Finite Difference Numerical Method Research Articles

Stability analysis study for the time-fractional Galilei invariant advection-diffusion model of distributive order using an efficient hybrid approach

Abstract In this manuscript, a new model of the time-fractional Galilei-invariant advection-diffusion model of distributed order is studied. An efficient hybrid numerical approach with high accuracy is used to estimate this equation. The finite difference numerical method is used to approximate the fractional operator in terms of the time variable and to approximate the integral term of distributed order, the Gaussian–Legendre integration is applied. To obtain a fully discrete numerical approach, we used a spectral element numerical approach, in which Legendre polynomials are used as the basis function. For the proposed numerical approach, the error and stability analysis are studied. For the efficiency of the numerical approach, some numerical examples are presented with graphs and tables.

Heat Transfer Characteristics of a Quiescent Fluid over a Moving Flat Surface

Optimizing thermal control methods and enhancing the efficiency of various engineering processes, such as the cooling of hot moving surfaces, is imperative. Therefore, it is crucial to investigate heat transfer and temperature distributions across moving surfaces in stationary fluids. In this study, the combined effects of viscous dissipation, temperature-dependent viscosity, and a moving surface in a quiescent fluid on the temperature and velocity profiles are investigated. A two-dimensional steady laminar boundary layer flow of an incompressible Newtonian fluid, driven by the movement of the surface where viscosity is an inverse function of temperature, is analyzed. The effects of flow parameters, including Reynolds number, Eckert number, Prandtl number, variable viscosity, and surface velocity on temperature and velocity profiles, are determined. The governing boundary layer partial differential equations are transformed into non-dimensional form and solved using the central finite difference numerical method, implemented in MATLAB software. The numerical results demonstrate that an increase in the Reynolds number and the variable viscosity parameter leads to an increase in the velocity profiles. Additionally, an increase in Reynolds number, Prandtl number, variable viscosity, and surface velocity leads to a decrease in temperature profiles. Finally, an increase in the Eckert number results in higher temperature profiles. Therefore, with suitable flow parameters, the temperature of the fluid can be regulated. These findings are useful in cooling hot sheets or metallic plates drawn over a quiescent fluid to obtain high-quality final products.

Theoretical analysis of the piston ring-pack for predicting friction and power loss in an internal combustion engine

The piston ring is considered an essential element in the operation of the internal combustion engine to improve its efficiency and reduce its emissions. It is resistant to high temperature and high-pressure gases in the combustion chamber. A one-dimensional analysis method of lubrication between piston rings and cylinder liner in mixed lubrication conditions is developed in this article by using the GT-Suite simulation software. We applied the finite difference numerical method to solve the Reynolds equation to obtain the variation during an engine cycle of the following parameters; film thickness, frictional force, and power loss. The solution results presented can be used to improve the design of the piston ring pack in internal combustion engines and is available for industry usage.

Time delayed fractional diabetes mellitus model and consistent numerical algorithm

The diabetes mellitus model (DMM) is explored in this study. Many health issues are caused by this disease. For this reason, the integer order DMM is converted into the time delayed fractional order model by fitting the fractional order Caputo differential operator and delay factor in the model. It is proved that the generalized model has the advantage of a unique solution for every time t. Moreover, every solution of the system is positive and bounded. Two equilibrium states of the fractional model are worked out i.e. disease free equilibrium state and the endemic equilibrium state. The risk factor indicator, R0 is computed for the system. The stability analysis is carried out for the underlying system at both the equilibrium states. The key role of R0 is investigated for the disease dynamics and stability of the system. The hybridized finite difference numerical method is formulated for obtaining the numerical solutions of the delayed fractional DMM. The physical features of the numerical method are examined. Simulated graphs are presented to assess the biological behavior of the numerical method. Lastly, the outcomes of the study are furnished in the conclusion section.

