Regular Perturbation Technique Research Articles

Theoretical Study of Forced Van Der Pol Oscillator Equation Using Multiple Two-timing Regular Parameter Perturbation and Asymptotic Expansion Techniques

This paper presents the theoretical study of forced Van der Pol oscillator equation. Oscillatory systems are studied to know measures that can reduce the amplitude of oscillation of the oscillatory system. Here, multiple two-timing regular parameter perturbation is applied since it is a kind of perturbation among other perturbation techniques that enables the study of the behaviour of a system under certain conditions. Asymptotic expansion technique was also applied. Excel Microsoft was used to analyse the uniformly valid asymptotic solution of the Van der Pol oscillator equation obtained. The uniformly valid asymptotic solution in the independent variable obtained, showed that damping alters the amplitude of the oscillatory system thereby affecting its motion. Increase in damping decreases the amplitude of oscillation of the system. With damping incorporated in the system though very small damping, the amplitude of oscillation reduces with time.

Entropy Generation For Non-Newtonian Fluid With Constant Viscosity

In this research, entropy generation for non-Newtonian fluid flow with constant viscosity is considered. The governing nonlinear equations of motion are solved analytically with regular perturbation techniques. Third grade fluid is employed to account for the non-Newtonian influence. The influence of some physical parameters involved in the analysis is studied. Results show that the parameter has the tendency of increasing the velocity of the fluid flow as well as the temperature of the cylindrical pipe. It is observed that as the Brinkman parameter increases, the cylindrical wall temperature is enhanced. Entropy generation number for both heat transfer and fluid friction for various values of and is examined. Results indicate that increase in these parameters increases the temperature of the cylinder, thereby increasing the entropy generation number.

LTNE effects on two‐layer configuration with throughflow

AbstractThe effect of throughflow on the beginning of convective motion in a fluid layer overlying a fluid‐saturated porous layer is examined theoretically in this article under the local thermal nonequilibrium model. We considered boundaries to be insulating to temperature perturbations. The normal mode approach is employed to analyze the governing equation for a composite configuration, with the solution sought through the regular perturbation technique to assess linear stability. The critical Rayleigh number, signifying the initiation of stationary convection, has been determined by solving the system of coupled equations. It is found that the system attains stability with increased porosity–conductivity ratios, regardless of the throughflow direction. This stability is consistently observed for depth ratios of 0.1 and 0.2. However, for higher depth ratios (0.5 and 1.0), the system exhibits dual nature stability when throughflow is present. Further, the stability of the system is enhanced by a graphical depiction illustrating how the throughflow parameter, porosity‐scaled conductivity ratio, depth ratio, Darcy number, and Prandtl number collectively influence the onset of convective motion.

Exploring pressure, temperature, and flow patterns in ciliated microfluidic systems

The integration of cilia-induced flow, nanofluids, and the inclusion of cobalt ferrite particles holds significant promise in fluid dynamics, heat transfer, and nanotechnology, offering potential breakthroughs in various technological and material applications. We explore the behavior of cilia-induced flow in a nanofluid confined within an annular domain, employing the Williamson fluid model to characterize the behavior of cobalt ferrite (CoFe2O4) nanoparticles. Our analysis is based on a mathematical treatment rooted in fundamental mass, momentum, and energy conservation principles while considering physical constraints (low Reynolds number and long wavelength) and adopting a dimensionless approach. By applying regular perturbation techniques, we derive series solutions for velocity and temperature profiles, providing insight into the complex interplay among cilia-generated flow, nanofluid properties, and the influence of Cobalt ferrite nanoparticles within the annular configuration. In particular, we uncovered clear correlations among cilia length, amplitude ratio, flow rate, and the Prandtl number with temperature distribution. Also, we observed substantial reductions in temperature trends under Weissenberg numbers and particle volume fractions.

