Set Of Partial Differential Equations Research Articles

Physics‐informed neural network method for modelling beam‐wall interactions

A mesh-free approach for modelling beam-wall interactions in particle accelerators is proposed. The key idea of our method is to use a deep neural network as a surrogate for the solution to a set of partial differential equations involving the particle beam, and the surface impedance concept. The proposed approach is applied to the coupling impedance of an accelerator vacuum chamber with thin conductive coating, and also verified in comparison with the existing analytical formula.

Modeling material transport regulation and traffic jam in neurons using PDE-constrained optimization

The intracellular transport process plays an important role in delivering essential materials throughout branched geometries of neurons for their survival and function. Many neurodegenerative diseases have been associated with the disruption of transport. Therefore, it is essential to study how neurons control the transport process to localize materials to necessary locations. Here, we develop a novel optimization model to simulate the traffic regulation mechanism of material transport in complex geometries of neurons. The transport is controlled to avoid traffic jam of materials by minimizing a pre-defined objective function. The optimization subjects to a set of partial differential equation (PDE) constraints that describe the material transport process based on a macroscopic molecular-motor-assisted transport model of intracellular particles. The proposed PDE-constrained optimization model is solved in complex tree structures by using isogeometric analysis (IGA). Different simulation parameters are used to introduce traffic jams and study how neurons handle the transport issue. Specifically, we successfully model and explain the traffic jam caused by reduced number of microtubules (MTs) and MT swirls. In summary, our model effectively simulates the material transport process in healthy neurons and also explains the formation of a traffic jam in abnormal neurons. Our results demonstrate that both geometry and MT structure play important roles in achieving an optimal transport process in neuron.

A comparative study of Ag–MgO/water and Fe[formula omitted]–CoFe[formula omitted]/EG–water hybrid nanofluid flow over a curved surface with chemical reaction using Buongiorno model

This investigation aims to scrutinize the flow phenomenon, thermal and concentration variations of two different hybrid nanofluids flow induced by a curved surface. The novelty of the work is to examine the heat and mass transport phenomena of Ag–MgO/water hybrid nanofluid and Fe3O4 –CoFe 2O4 /water–ethylene glycol hybrid nanofluid under the influences of viscous dissipation, heat generation/absorption and chemical reaction using Buongiorno model. Suitable similarity transformations are implemented to convert the set of partial differential equation (PDEs) into non-linear ordinary differential equations (ODEs), and then shooting based Runge–Kutta–Fehlberg method of fourth fifth order (RKF45) via MATLAB is executed to achieve the numerical solution. The solution is discussed through graphs which provide a detailed physical clarification of each profile in terms of involved parameters effects. Also, the comparison of heat and mass transfer rates in both hybrid nanofluids is deliberated with the assistance of tables. It is noticed that the temperature profiles for both hybrid nanofluids grow high with a boost in the values of the viscous dissipation, heat generation/absorption and thermophoresis parameters. Moreover, the increasing values of thermophoresis parameter and chemical reaction parameter improve the mass transfer rate of both Ag–MgO/water hybrid nanofluid and Fe3O4 –CoFe 2O4 /water–ethylene glycol hybrid nanofluid. The proposed investigation shows the importance of these hybrid nanofluids in chemical, manufacturing and nuclear plants, solar technologies, etc.

Computational Analysis of Nanoparticle Shapes on Hybrid Nanofluid Flow Due to Flat Horizontal Plate via Solar Collector.

The present work discusses the 2D unsteady flow of second grade hybrid nanofluid in terms of heat transfer and MHD effects over a stretchable moving flat horizontal porous plate. The entropy of system is taken into account. The magnetic field and the Joule heating effects are also considered. Tiny-sized nanoparticles of silicon carbide and titanium oxide dispersed in a base fluid, kerosene oil. Furthermore, the shape factors of tiny-sized particles (sphere, bricks, tetrahedron, and platelets) are explored and discussed in detail. The mathematical representation in expressions of PDEs is built by considering the heat transfer mechanism owing to the effects of Joule heating and viscous dissipation. The present set of PDEs (partial differential equations) are converted into ODEs (ordinary differential equations) by introducing suitable transformations, which are then resolved with the bvp4c (shooting) scheme in MATLAB. Graphical expressions and numerical data are obtained to scrutinize the variations of momentum and temperature fields versus different physical constraints.

