Transient Solution Research Articles

Hydrodynamic heat transfer in solids

In solids, hydrodynamics have been observed in both amorphous and crystalline materials under a wide range of conditions. However, its fundamental physics is still unclear. Recent advancements in nanotechnology, biomedicine and electronics have reinvigorated interest in research on hydrodynamic solids. Here, we established the hydrodynamic transport in solids from solid mechanics. We showed that isentropically, a solid becomes superfluid locally in space and time when subjected to periodic motions, and at resonance, the first and second sound are generated. We validated the speed of the second sound against the velocity of S waves, and the agreement is satisfactory. From the concept of the boundary of the second law, we established the boundary conditions for the governing equations of the supersolid and obtained the transient and steady-state solutions of the supersolid. The theory agrees well with the measured flash temperature of the bulk metallic glass (BMG), which elucidates the physics why thermocouples do not measure flash temperature accurately. The finding shall lead to the review of the adequacy of instrumentation where temperature measurement is critical. The validation of the steady-state solution was carried out by comparison to the interfacial thermal conductance (ITC) on solid-solid interfaces. The first dataset has metal-metal interfaces with variable pressures, and the second dataset has metal-dielectric interfaces with variable temperatures, whose ITC differs by 4 orders of magnitude. We discuss the physics of this difference. We believe that the theory of hydrodynamics in solids provides insights for a vast number of disciplines and that the methods and solutions can help practitioners.

New Green’s Function Method for Solving Underground Oil Seepage Problems

Abstract Compared with the variables separation method and integral-transform method, Green’s function method is much more applicable and thus of great practical significance. Based on the general steps of Green’s function method, a new method is originally raised to get Green’s function in oil and gas flow problems through underground formations. After that, the transient pressure solution is get for flow in a spherical reservoir by using this method. This case enlarged the application of Greenʼs method function, and this work can also be a good foundation for the solution of more and more complicated flow problems in oil and gas reservoirs.

Analytical evaluation of steady-state solute distribution in through-diffusion and membrane behavior test under non-perfectly flushing boundary conditions

The through-diffusion and membrane behavior testing procedure using a closed-system apparatus has been widely used for concurrent measurement of diffusion and membrane efficiency coefficients of low-permeability clay-based barrier materials. However, the common assumption of perfectly flushing conditions at the specimen boundaries could induce errors in analyses of the diffusion coefficients and membrane efficiencies. In this study, an innovative pseudo three-dimensional (3D) analytical method was proposed to evaluate solute distribution along the boundary surfaces of the soil-porous disks system, considering the non-perfectly flushing conditions. The results were consistent with numerical models under two scenarios considering different inflow/outflow positions. The proposed model has been demonstrated to be an accurate and reliable method to estimate solute distributions along the boundaries. The calculated membrane efficiency coefficient and diffusion coefficient based on the proposed analytical method are more accurate, resulting in up to 50% less relative error than the traditional approach that adopts the arithmetic mean value of the influent and effluent concentrations. The retardation factor of the clay specimen also can be calculated with a revised cumulative mass approach. Finally, the simulated transient solute transport matched with experimental data from a multi-stage through-diffusion and membrane behavior test, validating the accuracy of the proposed method.

Dynamic model and application analysis of flow field interference between marine risers based on improved shielding effect model

With the development of petroleum production engineering towards the deep sea, the collision of adjacent risers in the riser array has become a problem of great concern. In order to better reflect the motion characteristics of the riser array system under the influence of ocean currents and wake effects, and to facilitate the research on active collision prevention control, this paper improves the Blevins wake effect model. And on this basis, a two-dimensional riser array system model is established by comprehensively considering internal flow, coupling between floater and the riser, lift force and other factors. Based on the above model, the transient solution of the system is carried out. The vibration analysis and collision analysis are carried out from the perspective of time domain and frequency domain. Moreover, the influence of the initial spacing between the risers, the top tension, and the natural frequency of the risers on the vibration and collision of the system is also studied in this paper.

Droplet motion with contact-line friction: long-time asymptotics in complete wetting

We consider the thin-film equation for a class of free boundary conditions modelling friction at the contact line, as introduced by E and Ren. Our analysis focuses on formal long-time asymptotics of solutions in the perfect wetting regime. In particular, through the analysis of quasi-self-similar solutions, we characterize the profile and the spreading rate of solutions depending on the strength of friction at the contact line, as well as their (global or local) corrections, which are due to the dynamical nature of the free boundary conditions. These results are complemented with full transient numerical solutions of the free boundary problem.

