Differential Transport Equations Research Articles

On asymptotics of the solution to the Cauchy problem for a singularly perturbed operator-differential transport equation with weak diffusion in the case of several space variables

A formal asymptotic expansion is constructed for the solution of the Cauchy problem for a singularly perturbed operator differential transport equation with small nonlinearities and weak diffusion in the case of several space variables. Under the conditions imposed on the data of the problem, the leading asymptotic term is described by the multidimensional generalized Burgers–Korteweg–de Vries equation. Under certain conditions, the remainder is estimated with respect to the residual.

Efficient general method for numerically modeling laser pulse propagation, overlap, and lifetime effects in amplifiers

An efficient numerical time-dependent general method is developed to address incoherent pulse overlap and lifetime effects in laser amplifiers. The alternating propagation-population laser energetics method (APPLE) has been validated against a semi-discrete coupled rate equation numerical method (SDRE) and analytic formalisms in bounding cases. APPLE is based on decoupled rates applied to a time-dependent framework where both space-time-dependent populations and pulse energetics are consistently updated in each time step. A significant advantage of APPLE lies in its conceptual simplicity, ease of implementation, and relatively small computational cost. SDRE tracks the populations through coupled rates and uses the method of lines to discretize the hyperbolic partial differential transport equations allowing for use of ordinary differential equation solvers. With reasonably sized mesh, we report both energetic and power pulse shape relative differences on the order of one percent between the models over a large range of initial conditions.

Asymptotics of solutions of the Cauchy problem for a singularly perturbed operator differential transport equation

Asymptotics of solutions of the Cauchy problem for a singularly perturbed operator differential transport equation

“Return to equilibrium” anisotropy model for non-equilibrium Reynolds stress closures

A long-standing question in the Reynolds-averaged modeling of turbulent flow is how to predict accurately flows with temporal or spatial unsteadiness. In this paper, we present a “return to equilibrium” approach to modeling the unsteady Reynolds stress anisotropy bij in terms of differential transport equations, the solutions to which are constrained by algebraic closures for bij in flows at equilibrium and by the corresponding rapid-distortion-theory solutions for flows far from equilibrium. When coupled with scale equations for the turbulent kinetic energy and the dissipation rate, these anisotropy evolution equations comprise a complete closure scheme for which no additional model coefficients are required. Evaluations of this closure scheme, in which two different existing equilibrium algebraic models for bij were employed, were carried out for homogeneous turbulence in flows with a step imposition of shear, with continuously oscillatory shear at different frequencies and with a prescribed time-dependent plane-strain rate. Good agreement was achieved between model predictions and experimental/simulation target data in all test cases, with either Girimaji's bij model or a structure-based bij model used as the equilibrium algebraic closure.

Modeling of an all-solid-state battery with a composite positive electrode

All solid-state batteries are considered as the most promising battery technology due to their safety and high energy density. This study presents an advanced mathematical model that accurately simulates the complex behavior of all-solid-state lithium-ion batteries with composite positive electrodes. The partial differential equations of ionic transport and potential dynamics in the electrode and electrolyte are solved and reduced to a low-order system with Padé approximation. Moreover, the imperfect contact and the electrical double layers at the solid-solid interface are also taken into consideration. Subsequent experiments are conducted for the blocked cell and half-cells to extract parameters. Next, the parameterized model is validated with extensive experimental data from NCM811/LPSC/Li4.4Si batteries, illustrating the superior capability of predicting cell voltage with an average RMSE of 19.5 mV for the discharging/charging phases and 2.8 mV for the end of relaxation under a total of 15 conditions. From the simulations, it can be concluded that the limiting factors for battery performance are overpotentials caused by concentration polarization within positive particles and interface reactions. Finally, through a parameter sensitivity analysis, we offer strategic guidelines for optimizing battery performance, thus enhancing the development efficiency of ASSBs.

