Double Perturbation Method Research Articles

Lie-group method solutions for a viscous flow in a dilating-squeezing permeable channel with velocity slip

The investigation focuses on the effects of wall velocity slip on the solution of a viscous, laminar, incompressible channel flow subjected to small-scaled contraction and expansion of the weakly permeable walls. The study of such flow systems is often contextual for fluid transport in biological organisms. In the considered flow configuration, the vertically moving porous walls enable the fluid to enter or exit with a constant rate. The tangential slip velocity of the flow at the porous walls is modeled with the Navier slip boundary condition. The flow dynamics inside the channel is governed by the full Navier–Stokes equations. The Lie symmetry analysis and the invariant method are adopted to reduce the number of independent variables in the system of governing equations. Consequently, a single fourth-order ordinary differential equation is obtained, which is solved analytically by the double perturbation method and the variation of iteration method. The solutions are compared for different arrangements. Furthermore, the approximated analytical solutions are likened to the numerical solutions obtained from a fourth-order Runge–Kutta solver embedding the Shooting method to check the accuracy. It is observed that the boundary layers are formed, and the flow rapidly turns near the walls, when suction and wall contraction coexist. Alternatively, if injection and wall expansion are paired, the flow adjacent to the walls is delayed. The existence of wall velocity slip advances the near-wall velocity and cuts down the speed of centerline velocity. It results in a change in the volumetric flow rate and shear rate. The overall pressure is also varied by higher wall velocity slip. The results are explored for different values of the permeation Reynold number and the dimensionless wall dilation rate to capture all possible impacts of the flow parameters. The current analysis rectifies the existing errors in the work of Boutros et al. [“Lie-group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability,”Appl. Math. Model. 31(6), 1092–1108 (2007)] with the no-slip boundary condition and discusses the overall influences of slip boundary condition on the Lie symmetry solution of the flow system.

Heat transfer characteristics of magnetohydrodynamic two fluid oscillatory flow in an inclined channel with saturated porous medium

This investigation aims to examine the hydromagnetic flow of two liquid flows of fully developed incompressible Newtonian fluid in an inclined channel through the porous medium accounting for radiative heat flux, Joule heating, and viscous dissipation. The walls of the channel are maintained at different temperatures. A double perturbation method is employed to derive analytical results for velocity and temperature. Graphical results are presented for various arising parameters such as Hartmann number, Grashoff number, ratio of viscosity parameter, thermal conductivity ratio, and porous parameter. Further, the results for mass flux are presented in a tabular form and discussed. Reduction in the velocity and temperature distribution is observed by enhancing Darcy dissipation and frequency parameter. As the angle of inclination increases, there is a rise in flow and heat distribution. With the strength of the magnetic field and the rise in the Reynolds number, the mass flux decreases. A comparative study is carried out with the previously published work and the results are found to be in good agreement.

Slit flow and thermal analysis of micropolar fluids in a symmetric channel with dynamic and permeable

The mass and thermal flow of micropolar fluid in a channel having permeable and moving walls perpendicular to the flow-direction is considered here. A mathematical formulation for this physical problem is made using the Eringen's micropolar fluid-thermal model. Closed form solutions for temperature, microspin of aciculate particle and velocity are derived using the double perturbation method followed by the similarity transformation for the Newtonian/micropolar fluids. The perturbation parameters are the Reynolds number (that controls the wall-injection/suction) and the wall dilation parameter (that controls the rate of flow through the pores). Usually the no-spin conditions are imposed on studying the micropolar fluid-flows, but we consider no-spin condition in this attempt in order to highlight effect of micropolar spin boundary layer in the vicinity of the walls of the channel. The convergence of the perturbation method is checked, that sounds well. The results obtained for particular case (Newtonian fluid) are compared with alike literature and they are found satisfactory. The results for velocity, temperature, shear stress, thermal transfer rate and spin are presented for different values of physical parameters quantitatively and qualitatively.

Lie group analysis for MHD squeezing flow of viscous fluid saturated in porous media

A two-dimensional laminar flow driven by fluid injection through porous surface which represent an incompressible fluid inside a filtration chamber during extraction of particles from the fluid is investigated. The study constructs a mathematical model that represents internal flow field during filtration process proficiently by using basic conservation laws of mass, momentum and energy. For better understanding of dynamics of the case study, solutions that lead to stable filtration process (balanced dynamical system) are obtained and consequently provide an insight of the important dynamics that lead to an optimal filtration process. The solution of partial differential equations representing the internal flow field similarity transformation based on Lie group method is utilized to reduce the system to ordinary differential equations. Thereafter, double perturbation method is employed to determine semi-analytical solutions of reduced system. Effects of various parameters that arise from the configuration (design) of the filter are presented graphically and analysed to show the connection between the case study and findings.