Transient PDF Solution of Nonlinear Stochastic Oscillator Subjected to Modulated Gaussian White Noise

In this paper, an extended exponential-polynomial-closure (EPC) procedure is presented to investigate the transient probability density function (PDF) of stochastic oscillator under the stimulation being modulated white noise based on the conventional EPC approach. By the presented EPC procedure, an evolutionary exponential–polynomial function is defined as the PDF solution and a set of evolutionary independent base functions are introduced to span a [Formula: see text] space. By integrating the residual error with those evolutionary functions spanning the [Formula: see text] space, a group of ordinary differential equations can then be formulated. Three nonlinear oscillators are analyzed numerically and the results comparisons of various techniques validate the presented EPC procedure in solution efficiency and accuracy, particularly in the PDF solution tails which play a crucial role in reliability analysis. The transient PDF solutions acquired by the presented EPC procedure match well with Monte Carlo simulation even in the PDF solution tails, whereas the solutions obtained by the equivalent linearization are deemed to be unacceptable for the strongly nonlinear oscillators subjected to modulated stimulation. Notably, the presented EPC procedure can highly increase the computational efficiency by about 1[Formula: see text]000 times in all the presented numerical examples in comparison to Monte Carlo simulation.

Prediction of oxygen distribution in the waste mass from an aeration well in bioreactor landfills.

The key to upgrade the efficiency of aerobic remediation of landfills is to determine the distribution characteristics of oxygen concentration during aerobic ventilation. This study discusses the distribution law of oxygen concentration with time and radial distance based on a single-well aeration test at an old landfill site. The transient analytical solution of the radial oxygen concentration distribution was deduced using the gas continuity equation and approximation of calculus and logarithmic functions. Oxygen concentration data from the field monitoring were compared with the results predicted by the analytical solution. The results indicated that the oxygen concentration initially increased and then decreased with prolonged aeration time. With an increase in radial distance, the oxygen concentration rapidly declined, followed by a gradual decrease. The influence radius of the aeration well increased slightly when the aeration pressure increased from 2 to 20 kPa. The field test data agreed with the analytical solution prediction results, preliminarily verifying the reliability of the oxygen concentration prediction model. Results from this study provide a basis of guidelines for the design, operation and maintenance management of a landfill aerobic restoration project.

Lack of robustness and accuracy of many numerical schemes for phase-field simulations

In this paper, we study the stability, accuracy and convergence behavior of various numerical schemes for phase-field modeling through a simple ODE model. Both theoretical analysis and numerical experiments are carried out on this ODE model to demonstrate the limitation of most numerical schemes that have been used in practice. One main conclusion is that the first-order fully implicit scheme is the only robust algorithm for phase-field simulations while all other schemes (that have been analyzed) may have convergence issue if the time step size is not exceedingly small. More specifically, by rigorous analysis in most cases, we have the following conclusions: (i) The first-order fully implicit scheme converges to the correct steady state solution for all time step sizes. In the case of multiple solutions, one of the solution branches always converges to the correct steady state solution. (ii) The first-order convex splitting scheme, which is equivalent to the first-order fully implicit scheme with a different time scaling, always converges to the correct steady state solution but may seriously lack numerical accuracy for transient solutions. (iii) For the second-order fully implicit and convex splitting schemes, for any time step size [Formula: see text], there exists an initial condition [Formula: see text], with [Formula: see text], such that the numerical solution converges to the wrong steady state solution. (iv) For [Formula: see text], all second-order schemes studied in this paper converge to the correct steady state solution although severe numerical oscillations occur for most of them if the time step size is not sufficiently small. (v) An unconditionally energy-stable scheme (such as the modified Crank–Nicolson scheme) is not necessarily better than a conditionally energy-stable scheme (such as the Crank–Nicolson scheme). Most, if not all, of the above conclusions are expected to be true for more general Allen–Cahn and other phase-field models.