Analytical and numerical solutions of radially symmetric aquifer thermal energy storage problems

Abstract Aquifer thermal energy storage (ATES) systems offer reduced energy costs, lower carbon emissions, and increased energy resilience. The feasibility, however, depends on several factors and usually require optimization. We study an ATES system with injection and extraction wells (cf. graphical abstract). The purpose of the investigation was to calculate the recovery factor of an ATES system with a cyclic repetition of injection and pumping. In the paper, we discuss analytical and numerical radial solutions of differential equations for heat transport in water-saturated porous media. A similar solution was obtained for a 2-D-horizontal confined aquifer with a constant radial flow. Numerical solutions were derived by using a high-resolution Lagrangian approach suppressing spurious oscillations and artificial dispersion. The numerical solution and the analytical solutions give consistent results and match each other well. The solutions describe instantaneous and delayed heat transfer between fluid and solid, as well as time-varying water flow. In hydrological terms, these solutions are relevant for a wide range of problems where groundwater reservoirs are utilized for extraction and storage (namely, irrigation; water supply; geothermal extraction).

Thermal analysis of magnetohydrodynamics (MHD) Casson fluid with suspended iron (II, III) oxide-aluminum oxide-titanium dioxide ternary-hybrid nanostructures

This study is carried out to enhance and analyze the thermal performance of non-Newtonian Casson fluid by immersing Ternary hybrid nanoparticles Fe3O4-Al2O3-TiO2 uniformly. To model the behaviour of such complex phenomena mathematically, a system of complex transport differential equations is developed by utilizing a non-Fourier heat transfer model for energy transport. The non-dimensional system of transport equations involving physical parameters is analyzed numerically by employing the Galerkin finite element methodology (G-FEM). The simulations predicted promising enhanced heat transfer characteristics in Ternary hybrid Casson fluid. The Casson fluid parameter has shown remarkable behavior on temperature distribution. Ternary hybrid nanoparticles have increased the overall thermal conductivity of the fluid and consequently, the temperature distribution is significantly raised. Hence ternary hybridity of nano-structures in Casson fluid is shown as an effective technique in improving the thermal performance of complex fluid.

Heat transfer characteristics in non‐Newtonian fluid flow due to a naturally permeable curved surface and chemical reaction

AbstractIn this research endeavor, Casson fluid flow and melting heat transfer due to a curved nonlinearly stretching sheet are investigated. The sheet is naturally permeable and the flow is considered in a porous medium. For flow in a porous medium, a modified Darcy's resistance term for Casson fluid is considered in the momentum equation. In the energy equation, heat transport characteristics, including viscous dissipation, are taken into account. Mass transport is also studied together with the impact of chemical reaction of higher order. The governing nonlinear partial differential equations of flow, heat, and mass transport are reduced to nondimensional ordinary differential equations using adequate similarity transformations and then solved numerically employing the bvp4c technique and Runge–Kutta fourth‐order method on MATLAB. The impacts of numerous occurring parameters on relevant fields (velocity field, temperature field, and concentration field) are depicted and discussed by plotting graphs. We concluded the curvature parameter, reduces the pace of the flow. The impacts of the stretching index, and melting parameter, are also found to reduce flow and temperature field. Furthermore, we noted that the reaction parameter, and its order, exhibit opposite impacts on the concentration field. Moreover, the numerical values of skin‐friction coefficient and Nusselt number calculated employing bvp4c and Runge–Kutta fourth‐order technique are expressed in tabular mode, and these are found in an excellent match. For validation of the results, skin‐friction coefficient values were computed using the Runge–Kutta fourth‐order technique and bvp4c solver, compared with the existing results, and a good agreement was found.

Indices for ventilation efficiency and their numerical computation

The paper addresses methods to evaluate the ventilation efficiency both in regard to air exchange and to contaminant removal in a room. Extra variables are introduced via transport differential equations which are solved by Computational Fluid Dynamics (CFD). Ventilation efficiency indices are defined and computed by alternative methods with an excellent agreement. For the numerical calculations a test chamber with a fume cupboard has been modelled and simulated. The paper discusses peculiarities in the calculation of the passive scalar equations to ensure mass conservation. Guidelines for optimal locations of the exhaust grilles of ventilation systems have been proposed to maximize ventilation efficiency.

