Regular Perturbation Technique Research Articles

Thermal Radiation, Chemical Reaction and Viscous Dissipation Effects on Unsteady MHD Flow of Viscoelastic Fluid Embedded in a Porous Medium

In this paper the effect of unsteady, incompressible, magneto hydrodynamics filled with electrically conducting viscoelastic fluid in an infinite vertical Couette porous channel wall embedded in a porous medium is analyzed. A uniform magnetic field is applied perpendicular to the channel wall. The temperature of the moving channel wall varies periodically and the temperature difference between the two infinite vertical channel walls is high due to thermal radiation. The Eckert number is the ratio of the kinetic energy of the flow to the temperature difference of the channel walls. The solution of the governing equations is obtained using regular perturbation techniques. These techniques are used to transform partial differential equations that are difficult to solve in closed form. These equations are reduced to a set of ordinary differential equations in dimensionless form so can be solved analytically. The effects of physical parameters Viz. Hartmann number, Viscoelastic parameter, Eckert number, Permeability of porous medium, Chemical reaction parameter, thermal Grashof number for heat transfer, modified Grashof number for mass transfer, frequency parameter and Schmidt number on flow parameters Viz., velocity, temperature and concentration has been discussed and shown graphically. The theoretical results have been supported by MATLAB code simulation study. The results show that velocity decreases with increasing values of frequency, Hartmann number and viscoelastic parameter but reverse effect is observed with temperature, thermal Grashof number, modified Grashof number and permeability of porous medium. Furthermore, The result shows that an increment in both thermal radiation parameter and Eckert number results in decrement of temperature near the moving porous channel wall while it approaches to a zero in the region close to the boundary layer of the stationary channel wall,. An increment in both chemical reaction and Schmidt number results in decreasing concentration. The velocity of fluid increases as Grashof number and modified Grashof number increases.

Effect of an inclined magnetic field on peristaltic flow of Bingham plastic fluid in an inclined symmetric channel with slip conditions

This paper studies the influence of an inclined magnetic field on peristaltic transport of incompressible Bingham plastic fluid in an inclined symmetric channel with heat transfer and mass transfer. Slip conditions for heat transfer and concentration are employed. The formulation of the problem is presented through, the regular perturbation technique for small Bingham number Bn is used to find the final expression of streamfunction, the flow rate, heat distribution and concentration distribution. The numerical solution of pressure rise per wave length is obtained through numerical integration because its analytical solution is impossible. Also the trapping phenomenon is analyzed. The effect of the physical parameters of the problem are discussed and illustrated graphically.

An asymptotic expansion method for geometric Asian options pricing under the double Heston model

Abstract The purpose of the paper is to provide an efficient method for the continuously monitored geometric Asian options under the double Heston model. By introducing two small parameters, we slightly modify the double Heston model. With singular and regular perturbation techniques, we derive the first-order asymptotic expansions for pricing geometric Asian options with fixed and floating strikes and provide the convergence of the asymptotic formulae. We also provide the Greeks of geometric Asian options. Numerical results verify the efficiency of the pricing method. We calibrate the modified model to real markets and examine the impacts of two-factor volatilities on geometric Asian option prices.

On the propagation of the non-similar wall jet flows with suction/injection

The present work is dedicated to the analytic analysis of the non-similar wall jet flows subject to various suction/injection profiles for the 1st time. To this end, it is employed regular perturbation technique to expand the solution about the Glauert origin. We have propagated the perturbed field up to the 15th order to identify the compatibility of the technique with the problem of interest as well as to explore some important features associated with the various suction/injection profiles. In the case of suction, the maximum normalized shear stress at the wall evolves as the exponent of the suction profiles varies; afterwards, a decrease in the quantity is expected so as to eventually annihilate the entire jet. Similarly, in the injection case, the exponent of the injection profiles plays a core role in the evolution of the normalized shear stress at the wall, the universal distance in which the quantity becomes zero and the recovery of the quantity after this critical location.