Anisotropic Darcy–Brinkman Magnetic Fluid Convection under the Influence of a Time-Dependent Sinusoidal Magnetic Field

The impact of the sinusoidal mode of a magnetic field involving time fluctuations on the threshold of the ferromagnetic smart liquid convection in a saturated permeable medium is investigated using the regular perturbation technique. The Darcy–Brinkman model with anisotropic permeability is used to describe the flow through porous media. The thermal anisotropy is implemented in the energy equation. The problem might be useful in thermal engineering applications such as dynamic loudspeakers and computer hard discs and in medical applications like the treatment of tumor cells and the cell separation, to name a few. The regular perturbation technique is based on the minimum amplitude of a magnetic field modulation, and the onset criterion is dealt with in terms of a correction in the critical Rayleigh number and wavenumber. The thermal Rayleigh number correction depends on the magnetic field modulation frequency, magnetic force, anisotropies, porosity, and Prandtl number. At moderate values of the magnetic field modulation frequency, the impact of various physical factors is perceived to be noteworthy. The influences of the magnetic mechanism, Prandtl number, porosity parameter, and Brinkman number are shown to augment the destabilizing effect of the magnetic field modulation for moderate values of the frequency of a modulation. However, the destabilizing effect of the magnetic field modulation is diminished due to an increase in the values of the mechanical anisotropy parameter and thermal anisotropy parameter. The study reveals that the effect of the magnetic field modulation could be exploited to control the convective instability in an anisotropic porous medium saturated by a ferromagnetic fluid.

Flow and heat transfer analysis on micropolar fluid through a porous medium between a clear and Al2O3−Cu/H2O in conducting field

The current study is focused on the flow of micropolar liquid in a saturated permeable medium sandwiched between clear and hybrid nanofluid filled in an inclined channel, with radiation and dissipation effects. Copper and Aluminium oxide nanoparticles is considered along with base (water) fluid. The solution for velocity, micro rotational velocity, and temperature are determined by applying the regular perturbation technique to the dimensionless version of the governing equations. The heat transport rate at the left y=−1 and right y=2 plate is also calculated, as well as shear stress expressed in terms of skin-friction coefficient and Nusselt number. The result obtained for various physical parameters are analyzed through several graphs and tables. Results reveal that an increase in the material parameter of micropolar fluid, the motion of the fluid flow decreases. The heat transport rate is enhanced and the velocity is degraded by the magnetic parameter. The Hybrid nanofluid temperature is greater when compared to clear and binary hybrid nanofluid. This work establishes the mechanisms for heat transport enhancement on micropolar liquid through a porous medium between a clear and hybrid nanofluid in conducting field.

Study of Heat and Mass Transfer of an Unsteady Magnetohydrodynamic (MHD) Nanofluid Flow Past a Vertical Porous Plate in the Presence of Chemical Reaction, Radiation and Soret Effects

This paper investigates the heat and mass transfer of an unsteady, MHD incompressible water-based nanofluid (Cu and TiO2) flow over a stretching sheet in a transverse magnetic field with thermal radiation Soret effects. The governing differential equations are transformed into a set of non-linear ordinary differential equations and solved using a regular perturbation technique with appropriate boundary conditions for various physical parameters. The effects of different physical parameters on the dimensionless velocity, temperature, and concentration profiles are depicted graphically and analyzed in detail. Favourable comparisons with previously published work on various exceptional cases of the problem are obtained. Finally, numerical values of the physical quantities, such as the local skin-friction coefficient, the local Nusselt number and the local Sherwood number, are presented in tabular form. Results describe that the velocity and temperature diminish with enhancing the thermal radiation. Concentration decreases with improving the chemical reaction. Both velocity and concentration are enhanced with increases of soret parameter. And also, water–based TiO2 nanofluids possess higher velocity than water-based Cu nanofluids.

Study of peristaltic activity in non-linear blood analysis of Williamson fluid in a microchannel

The present investigation is modeled to study the effect of homogeneous and heterogeneous reactions on the peristaltic blood transportation of electrically conducting Williamson fluid in an asymmetric microchannel under velocity slip conditions. The mechanism of heat transfer is also examined in conjunction with viscous dissipation and thermal radiation. Implementation of small Reynolds number and large wavelength concepts leads to a reduction in the complexity of the system. Series solutions for velocity, temperature, stream function, and concentration fields are found using the regular perturbation technique. Using Mathematica software, Simpson’s rule was used to evaluate the pumping features such as pressure rise and frictional force. The effects of embedded parameters are portrayed and discussed by using a graphical approach. The skin friction and Nusselt number at the channel walls are established for a choice of assessments of significant parameters and which are exhibited in a tabular manner. It is observed that an enhancement in the values of Weissenberg number fluid velocity reduces at the lower channel wall whereas it increases at the channel upper wall.