Modeling and simulation of suction-driven flow and heat transfer with temperature-dependent thermal conductivity in a porous channel

In this paper, the heat transfer equation is coupled to the Navier-Stokes equations for a steady flow induced by uniform suction on two porous walls kept at different temperatures. The incompressibility of the fluid and the fact that the velocity field has two components lead to introduce the stream function in the governing equations. A similarity method is used in order to transform the resulting set of partial differential equations satisfied by the stream function and temperature into two ordinary differential equations for the same problem. The analysis is focused on the description of the behaviors of the normal and axial velocities, the streamlines, and temperature through multiple solution branches under different values of the main control parameters of the problem which are the Reynolds number, the Péclet number and the sensitivity of the thermal conductivity to the variations of temperature.

Melting effect on Cattaneo–Christov and thermal radiation features for aligned MHD nanofluid flow comprising microorganisms to leading edge: FEM approach

An investigation for fluctuating temperature with Cattaneo–Christov features and self-motive bioconvective microbes immersed in the water based nanofluid has been observing with excision/accretion of the leading edge. A set of partial differential equations is transferring by implementing the similarity measures to an ordinary differential pattern. A finite element technique is employed to establish the numerical results of the set of non-linear system of equations via Matlab programming. The numerical solution's reliability and validity are determined by ascertaining convergence specifications and comparing them to established novel solutions. The formulation of boundary layer configurations for microbes propagation, fluid temperature, the volume fraction of nano-inclusions, and the fluid velocity with varying persuasive parameters have discussed. The main findings of the prevailing assessment are that an increasing trend in the ablation/slash accretion in leading-edge declines in the non-dimensional melting factor of fluid velocity and temperature, while a decreasing trend has seen with the enhancing impact of ablation/slash accretion at the leading edge that represents the flow velocity deceleration. Furthermore, a significant result is that an increase in the non-dimensional melting factor enhances the density of the liquid.

Numerical study of simultaneous transport of heat and mass transfer in Maxwell hybrid nanofluid in the presence of Soret and Dufour effects

The article aims to investigate the influence of copper (Cu) and a combination of copper (Cu) and aluminum oxide (Al2O3) on the simultaneous transfer of heat and mass in sodium alginate liquid moving over a circular pipe. This transport activity is modeled by the use of conservation laws with correlations for physical quantities of Cu, Al2 O3, and sodium alginate. Through cylindrical coordinates formulation, the set of partial differential equations is obtained. These models are solved numerically by the finite element method (FEM). The relaxation time associated with momentum diffusion in Maxwell fluid plays role in controlling the viscous region. Moreover, momentum relaxation time in Cu/sodium alginate is strong than that in Cu–Al2O3/sodium alginate. It is noticed from simulations that particles of Cu/ sodium alginate have a greater velocity than the velocity of Cu–Al2O3/sodium alginate. Therefore, distortion of magnetic lines by the flow of Cu/sodium alginate is more than the distortion of magnetic lines by the flow of Cu–Al2O3/sodium alginate. The rise in the thermal conductivity of sodium alginate due to simultaneous dispersion of Cu and Al2O3 is more than the rise in thermal conductivity of sodium alginate. Thus for maximum HT, the simultaneous dispersion of Cu and Al2O3 in sodium alginate is recommended.

A mode-matching method for the prediction of stick-slip relative motion of two elastic rods in frictional contact

This paper presents a computationally efficient mode-matching method to predict the relative axial motion of two elastic rods in frictional contact. The motion is of the stick-slip type and is non-uniform along the rods. The proposed method utilizes the piecewise linearity of the problem in time and space. The original set of nonlinear partial differential equations describing the dynamics of the coupled system is first reduced to a system of linear, per time interval, ordinary differential equations by means of modal decomposition. The global modes are used for one of the two rods, while for the other rod, different modes are identified per time interval based on the regions in stick or slip phase. Subsequently, the system response is obtained by combining the piecewise linear solutions. A comparison of the solution method proposed with standard numerical techniques shows its advantage both in terms of computational time and accuracy. Numerical examples demonstrate the capability of the method to analyse cases involving either harmonic- or impact-type forces that drive the relative motion. Although the discussion in this paper is limited to the one-dimensional configuration, the approach is generic and can be extended to problems in more dimensions.