Heat transfer analysis on electroosmotic flow via peristaltic pumping in non-Darcy porous medium

A theoretical model is developed to simulate electroosmotic transport via peristaltic motion in non-Darcy porous medium. The flow analysis is done under the magnetohydrodynamic environment. Peristaltic flow is induced by a traveling wave along the channel wall. Under the creeping flow constraints, low Reynolds number and long wavelength approximations are assumed to simplify the governing equations. Debye-Hückel linearization is also employed. A regular perturbation technique is applied to obtain an approximate solution by considering small Forchheimer number cF as a perturbation parameter. The results are presented graphically and effects of pertinent parameters on pumping characteristics, thermal characteristics and Nusselt number are examined. The model and associated results may encourage biomedical engineers to design and develop such biomicrofluidics devices which may help in transporting physiological fluids.

Analytical solutions for longitudinal-plane motion of hypersonic skip-glide trajectory

The analytical solutions for longitudinal-plane motion of hypersonic skip-glide trajectory are proposed by perturbation theory. First, the analytical solution of downrange distance is approximated by the outer solutions of a two-timescale singular perturbation system where the solution error is less than 2%. Then, the zeroth-order solutions and first-order corrections of altitude and flight-path angle are developed using regular perturbation techniques which could decompose the coupled and nonlinear system into a sequence of linear problems to solve in order. Because the first-order system for altitude and flight-path angle is a linear time-varying system that cannot be solved by traditional method such as Laplace transform, two methods are put forward to simplify the system: spectral decomposition method (SDM) and linear transformation method (LTM). SDM assumes the system to be time invariant and then uses spectral decomposition to solve it. LTM introduces a small compensation into the system such that the system can be decoupled into two independent first-order differential equations by linear transformation. Then, by combining the advantages of the two kinds of solutions, more accurate integrated analytical solutions are proposed. The simulation results show that the analytical solutions are in good agreement with the numerical solutions. In addition, the computation time of the analytical solutions is about two orders of magnitude less than that of the numerical solutions.

An analytical approximation method for pricing barrier options under the double Heston model

The purpose of the paper is to provide an efficient pricing method for single barrier options under the double Heston model. By rewriting the model as a singular and regular perturbed BS model, the double Heston model can separately mimic a fast time-scale and a slow time-scale. With the singular and regular perturbation techniques, we analytically derive the first-order asymptotic expansion of the solution to a barrier option pricing partial differential equation. The convergence and efficiency of the approximate method is verified by Monte Carlo simulation. Numerical results show that the presented asymptotic pricing method is fast and accurate.

Semi-analytical solution of nonlinear dynamic behaviour for fully saturated porous media

In this study, a semi-analytical solution of the constitutive nonlinear differential equations of fully saturated porous media is presented. Also, the nonlinear dynamic behaviour of material and geometry is considered. The u–w–p formulation of fully coupled dynamic field is used to simulate the pore fluid and soil skeleton responses with the change of pore pressure. The governing equations with the corresponding boundary conditions were solved analytically using regular perturbation technique. The parametric study was utilized to investigate the effect of nonlinearity of material and geometry on the pore pressure, total stress, solid and fluid displacements. Results are illustrated graphically for the presented semi-analytical model. It was found that the values of frequencies of applied load and level of permeability had a great effect on the total stress distribution, pore pressure, solid displacement and fluid displacement.

Effect of Exponentially Temperature-Dependent Viscosity on the Onset of Penetrative Ferro-Thermal-Convection in a Saturated Porous Layer via Internal Heating

The effect of viscosity depending exponentially on temperature on the onset of penetrative ferro-thermal-convection (FTC) in a saturated horizontal porous layer in the presence of vertical magnetic field is investigated. The bounding surface of the ferrofluid layer is considered to be rigid-rigid and insulated to temperature perturbations. The resulting eigenvalue problem is solved numerically using the Galerkin technique and also analytically by a regular perturbation technique with wave number as a perturbation parameter. The analytical and numerical results are found to be concurrence. The characteristics of stability of the system are strongly dependent on the viscosity parameter B. The effect of B on the onset of ferroconvection in a porous layer is dual in nature depending on the choices of physical parameters and a sublayer starts to form at higher values of B. Whereas, increase in magnetic number M1 and the Darcy number Da is to advance the onset of ferroconvection in a porous layer. The nonlinearity of fluid magnetization M3 is found to have no influence on the onset of ferroconvection.