Asymptotic solutions for the asymmetric flow in a channel with porous retractable walls under a transverse magnetic field

The self-similarity solutions of the Navier-Stokes equations are constructed for an incompressible laminar flow through a uniformly porous channel with retractable walls under a transverse magnetic field. The flow is driven by the expanding or contracting walls with different permeability. The velocities of the asymmetric flow at the upper and lower walls are different in not only the magnitude but also the direction. The asymptotic solutions are well constructed with the method of boundary layer correction in two cases with large Reynolds numbers, i.e., both walls of the channel are with suction, and one of the walls is with injection while the other one is with suction. For small Reynolds number cases, the double perturbation method is used to construct the asymptotic solution. All the asymptotic results are finally verified by numerical results.

Analytical study on motion and shape of creeping Boger drops falling through viscoelastic media

In the present study, an analytical solution based on a double perturbation method is obtained to simulate the creeping motion of a Boger drop descending in an immiscible viscoelastic quiescent medium. The perturbation parameters are considered to be the Deborah number defined based on the relaxation times of interior and exterior media and the Capillary number. The Oldroyd-B model is implemented as a constitutive equation for both interior and exterior phases. The analytical solutions are obtained up to second order of perturbation expansion for both phases. The motion of the droplet in the surrounding medium is considered to be creeping. Effects of different parameters including, Deborah numbers, viscosity ratios, elasto-capillary number, and viscosity ratio between the drop and external flows are investigated on the shape, motion, and stream function in detail. In contrast to the generally spherical shape of creeping drops in Newtonian regimes, it is shown that the effect of elasticity in a non-Newtonian scenario can feature prolate or oblate shapes. It is shown that an increment in elastic properties of interior fluid is followed by decrement in the terminal velocity of the droplet, while an increment in elasticity of exterior fluid has a contrary effect and increases the terminal velocity. The results have shown suitable agreements with previously reported experimental results.

A double perturbation method of postbuckling analysis in 2D curved beams for assembly of 3D ribbon-shaped structures

Mechanically-guided 3D assembly based on controlled, compressive buckling represents a promising, emerging approach for forming complex 3D mesostructures in advanced materials. Due to the versatile applicability to a broad set of material types (including device-grade single-crystal silicon) over length scales from nanometers to centimeters, a wide range of novel applications have been demonstrated in soft electronic systems, interactive bio-interfaces as well as tunable electromagnetic devices. Previously reported 3D designs relied mainly on finite element analyses (FEA) as a guide, but the massive numerical simulations and computational efforts necessary to obtain the assembly parameters for a targeted 3D geometry prevent rapid exploration of engineering options. A systematic understanding of the relationship between a 3D shape and the associated parameters for assembly requires the development of a general theory for the postbuckling process. In this paper, a double perturbation method is established for the postbuckling analyses of planar curved beams, of direct relevance to the assembly of ribbon-shaped 3D mesostructures. By introducing two perturbation parameters related to the initial configuration and the deformation, the highly nonlinear governing equations can be transformed into a series of solvable, linear equations that give analytic solutions to the displacements and curvatures during postbuckling. Systematic analyses of postbuckling in three representative ribbon shapes (sinusoidal, polynomial and arc configurations) illustrate the validity of theoretical method, through comparisons to the results of experiment and FEA. These results shed light on the relationship between the important deformation quantities (e.g., mode ratio and maximum strain) and the assembly parameters (e.g., initial configuration and the applied strain). This double perturbation method provides an attractive route to the inverse design of ribbon-shaped 3D geometries, as demonstrated in a class of helical mesostructures.

A Double Perturbation Method for Reducing Dynamical Degradation of the Digital Baker Map

The digital Baker map is widely used in different kinds of cryptosystems, especially for image encryption. However, any chaotic map which is realized on the finite precision device (e.g. computer) will suffer from dynamical degradation, which refers to short cycle lengths, low complexity and strong correlations. In this paper, a novel double perturbation method is proposed for reducing the dynamical degradation of the digital Baker map. Both state variables and system parameters are perturbed by the digital logistic map. Numerical experiments show that the perturbed Baker map can achieve good statistical and cryptographic properties. Furthermore, a new image encryption algorithm is provided as a simple application. With a rather simple algorithm, the encrypted image can achieve high security, which is competitive to the recently proposed image encryption algorithms.

Quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay amplitude

Quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay amplitude is investigated in this paper. The case where the delay amplitude in the position is modulated with a certain amplitude and a resonant frequency is considered. Application of the double perturbation method enables approximation of the amplitudes of quasi-periodic vibrations of the oscillator near a parametric resonance for which the frequency of the delay amplitude modulation is near twice the natural frequency of the oscillator. Analytical approximations supported by numerical simulations provide the regions in parameter space where quasi-periodic vibrations exist. Results show that the modulation of the delay amplitude in the position not only gives birth to quasi-periodic vibrations in a large range of parameters, but also with broadband large-amplitude covering a wide range of excitation frequency.