The impacts of shape factors in a chemically reacting two-passage vertical channel filled with kerosene based graphene oxide and MoS2 mixture in a porous medium

The present investigation is carried out to examine the effect of shape factors on the heat and mass transfer flow of nanofluid (Kerosene-Graphene Oxide) and hybrid nanofluid (Kerosene- Molybdenum disulphide and Graphene Oxide) in the presence of thermal radiation and species diffusion in an inclined two-phase porous channel. The Darcy and viscous dissipation effects are incorporated. The properties of hybrid and nanofluids are studied using nanoparticle volume fraction. The analytical solution for the governing equations is obtained by the regular perturbation technique. The significance of physical parameters on the flow dynamics is examined using a visual representation of the solutions. Various entrenched parameters are used in the calculation of the Nusselt number and shear stress at the surfaces. The significant result of this study is the maximum enhancement of heat transfer and fluid velocity in the hybrid nanofluid region with lamina-shaped nanoparticles. An increase in the radiation parameter and the volume fraction of graphene oxide nanoparticles decreases the temperature while opposite phenomenon is observed for MoS2 nanoparticles. This study has been validated by prior research to demonstrate its efficiency.

Impact of Variable Liquid Properties on Peristaltic Transport of Non-Newtonian Fluid Through a Complaint Non-Uniform Channel

The present investigation emphasises a new attempt at the peristaltic mechanism of Eyring Powell fluid through a non-uniform channel. The analysis is performed in the presence of wall properties under the influence of variable liquid properties, and the flow problem is mathematically developed. The channel walls are subjected to no-slip conditions with long-wavelength and low Reynolds-number approximations employed in the study. The nonlinear governing equations are normalised using relevant non-dimensional parameters, and the solutions are obtained with the help of a regular perturbation technique. The influence of pertinent physical parameters of interest, such as velocity, temperature, concentration and streamlines, are represented graphically. The investigation reveals that the material parameters and elastic parameters of the Eyring Powell fluid model strongly affect the velocity and temperature profiles and that the opposite behaviour has been observed in the material parameters.

The peristaltic flow for Carreau fluid through an elastic channel

Abstract The purpose of this study is to investigate the effect of an elastic wall on the peristaltic flow of Carreau fluid between two concentric cylinders, where the inner tube is cylindrical with an inelastic wall and the outer wall is a regular elastic sine wave. For this problem, cylindrical coordinates were used, and a short wavelength “relative to channel width for its length,” as well as the governing equations of Carreau fluid in the Navier–Stokes equations. Then, the analytical solution has been investigated by using the regular perturbation technique. The solutions obtained by this perturbation are up to the fourth-order in dimensionless Weissenberg number ( W e {W}_{{\rm{e}}} ). The performed computations of various parameter values such as velocity, shear stress, and wave frame streamlines are discussed in detail for different values of the Weissenberg number ( W e {W}_{{\rm{e}}} ). The obtained results demonstrate that the fluid velocity increases with the increase in the value of W e {W}_{{\rm{e}}} and some features of the wall, while the opposite behavior is observed with the increase in other features of the wall. Hence, the presented numerical analysis reveals many aspects of the flow by considering a non-Newtonian Carreau fluid model, and the presented model can be equally applicable to other bio-mathematical studies. The results were evaluated using the Mathematica software program. The Mathematica program was used by entering various data for the parameters, where the program showed the graphs, then the effect of these parameters became clear and the results were mentioned in the conclusion.

Thermal Radiation and Mass Transfer Analysis in an Inclined Channel Flow of a Clear Viscous Fluid and H2O/EG-Based Nanofluids through a Porous Medium

Nanofluid flow has acquired various interesting dimensions with the advent of several novel approaches to studying thermophysical properties. The present work focuses on a comparative study of clear viscous and nanofluid (EG−Al2O3, EG−Zr, H2O−Al2O3, H2O−Zr) flow in a two-phase inclined channel saturated with a porous medium in the presence of thermal radiation, species diffusion, and viscous and Darcy dissipation effects. The controlling equations of the flow model were solved analytically using the regular perturbation technique. The graphical solutions are used to examine the impacts of physical parameters on the most significant flow features. Surface graphs with distinct entrenched parameters represent heat transfer rates and shear stresses on plates. The resulting heat transfer was enhanced by raising the thermal and solute buoyancy strengths, while thermal radiation had the opposite outcome. This enhancement of temperature was maximum for water–zirconium and minimum for ethylene glycol–aluminum oxide nanofluid. The concentration of the entire fluid medium is reduced by decreased mass diffusivity. The enhancement of temperature and velocity is found to be maximum in the nanofluid region and clear fluid region, respectively. This study is validated with previously published works to demonstrate its effectiveness.