Effect of temperature-dependent properties on electroosmotic mobility at arbitrary zeta potentials

• Electroosmotic mobility and volumetric flow rate as temperature-dependent zeta potential are determined. • Analytical solutions for electric and hydrodynamic fields by assuming temperature-dependent zeta potentials are obtained. • The zeta potential is obtained as a function of the temperature field in the electroosmotic flow . • Non-isothermal electroosmotic flow at arbitrary zeta potentials is theoretically analyzed. A theoretical analysis to determine the electroosmotic mobility in an electroosmotic flow (EOF) in a microchannel at arbitrary zeta potentials is conducted in this study. As an important characteristic in this work, we consider that the wall zeta potentials of the microchannel and the viscosity and electrical conductivity of the electrolyte solution vary with temperature. The flow and the electric and temperature fields are obtained using lubrication approximation theory (LAT) together with the application of the regular perturbation technique. The electroosmotic mobility is evaluated, showing an increase higher than 18% (for the values of the physical properties used in this work) when physical properties, including the zeta potential of the microchannel walls, are considered as temperature-dependent functions compared with the isothermal case. Additionally, we show that the volumetric flow rate is drastically influenced when the zeta potential varies with temperature.

Influence of Thermodiffusion on Time Dependent Casson Fluid Flow past a Wavy Surface

The characteristics of unsteady motion of Casson fluid over a wavy surface with magnetic field is studied. The energy and diffusion equations are encompassed with radiation, heat source and thermo diffusion effects. The flow governing equations have been solved using regular perturbation technique. MATLAB bvp4c package is for discerning the impact of governing parameters on the usual flow profiles. various pertinent parameters on velocity, temperature and concentration profiles of the Casson fluid flow. The frictional factor, heat and mass transfer coefficients are tabulated for the same parameters. From this analysis, it is worth mentioning that a hike in fluid viscosity slowdowns the velocity profiles but the result is opposite with Soret effect. Also, accelerating values of radiation parameter uplifts the rate of mass and heat transport but controls the shear stress.

Unsteady Peristaltic Transport of a Particle-Fluid Suspension: Application to Oesophageal Swallowing

Abstract This model characterises the flow behaviour of suspended particles when swallowing through oesophagus. Transport of particle-fluid mixture induced by dilating peristaltic waves on a circular cylindrical tube was considered for investigation. Unsteady closed-form solutions for pressure gradient, velocity and stream function are obtained by applying regular perturbation technique up to the first order of wavenumber (the ratio of the tube radius to the wavelength). It is observed that the axial velocity of the fluid is greater than that of the solid particles almost everywhere. However, at the wall the axial velocity of the fluid is zero due to the no-slip condition imposed on it; but the suspended particulate material has non-zero positive axial velocity. Thus, the axial velocity of the suspended particles near the tube wall is more than that of the fluid velocity. It is further observed that the axial velocity is negative in the regions close to maximum occlusions giving way to instantaneous backward flow. It is also inferred that the maximum axial velocity of the particle-fluid suspension with non-zero wavenumber is more than that of the particle-free fluid with zero wavenumber. We examined the effect of volume fraction of suspended particulate matter on pressure gradient and velocity. An increase in the volume fraction diminished the pumping performance and also the axial and radial velocities. The results are also compared with those which were obtained for zero wavenumber. The outcome of the investigation endorses the doctors’ advice to patients suffering from achalasia, oesophageal stricture and oesophageal tumors to take liquid or food items with lesser solid contents. Streamline patterns are changed by increasing the flow rate while trapping occurs at high flow rates.

Effect of Thermal Radiation, Chemical Reaction and Viscous Dissipation on MHD Flow

This study examines the effect of thermal radiation, chemical reaction and viscous dissipation on a magnetohydro- dynamic flow in between a pair of infinite vertical Couette channel walls. The momentum equation accounts the effects of both the thermal and the concentration buoyancy forces of the flow. The energy equation addresses the effects of the thermal radiation and viscous dissipation of the flow. Also, the concentration equation includes the effects of molecular diffusivity and chemical reaction parameters. The gray colored fluid considered in this study is a non-scattering medium and has the property of absorbing and emitting radiation. The Roseland approximation is used to describe the radiative heat flux in the energy equation. The velocity of flow transforms kinetic energy into heat energy. The increment of the velocity due to internal energy results in heating up of the fluid and consequently it causes increment of the thermal buoyancy force. The Eckert number being the ratio of the kinetic energy of the flow to the temperature difference of the channel walls is directly proportional to the thermal energy dissipation. It can be observed that increasing the Eckert number results in increasing velocity. A uniform magnetic field is applied perpendicular to the channel walls. The temperature of the moving wall is high enough due to the presence of thermal radiation. The solution of the governing equations is obtained using regular perturbation techniques. These techniques help to convert partial differential equations to a set of ordinary differential equations in dimensionless form and thus they are solved analytically. The following results are obtained: from the simulation study it is observed that the flow pattern of the fluid is affected due to the influence of the thermal radiation, the chemical reaction and viscous dissipation. The increment in the Hartmann number results in the increment of the Lorentz force but a decrement in velocity of the flow. An increment in the radiative parameter results in a decrement in temperature. An increment in the Prandtl number results in a decrement in thermal diffusivity. An increment in both the chemical reaction parameter and molecular diffusivity results in a decrement in concentration.