Tangent hyperbolic MHD nanoliquids on non-isothermal stretched sheets: Analyzing the impact of transport parameters, variable fluid properties and convective boundary conditions

The key focus of this research is to look at how tangent hyperbolic magnetohydrodynamic (MHD) nano-liquids move in a non-isothermal coagulated stretched sheet. In this study, we intend to explore how fluid behavior responds to alternations in transport physical parameters. The coagulated sheet is subjected to two distinct boundary conditions to inspect the heat and mass transport rate of the nano liquid. The convective heat boundary condition (CBC) and mass-convective boundary condition (MCBC) are employed to specify the physical conditions at the boundaries of the problem domain. The boundary conditions, responsible for regulating heat flow and mass transfer rates, play a crucial role in shaping the liquid’s behavior. Using a similarity solution, the partial differential equation describing the flow of mass and heat within the system transforms into an interconnected network of non-linear ordinary differential equations. These mathematical equations are subsequently figured out using a numerical finite difference method. This study additionally examines the correlation between thermophoresis and Brownian motion, two fundamental concepts in the dynamics of colloidal suspensions. The study’s findings indicate that the temperature profile increases in all scenarios when the variable thermal conductivity and variable viscosity parameters are increased. In contrast, for the same parameters, the velocity profile is decreased. Further, increasing wall thickness reduces heat dissipation; consequently, the temperature profile similarly affects the velocity power index. The Biot number improves the rates of temperature transfer. These results underscore the significant influence of these parameters in predicting the behavior of MHD tangent hyperbolic nanofluids. This study elucidates the intricate interaction among different physical parameters in the dynamics of nano liquids, which holds significant implications for various industrial applications.

Coupled heat and fluid transport in pulled extrusion of optical fiber preforms

In the fabrication of optical fibers, the viscosity of the glass varies dramatically with temperature so that heat transfer plays an important role in the deformation of the fiber geometry. To date, models of fiber drawing processes have either assumed isothermal conditions or that advection is the dominant heat-transfer mechanism, with conduction therefore neglected. While advective heat transfer and neglect of conduction are valid in the fast process of drawing a preform to a fiber, heat conduction is also important in the slow process of extruding a fiber preform. In this paper, we derive and solve, in the context of pulled extrusion, a coupled heat and fluid flow model that incorporates heat conduction. The dramatic variation of viscosity with temperature means that this problem is extremely challenging to solve via standard numerical techniques and we therefore develop a novel finite-difference numerical solution method that proves to be highly robust. We use this method to show that conduction significantly affects the size of internal holes at the exit of the device and hence has important consequences for the process of manufacturing optical fibers.

Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations

An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at t=0. In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the L2 norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.

Seal dynamics of supercritical carbon dioxide turbine with the uncertain rotor nonlinear motion

The actual rotor motion and seal dynamics are nonlinear and uncertain. The uncertain effects include the seal flow and the rotor system parameters. While the influence of system parameters makes it more difficult to accurately predict the seal dynamics. A rotor nonlinear whirling model was established to achieve the synchronous solution of rotor motion and seal simulation. The coupling effect between rotor system parameters and seal aerodynamic performance was adopted. The actual seal dynamics for SCO2 turbine were obtained and evaluated by the numerical finite difference method. In the rotor nonlinear whirling model, the rotor has the uncertain continuous free motion. The influence of the system main parameters, such as elastic recovery stiffness and unbalanced mass force, on seal dynamics is revealed. And the dynamic coefficients beyond the artificial specific disturbance frequency are shown. The results show that the rotor nonlinear whirling can better predict the seal dynamics. The rotation speed, elastic recovery stiffness and unbalanced mass eccentricity have different significant effects on seal dynamics.

Numerical Analysis of Low-Enthalpy Deep Geothermal Energy Extraction Using a Novel Gravity Heat Pipe Design

Geothermal energy, derived from the Earth’s internal heat, can be harnessed due to the geothermal gradient between the Earth’s interior and its surface. This heat, sustained by radiogenic decay, varies across regions, and is highest near volcanic areas. In 2020, 108 countries utilised geothermal energy, with an installed capacity of 15,950 MWe for electricity and 107,727 MWt for direct use in 2019. Low-enthalpy sources require binary systems for power production. Open-loop systems face issues like scaling, difficult water treatment, and potential seismicity, while closed-loop systems, using abandoned petroleum or gas wells, reduce costs and environmental impacts greatly. The novel geothermal gravity heat pipe (GGHP) design eliminates parasitic power consumption by using hydrostatic pressure for fluid circulation. Implemented in an abandoned well in north-east (NE) Slovenia, the GGHP uses a numerical finite difference method to model heat flow. The system vaporises the working fluid in the borehole, condenses it at the surface, and uses gravitational flow for circulation, maintaining efficient heat extraction. The model predicts that continuous maximum capacity extraction depletes usable heat rapidly. Future work will explore sustainable heat extraction and potential discontinuous operation for improved efficiency.

