Kutta Integration Research Articles

A μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-mode approach for exponential integrators: actions of φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions of Kronecker sums

We present a method for computing actions of the exponential-like φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions for a Kronecker sum K of d arbitrary matrices Aμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\mu $$\\end{document}. It is based on the approximation of the integral representation of the φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions by Gaussian quadrature formulas combined with a scaling and squaring technique. The resulting algorithm, which we call phiks, evaluates the required actions by means of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-mode products involving exponentials of the small sized matrices Aμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\mu $$\\end{document}, without forming the large sized matrix K itself. phiks, which profits from the highly efficient level 3 BLAS, is designed to compute different φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions applied on the same vector or a linear combination of actions of φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions applied on different vectors. In addition, thanks to the underlying scaling and squaring techniques, the desired quantities are available simultaneously at suitable time scales. All these features allow the effective usage of phiks in the exponential integration context. In fact, our newly designed method has been tested in popular exponential Runge–Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions. The numerical experiments with discretized semilinear evolutionary 2D or 3D advection–diffusion–reaction, Allen–Cahn, and Brusselator equations show the superiority of the proposed μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-mode approach.

Efficient simulation of complex Ginzburg–Landau equations using high-order exponential-type methods

In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg–Landau equations on Cartesian product domains with homogeneous Dirichlet/Neumann or periodic boundary conditions. To this aim, we employ for the time integration high-order exponential methods of splitting and Lawson type with constant time step size. These schemes enjoy favorable stability properties and, in particular, do not show restrictions on the time step size due to the underlying stiffness of the models. The needed actions of matrix exponentials are efficiently realized by using a tensor-oriented approach that suitably employs the so-called μ-mode product (when the semidiscretization in space is performed with finite differences) or with pointwise operations in Fourier space (when the model is considered with periodic boundary conditions). The overall effectiveness of the approach is demonstrated by running simulations on a variety of two- and three-dimensional (systems of) complex Ginzburg–Landau equations with cubic or cubic-quintic nonlinearities, which are widely considered in literature to model relevant physical phenomena. In fact, we show that high-order exponential-type schemes may outperform standard techniques to integrate in time the models under consideration, i.e., the well-known second-order split-step method and the explicit fourth-order Runge–Kutta integrator, for stringent accuracies.

Study of a Numerical Integral Interpolation Method for Electromagnetic Transient Simulations

In the fixed time-step electromagnetic transient (EMT)-type program, an interpolation process is applied to deal with switching events. The interpolation method frequently reduces the algorithm’s accuracy when dealing with power electronics. In this study, we use the Butcher tableau to analyze the defects of linear interpolation. Then, based on the theories of Runge–Kutta integration, we propose two three-stage diagonally implicit Runge–Kutta (3S-DIRK) algorithms combined with the trapezoidal rule (TR) and backward Euler (BE), respectively, with TR-3S-DIRK and BE2-3S-DIRK for the interpolation and synchronization processes. The proposed numerical integral interpolation scheme has second-order accuracy and does not produce spurious oscillations due to the size change in the time step. The proposed method is compared with the critical damping adjustment method (CDA) and the trapezoidal method, showing that it does not produce spurious numerical oscillations or first-order errors.

Anti-symmetric and positivity preserving formulation of a spectral method for Vlasov-Poisson equations

We analyze the anti-symmetric properties of a spectral discretization for the one-dimensional Vlasov-Poisson equations. The discretization is based on a spectral expansion in velocity with the symmetrically weighted Hermite basis functions, central finite differencing in space, and an implicit Runge Kutta integrator in time. The proposed discretization preserves the anti-symmetric structure of the advection operator in the Vlasov equation, resulting in a stable numerical method. We apply such discretization to two formulations: the canonical Vlasov-Poisson equations and their continuously transformed square-root representation. The latter preserves the positivity of the particle distribution function. We derive analytically the conservation properties of both formulations, including particle number, momentum, and energy, which are verified numerically on the following benchmark problems: manufactured solution, linear and nonlinear Landau damping, two-stream instability, bump-on-tail instability, and ion-acoustic wave.