Computation of magnetohydrodynamic electro-osmotic modulated rotating squeezing flow with zeta potential effects

Motivated by exploring novel intelligent functional materials deployed in electromagnetic squeezing flows such systems, a comprehensive mathematical model is developed to investigate the squeezing flow of a smart viscous ionic magneto-tribological fluid with zeta potential effects, intercalated between two parallel plates rotating in unison, under the simultaneous application of electric and magnetic fields. The lower disk permits lateral mass flux (suction or injection). Continuity and momentum equations are represented in the proposed three-dimensional mathematical model by a set of partial differential equations. The electro-viscous effects resulting from distorted electric double-capacity flow fields are comprehensively examined for various intensities of applied plate motion. The formulation features a more robust approach to the traditional Poisson-Boltzmann equation model. A similarity transformation is used to translate the governing equations into ordinary differential equations, which are then numerically solved with appropriate boundary conditions at the disks using MATLAB software. Via graphical visualization of velocity profiles, pressure gradients and the upper wall coefficient of skin friction, several characteristics of squeezing flow are analysed. The computations show that there is a rise in pressure near the plate walls and a fall in pressure in the centre with increment in rotational, electroosmosis, electric field, and magnetic parameters. However, by selecting the appropriate squeezing velocity, the viscous drag on the lower plate can be effectively reduced. It is also observed that as the disk (wall) suction parameter increases, both radial and transverse velocities are damped. The current study generalizes previous investigations with the novelty of rotation and also pressure gradient computations. Furthermore, it provides a useful benchmark for alternative numerical simulations.

Effectiveness of Newtonian Heating on Magneto-Free Convective Flow of Polar Nanofluid across a Solid Sphere

This paper explains the free convective flowing of micropolar nanofluid through a solid sphere with Newtonian heating and the magnetic field influence. Sets of partial differential equations are converted by using convenient transformations to ordinary differential equations. The system of similar and nonsimilar equations is solved numerically using the Runge–Kutta–Fehlberg method (RKF45) using MAPLE software (version 20).The numerical results are validated by comparison with previously published works, and excellent agreement is found between them. The influence of the magnetic field parameter, solid volume fraction, and micropolar parameter on velocity, temperature, and angular velocity profiles are shown graphically. In addition, both the skin friction coefficient and Nusselt number are also discussed. It is found that the skin friction increases with an increase in the solid volume fraction of both nanoparticles and Newtonian heating and micropolar parameters. In addition, the magnetic field reduces both the skin friction and the Nusselt number. Moreover, the solid volume fraction and Newtonian heating parameter enhance the Nusselt number.

Numerical solution of MHD, Soret, Dufour, and thermal radiation contributions on unsteady free convection motion of Casson liquid past a semi‐infinite vertical porous plate

AbstractIn this paper, the unsteady motion of Casson liquid over a half‐infinite penetrable vertical plate with MHD, thermal radiation, Soret, and Dufour contributions have been explored numerically. In the physical geometry, the Casson liquid flows to the layer from the penetrable vertical plate. At the layer, Casson liquid is set into motion and the flow equations are illustrated using coupled partial differential equations (PDEs). This set of PDEs is simplified to form dimensionless PDEs with the use of normal nondimensional transformation. The controlling parameters' effects on the working fluid are extensively discussed on velocity, concentration, and temperature and presented graphically. Computational values of Nusselt plus Sherwood number and skin friction for controlling parameters are depicted in a tabular form. Our outcomes show that a raise in the Casson term depreciates the velocity because of the magnetic parameter influence on the fluid flow. The Soret parameter was found to accelerate the skin friction along with the Sherwood number coefficients. An incremental value of the Dufour parameter was detected to hike the skin friction alongside the Nusselt number. Results of this study were found to be in conformity with previously published work.

Temperature-dependent variable viscosity and thermal conductivity effects on non-Newtonian fluids flow in a porous medium

Purpose The purpose of this paper is to consider the simultaneous flow of Casson Williamson non Newtonian fluids in a vertical porous medium under the influence of variable thermos-physical parameters. Design/methodology/approach The model equations are a set of partial differential equations (PDEs). These PDEs were transformed into a non-dimensionless form using suitable non-dimensional quantities. The transformed equations were solved numerically using an iterative method called spectral relaxation techniques. The spectral relaxation technique is an iterative method that uses the Gauss-Seidel approach in discretizing and linearizing the set of equations. Findings It was found out in the study that a considerable number of variable viscosity parameter leads to decrease in the velocity and temperature profiles. Increase in the variable thermal conductivity parameter degenerates the velocity as well as temperature profiles. Hence, the variable thermo-physical parameters greatly influence the non-Newtonian fluids flow. Originality/value This study considered the simultaneous flow of Casson-Williamson non-Newtonian fluids by considering the fluid thermal properties to vary within the fluid layers. To the best of the author’s knowledge, such study has not been considered in literature.

