Energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media

In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational response.Unlike the first-order PDE-ODE governing equations considered previously in Bokil et al.(J Comput Phys 350:420–452,2017)and Lyu et al.(J Sci Comput 89:1–42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlinear dispersion in the material.The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is further applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order accuracy.Indeed with d_(E)as the number of the components of the electric field,only a d_(E)×d_(E)nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small nonlinear systems are completely decoupled over one time step,rendering very high parallel efficiency.We evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accuracy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.

High-Order Spatial FDTD Solver of Maxwell’s Equations for Terahertz Radiation Production

We applied a spatial high-order finite-difference-time-domain (HO-FDTD) scheme to solve 2D Maxwell’s equations in order to develop a fluid model employed to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma. We examined the performance of the applied scheme, in this context, we implemented the developed model to study selected phenomena in terahertz radiation production, such as the excitation energy and conversion efficiency of the produced THz radiation, in addition to the influence of the pulse chirping on properties of the produced radiation. The obtained numerical results have clarified that the applied HO-FDTD scheme is precisely accurate to solve Maxwell’s equations and sufficiently valid to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma.

Influences of double diffusion upon radiative flow of thin film Maxwell fluid through a stretching channel

This work explores the influence of double diffusion over thermally radiative flow of thin film hybrid nanofluid and irreversibility generation through a stretching channel.The nanoparticles of silver and alumina have mixed in the Maxwell fluid(base fluid).Magnetic field influence has been employed to channel in normal direction.Equations that are going to administer the fluid flow have been converted to dimension-free notations by using appropriate variables.Homotopy analysis method is used for the solution of the resultant equations.In this investigation it has pointed out that motion of fluid has declined with growth in magnetic effects,thin film thickness,and unsteadiness factor.Temperature of fluid has grown up with upsurge in Brownian motion,radiation factor,and thermophoresis effects,while it has declined with greater values of thermal Maxwell factor and thickness factor of the thin film.Concentration distribution has grown up with higher values of thermophoresis effects and has declined for augmentation in Brownian motion.

Casson hybrid nanofluid flow over a Riga plate for drug deliveryapplications with double diffusion

Casson fluid-mediated hybrid nanofluids are more effective at transferring heat than traditional heat transfer fluids in terms of thermal conductivity.Heat exchangers,cooling systems and other thermal management systems are ideal for use with Casson fluids.Precise control of the flow and release of medication is necessary when using Casson fluids in drug delivery systems because of their unique rheological properties.Nanotechnology involves the creation of nanoparticles that are loaded with drugs and distributed in Casson fluid-based carriers for targeted delivery.In this study,to create a hybrid nanofluid,both single-walled carbon nanotubes(SWCNTs)and multi-walled carbon nanotubes(MWCNTs)are dispersed in a Casson fluid with Fourier’s and Fick’s laws assumptions.The Casson fluid is suitable for various engineering and medical applications due to the enhancement of heat transfer and thermal conductivity by the carbon nanotubes.Our objective is to understand how SWCNTs and MWCNTs impact the flow field by studying the flow behavior of the Casson hybrid nanofluid when it is stretched against a Riga plate.The Darcy-Forchheimer model is also used to account for the impact of the porous medium near the stretching plate.Both linear and quadratic drag terms are taken into account in this model to accurately predict the flow behavior of the nanofluid.In addition,the homotopy analysis method is utilized to address the model problem.The outcomes are discussed and deliberated based on drug delivery applications.These findings shed valuable light on the flow characteristics of a Casson hybrid nanofluid comprising SWCNTs and MWCNTs.It is observed that the incorporation of carbon nanotubes makes the nanofluid a promising candidate for medical applications due to its improved heat transfer properties.