Theoretical Analysis of Roll-over-web Coating of a Non-Newtonian Polymer Using Lubrication Approximation Theory

Roll-coating process plays an important role in many industries for its practical applications such as paint, PVC coated fabrics and plastic industries. There are several roll-to-roll coating methods including forward and reverse roll-coating. However, the roll-over-web coating study of the Rabinowitsch model is presented in this paper. The flow equations for the problem are developed and converted into dimensionless form with the help of dimensionless variables and then finally simplified by a well-known lubrication approximation theory. We employ the regular perturbation technique to get analytical expressions for velocity, pressure, and pressure gradient. Engineering quantities such as power input function and roll-separating force are calculated by Runge-Kutta method. The dimensionless Rabinowitsch parameter effect on velocity, pressure, pressure gradient, load-carrying force, and power input are shown graphically. It is interesting to note that for the shear thickening case, the Rabinowitsch model predicts 35% higher pressure, while in the shear thinning case it predicts 29% less pressure in the nip region when compared to the Newtonian model. The force and power show a decreasing trend on increasing the dimensionless Rabinowitsch parameter a. Moreover, the separation point shifts right of its Newtonian value when fluid behaves like shear thickening and volumetric flow rate increases which causes the coating thickness to increase.

Onset of double diffusive surface-tension driven convection in fluid layer overlying a layer of anisotropic porous layer with soret effect

The onset of double-diffusive surface tension-driven convective motion in a fluid layer overlying a fluid-saturated anisotropic porous layer is investigated analytically in the presence of the soret effect. We considered boundaries to be insulating to temperature perturbations. The governing equation that satisfies the composite system is analyzed by the normal mode approach and solved by the regular perturbation technique for linear stability. By solving coupled equations, a mathematical expression for the critical Marangoni number is obtained. Under the effect of the anisotropy, soret parameters and the impact of various physical parameters on the start of convective motion is illustrated graphically, and the stability system is investigated.

IMPACT OF VARIABLE SLIP AND WALL PROPERTIES ON PERISTALTIC FLOW OF EYRING–POWELL FLUID THROUGH INCLINED CHANNEL: ARTIFICIAL INTELLIGENCE BASED PERTURBATION TECHNIQUE

This paper emphasizes a new attempt at the peristaltic transport of Eyring–Powell fluid through a uniform channel. The analysis is performed in the presence of wall properties under the influence of variable liquid properties, the flow problem is mathematically developed. The channel walls are subjected to velocity and thermal slip conditions with long-wavelength and low Reynolds number approximations employed in the study. The governed nonlinear partial differential equations are normalized by using relevant nondimensional parameters and the solutions are obtained with the help of a regular perturbation technique. The effects of physical parameters of interest such as velocity, temperature, concentration, and streamlines are represented graphically. The investigations reveal that the material parameters and elastic parameters of the Eyring–Powell fluid model play a significant role by affecting the velocity and temperature profiles.

Numerical Examination on Impact of Hall Current on Peristaltic Flow of Eyring-Powell Fluid under Ohmic-Thermal Effect with Slip Conditions

Aims:: This article is intended to investigate and determine the combined impact of Slip and Hall current on Peristaltic transmission of Magneto-hydrodynamic (MHD) Eyring- Powell fluid. Background: The hall term arises, taking strong force-field under consideration. Velocity, thermal, and concentration slip conditions are applied. The energy equation is modeled by considering the Joule-thermal effect. To observe the non-Newtonian behavior of the fluid, the constitutive equations of Eyring-Powell fluid are encountered. Objective: Flow is studied in a wave frame of reference traveling with the wave's velocity. The mathematical modeling is done by utilizing adequate assumptions of long wavelength and low Reynolds number. Method: The closed-form solution for momentum, temperature, and concentration distribution is computed analytically using the regular perturbation technique for the small fluid ter(A). Results: Graphical results are presented and discussed in detail to analyze the behavior of sundry parameters on flow quantities (i.e., velocity, temperature, and concentration profile). It is noticed that Powell-Eyring fluid parameters (A,B) have a significant role in the outcomes. Conclusion: The fluid parameter A magnifies the velocity profile, whereas the other fluid parameter B shows the opposite behavior.