Sedimentation of a surfactant-laden drop under the influence of an electric field

The sedimentation of a surfactant-laden deformable viscous drop acted upon by an electric field is considered theoretically. The convection of surfactants in conjunction with the combined effect of electrohydrodynamic flow and sedimentation leads to a locally varying surface tension, which subsequently alters the drop dynamics via the interplay of Marangoni, Maxwell and hydrodynamic stresses. Assuming small capillary number and small electric Reynolds number, we employ a regular perturbation technique to solve the coupled system of governing equations. It is shown that when a leaky dielectric drop is sedimenting in another leaky dielectric fluid, the Marangoni stress can oppose the electrohydrodynamic motion severely, thereby causing corresponding changes in the internal flow pattern. Such effects further result in retardation of the drop settling velocity, which would have otherwise increased due to the influence of charge convection. For non-spherical drop shapes, the effect of Marangoni stress is overcome by the ‘tip-stretching’ effect on the flow field. As a result, the drop deformation gets intensified with an increase in sensitivity of the surface tension to the local surfactant concentration. Consequently, for an oblate type of deformation the elevated drag force causes a further reduction in velocity. For similar reasons, prolate drops experience less drag and settle faster than the surfactant-free case. In addition to this, with increased sensitivity of the interfacial tension to the surfactant concentration, the asymmetric deformation about the equator gets suppressed. These findings may turn out to be of fundamental significance towards designing electrohydrodynamically actuated droplet-based microfluidic systems that are intrinsically tunable by varying the surfactant concentration.

Effect of peristaltic flow of a third grade fluid in a tapered asymmetric channel

Peristaltic transport analysis of a third order fluid in a tapered asymmetric channel is made. The tapered asymmetric channel has been produced by choosing the peristaltic wave train on the narrow walls to have different amplitudes and phase. Long wavelength and low Reynolds number approach have been employed. The expressions for the horizontal velocity, stream function, pressure rise and frictional forces are obtained by a regular perturbation technique. Numerical computations have also been performed for the pressure rise, frictional force and amplitude of the shear stress and the effects of rheology parameter, amplitudes and phase difference of walls on the flow characteristics are further discussed in detail.

Dufour Effects on Unsteady MHD Free Convective Flow of Two Immiscible Fluid in a Horizontal Channel Under Chemical Reaction and Heat Source

The effect of Dufour on Unsteady MHD convective immiscible fluid flow passing through a horizontal channel in the presence of Heat, mass transfer, chemical reaction and Heat source consumption are studied here. The non-linear equations are solved with the appropriate boundary conditions. The governing equations of the flow are solved by a regular Perturbation technique. Then the effects of various flow parameters like Grashof numbers for heat and mass transfer, Prandtl number, Viscosity ratio, conductivity ratio, Radiative parameter, Schmidt number, Dufour effect etc. on the velocity field have been solved graphically and analyzed quantitatively.

English

In this paper, unsteady two-dimensional convective heat and mass transfer flow of a viscous, incompressible, electrically conducting optically thin fluid which is bounded by a vertical infinite plane surface was considered. A uniform applied homogeneous magnetic field is considered in the transverse direction with first order chemical reaction. An analytical solution for two-dimensional oscillatory flow on unsteady mixed convection of an incompressible viscous fluid, through a porous medium bounded by an infinite vertical plate in the presence of chemical reaction and thermal radiation are presented. The surface absorbs the fluid with a constant suction and the free stream velocity oscillates about a constant mean value. The resulting nonlinear partial differential equations were transformed into a set of ordinary differential equations using two-term series. The closed form solutions for velocity, temperature, concentration, skin friction, Nusselt number, and Sherwood number have been obtained, using the regular perturbation technique. Numerical evaluation of the analytical solutions was performed and the results are presented in tabular and graphical form. This illustrates the influence of the various parameters involved in the problem on the solutions. Key words: Magnetohydrodynamics (MHD) flow, heat and mass transfer, oscillatory flow, thermal radiation, chemical reaction.