Aspects of Cattaneo-Christov heat flux in nonlinear radiative ternary, hybrid, and single mass diffusion past stretching surface; A comparative study

The main aim of this study is to analyze the convective heat transport in hydromagnetic TiO2 water nanofluid (NF), TiO2−SWCNT water hybrid nanofluid (HNF) and TiO2−SWCNT−MWCNT water ternary hybrid nanofluid flow (THNF) under the influence of Cattaneo-Christov (CC) heat flux and nonlinear thermal radiation over a stretching sheet. This study will help in environmental purification, to decompose harmful substances and other applications that require cooling. Furthermore, consideration of CC heat flux is important for a finite speed heat flow to reduce system disorder. The system of governing partial differential (PDEs) equations for the heat transport within THNF flow is transformed into the system of ordinary differential equations (ODEs) and solved numerically via the shooting technique with Runge-Kutta-4 method. The flow behaviors for Lorentz force and volume fraction of TiO2(φ1), SWCNT(φ2) and MWCNT(φ3) nanomaterials are portrayed for both no slip (α=0) and velocity slip (α=0.5) conditions. It is concluded that the velocity is more significant in the case of no slip condition when one, two and three different nanomaterials are dispersed in the base fluid water. It is further observed that the heat and momentum transport is more significant in the case when we consider TiO2−SWCNT−MWCNT water THNF flow as compared to TiO2 water NF, TiO2−SWCNT water HNF flow over a stretching surface. Our results show that the present analysis on heat transport in the THNF is more significant as compared to the THNF taken in the study given by Majunatha et al. [14].

Asymptotics of the Solution to the Cauchy Problem for a Singularly Perturbed Operator Differential Transport Equation with Weak Diffusion

Formal asymptotic expansions of the solution to the Cauchy problem for a singularly perturbed operator differential transport equation with weak diffusion and small nonlinearity are constructed in the critical case. Under certain conditions imposed on the data of the problem, an asymptotic expansion of the solution is constructed in the form of series in powers of a small parameter with coefficients depending on stretched variables. Problems for determining all terms of the asymptotic expansion are obtained. It is shown that the leading term of the solution asymptotics is determined by solving Cauchy problems for a parabolic Burgers-type equation and, under certain conditions, for a Korteweg–de Vries–Burgers type equation. The remainder terms are estimated with respect to the residual.

Asymptotics of the Solution to the Cauchy Problem for a Singularly Perturbed Operator Differential Transport Equation with Weak Diffusion

Asymptotics of the Solution to the Cauchy Problem for a Singularly Perturbed Operator Differential Transport Equation with Weak Diffusion

A turbulence modeling study of the precessing vortex core in a pipe flow

The phenomenon of precessing vortex core is observed experimentally when swirl is imparted on an axial flow in a pipe. It manifests as a coherent structure in the form of a helical vortex of regular wavelength whose axis is coincident with the pipe’s axis. The most striking consequence of this pattern of flow is the generation of periodic fluctuations in the streamwise distribution of the wall static pressure and skin friction. While the prediction of the precessing vortex has proved possible with large-eddy simulations, there is no record of this phenomenon being captured in great detail by Reynolds-averaged Navier-Stokes methods utilizing turbulence models to close the time-averaged equations. The purpose of the research reported here was to determine whether the precessing vortex core and its impact on conditions at the wall can be captured using this approach. The turbulence model used was of the Reynolds-stress transport type which involves the solution of a differential transport equation for each of the six non-zero components of the Reynolds stress tensor. Previous studies in which such models were used in the prediction of rotating and swirling flows have shown that their performance is largely determined by the way in which the difficult fluctuating pressure-strain correlations that appear in these equations are modeled. To this end, four very different alternative models for these correlations were assessed by comparisons with experimental data for a swirling pipe flow at relatively high swirl number. It was found that the models do indeed capture the precessing vortex revealing it to be in the form of an exceptionally well-defined double vortex. It was also found that the expected periodic fluctuations in static pressure and wall skin friction, not previously obtained in turbulence model studies, are captured by the present closures.

Dynamics of heat transport in flow of non‐linear Oldroyd‐B fluid subject to non‐Fourier's theory

AbstractTo study non‐Fourier heat and mass transport in the stagnation point flow of magnetized Oldroyd‐B fluid due to stretching cylinder, the Cattaneo–Christov heat flux model is used in this investigation. Further, as the controlling agents for thermal and solutal transport in the fluid flow, the heat generation/absorption source and chemical reaction are also considered. The formulations of a such physical phenomenon are going to form the PDEs. Through appropriate similarity variables, these governing partial differential equations for flow and energy transport are converted into the ordinary differential. The analytical series solutions are obtained through the use of a homotopic approach. The graphical upshots are conducted for velocity fields, temperature, and concentration distributions. In addition, energy transport analysis is performed for two kinds of surface heating mechanisms, namely the prescribed surface temperature (PST) and constant wall temperature (CWT). The outcomes of the current investigation revealed that a higher rate of heat transfer is observed in the case of CWT as compared to PST. Moreover, the increasing values of thermal and solutal relaxation time parameters reduce the heat and mass transport in the fluid flow, respectively.