Highly confined epsilon-near-zero and surface phonon polaritons in SrTiO3 membranes

Recent theoretical studies have suggested that transition metal perovskite oxide membranes can enable surface phonon polaritons in the infrared range with low loss and much stronger subwavelength confinement than bulk crystals. Such modes, however, have not been experimentally observed so far. Here, using a combination of far-field Fourier-transform infrared (FTIR) spectroscopy and near-field synchrotron infrared nanospectroscopy (SINS) imaging, we study the phonon polaritons in a 100 nm thick freestanding crystalline membrane of SrTiO3 transferred on metallic and dielectric substrates. We observe a symmetric-antisymmetric mode splitting giving rise to epsilon-near-zero and Berreman modes as well as highly confined (by a factor of 10) propagating phonon polaritons, both of which result from the deep-subwavelength thickness of the membranes. Theoretical modeling based on the analytical finite-dipole model and numerical finite-difference methods fully corroborate the experimental results. Our work reveals the potential of oxide membranes as a promising platform for infrared photonics and polaritonics.

Topological nano-switches in higher-order topological insulators

We consider multi-terminal transport through a flake of rectangular shape of a two-dimensional topological insulator in the presence of an in-plane magnetic field. This system has been shown to be a second-order topological insulator, thus exhibiting corner states at its boundaries. The position of the corner states and their decay length can be controlled by the direction of the magnetic field. In the leads we assume that the magnetic field is absent and therefore we have helical one-dimensional propagating states characteristic of the spin-Hall effect. Using a low-energy effective Hamiltonian we show analytically that, in a two-terminal setup, transport can be turned on and off by a rotation of the in-plane magnetic field. Similarly, in a three terminal configuration, the in-plane magnetic field can be used to turn on and off the transmission between neighbouring contacts, thus realising a directional switch. Analytical calculations are supplemented by a numerical finite-difference method. For small values of the Fermi energy and field strength, the analytical results agree exceptionally well with the numerics. The effect of disorder is also addressed in the numerical approach. We find that the switching functionality is remarkably robust to the presence of strong disorder stemming from the topological nature of the states contributing to the electron transport.

Pulsatile flow of blood and Cattaneo Christov heat flux model in diseased arteries with variable properties

ABSTRACT The current effort is executed to interpret the heat transmission of combined stenosis and aneurysm in an artery, taking into account variable properties such as viscous dissipation and thermal radiation. We reduced the two directional energy and momentum equations to unidirectional flow by normalizing them under the mild stenotic assumption. After that, a mathematical framework for the physical problem is developed. An intricate nonlinear set of partial differential equations results from this formulation. A suitable numerical method of finite difference is employed on consequent equations to solve them. Afterward, graphical results predict the region of flow and the influence of involved parameters is shown in pictorial form. Furthermore, these solutions are employed to test how certain parameters, like streamlines, flow rate, and wall shear stress, affect blood flow. Using streamlines, we will also observe the general behavior of blood flow patterns.

Generation Of Scaled Long-Period Ship Waves In A Pump-Driven Flume

Experimental investigations on the generation of long-period, primary ship waves using a pump-driven technique in a closed-circuit flume are presented. Long-period ship waves can have a major impact on river banks and engineering structures. As larger ship dimensions accompanied by a corresponding rise in loads are expected in the future, the characterisation of long-period ship waves and understanding their impact on bank protection and engineering structures is crucial, especially for appropriate design. Our study combines a closed-loop PID controlled wave generation with a numerical finite difference method (FDM) model. This setup offers the advantage of generating repeatable waves at arbitrary positions along the test section of the laboratory flume. The experiments conducted demonstrate a good reproduction of a scaled long-period ship wave signal (1:10) in a distance of approximately 9 m from the flume inlet. The combination of the PID controlled pump-driven technique with a numerical model provides a solid basis for further studies on ship-induced loads, with the aim to predict future wave loads, better understand the wave-structure-interaction and to mitigate potential damage.