Shock waves in a strongly coupled inhomogeneous dusty plasma

Nonlinear theory for certain dust acoustic waves has been developed for a strongly coupled inhomogeneous collisionless dusty plasma containing dust particles of different sizes. The generalized hydrodynamic set of partial differential equations is considered to describe the fluid model of the present nonlinear system. A reductive perturbation technique is employed on this nonlinear set of equations to derive a variable coefficient Korteweg-deVries (KdV)–Burgers equation with a forcing term. The generalized expansion method is performed to obtain an analytic expression for the solution. This analytic solution describes the monotonic kink-type shock waves, due to the dominant nature of the dissipation term over the dispersion term. In addition to the effects of inhomogeneity and grain size distribution, the effects of nonthermal ions, dust–dust correlation and ion temperature on the wave propagation are investigated. Also, a steady-state monotonic shock solution of the KdV-Burgers equation is obtained numerically. The obtained results are discussed in the context of some dusty plasmas, previously reported experimentally or theoretically.

Study on time-dependent Oldroyd-B fluid flow over a convectively heated surface with Cattaneo-Christov theory

The time-dependent two-dimensional Oldroyd-B fluid flow is investigated in the presence of generalized Fourier's and Fick's laws. The mechanism of heat and mass transport is studied using the Cattaneo-Christov (CC) double diffusion theory, which characterizes the thermal and solutal relaxation factors. In addition, the temperature-dependent thermal conductivity of the fluid is taken into consideration. The modified Fourier's and Fick's approach is used to develop a set of partial differential equations for the flow of an Oldroyd-B fluid as well as thermal and solutal transport. By using the suitable similarity transformations, the governing equations are converted into an ordinary differential equation. To visualize the effects of dimensionless physical constraints on flow and energy transport phenomenon, in the domain [0, ) numerical integration is performed with the help of (Midrich scheme) in Maple software to solve the highly non-linear ordinary differential equations. The effects of various arising parameters can be seen in the graphs of velocities, temperature, and concentration. According to the findings, the thermal and solutal penetration depth declines for the growing values of flow controlling parameter and Lewis number, respectively. The current results are very similar to those found in the literature.

Selected Mathematical Models Describing Flow in Gas Pipelines

The main aim of simulation programs is to study the behavior of gas pipe networks in certain conditions. Solving a specified set of differential equations describing transient (unsteady) flow in a gas pipeline for the adopted parameters of load and supply will help us find out the value of pressure or flow rate at selected points or along selected sections of the network. Transient gas flow may be described by a set of simple or partial differential equations classified as hyperbolic or parabolic. Derivation of the mathematical model of transient gas flow involves certain simplifications, of which one-dimensional flow is most important. It is very important to determine the conditions of pipeline/transmission network operation in which the hyperbolic model and the parabolic model, respectively, should be used. Parabolic models can be solved numerically in a much simpler way and can be used to design simulation programs which allow us to calculate the network of any structure and any number of non-pipe elements. In some conditions, however, they describe the changes occurring in the network less accurately than hyperbolic models do. The need for analysis, control, and optimization of gas flows in high-pressure gas pipelines with complex structure increases significantly. Very often, the time allowed for analysis and making operational decisions is limited. Therefore, efficient models of unsteady gas flows and high-speed algorithms are essential.

Implicit implementation of the nonlocal operator method: an open source code

In this paper, we present an open-source code for the first-order and higher-order nonlocal operator method (NOM) including a detailed description of the implementation. The NOM is based on so-called support, dual-support, nonlocal operators, and an operate energy functional ensuring stability. The nonlocal operator is a generalization of the conventional differential operators. Combined with the method of weighed residuals and variational principles, NOM establishes the residual and tangent stiffness matrix of operate energy functional through some simple matrix without the need of shape functions as in other classical computational methods such as FEM. NOM only requires the definition of the energy drastically simplifying its implementation. The implementation in this paper is focused on linear elastic solids for sake of conciseness through the NOM can handle more complex nonlinear problems. The NOM can be very flexible and efficient to solve partial differential equations (PDEs), it’s also quite easy for readers to use the NOM and extend it to solve other complicated physical phenomena described by one or a set of PDEs. Finally, we present some classical benchmark problems including the classical cantilever beam and plate-with-a-hole problem, and we also make an extension of this method to solve complicated problems including phase-field fracture modeling and gradient elasticity material.