非傅里叶效应对Maxwell流体在垂直槽道中热对流不稳定性的影响

基于Maxwell流体在科学、工程和技术等各个领域的重要性,并且结合非傅里叶热传导效应在许多不同研究领域的广泛关注,文章研究了非傅里叶效应对Maxwell流体在垂直槽道中热对流不稳定性的影响.将传统傅里叶热传导模型加入一个包含热松弛时间的新瞬态项,得到了非傅里叶热传导方程.使用切比雪夫配置法数值求解了一个广义特征值问题,得到了时间增长率和中性稳定曲线随各参数的变化情况.结果表明,时间增长率会随着松弛时间参数Λ增加而增加.中性稳定曲线表明,在波数相对较小的范围内,松弛时间的影响几乎可以忽略.但是当波数较大时,松弛时间对热对流不稳定性的影响逐渐加剧.这表明Maxwell流体的弹性效应增强了垂直管道内的热对流不稳定性.对于傅里叶流体,中性稳定性曲线不会随着Prandtl数变化.另一方面,非傅里叶效应增强了热对流不稳定性,并且新的双曲型热传导方程使得中性稳定曲线产生了波动现象.进一步研究表明,这种波动会随着Prandtl数的增加而增强.当Prandtl数超过一定临界值时,中性稳定曲线同时出现稳态分支和振荡分支,而且在振荡分支中,不稳定性显著增强.

非均匀磁场下Maxwell磁纳米流体的拉伸流动与磁扩散分析

磁性纳米颗粒可以提升聚合物的导电性和导热性等性能,被广泛应用于机械、生物医学、能源存储等领域.当外界施加非均匀磁场时,感应磁场在高Reynolds数的情况下不可忽略.为探究磁性纳米颗粒对层流边界层内黏弹性流体非稳态拉伸流动与磁扩散的影响,将时间分布阶Maxwell本构方程与动量方程耦合,建立了二维不可压缩Maxwell磁纳米流体的速度与磁扩散偏微分方程组.采用有限差分法进行数值分析,通过控制磁性纳米颗粒种类、体积分数和磁参数大小,分析了流体的速度和感应磁场在边界层内的分布.研究发现:在熔融聚合物中添加Fe2O3纳米颗粒后,流体的速度、感应磁场最大,速度和磁边界层的厚度最厚;Maxwell纳米流体的松弛时间参数增大,速度与磁扩散均减小;另外,随着磁参数增大,流体的速度边界层厚度减小,磁边界层厚度增大;Fe3O4纳米颗粒的体积分数越大,流体流动越快,感应磁场越小.因此,非均匀磁场下在聚合物中添加磁性纳米颗粒的研究,为改善材料的性能给予了可参考的数据.

Chebyshev谱方法研究非稳态Maxwell流体在轴向余弦振荡圆柱上的斜驻点流动

研究了非稳态Maxwell流体斜撞击轴向余弦振荡圆柱的斜驻点流动.首先,基于斜驻点流动特性,在柱面坐标系下求得关于压力的二阶常微分方程,对压强进行修正,建立了非稳态Maxwell流体在振荡圆柱上斜驻点流动的边界层模型.接着,合理的相似变换将模型转化,使用Chebyshev谱方法求得模型的数值解.结果表明,在贴近圆柱表面的流体随着圆柱体做周期性运动;圆柱的曲率越大越会使在同一时刻同一位置处的流体质点的速度越大;相反,非稳态参数及流体的记忆特性也会在更靠近圆柱壁面处阻碍流体流动.

PLANE SURFACE SUDDENLY SET IN MOTION IN A VISCOELASTIC FLUID WITH FRACTIONAL MAXWELL MODEL

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced.The flow near a wall suddenly set in mo- tion is studied for a non-Newtonian viscoelastic fluid with the fractional Maxwell model.Exact solutions of velocity and stress are obtained by using the discrete in- verse Laplace transform of the sequential fractional derivatives.It is found that the effect of the fractional orders in the constitutive relationship on the flow field is signif- icant.The results show that for small times there are appreciable viscoelastic effects on the shear stress at the plate,for large times the viscoelastic effects become weak.

Unsteady rotating flows of a viscoelastic fluid with the fractional Maxwell model between coaxial cylinders

The fractional calculus is used in the constitutive relationship model of viscoelastic fluid. A generalized Maxwell model with fractional calculus is considered. Based on the flow conditions described, two flow cases are solved and the exact solutions are obtained by using the Weber transform and the Laplace transform for fractional calculus.