The Impact of Velocity Update Frequency on Time Accuracy for Mantle Convection Particle Methods

AbstractComputing the velocity field is an expensive process for mantle convection codes. This has implications for particle methods used to model the advection of quantities such as temperature or composition. A common choice for the numerical treatment of particle trajectories is classical fourth‐order explicit Runge–Kutta (ERK4) integration, which involves a velocity computation at each of its four stages. To reduce the cost per time step, it is possible to evaluate the velocity for a subset of the four time integration stages. We explore two such alternative schemes, in which velocities are only computed for: (a) stage 1 on odd‐numbered time steps and stages 2–4 for even‐numbered time steps, and (b) stage 1 for all time steps. A theoretical analysis of stability and accuracy is presented for all schemes. It was found that the alternative schemes are first‐order accurate with stability regions different from that of ERK4. The efficiency and accuracy of the alternate schemes were compared against ERK4 in four test problems covering isothermal, thermal, and thermochemical flows. Exact solutions were used as reference solutions when available. In agreement with theory, the alternate schemes were observed to be first‐order accurate for all test problems. Accordingly, they may be used to efficiently compute solutions to within modest error tolerances. For small error tolerances, however, ERK4 was the most efficient.

Gauss–Newton Runge–Kutta integration for efficient discretization of optimal control problems with long horizons and least-squares costs

This work proposes an efficient treatment of continuous-time optimal control problems with long horizons and nonlinear least-squares costs. In particular, we present the Gauss–Newton Runge–Kutta (GNRK) integrator which provides a high-order cost integration. Crucially, the Hessian of the cost terms required within an SQP-type algorithm is approximated with a Gauss–Newton Hessian. Moreover, L2 penalty formulations for constraints are shown to be particularly effective for optimization with GNRK. An efficient implementation of GNRK is provided in the open-source software framework acados. We demonstrate the effectiveness of the proposed approach and its implementation on an illustrative example showing a reduction of relative suboptimality by a factor greater than 10 while increasing the runtime by only 10%.

Influence of suction and injection on the electrical magnetohydrodynamic behavior of nano-fluid in a nonlinear radiative flow over an extending surface

The present communication aims to investigate the behavior of a radiative electrical MHD Casson nanofluid flowing over a stretched surface, under the influence of nonlinear thermal radiation, and suction/injection effects. The governing equations include parameters for temperature and concentration, which are adjusted based on the dependence of viscosity and thermal conductivity, thus enhancing the fluid flow properties. By using similarity transformations, the system of PDEs is converted to an ODE, which is then numerically solved using the well-known fourth-order Runge–Kutta integration strategy based on the shooting method. The impacts of operating parameters on velocity, temperature, and concentration profiles were explained and examined through tables and graphs. Temperature and concentration are increased as the injection ( S < 0 ) increases . However, they show decrement when the suction ( S > 0 ) increases . The novelty of this study focuses on the mathematical development of the flow problem with significant results and also investigating the impact of electric field on velocity of the fluid flow and shear stress. These results are applicable in manufacturing of stretchable materials.

Adaptive Rational Krylov Methods for Exponential Runge–Kutta Integrators

Adaptive Rational Krylov Methods for Exponential Runge–Kutta Integrators

Non-Cooperative LEO Satellite Orbit Determination Using Single Station for Space-Based Opportunistic Positioning

Space-based opportunistic positioning is a crucial component of resilient positioning, navigation, and timing (PNT) systems, and it requires the acquisition of orbit information for non-cooperative low Earth orbit (LEO) satellites. Traditional methods for orbit determination (OD) of non-cooperative LEO satellites have difficulty in achieving a balance between reliability, hardware costs, and availability duration. To address these challenges, this study proposes a framework for single-station orbit determination of non-cooperative LEO satellites. By utilizing signals of opportunity (SOPs) captured by a single ground station, the system performs initial orbit determination (IOD), precise orbit determination (POD), and orbit prediction (OP), enabling the long-term determination of satellite positions and velocities. Under the proposed framework, the reliability and real-time performance are dependent on the initial orbit determination and the orbit calculation based on the dynamical model. To achieve initial orbit determination, a three-step algorithm is designed. (1) An improved search method is employed to estimate a coarse orbit using single-pass Doppler measurements. (2) Data association is conducted to obtain multi-pass Doppler observations. (3) The least squares (LS) is implemented to determine the initial orbit using the associated multi-pass Doppler measurements and the coarse orbit. Additionally, to enhance computational efficiency, two fast orbit calculation algorithms are devised. These algorithms leverage the numerical stability of the Runge–Kutta integrator to reduce computations and exploit the strong correlation among nearby time intervals of orbits with small eccentricities to minimize redundant calculations, thereby achieving orbit calculation efficiently. Finally, through positioning experiments, the determined orbits are demonstrated to have accuracy comparable to that of two-line elements (TLE) updated by the North American Aerospace Defense Command (NORAD).