Analytical trajectory prediction for skip re-entry of lifting vehicle

An analytical trajectory prediction method is put forward for skip re-entry of lifting vehicles. First, the 3-D analytical solutions of skip re-entry trajectory are developed, which are crucial to realize the analytical trajectory prediction. To facilitate the derivation, an auxiliary geocentric inertial frame is established to deal with the strong nonlinear problem caused by the Earth’s rotation. Thereafter, the analytical solutions of the downrange, crossrange, and velocity are approximated by the outer solutions of a two-time scale singular perturbation. The analytical solutions of the altitude and flight-path angle are developed using regular perturbation techniques which can decompose the strongly coupled and nonlinear system into a sequence of linear problems. Second, since the analytical solutions are with respect to the lift-to-drag ratio and bank angle, a method based on the EKF technique is proposed to estimate the lift-to-drag ratio and bank angle of the re-entry vehicle. Third, the analytical trajectory prediction schemes for the lifting re-entry vehicle with multiple bank reversals are put forward. The simulations show that the proposed analytical solutions are in good agreement with the numerical solutions, but the computation time of the analytical solutions is about two orders of magnitude less than that of the numerical solutions. In addition, the superior performance of the proposed analytical trajectory prediction method is well demonstrated by the simulation results.

Thermomagnetic ferroconvection in an anisotropic permeable layer exposed to a modulated magnetic field

The impact of a sinusoidal mode of magnetic field involving time-dependent on the threshold of magnetic smart liquid advection in a saturated Darcy-permeable framework is investigated using a regular perturbation technique. Anisotropic permeability and thermal anisotropy are used to describe the flow through permeable medium. The regular perturbation technique is based on minimum amplitude of time-fluctuated magnetic field, the threshold condition is computed with regard to correction in a critical Rayleigh number and wavenumber. Correction in Rayleigh number is identified by modulating the magnetic field, modulation frequency, magnetic parameter, mechanical anisotropy, thermal anisotropy and Vadasz number. At intermediate frequency values, the impact of various physical factors is perceived to be noteworthy. It is found that by fine tuning the frequency of magnetic field modulation, we can either accelerate or postpone the onset of ferroconvection. The most sophisticated scientific application packages, Wolfram Mathematica 11.3 is used to extract the numerical values as well as plotting graphs. The problem sheds some light on convective heat transfer mechanisms in ferromagnetic fluid with time-varying magnetic field.

Local Thermal Non-Equilibrium Dominant Darcy-Rayleigh-Benard-Magneto-Marangoni Convection in a Composite Layer

The domination by Local Thermal Non-Equilibrium (LTNE) with regard to the onset of Darcy-Rayleigh-Benard-Marangoni (DRBM) convection swayed by magnetic field in a composite layer set-up is studied pertinent to incompressible fluid. The precinct above the fluid layer is presumed to be free and that below the porous layer is presumed to be rigid. Regular perturbation technique is exercised on the acquired problem to achieve the analytical solution taking adiabatic-adiabatic conditions into account at the boundaries. Effects of parameters such as solid phase thermal expansion ratio, solid phase thermal diffusivity ratio and inter-phase thermal diffusivity ratio that influences LTNE are discussed. The impact of change in measures of variables viz. fluid phase thermal expansion ratio, fluid phase thermal diffusivity ratio, porous parameter, thermal ratio, Chandrasekhar number and Marangoni number with respect to both LTNE and LTE situations are together explored and portrayed graphically. It is to be noted that for smaller values of fluid phase thermal expansion ratio and fluid phase thermal diffusivity ratio, the LTNE effects are prominent and cannot be ignored.

Outcomes of Uniform as well as Non-Uniform Temperature Profiles on the Onset of Double Diffusive Magneto-Darcy-Rayleigh-Benard Convection in a Two Layer set up in the presence of Local Thermal Non-Equilibrium

Outcomes of uniform as well as non-uniform temperature profiles that is linear, parabolic, inverted parabolic, Piecewise Linear Profile Heated from Below (PLHB), Piecewise Linear Profile Cooled from Above (PLCA) and Step Function (SF) temperature profile on the onset of Double Diffusive Magneto-Darcy-Rayleigh-Benard (DDMDRB) Convection in a two layer set up in the presence of Local Thermal Non-Equilibrium with incompressible fluid horizontally surrounded by adiabatic rigid boundaries. Analytical solution to the attained problem is accomplished by means of regular perturbation technique. The consequences of altering the parameters namely fluid phase thermal expansion ratio, solid phase thermal expansion ratio, solid phase thermal diffusivity ratio, inter-phase diffusivity ratio, solute Rayleigh number, Chandrasekhar number, thermal ratio and porous parameter have been elucidated graphically.