Numerical-analytical approach to solving problems of non-stationary thermal conductivity of a non-thin annular plate

This paper considers the first stage of calculating the initial boundary value problem of non-stationary thermal conductivity of cylindrical bodies using a modified method of lines, namely dimension reduction of the original differential equations, initial and boundary conditions. The original equations of thermal conductivity are defined in a cylindrical coordinate system in a spatial setting. An object is a cylindrical body with commensurate dimensions. This area of research is relevant, because when calculating the load bearing elements of structures to thermal effects, the first step is to determine the distribution of temperature fields. Boundary conditions are considered as conditions of convective heat transfer, which by means of boundary transition are transformed into boundary conditions of the first and second types. Dimension reduction with respect to spatial coordinate is performed by the Bubnov-Galorkin-Petrov projection method using local basis functions. These functions are called "cover" functions, which are related to the lines drawn on the domain of the task. Normalized trigonometric series are used to reduce the dimensionality of the equations with respect to circular coordinate. All transformations are performed in index form. In addition to differential equations, the projection method reduces the dimensionality of the initial and boundary conditions. In this paper, the most optimal form of writing reduced equations is determined, which provides the ease of reducing the dimensionality of the original differential equations. Initial and boundary conditions take into account the impact of the environment. All this makes it possible to set a reduced initial boundary value problem for further calculation by numerical finite difference methods.

Bifurcation and asymptotics of cubically nonlinear transverse magnetic surface plasmon polaritons

Linear Maxwell equations for transverse magnetic (TM) polarized fields support single frequency surface plasmon polaritons (SPPs) localized at the interface of a metal and a dielectric. Metals are typically dispersive, i.e. the dielectric function depends on the frequency. We prove the bifurcation of localized SPPs in dispersive media in the presence of a cubic nonlinearity and provide an asymptotic expansion of the solution and the frequency. The problem is reduced to a system of nonlinear differential equations in one spatial dimension by assuming a plane wave dependence in the direction tangential to the (flat) interfaces. The number of interfaces is arbitrary and the nonlinear system is solved in a subspace of functions with the H1-Sobolev regularity in each material layer. The corresponding linear system is an operator pencil in the frequency parameter due to the material dispersion. The studied bifurcation occurs at a simple isolated eigenvalue of the pencil. For geometries consisting of two or three homogeneous layers we provide explicit conditions on the existence of eigenvalues and on their simpleness and isolatedness. Real frequencies are shown to exist in the nonlinear setting in the case of PT-symmetric materials. We also apply a finite difference numerical method to the nonlinear system and compute bifurcating curves.

Computational modeling of microalgal biofilm growth in heterogeneous rotating algal biofilm reactors (RABRs) for wastewater treatment

Rotating algal biofilm reactors (RABRs) are innovative systems designed to cultivate microalgae biofilms efficiently. In this paper, we have developed a novel mathematical model to accurately capture the growth dynamics of algae biofilms within RABR. By considering the spatial heterogeneity of the RABR, we introduce a PDE-based model that addresses the spatial variations across the substratum, enabling a more accurate simulation of biofilm growth in RABRs. The photosynthesis process is modeled through reactive kinetics, driving the growth of the algae biofilm. To analyze the system's behavior, we employ finite difference numerical methods to solve the complex PDE model. We then conduct extensive numerical simulations to understand algae biofilm growth in the RABR environment under various operational factors and environmental conditions. One primary focus in these simulations is to investigate the impact of various harvesting strategies, harvesting frequencies, light intensity, and light exposure on the overall biomass productivity of the algae biofilm. The numerical results provide valuable insights into optimizing algae biofilm growth and designing harvesting techniques in RABR systems. Our proposed novel mathematical model provides an effective platform for the theoretical investigation and design of RABRs for wastewater treatment.