Comparative Mathematical Study of Blood Flow Through Stenotic and Aneurysmatic Artery with the Presence and Absence of Blood Clots

Numerical predictions of blood flow and hemodynamic properties through stenosis and aneurysm artery have been studied in the presence of blood clots at the constricted area. The finite element method has been used to solve the partial differential equations of continuity, momentum, Oldroyd-B, and bioheat transport in cartesian coordinates systems. The present investigation carries the potential to compute blood velocity, pressure, and drag coefficients with significance at the throat of stenosis and aneurysm. The models have also been employed to study simulation, blood clots, and hemodynamic characteristics for all modifications. The impact of shearthinning on blood flow is intensified compared to the viscoelastic properties. It is found that the maximum effect of the drag coefficient is visible at the hub of stenotic for nonclotting models. The highest pressure and the lowest velocity are gained for the presence of blood clots at the constraint area. The impact of stenosis and aneurysm artery, drag coefficient, and blood clots on blood flow is the main physical outcome that may be reported in medical science to identify atherosclerosis diseases. The quantitative analysis has been completed numerically with the physiological significance of hemodynamic factors of blood flow which shows the validity of the present model.

Convective transport via thermophoresis and brownian forces in MHD Hiemenz stagnation-point flow

The heat and mass transfer in normal impingement of the Hiemenz stagnation-point flow of a viscous nanofluid on a bidirectional stretching sheet under the influence of the magnetic field is investigated. In a recent study, Weidman (Meccanica, vol. 53, p.833, 2018) has analyzed the phenomenon of the Hiemenz stagnation-point flow over the biaxially stretching sheet. Dual solutions have been found in this study for the various values of [Formula: see text] (streamwise stretching rate) and [Formula: see text] (cross-stream stretching rate). In the present work, the mechanism of thermal and solutal energy transport in the Hiemenz stagnation-point flow of nanoliquid is studied in the view of thermophoretic and Brownian forces. The solutions of similar ordinary differential equations (ODEs) for flow and energy transport in viscous nanofluid are computed with help of bvp4c numerical technique. The impact of dimensionless parameters on flow field and energy transport is presented graphically and discussed with physical reasoning. The outcomes reveal that the higher magnitude of Lorentz force declines the flow field in cross-stream and vertical directions, but conflicting behavior is noted for streamwise velocity component in the case of [Formula: see text] and [Formula: see text]. Both thermophoresis and Brownian motion of nanoparticles enhance the energy transport in Hiemenz flow of viscous fluid. The results of the present study are validated through the shear stress parameters with a previously published study.

Laser light-induced deformation of free surface of oil due to thermocapillary Marangoni phenomenon: Experiment and computational fluid dynamics simulations

Remote light-induced free liquid surface deformation has been studied in various systems for decades. One of the mechanisms able to do this task is driven by the thermocapillary Marangoni effect. The strength of the light–matter interaction, which is usually weak, here is amplified by the light absorption and heat production that changes surface tension. Here, we report on an experimental study aimed at dynamical aspects of the deformation induced under conditions of chopped laser excitation light. The light-induced deformations are usually in the range of several micrometers. Therefore, we applied the interferometric technique to measure deformation profiles in real time. Experiments were performed in the shallow bath of the rapeseed oil with an azo-dye and excited with 514.5 nm and probed with 650 nm coherent laser beams, respectively. The mechanism of deformation driven by Marangoni effect was carefully modeled in 3D by computational fluid dynamic numerical simulations within the COMSOL Multiphysics package. The adaptive mesh technique used in the simulation together with solving the time-dependent coupled Navier–Stokes and heat transport differential equations allowed us to replicate the experimental findings. A satisfactory agreement between the results of the simulations and those of the experiment in terms of the dynamics, shape, and depth of the deformation has been obtained. The toroidal-like whirls accompanying the thermocapillary Marangoni effect were identified by the simulation results. We then experimentally proved that these toroidal-like vortices, which accompany laser heating in dyed oil, formed a kind of novel hydrodynamic trap, in the center of their quiet zone, in which microcrystals can be trapped.