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

The dynamic characteristics of the fractional Duffing system with nonlinear time delay term are studied according to the average method and the harmonic balance method. First, the steady‐state solutions of the fractional Duffing system with a nonlinear time delay term are obtained by applying the average method and the harmonic balance method. The analytical expressions of the amplitude–frequency characteristic curves under the average method and the harmonic balance method are also established. Second, the stability criterion of the system is obtained according to the indirect method of studying the motion stability problem, and the concept of the stability condition parameter is proposed in this process. Third, the transient solution of the fractional Duffing system with a nonlinear time delay term is obtained utilizing the average method, and the approximate analytical solution of the fractional Duffing system with nonlinear time delay term is obtained. Finally, the amplitude–frequency characteristic curves of the different fractional differential terms, coefficients of different fractional differential terms, magnitude of different time delay quantities, and feedback strength of different delay are analyzed by numerical simulation. Furthermore, the amplitude–frequency characteristic curves obtained by the average method and the harmonic balance method under different order of fractional differential term are compared and analyzed by numerical simulation. The time sequence and phase diagrams of the system under different order of fractional differential term and different time delay quantities are analyzed by numerical simulation. In addition, the correctness of the analytical analysis is verified by comparing the analytical results with the numerical simulation results under the average method and the harmonic balance method, respectively, by numerical simulation.

Transient radiative transfer in two graded – index slab adhered by an infinitely thin vacuum via the lattice Boltzmann method

The lattice Boltzmann method is presented to solve the radiative transport of both diffuse and collimated problems in inhomogeneous and semitransparent two – layered slab with graded refractive indices. The two layers are bounded by Fresnel's walls and separated by an infinitely thin vacuum interlayer with negligible thickness to represent the case with surface roughness or assembling of two layers using a transparent binder or intumescent resin. The high – order discrete ordinates is used to simulate the angular redistribution due to variable refractive indices and multiple reflections at the boundaries and interface. The accuracy and efficiency of the methodology is firstly tested for transient and steady state solution in two – layered slab with constant or variable refractive indices. Effects of the distributions of refractive indices and surrounding, scattering albedos, and the optical thicknesses on the transient radiative signals are examined. Calculation results show that, these parameters have a significant influence on radiative transfer and the angular intensities at different few times levels are presented to illustrate the effects of the interface characteristics, and the discontinuous nature of the intensity at the interface is discussed. The developed LBM is a promising, efficient and simple solver to perform the radiative transport in inhomogeneous layered media with graded – index functions.

Pressure transient behaviors for horizontal wells with well interferences, complex fractures and two-phase flow

During the production of multi-stage, multi-well pads in shale reservoirs, there are common phenomena of well interferences, complex fracture generation, and multi-phase flow for the multi-well pad. However, there still lacks an understanding on the pressure transient behaviors with those phenomena. To narrow this gap, the pressure transient behaviors of oil-water two-phase flow of fractured horizontal wells with complex fractures and well interferences are studied by a numerical pad-well model, based on PEBI grids and discrete fracture model (DFM).The pressure transient solutions of fractured horizontal wells with complex fractures and well interferences have been demonstrated by model verifications and field applications. The results show that the flow regime of interference effects is stronger when the child well has a larger production rate, a smaller well spacing, more secondary fractures, longer secondary fractures, and higher fracture conductivity. It is also found that the formation pressure loss increases with increasing oil phase viscosity and oil volume factor when the viscosity of water is constant. The oil/water volume factor mainly affect the well testing interpretation of parameters such as wellbore reservoir coefficient. What's more, the field application shows that the well testing data are interpreted and the fracture parameters are evaluated successfully with well interferences, while the single-well testing model without considering well interference cannot match with field data at the late stage and the estimation errors will occur. This work is meaningful to understand the well interference and to evaluate the fracture parameters in multi-stage, multi-well pad with multi-phase flow.

Semi-analytical solution of single-well push–pull test under transient flow conditions

The single-well push–pull (SWPP) test has been extensively used in parameter estimation models. Numerous analytical solutions to the problem are available, all of which consider solute transport in a non-uniform flow field from steady radial flow created by the SWPP test. However, the non-uniform flow velocity is based on the Thiem equation and varies in a spatial manner, rather than a temporal manner. No analytical solution has been previously described for the solute transport equation of SWPP under fully transient flow. In this study, the generalized integral transform technique (GITT) was used to develop a new semi-analytical solution for solute transport in the SWPP test under fully transient flow. Four phases of the SWPP test were included: injection, chaser, rest, and extraction. With the proposed solution, the differences between a transient flow SWPP model solution and a piecewise steady-state flow SWPP model solution were non-negligible; such differences increased with decreases in the dimensionless parameter related to aquifer flow properties, which is proportional to transmissivity, aquifer thickness, and porosity, but inversely proportional to the storage coefficient and pumping rate. Additionally, long tails of BTCs are characteristic of type curves under the transient flow model. The long tails of BTCs under transient flow could result in overestimation of dispersivity if the SWPP model uses piecewise steady-state flow, rather than transient flow, for parameter estimation. The proposed semi-analytical solution provides a useful tool for curve-fitting during parameter estimation when the effects of transient flow are significant. The proposed solution can also serve as a performance comparison for testing numerical solutions to the SWPP test.

