  
Entropy Analysis ◽  
Slip Effects ◽  
The Heat Transfer Coefficient ◽  
Increasing Function ◽  
Perturbation Solutions

The magnitude of shear stress at the ciliated wall is considered as the measure of efficiency of cilia beatings as it describes the momentum transfer between the medium and the cilia. Under high shear rate, some non-Newtonian fluids behave as visco-inelastic fluids. We consider here a ciliated channel coated with Prandtl fluid, a visco-inelastic fluid, with Hartmann layer under momentum and thermal slip effects. The flow in the channel is produced due to beatings of cilia that obey an elliptic path of motion in the flow direction. An entropy analysis of the flow is also conducted in wave frame. After introducing lubrication approximations in the governing equation, the perturbation solutions are calculated. The data for pressure rise per metachronal wavelength and frictional force at the ciliated wall are obtained by numerical integration. The analysis reveals that the higher values of cilia length and velocity slip parameters support fluid flow near the channel wall surface. Fluid temperature is an increasing function of thermal slip but a decreasing function of cilia length and slip parameters. Entropy in the channel can be minimized with an increase in cilia length and slip effect at the boundary. The magnitude of the heat transfer coefficient decreases by taking the substantial slippage and tiny cilia in length at the microchannel wall.

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