Dispersion coefficient in an electro-osmotic flow of a viscoelastic fluid through a microchannel with a slowly varying wall zeta potential

The dispersion coefficient of a passive solute in a steady-state pure electro-osmotic flow (EOF) of a viscoelastic liquid, whose rheological behaviour follows the simplified Phan-Thien–Tanner (sPTT) model, along a parallel flat plate microchannel, is studied. The walls of the microchannel are assumed to have modulated and low $\unicode[STIX]{x1D701}$ potentials, which vary slowly in the axial direction in a sinusoidal manner. The flow field required to obtain the dispersion coefficient was solved using the lubrication approximation theory (LAT). The solution of the electric potential is based on the Debye–Hückel approximation for a symmetric $(z:z)$ electrolyte. The viscoelasticity of the fluid is observed to notably amplify the axial distribution of the effective dispersion coefficients due to the variation in the $\unicode[STIX]{x1D701}$ potentials of the walls. The problem was formulated for two cases: when the Debye layer thickness (EDL) was on the order of unity (thick EDL) and in the limit where the thickness of the EDL was very small compared with the height of the microchannel (thin EDL limit). Due to the coupling between the nonlinear governing equations and the sPTT fluid model, they were replaced by their approximate linearized forms and solved in the limit of $\unicode[STIX]{x1D700}\ll 1$ using the regular perturbation technique. Here $\unicode[STIX]{x1D700}$ is the amplitude of the sinusoidal function of the $\unicode[STIX]{x1D701}$ potentials. Additionally, the numerical solution of the simplified governing equations was also obtained for $\unicode[STIX]{x1D700}=O(1)$ and compared with the approximate solution, showing excellent agreement for $0\leqslant \unicode[STIX]{x1D700}\leqslant 0.3$. Note that the dispersion coefficient primarily depends on the Deborah number, on the ratio of the half-height of the microchannel to the Debye length, and on the assumed variation in the $\unicode[STIX]{x1D701}$ potentials of the walls.

A non-linear study of fluctuating fluid flow on MHD mixed convection through a vertical permeable plate

In this paper, an analytical solution for an unsteady (independent of time), MHD mixed convection, two-dimensional (x and y), laminar, viscous flow of an incompressible fluid through a vertical permeable plate in a porous medium was developed with these assumptions:(i) the suction velocity (which is normal to the plate)and the free stream velocity both fluctuate with respect to time with a fixed mean; (ii) the wall temperature is constant;(iii) difference between the temperature of the plate and the free stream is moderately large due to the free convection currents. Based on the physical configuration of the model, the governing equations are derived and are non-dimensionalize using dimensionless parameters. The resultant nonlinear partial differential equations are solved using double regular perturbation technique analytically. The results are computed numerically to understand the behaviour of the fluid (i.e., effects of MHD, viscosity, body force etc.) for various non-dimensional parameters involving like Grashof number Gr, Prandtl number Pr, Hartmann number M, Eckert number E, the Viscous ratio λ and so on for velocity and temperature. These results are found to be in good agreement with known results available in the literature in the absence of few physical parameters. The numerical values of the above said flow is discussed through graphs on velocity and temperature.

The Linear and Non-linear Study of Effects of Temperature Modulation on Double Diffusive Convection in Oldroyd-B Liquids

The problem aims to find the effects of temperature modulation in viscoelastic fluids, namely Oldroyd-B liquids, subjected to double diffusive convection. Both linear as well as a non-linear study has been carried out. A regular perturbation technique has been employed to determine the correction Rayleigh number. The results show that the Lewis number and strain retardation parameter induces stability to the system whereas stress relaxation parameter destabilizes the system for synchronous modulation. Truncated Fourier series represents the non-linear analysis. Nusselt number and Sherwood number are used to quantify the heat and mass transfer respectively. Strain retardation parameter and the Lewis number decrease the heat and mass transfer for synchronous modulation while stress relaxation parameter increases them. The opposite results are obtained for asynchronous and lower wall modulations. Modulation is shown mostly to give rise to super-critical motion.