Mechanisms of Diffusion thermo and Thermal diffusion on MHD Mixed Convection Flow of Casson fluid over a vertical cone with porous material in the presence of thermophoresis and a Brownian motion

In this present article, we analyzed the Effects of Diffusion thermo and Thermal Diffusion on magnetohydrodynamic (MHD) mixed convection flow for Casson nanofluid is deliberated a vertical cone with porous material. The modeled equations are transformed into a set of non-linear ODEs by employing similar transformable variables. These equations are then solved numerically using the shooting method, through the fourth-order Runge–Kutta integration procedure. Effects of some prominent physical parameters, such as diffusion thermo, Prandtl number, thermophoresis parameter, and magnetic parameter on the velocity, temperature, and concentration profiles are discussed graphically and numerically. Numerical calculations and graphs are used to illustrate the important features of the solution on fluid flow velocity, heat, and mass transfer characteristics under different quantities of parametric circumstances entering into the problem. Moreover, we computed the physical variables such as the coefficient of shear stress, rate of heat, and mass transfer. To establish the veracity of our present results, we compared them to previously published research and found substantial concordance.

Quantifying and eliminating the time delay in stabilization exponential time differencing Runge–Kutta schemes for the Allen–Cahn equation

Although the stabilization technique is favorable in designing unconditionally energy stable or maximum-principle-preserving schemes for gradient flow systems, the induced time delay is intractable in computations. In this paper, we propose a class of delay-free stabilization schemes for the Allen–Cahn gradient flow system. Considering the Fourier pseudo-spectral spatial discretization for the Allen–Cahn equation with either the polynomial or the logarithmic potential, we establish a semi-discrete, mesh-dependent maximum principle by adopting a stabilization technique. To unconditionally preserve the mesh-dependent maximum principle and energy stability, we investigate a family of exponential time differencing Runge–Kutta (ETDRK) integrators up to the second-order. After reformulating the ETDRK schemes as a class of parametric Runge–Kutta integrators, we quantify the lagging effect brought by stabilization, and eliminate delayed convergence using a relaxation technique. The temporal error estimate of the relaxation ETDRK integrators in the maximum norm topology is analyzed under a fixed spatial mesh. Numerical experiments demonstrate the delay-free and structure-preserving properties of the proposed schemes.

Two new classes of exponential Runge–Kutta integrators for efficiently solving stiff systems or highly oscillatory problems

We note a fact that stiff systems or differential equations that have highly oscillatory solutions cannot be solved efficiently using conventional methods. In this paper, we study two new classes of exponential Runge–Kutta (ERK) integrators for efficiently solving stiff systems or highly oscillatory problems. We first present a novel class of explicit modified version of exponential Runge–Kutta (MVERK) methods based on the order conditions. Furthermore, we consider a class of explicit simplified version of exponential Runge–Kutta (SVERK) methods. Numerical results demonstrate the high efficiency of the explicit MVERK integrators and SVERK methods derived in this paper compared with the well-known explicit ERK integrators for stiff systems or highly oscillatory problems in the literature.

Large-Strain Vibrations of Axially Stretched Hyperelastic Rods

Fast-moving soft robotics usually suffer large-strain deformation with a relatively high velocity. To improve their performance, it is necessary to study the underly dynamical mechanism. However, the inherent strong material nonlinearity brings considerable difficulties in modeling and calculating the soft structure’s nonlinear dynamics. This work focuses on the large-strain vibrations of the hyperelastic rod under an axial tensile force. Several typical hyperelastic models, including the neo-Hookean, Mooney, and Yeoh models, are used to derive the equations governing the rod’s dynamics. The equations based on these models are relatively complex, and their corresponding numerical calculation diverges in some cases. Therefore, a reference stretch ratio(s) polynomial fitting (RSRPF) model is proposed to derive the concise governing equation. Such a governing equation is effectively solved by the combination of the Galerkin discretization and the fourth-order Runge–Kutta integration algorithm. The correctness of the developed model and the employed algorithm is verified by comparing with the previous data of a hyperelastic membrane. In most cases, the rod periodically vibrates under a periodic tensile force. However, when the structural damping is small and the load frequency approaches the resonant frequency, the hyperelastic rod may quasi-periodically vibrate. The revealed large-strain vibration mechanism is supposed to be useful for guiding the dynamical design of soft structures.