Fracture prediction and application in carbonate reservoirs based on the rescaled range analysis- finite difference numerical analysis methods

For carbonate reservoirs, the study of fractures is crucial. In this paper, the identification and prediction of fractures in carbonate reservoirs are carried out by applying R/S-FD numerical analysis method by using core, thin section and imaging logging data. The analysis concludes that the AC, CAL, DEN, and GR curves are very sensitive to the response of fracture development segments as calculated by the rescaled range analysis (R/S) analysis method, but some fractures are still misidentified. After finite difference (FD) calculation on the basis of R/S calculation, high-angle structural fractures can be accurately identified, and the R/S-FD fracture identification results of different curves can be verified with the results of core and imaging logging, and the fracture identification accuracy can reach 71.69%. The number of fracture development segments is positively correlated with the cumulative oil production, the number of producing layers and the cumulative thickness of producing layers, and the wells with more fracture development segments have higher cumulative oil production, more producing layers and larger cumulative thickness of producing layers. This method improves the accuracy of fracture identification in carbonate reservoirs, which is of great significance for oil and gas exploration.

Numerical modeling of Carreau–Yasuda fluid with induced magnetic field in a heated curved channel having ciliary walls

Cilia motion is commonly located in mammalian cells of living organisms which continually grow and feature conspicuously. The ciliary apparatus is associated with cell cycle movement and proliferation, and cilia play a dynamic part in human and animal growth and everyday life. The motivation of these applications, this article is to present a metachronal wave analysis for non-Newtonian Carreau–Yasuda fluid inside a curved channel with ciliated walls. Hall effect is encountered in the momentum equations to measure the magnetic field while the viscous dissipation and thermal radiation terms are considered to address the energy analysis. The curvilinear coordinates are used due to the curved nature of flow geometry in the derivation of flow equations. The computations for axial velocity, pressure rise, temperature field, mass concentration, and stream function are carried out under low Reynolds number and long wavelength approximation in the wave frame of reference by utilizing appropriate numerical implicit finite difference method (FDM). The implementation of numerical procedure and graphical representation of the computations are accomplished using MATLAB language. Furthermore, the skin friction and Nusselt number at the channel walls are determined for a variety of critical parameter assessments.

Heat and mass transfer characteristics of vertical falling film cylindrical plastic surfaces under partial wetting conditions for liquid desiccant regeneration

The regenerator plays a dominant role in the overall energy-economic performance of liquid desiccant systems. Nevertheless, very few studies have been reported on falling film liquid desiccant regenerators. Most of the previous studies targeted experimental/numerical modelling needs of regeneration process under design conditions. In the current research workexperimental investigation of regeneration process is carried out to study the coupled heat and mass transfer process for adiabatic regeneration of LiCl liquid desiccant over outer surface of vertical Plain and Modified polypropylene cylindrical surfaces under design (complete wetting) and part load (partial wetting) operating conditions. Modified cylindrical surfaces are developed to compensate the poor wetting of the plastic surfaces. New generalised one-dimensional numerical philosophy is proposed to accommodate the need of part load operating conditions by inclusion of actual wetting of the working surface as well as convective heat transfer happening from dry patches of solid surface to the flowing air. The heat and mass transfer coefficients are determined by solving the governing coupled heat and mass balance non-linear differential equation using the numerical finite difference method. It is found that the Modified surface offered an average improvement of 35.8% in the mass transfer coefficient under partial wetting conditions. New Sh and Nu number correlations are proposed by incorporating heat and mass transfer driving potentials and new parameter wetting factor. The developed correlations predicted the change in humidity and temperature across the regenerator with an average error of 6.2% and 7.5%, whereas from the existing two correlations found in the literature, the first one predicted the same experimental observations with an average error of 54.6% and 36.2%, while second one predicted with an average error of 77.3% and 22.0%, respectively. The results highlighted that the assumption of complete wetting of working surface under part load operating conditions leads to significant error in prediction of air outlet conditions. The proposed correlations will be helpful for numerical modelling and simulation of falling film regenerators under wide range of liquid loading conditions.

Dynamics of the black soliton in a regularized nonlinear Schrödinger equation

We consider a family of regularized defocusing nonlinear Schrödinger (NLS) equations proposed in the context of the cubic NLS equation with a bounded dispersion relation. The time evolution is well-posed if the black soliton is perturbed by a small perturbation in the Sobolev space H s ( R ) H^s(\mathbb {R}) with s > 1 2 s > \frac {1}{2} . We prove that the black soliton is spectrally stable (unstable) if the regularization parameter is below (above) some explicitly specified threshold. We illustrate the stable and unstable dynamics of the perturbed black solitons by using the numerical finite-difference method. The question of orbital stability of the black soliton is left open due to the mismatch of the function spaces for the energy and momentum conservation.