An Improved Load Distribution Model for Gear Transmission in Thermal Elastohydrodynamic Lubrication

The gear drive generally operates in elastohydrodynamic lubrication (EHL) contacts, and the existence of oil film effectively reduces wear and improves transmission stability. However, little research has been devoted to studying the effect of lubrication characteristics on load distribution of gear transmissions. In order to investigate the coupling effect between the lubrication behavior and load distribution, an analytical load distribution model suitable for EHL contact spur gear pairs is proposed. The non-Newtonian transient thermal EHL solution, flexibility of meshing teeth, structural coupling deformation of the gear body and extended tooth contact are considered in the deformation compatibility condition for iteratively solving the load distribution. A parametric analysis is performed to determine the influence of load torque and rotation speed on load sharing ratio and loaded static transmission error. The transient lubrication behaviors based on the proposed load distribution model is compared with that obtained from the traditional model. A series of comparisons with different models demonstrated the correctness, significance and generality of the present model. The results show that it is necessary to consider the thermal EHL calculation into the iterative solution procedure of load distribution model for EHL contact gear pairs. The proposed model is a useful supplement for an accurate study of thermal EHL characteristics of gear transmissions.

Three-dimensional mesoscopic simulation of radiative transfer in graded index media

Light propagation in three-dimensional (3D) graded index (GRIN) media has a complex spatial path, making it difficult to solve thermal radiative transfer problems. In this paper, a mesoscopic lattice Boltzmann method (LBM) that implements simple steady-state and transient radiative transfer solutions in 3D GRIN media is proposed. During the construction of the lattice Boltzmann (LB) model, the concept of the “local radiative transfer direction”, which addresses complex curved trajectories in 3D GRIN media simply, is introduced. The radiative transfer equation of 3D GRIN media can be strictly recovered from the LB model by adopting the Chapman-Enskog analysis. Numerical results indicate that radiative transfer problems in 3D GRIN media can be solved effectively by the LBM. Additionally, the influences of different optical parameters on steady-state and transient radiative transfer in 3D GRIN media are discussed. The LBM is a promising mathematical tool for radiative transfer in 3D GRIN media.

A fitness trade-off explains the early fate of yeast aneuploids with chromosome gains

The early development of aneuploidy from an accidental chromosome missegregation shows contrasting effects. On the one hand, it is associated with significant cellular stress and decreased fitness. On the other hand, it often carries a beneficial effect and provides a quick (but typically transient) solution to external stress. These apparently controversial trends emerge in several experimental contexts, particularly in the presence of duplicated chromosomes. However, we lack a mathematical evolutionary modeling framework that comprehensively captures these trends from the mutational dynamics and the trade-offs involved in the early stages of aneuploidy. Here, focusing on chromosome gains, we address this point by introducing a fitness model where a fitness cost of chromosome duplications is contrasted by a fitness advantage from the dosage of specific genes. The model successfully captures the experimentally measured probability of emergence of extra chromosomes in a laboratory evolution setup. Additionally, using phenotypic data collected in rich media, we explored the fitness landscape, finding evidence supporting the existence of a per-gene cost of extra chromosomes. Finally, we show that the substitution dynamics of our model, evaluated in the empirical fitness landscape, explains the relative abundance of duplicated chromosomes observed in yeast population genomics data. These findings lay a firm framework for the understanding of the establishment of newly duplicated chromosomes, providing testable quantitative predictions for future observations.

Effects of thermal dispersion on transient free convective flow in vertical porous channel filled with porous material

In this work, the effect of suction/injection on transient free convective flow in vertical porous (suction/injection on the channel surfaces) channel filled with porous material in the presence of thermal dispersion was studied. The Boussinesq assumption is applied and the nonlinear governing equations of motion and energy are developed. The time dependent problem is solved using implicit finite difference method while steady state problem is solved by perturbation technique method. The solution obtained is graphically represented and the effects of suction/injection, time, Darcy number, thermal dispersion, and Prandtl number on the fluid flow and heat transfer characteristics. During the course of computation, an excellent agreement was found between the well-known steady state solutions sand transient solutions at large value of time. Furthermore, the time required to reach steady state velocity and temperature field strongly dependent on suction/injection parameter, Prandtl number and thermal dispersion parameter. The introduction of suction/injection has distorted the symmetric nature of the flow formation.