The Damped Nonlinear Quasiperiodic Mathieu Equation Near 2:2:1 Resonance

We investigate the damped cubic nonlinear quasiperiodic Mathieu equation d 2x dt 2 + (δ + e cos t + eμ cos ωt)x + eμc dx dt + eμγ x 3 = 0 in the vicinity of the principal 2:2:1 resonance. By using a double perturbation method which assumes that both e and μ are small, we approximate analytical conditions for the existence and bifurcation of nonlinear quasiperiodic motions in the neighborhood of the middle of the principal instability region associated with 2:2:1 resonance. The effect of damping and nonlinearity on the resonant quasiperiodic motions of the quasiperiodic Mathieu equation is also provided. We show that the existence of quasiperiodic solutions does not depend upon the nonlinearity coefficient γ , whereas the amplitude of the associated quasiperiodic motion does depend on γ .

Coupling of geminal protons in nuclear magnetic resonance

The J HHgem coupling constants of six polyatomic molecules were calculated ab initio by the double perturbation method in a basis of localized SCF orbitals. Leading terms of different orders were retained. The agreement with experimental values is satisfactory.

A double expansion method for calculating molecular properties

A double perturbation method is proposed for the calculation of molecular energies over a whole range of internuclear separations. An expression is derived which gives the energy in terms of atomic energies together with energy contributions from the ligands. The energy is written in a form which enables calculations to be performed on a whole sequence of iso-electronic molecular species. The method is specifically aimed to deal with those symmetrical molecules which contain one heavy atom surrounded by a number of identical lighter atoms. As an illustration of the general formalism the ground state of two-electron diatomic ions is considered.

A double expansion method for calculating molecular properties

A double perturbation method is proposed for the calculation of molecular energies over a whole range of internuclear separations. An expression is derived which gives the energy in terms of atomic energies together with energy contributions from the ligands. The energy is written in a form which enables calculations to be performed on a whole sequence of iso-electronic molecular species. The method is specifically aimed to deal with those symmetrical molecules which contain one heavy atom surrounded by a number of identical lighter atoms. As an illustration of the general formalism the ground state of two-electron diatomic ions is considered.

Double perturbation method on LCAO-SCF functions

A perturbation theory on LCAO-SCF functions is developed, by the use of double-perturbation methods. First and second order expressions for the energy are obtained.

Electron-Coupled Proton Spin-Spin Interaction in Saturated Hydrocarbon Molecule. I

Electron coupled proton spin-spin interactions in saturated hydrocarbon molecules. J HH' , are investigated theoretically. The general formula for J HH' is derived by double perturbation method, in which \(\mathfrak{H}_{F}\) (the Fermi interactions between electrons and proton-spins) and \(\mathfrak{H}_{1}\) (correlations between electrons) are taken as perturbations. The calculation is performed up to first order in \(\mathfrak{H}_{1}\). It is shown that the zeroth order term in \(\mathfrak{H}_{1}\) becomes nearly equal to zero under the approximation, in which energy differences between the ground state and the excited states are replaced by a suitable mean. Therefore, J HH' calculated by the molecular orbital method without configuration interactions is a part of J HH' , which is different from the one calculated by the valence bond method with the above approximation. The order of magnitudes of the zeroth and the first order term in \(\mathfrak{H}_{1}\) are compared in the case of methane molecule.

Spin-Spin Interactions in Nuclear Magnetic Resonance. Contact Contribution

A method for calculating the contact contribution to spin-spin interactions between nuclei in nonaromatic molecules is described. The method is based on the valence bond model. The equivalent Hamiltonian of the Dirac vector model is used for a perturbation calculation in a representation where the total spin of the two electrons in each bond in the molecule is a good quantum number. The Ramsey-Purcell contact term in the interaction is calculated by a double perturbation method where only terms linear in the electron-nuclear interactions are considered but the perturbation is carried to higher order in the exchange integrals. In this way the interaction constants can be obtained explicitly in terms of the exchange integrals, and the calculation of the valence bond wave functions is avoided. It is then possible to see how the signs and magnitudes of the interaction constants depend on these integrals and what the dominant interaction mechanisms are. The perturbation series is evaluated explicitly with the electron nuclear contact interaction and the results are compared with the results of a similar calculation based on Ramsey's closure formula. In this way the objections against the closure procedure can be avoided. It is shown that Ramsey's formula can be used when different average energy separations are used for terms of different order in the expansion. The results of Ramsey for H2 and of Karplus et al. for hydrogen atoms separated by two and three bonds and the justification of the method of calculation used by them are discussed in detail. It is found that only perturbation terms of one type are important in each of the cases discussed by these authors and therefore their procedure can be justified. The results previously obtained by the author for spin-spin interactions in the allyl group are also discussed.