Stability criteria and convective mass transfer from the falling spherical drops, part II: Herschel–Bulkley fluids

AbstractThe combined effects of yield stress, shear‐thinning, and shear‐thickening fluid behaviour are investigated when a drop is falling in a Herschel–Bulkley fluid. The constitutive relation for Herschel–Bulkley fluids is regularized using the Papanastasiou regularization method. The governing partial differential equations for mass, momentum, and species transport are solved spanning a wide range of dimensionless numbers as Reynolds number, ; Schmidt number (10); Bingham number, ; viscosity ratio (0.1 and 10); and power‐law index, . The velocity field and mass transfer characteristics are expressed using streamlines, velocity contours, concentration contours, and sheared and un‐sheared regions, while the surface averaged gross engineering quantities are described as a drag coefficient, yield‐stress parameter, and Sherwood number. All else being equal, sheared regions in shear‐thinning fluids are observed to be larger with respect to the shear‐thickening fluids at finite Reynolds numbers. In the fully plastic flow limit, the yield stress effects dominate in the flow field, and therefore, the critical yield‐stress parameter is observed to be independent of shear‐thinning and shear‐thickening fluid behaviours. However, in the viscoplastic limit (finite Bingham number), shear‐thinning fluid always requires a larger value of yield stress to be static in the fluid with reference to shear‐thickening fluids. The new set of dimensionless parameters are defined based on the effective viscosity scales, and the predictive correlations are put forward for both drag coefficient and Sherwood number.

Chemical entropy generation and second-order slip condition on hydrodynamic Casson nanofluid flow embedded in a porous medium: a fast convergent method

The chemical entropy generation analysis is an approach to optimize the performance of different thermal systems by investigating the related irreversibility of the system. The influences of second-order slip with the chemical reaction on the boundary layer flow and heat transfer of a non-Newtonian nanofluid in a non-Darcian porous medium have been investigated numerically. Simultaneous solutions are presented for first and second-order velocity slips. The second-order boundary conditions serve as a closure of a system of the continuity, transport, and energy differential equations. The current work differs from the previous studies in the application of a new second-order slip velocity model. The Casson fluid model is applied to characterize the non-Newtonian fluid behavior. The effect of the second slip parameter on the present physical parameters was discussed through graphs and it was found that this type of slip is a very important one to predict the investigated physical model. The present study provides two fast convergent methods on the semi-infinite interval, namely Chebyshev collocation method and optimal homotopy analysis method are used to analyze the fluid flow, heat, and mass transport. Compared with available analytical and numerical solutions, current methods are effective, quickly converging, and with great accuracy. It was shown that the account for the second-order terms in the boundary conditions noticeably affects the fluid flow characteristics and does not influence on the heat transfer characteristics.

Temporal Spectrum of a Scattered Electromagnetic Waves in the Conductive Collision Turbulent Magnetized Plasma

Using WKB method the peculiarities of the temporal spectrum of an ordinary and extraordinary electromagnetic wave scattered in a weakly randomly inhomogeneous three-dimensional nonstationary and conductive magnetized plasma are investigated. On the basis of the stochastic differential transport equation for the frequency fluctuation the broadening and the displacement of the temporal spectrum for both waves are obtained for the polar terrestrial ionosphere. These statistical characteristics contains anisotropic parameters: velocity of a plasma stream, conductivities of the ionosphere, elongated electron density irregularities are characterized by the anisotropy factor and inclination angle with respect to the geomagnetic lines of forces. The analysis of the power spectrum of the waves as a function of the propagation distance and the nondimensional frequency parameter containing the carrier frequency and characteristic temporal scale of electron density fluctuations are carried out. Analytical and numerical calculations have shown that the terrestrial conductivity and anisotropy factors exert a substantial influence on evaluation of the temporal spectrum than the inclination angle. It was found that the wave spectrum increases initially as the square root of the propagation distance, but at large distances it approaches a limiting value. Statistical moments of a scattered ordinary and extraordinary waves do not depend on an absorption sign and valid for both absorptive and active media.