Mathematical analysis of nonlinear models and their role in dynamics

Micropolar fluids provide a more accurate description of fluid behavior at small scales, where classical continuum models may fail to capture complicated microscale interactions. These interactions become significant in situations involving non-Newtonian fluids, complex geometries, and materials with internal structures. Unlike traditional fluids, micropolar fluids consider the effects of micro-rotations and micro-deformations. The objective of this exploration is to investigate the characteristics of flow in a micropolar-Casson fluid that is doubly stratified, and is induced by a stretching sheet. Additionally, the study investigates the transfer of both thermal energy and mass species over a two-dimensional porous medium. By utilizing suitable similarity transformations, a system of nonlinear ordinary differential equations (ODEs) is derived by transforming the governing partial differential equations (PDEs) that describe the physics of the problem. The combination of a fourth-order Runge–Kutta integration scheme and the Shooting method is utilized to solve these equations. Various graphs and tables are utilized to illustrate the impact of physical parameters on the dimensionless quantities. Our findings exhibit strong conformity with the existing results in literature for specific scenarios. Furthermore, the outcomes highlight that the velocity and micro-rotation are enhanced by the material parameter.

Study of Nonlinear Aerodynamic Self-Excited Force in Flutter Bifurcation and Limit Cycle Oscillation of Long-Span Suspension Bridge

This article establishes a nonlinear flutter system for a long-span suspension bridge, aiming to analyze its supercritical flutter response under the influence of nonlinear aerodynamic self-excited force. By fitting the experimental discrete values of flutter derivatives using the least squares method, a polynomial function of flutter derivatives with respect to reduced wind speed is obtained. Flutter critical value is determined by the linear matrix eigenvalues of a state-space equation. The occurrence of a supercritical Hopf bifurcation in the nonlinear system is determined by the Jacobian matrix eigenvalues of the state-space equation and the system’s vibrational response at the critical state. The vibrational response of the supercritical state is obtained through Runge–Kutta integration, revealing the presence of stable limit cycle oscillation (LCO) and unstable limit cycle oscillation in the system, and through analyzing the relationship between the LCO amplitude and wind speed. Considering cubic nonlinear damping and stiffness, the effects of different factors on the nonlinear flutter system are analyzed.

Effects of Activation Energy Diffusion Thermo and Nonlinear Thermal Radiation on Mixed Convection Casson Fluid Flow Through a Vertical Cone with Porous Media in the Presence of Brownian Motion and Thermophoresis

In this paper, analyze the impact of Activation energy, Diffusion thermo and nonlinear thermal radiation of mixed convection casson fluid flow through a vertical cone with porous material in the presence of Brownian and thermophoresis has been studied. Numerical techniques, namely Runge–Kutta integration and the shooting method, were applied to obtain solutions to similarity-transformed equations governing continuity, momentum, energy and concentration. The study examined the distributions of flow velocity, temperature and concentration under the interactive effects of the fluid. The results revealed that the activation energy of the Arrhenius equation plays an important role in fluid transport mechanisms within a chemically reactive system involving the Soret effect for a low-Schmidt-number fluid. When the activation energy parameter E was greater within the range 0 &lt; E &lt; 5, the wall shear stress was stronger, heat transfer rate increased, and mass transfer rate decreased.

Analysis of chemically reactive squeezing flow of silica and titania hybrid nanoparticles in water-based medium between two parallel plates with higher order slip

Considering the significance of hybrid nanofluid over convectional fluid in terms of high rate of heat transfer, nanoscience and nanotechnology has been enhanced by dispersing nanoparticles in the base fluid to obtain new material with series of properties and applications. In this study, the analysis of chemically reactive squeezing flow conveying silica and titanium dioxide nanoparticles in water-based medium across two parallel plates with higher order velocity slip is carried out employing three different chemical kinetics for exothermic/endothermic reactions. The system of partial differential equations resulting from the fluid model assumed the ordinary differential form in alliance with appropriate similarity variables. The modified ordinary differential equations is simulated numerically in MATLAB software package using fourth order Runge–Kutta integration scheme in line with shooting techniques. The tested validity for limited case conform to preceding reports in the literature. The outcomes from the scrutiny uncovered in tables and graphs revealed that the velocity and temperature distributions of the hybrid nanofluid decrease steadily as first order slip factor varies from 0.2 to 1.0 but increase with second order slip factor at all levels of chemical kinetics. Moreover, for exothermic reaction, the rate of heat transfer decreases at the lower plate by −221.923% with increasing value of activation energy parameter when m=−2,0,0.5 but converse is the case in endothermic reaction as the rate of heat transfer increases by 106.382%.