Complex Maxwell’s Equations

Maxwell’s equations in electromagnetism can be categorized into three dis-tinct groups based on the electromagnetic source when employing quaterni-ons. Each group represents a self-contained system in which Maxwell’s equations are applied and validated concurrently, in contrast to the previous approach that did not account for this. It has been noted that the formulation of these Maxwell equations ultimately results in the formulation of Max-well’s equations utilizing the scalar function.

Numerical solution of oscillatory flow of Maxwell fluid in a rectangular straight duct

A numerical analysis is presented for the oscillatory flow of Maxwell fluid in a rectangular straight duct subjected to a simple harmonic periodic pressure gradient.The numerical solutions are obtained by a finite difference scheme method.The stability of this finite difference scheme method is discussed.The distributions of the velocity and phase difference are given numerically and graphically.The effects of the Reynolds number,relaxation time,and aspect ratio of the cross section on the oscillatory flow are investigated.The results show that when the relaxation time of the Maxwell model and the Reynolds number increase,the resonance phenomena for the distributions of the velocity and phase difference enhance.

MHD graphene-polydimethylsiloxane Maxwell nanofluid flow in a squeezing channel with thermal radiation effects

The magnetohydrodynamic (MHD) graphene-polydimethylsiloxane (PDMS) nanofluid flow between two squeezing parallel plates in the presence of thermal radiation effects is investigated. The energy efficiency of the system via the Bejan number is studied extensively. The governing partial differential equations are converted by using the similarity transformations into a set of coupled ordinary differential equations. The set of these converted equations is solved by using the differential transform method (DTM). The entropy generation in terms of the Bejan number, the coefficient of skin-friction, and the heat transfer rate is furthermore investigated under the effects of various physical parameters of interest. The present study shows that the Bejan number, the velocity and thermal profiles, and the rate of heat transfer decrease with a rise in the Deborah number De while the skin-friction coefficient increases. It is also observed that the entropy generation due to frictional forces is higher than that due to thermal effects. Thus, the study bears the potential application in powder technology as well as in biomedical engineering.

Variable fluid properties and variable heat flux effects on the flow and heat transfer in a non-Newtonian Maxwell fluid over an unsteady stretching sheet with slip velocity

The effects of variable fluid properties and variable heat flux on the flow and heat transfer of a non-Newtonian Maxwell fluid over an unsteady stretching sheet in the presence of slip velocity have been studied. The governing differential equations are transformed into a set of coupled non-linear ordinary differential equations and then solved with a numerical technique using appropriate boundary conditions for various physical parameters. The numerical solution for the governing non-linear boundary value problem is based on applying the fourth-order Runge-Kutta method coupled with the shooting technique over the entire range of physical parameters. The effects of various parameters like the viscosity parameter, thermal conductivity parameter, unsteadiness parameter, slip velocity parameter, the Deborah number, and the Prandtl number on the flow and temperature profiles as well as on the local skin-friction coefficient and the local Nusselt number are presented and discussed. Comparison of numerical results is made with the earlier published results under limiting cases.

MHD flow of upper-convected Maxwell fluid over porous stretching sheet using successive Taylor series linearization method

This paper investigates the magnetohydrodynamic （MHD） boundary layer flow of an incompressible upper-convected Maxwell （UCM） fluid over a porous stretching surface. Similarity transformations are used to reduce the governing partial differential equations into a kind of nonlinear ordinary differential equations. The nonlinear prob- lem is solved by using the successive Taylor series linearization method （STSLM）. The computations for velocity components are carried out for the emerging parameters. The numerical values of the skin friction coefficient are presented and analyzed for various parameters of interest in the problem.

Unsteady flow of viscoelastic fluid between two cylinders using fractional Maxwell model

The unsteady flow of an incompressible fractional Maxwell fluid between two infinite coaxial cylinders is studied by means of integral transforms. The motion of the fluid is due to the inner cylinder that applies a time dependent tor- sional shear to the fluid. The exact solutions for velocity and shear stress are presented in series form in terms of some generalized functions. They can easily be particularized to give similar solutions for Maxwell and Newtonian fluids. Fi- nally, the influence of pertinent parameters on the fluid motion, as well as a comparison between models, is highlighted by graphical illustrations.