Contractivity-based variable gain dynamic motion control for a laser beam steering system: Synthesis and performance analysis

This paper deals with the synthesis and experimental performance evaluation of a contractivity-based nonlinear dynamic motion control scheme for a Laser-Beam Steering (LBS) system, which includes a saturated integral action and a variable gain. The variable gain, in the control law, is used to discriminate between “signal” and “noise” in the velocity measurements, allowing to do a trade-off between the low-frequency tracking and disturbance rejection properties and high-frequency measurement noise amplification, an effect known as waterbed effect. Then, the contractivity-based framework handles the stabilization problem together with the closed-loop performance, allowing one to generalize key properties of linear control systems to analyze transient and steady-state solutions performances in the nonlinear case. The proposed control scheme is evaluated on an experimental platform for the set-point regulation and trajectory tracking problems under different scenarios. Moreover, the effectiveness of the proposed control scheme is compared with linear controllers for the LBS system available in the literature.

Acoustically driven translation of a single bubble in pulsed traveling ultrasonic waves

The acoustic radiation force has been proven as an effective mechanism for displacing particles and bubbles, but it has been mainly applied in a standing wave mode in microfluidics. Alternatively, the use of pulsed traveling acoustic waves could enable new options, but its transient dynamic, which entails the additional complexities of pulse timing, reflections, and the type of waveform, has not yet been fully investigated. To better understand these transient effects, a transient numerical solution and an experimental testbed were developed to gain insights into the displacement of microbubbles when exposed to on- and off-periods of pulsed traveling waves. In this study, a practical sinusoid tone burst excitation at a driving frequency of 0.5 MHz is investigated. Our numerical and experimental results were found to be in good agreement, with only a 13% deviation in the acoustically driven velocity. With greater detail from the numerical solution at a sampling rate of 1 GHz, the fundamental mechanism for the bubble translation was revealed. It was found that the added mass force, gained through the on-period of the pulse, continued to drive the bubble throughout the off-period, enabling a large total displacement, even in the case of low duty-cycle (2%) pulsing. In addition, the results showed greater translational velocity is possible with a lower number of cycles for the same input acoustic energy (constant duty cycle and acoustic pressure amplitude). Overall, this study proposes a new, practical, and scalable approach for the acoustic manipulation of microbubbles for scientific, biomedical, and industrial applications.

Electro-Thermo-Convection of a Dielectric Liquid in the External DC and AC Electric Fields

The electro-thermo-convection of a dielectric liquid in a horizontal capacitor is investigated under the autonomous charge injection from the cathode and heating from above. In the case of a DC electric field, the linear stability analysis is carried out, and the thresholds of monotonic and oscillatory instability are determined. The finite difference method is used for the numerical simulation of the nonlinear behavior of electro-thermo-convective patterns: stationary convection and traveling waves. In the case of AC, electric field transient and permanent oscillations are analyzed. Two types of stable solutions are found. The modulated traveling waves are characterized by the quasiperiodic oscillations of convective characteristics. Another solution is modulated electroconvection (MEC). The patterns of MEC oscillate around some average flow synchronously with the external AC field and do not move laterally. The average intensity of convective mixing in modulated traveling waves is several times less than in modulated electroconvection. The spatiotemporal evolution of the stream function, temperature, and charge distributions for different types of transient and permanent solutions are analyzed.

Effects of Soil Type and Thermal Boundary on Predicting Temperature Profiles and Groundwater Fluxes.

In the last few years, several articles have studied heat as a groundwater tracer and developed analytical geothermal solutions to predict the subsurface temperature and groundwater fluxes. These solutions can be sorted into steady-state and transient solutions. The steady-state solutions cannot describe the time-varying subsurface temperature, while the transient solutions ignore subsurface thermal boundary effects. Moreover, soil type may be another crucial factor significantly affecting the prediction results. This study compares six existing classical analytical solutions to examine the effects of soil types and subsurface thermal boundaries on simulating temperature-depth profiles and estimating groundwater fluxes. Several synthetic cases are built by considering the common soil types, sand and clay, to demonstrate their effects on predicting the profiles. A field case is used to show the effect of subsurface thermal boundaries on the groundwater flux estimated by an inverse approach. The study results indicate that the soil types have significant influences on simulating the profiles, and the influences grow with time. Some existing solutions may give inaccurate estimations of the field groundwater flux since they merely consider the heat source from the temperature variations on the ground surface but ignore possible thermal boundary effects in the subsurface. These findings will be valuable to those applying heat as a tracer to investigate infiltration.