Analytical Study of MHD Mixed Convection Flow for Maxwell Nanofluid Through a Vertical Cone with Porous Material in the Presence of Variable Thermal Conductivity and Soret, Dufour Effects

The key purpose of the existing article is to discuss the magnetohydrodynamic (MHD) mixed convection flow for Maxwell nanofluid is deliberated a vertical cone with porous material. A variable thermal conductivity and Dufour, Soret effects are also taken into consideration. The modeled equations are transformed into a set of non-linear ODEs by employing similar transformable variables. These equations are then solved numerically using the shooting method, through the fourth-order Runge–Kutta integration procedure. Effects of some prominent physical parameters, such as diffusion thermo, Prandtl number, thermophoresis parameter, and magnetic parameter on the velocity, temperature, and concentration profiles are discussed graphically and numerically. The main outcomes of this investigation are that Velocity slows down with augmentation in Maxwell and magnetic parameters. Temperature increases with radiation and thermophoresis parameters and reduces with growing values of Prandtl number and Brownian motion parameters. The values of skin-friction coefficient, Nusselt number coefficient and Sherwood number coefficient are presented in table. A comparison with previously reported data is made and an excellent agreement is noted.

Physics-informed deep learning for three-dimensional transient heat transfer analysis of functionally graded materials

We present a physics-informed deep learning model for the transient heat transfer analysis of three-dimensional functionally graded materials (FGMs) employing a Runge–Kutta discrete time scheme. Firstly, the governing equation, associated boundary conditions and the initial condition for transient heat transfer analysis of FGMs with exponential material variations are presented. Then, the deep collocation method with the Runge–Kutta integration scheme for transient analysis is introduced. The prior physics that helps to generalize the physics-informed deep learning model is introduced by constraining the temperature variable with discrete time schemes and initial/boundary conditions. Further the fitted activation functions suitable for dynamic analysis are presented. Finally, we validate our approach through several numerical examples on FGMs with irregular shapes and a variety of boundary conditions. From numerical experiments, the predicted results with PIDL demonstrate well agreement with analytical solutions and other numerical methods in predicting of both temperature and flux distributions and can be adaptive to transient analysis of FGMs with different shapes, which can be the promising surrogate model in transient dynamic analysis.

Local Non-Similar Solution for Non-Isothermal Electroconductive Radiative Stretching Boundary Layer Heat Transfer with Aligned Magnetic Field

Continuous two-dimensional boundary layer heat transfer in an electroconductive Newtonian fluid from a stretching surface that is biased by a magnetic field aligned with thermal radiation is the subject of this study. The effects of magnetic induction are induced because the Reynolds number is not small. The sheet is traveling with a temperature and velocity that are inversely related to how far away from the steady edge it is from the plane in which it is traveling. We also imposed external velocity u=uex=Dxp in the boundary. The necessary major equations are made dimensionless by the local non-similarity transformation and become a system of non-linear ordinary differential equations after being transformed from non-linear partial differential equations. The subsequent numerical solution of the arisen non-dimensional boundary value problem utilizes a sixth-order Runge–Kutta integration scheme and Nachtsheim–Swigert shooting iterative technique. A good correlation is seen when the solutions are compared to previously published results from the literature. Through the use of graphical representation, the physical impacts of the fluid parameters on speed, induced magnetic field, and temperature distribution are carried out. Furthermore, the distributions for skin friction coefficient and local Nusselt number are also studied for different scenarios. The skin friction coefficient and local Nusselt number are observed to increase with greater values of the temperature exponent parameter and velocity exponent parameter. However, as heat radiation increases, the local Nusselt number decreases even though temperatures are noticeably higher. The study finds applications in magnetic polymer fabrication systems.