Simultaneous effects of magnetic field and space porosity on compressible Maxwell fluid transport induced by a surface acoustic wave in a microchannel

Peristaltic motion induced by a surface acoustic wave of a viscous, compressible and electrically conducting Maxwell fluid in a confined parallel-plane microchannel through a porous medium is investigated in the presence of a constant magnetic field. The slip velocity is considered and the problem is discussed only for the free pumping case. A perturbation technique is employed to analyze the problem in terms of a small amplitude ratio. The phenomenon of a “backward flow” is found to exist in the center and at the boundaries of the channel. In the second order approximation, the net axial velocity is calculated for various values of the fluid parameters. Finally, the effects of the parameters of interest on the mean axial velocity, the reversal flow, and the perturbation function are discussed and shown graphically. We find that in the non-Newtonian regime, there is a possibility of a fluid flow in the direction opposite to the propagation of the traveling wave. This work is the most general model of peristalsis created to date with wide-ranging applications in biological, geophysical and industrial fluid dynamics.

Coupling model for unsteady MHD flow of generalized Maxwell fluid with radiation thermal transform＊

This paper introduces a new model for the Fourier law of heat conduction with the time-fractional order to the generalized Maxwell fluid. The flow is influenced by magnetic field, radiation heat, and heat source. A fractional calculus approach is used to establish the constitutive relationship coupling model of a viscoelastic fluid. We use the Laplace transform and solve ordinary differential equations with a matrix form to obtain the velocity and temperature in the Laplace domain. To obtain solutions from the Laplace space back to the original space, the numerical inversion of the Laplace transform is used. According to the results and graphs, a new theory can be constructed. Comparisons of the associated parameters and the corresponding flow and heat transfer characteristics are presented and analyzed in detail.

Unsteady peristaltic transport of Maxwell fluid through finite length tube:application to oesophageal swallowing

This paper analytically investigates the unsteady peristaltic transport of the Maxwell fluid in a finite tube. The walls of the tube are subjected to the contraction waves that do not cross the stationary boundaries. The analysis is carried out by a long wavelength approximation in the non-dimensional form. The expressions for the axial and radial velocities are derived. The pressures across the wavelength and the tubelength are also estimated. The reflux phenomenon is discussed, which culminates into the determination of the reflux limit. Mathematical formulations are physically interpreted for the flow of masticated food materials such as bread and white eggs in the oesophagus. It is revealed that the Maxwell fluids are favorable to flow in the oesophagus as compared with the Newtonian fluids. This endorses the experimental finding of Takahashi et al. （Takahashi, T., Ogoshi, H., Miyamoto, K., and Yao, M. L. Viscoelastic properties of commercial plain yoghurts and trial foods for swallowing disorders. Rheology, 27, 169- 172 （1999））. It is further revealed that the relaxation time does not affect the shear stress and the reflux limit. It is found that the pressure peaks are identical in the integral case while different in the non-integral case.

Borehole guided waves in a non-Newtonian(Maxwell) fluid-saturated porous medium

The property of acoustic guided waves generated in a fluid-filled borehole surrounded by a non-Newtonian （Maxwell） fluid-saturated porous formation with a permeable wall is investigated. The influence of non-Newtonian effects on acoustic guided waves such as Stoneley waves, pseudo-Rayleigh waves, flexural waves, and screw waves propagations in a fluid-filled borehole is demonstrated based on the generalized Biot-Tsiklauri model by calculating their velocity dispersion and attenuation coefficients. The corresponding acoustic waveforms illustrate their properties in time domain. The results are also compared with those based on generalized Biot＇s theory. The results show that the influence of non-Newtonian effect on acoustic guided wave, especially on the attenuation coefficient of guided wave propagation in borehole is noticeable.

Transient flows of Maxwell fluid with slip conditions

Two fundamental flows, namely, the Stokes and Couette flows in a Maxwell fluid are considered. The exact analytic solutions are derived in the presence of the slip condition. The Laplace transform method is employed for the development of such solutions. Limiting cases of no-slip and viscous fluids can be easily recovered from the present analysis. The behaviors of embedded flow parameters are discussed through graphs.