Aircraft Icing Severity Evaluation

Aircraft icing refers to the ice buildup on the surface of an aircraft flying in icing conditions. The ice accretion on the aircraft alters the original aerodynamic configuration and degrades the aerodynamic performances and may lead to unsafe flight conditions. Evaluating the flow structure, icing mechanism and consequences is of great importance to the development of an anti/deicing technique. Studies have shown computational fluid dynamics (CFD) and machine learning (ML) to be effective in predicting the ice shape and icing severity under different flight conditions. CFD solves a set of partial differential equations to obtain the air flow fields, water droplets trajectories and ice shape. ML is a branch of artificial intelligence and, based on the data, the self-improved computer algorithms can be effective in finding the nonlinear mapping relationship between the input flight conditions and the output aircraft icing severity features.

Depth-averaged coupling of submerged granular deformation with fluid flow: An augmented HLL scheme

Debris flow in or towards water bodies, and bed load transport in open channels manifest from the interaction of fluid with deformable granular media at varied spatiotemporal scales. The present study aims to formulate a robust two-dimensional coupled depth-averaged framework for simulation of submarine deformation of non-cohesive granular media and the corresponding effect on water and the otherwise. The unified framework consisting of conditional hyperbolic set of partial differential equations has been conceptualized as two concurrent yet independent systems interacting through source/sink terms comprising of drag force, shear and water-surface gradient. The system is solved on a regular finite volume grid through a robust modification of the Harten-Lax-Van Leer (HLL) scheme. The interaction terms are embedded into the interfacial fluxes, where additional closure employs the effect of dry-wet zone transition and flow arrest for both modules. Various aspects of the proposed framework are validated against an array of test cases ranging from landslide induced tsunami to dambreak induced scour. The last representative scenario emphasizes the potential applicability of the model for real-time extension to field-scale problems. A few one-dimensional approximations are explored in the Appendix to calculate relative velocity between interstitial fluid and granular media for computation of drag force.

Stability results for the functional differential equations associated to water hammer in hydraulics

There is considered a system of two sets of partial differential equations describing the water hammer in a hydroelectric power plant containing the dynamics of the tunnel, turbine penstock, surge tank and hydraulic turbine. Under standard simplifying assumptions (negligible Darcy–Weisbach losses and dynamic head variations), a system of functional differential equations of neutral type, with two delays, can be associated to the aforementioned partial differential equations and existence, uniqueness and continuous data dependence can be established. Stability is then discussed using a Lyapunov functional deduced from the energy identity. The Lyapunov functional is "weak" i.e. its derivative function is only non-positive definite. Therefore only Lyapunov stability is obtained while for asymptotic stability application of the Barbashin–Krasovskii–LaSalle invariance principle is required. A necessary condition for its validity is the asymptotic stability of the difference operator associated to the neutral system. However, its properties in the given case make the asymptotic stability non-robust (fragile) in function of some arithmetic properties of the delay ratio.

Fault-Tolerant Control of an Offshore Ocean Thermal Energy Conversion System with Output Constraints

A fault-tolerant control method is investigated for an offshore ocean thermal energy conversion (OTEC) system with constraints in this work. In the system modelling, the distributed parameter system is used to build the OTEC system, which is expressed by a set of partial differential equations (PDEs) and ordinary differential equations (ODEs). Moreover, a boundary fault-tolerant control scheme is proposed to solve actuator failures. Meanwhile, according to the control scheme, system vibration suppressing and system stability can be realized. Two proposed coupled controllers are designed to solve actuator faults while considering the output nonlinearity of the OTEC system. In addition, the stability of the OTEC system based on barrier Lyapunov function is proven by simulation results, which verifies the effectiveness of the proposed coupled fault-tolerant-boundary control method. Finally, the proposed method can effectively improve the performance and safety of ocean thermal energy conversion in the OTEC system.