Time-dependent Boundary Conditions Research Articles

Dynamical behaviour of a logarithmically sensitive chemotaxis model under time-dependent boundary conditions

Abstract This article studies the dynamical behaviour of classical solutions of a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, with time-dependent boundary conditions. It is shown that under suitable assumptions on the boundary data, solutions starting in the $H^2$ -space exist globally in time and the differences between the solutions and their corresponding boundary data converge to zero as time goes to infinity. There is no smallness restriction on the magnitude of the initial perturbations. Moreover, numerical simulations show that the assumptions on the boundary data are necessary for the above-mentioned results to hold true. In addition, numerical results indicate that the solutions converge asymptotically to time-periodic states if the boundary data are time-periodic.

Forced flow reversal in ferrofluidic Couette flow via alternating magnetic field

Time-dependent boundary conditions are very common in natural and industrial flows and by far no exception. An example of this is the movement of a magnetic fluid forced due to temporal modulations. In this study, we used numerical methods to examine the dynamics of ferrofluidic wavy vortex flows (WVF2, with dominant azimuthal wavenumber m=2) in the counter-rotating Taylor–Couette system, which was subjected to time-periodic modulation/forcing in a spatially homogeneous magnetic field. In the absence of a magnetic field, all WVF2 states move in the opposite direction to the rotation of the inner cylinder, they are retrograde. However, when strength or frequency of the alternating magnetic field increases, the motion direction of the flow pattern changes. Thus, the alternating field provides a precise and controllable key parameter for triggering the system response and controlling the flow. Aside, we also observed intermittent behavior when one solution became unstable, leading to random transitions in both, the transition time and toward the different final solutions. Our findings suggest that, in ferrofluids, flow pattern reversal can be induced by varying a magnetic field in a controlled manner, which may have applications in the development of modern fluid devices in laboratory experiments. These findings provide a framework to study other types of magnetic flows driven by time-dependent forcing.

Moving mirrors, OTOCs and scrambling

We explore the physics of scrambling in the moving mirror models, in which a two-dimensional CFT is subjected to a time-dependent boundary condition. It is well-known that by choosing an appropriate mirror profile, one can model quantum aspects of black holes in two dimensions, ranging from Hawking radiation in an eternal black hole (for an “escaping mirror”) to the recent realization of Page curve in evaporating black holes (for a “kink mirror”). We explore a class of OTOCs in the presence of such a boundary and explicitly demonstrate the following primary aspects: First, we show that the dynamical CFT data directly affect an OTOC and maximally chaotic scrambling occurs for the escaping mirror for a large-c CFT with identity block dominance. We further show that the exponential growth of OTOC associated with the physics of scrambling yields a power-law growth in the model for evaporating black holes which demonstrates unitary dynamics in terms of a Page curve. We also demonstrate that, by tuning a parameter, one can naturally interpolate between an exponential growth associated with scrambling and a power-law growth in unitary dynamics. Our work explicitly exhibits the role of higher-point functions in CFT dynamics as well as the distinction between scrambling and Page curve. We also discuss several future possibilities based on this class of models.

On the use of fast projection methods with unsteady velocity boundary conditions

This article aims at clarifying and amending a previously published result on the use of pseudo-pressure approximations for fast incompressible Navier-Stokes solvers with time-dependent boundary conditions. Fast projection methods were proposed to speed up high-order incompressible Navier-Stokes solvers by replacing pressure projections with simple approximations. A generalized approximation theory was proposed in [1], in which the time-dependent boundary conditions were ignored. In this comment, we investigate the effect of unsteady boundary conditions on the order of accuracy and demonstrate that the pressure approximations derived in [1] hold for all RK2 integrators, while RK3 approximations are only third order for certain sets of coefficients. We show that third order is lost for other cases and shed light on why order is broken. This work serves as a reference for the treatment of unsteady boundary conditions for fast-projection methods. Numerical validation is performed on a well established channel flow problem with a variety of unsteady inlet conditions for a wide range of explicit RK2 and RK3 integrators.

Theoretical study of solidification phase change heat and mass transfer with thermal resistance and convection subjected to a time-dependent boundary condition

The solidification of phase-change materials (PCMs) is a key process that occurs commonly in materials science and metallurgy, such as in the casting of alloys and energy management systems. There is a lot of literature in this area that assumes the PCMs are in close contact with the heat source or sink. However, a non-freezing wall frequently encloses them in practical situations. This work presents a phase change problem that describes the solidification of a semi-infinite PCM with thermal resistance. We assume that time-dependent heat flux drives the solidification process. The PCM first convert into mush and then into solid, which leads to a three-region problem. The current study accounts for both conduction as well as convection heat transfer mechanisms. Unfortunately, the exact solution to such problems with time-dependent flux-type boundary conditions may not be possible. Thus, there is considerable interest in deriving the analytical solution. The space–time transformation yields the analytical solution to the problem. A numerical example of Al−Cu alloy with 5%Cu is presented to demonstrate the current study. Thermal resistance shows a pronounced impact on the temperature field. Lower thermal resistance offers faster solidification rate. It is found that as the heat transfer constant increases, the rate of propagation of solid–mush and mush–solid interfaces gets enhanced. In addition, the growth of thermal resistance is of linear nature, with variation in the value of Q. The solidified region has higher concentration than the mush region. The current study is applicable to both eutectic systems and solid solution alloys.

NUMERICAL SIMULATION OF UNSTEADY SIGNAL TRANSDUCTION DURING THE FORMATION OF INVADOPODIA USING LEVEL SET METHOD

The finger-like protrusions formed by an invasive cancer cell known as invadopodia is actively observed recently because of this formation on the cancer invasion. The signal on the interface which is stimulated upon contact between epidermal growth factor receptor and ligand, is investigated to be the start of invadopodia formation. In this research, a model of invadopodia formation with signal variable is formulated in two dimensions. The plasma membrane is assumed to be free boundary. The signal is in an unsteady state. The equation of signal is represented by heat-like equation with time-dependent boundary condition. Trigonometry types of the boundary condition which are sine and cosine function are tested to identify the most suitable boundary condition represented the free boundary. The plasma membrane is considered as zero level set function. The membrane moves by the velocity of the signal inside the cell. To handle the free boundary problem, the level set method combining features of ghost method, interpolation and extrapolation method is applied to solve the model numerically. Our result shows that the free boundary is moved at different positions and seen to move inward meaning that the boundary has shrunk. Cosine function is discovered to fit the boundary conditions since the boundary solutions are stable across the time. The computation of the signal density profiles displayed highest density on the membrane.

Efficient exponential Rosenbrock methods till order four

In a previous paper, a technique was described to avoid order reduction with exponential Rosenbrock methods when integrating initial boundary value problems with time-dependent boundary conditions. That requires to calculate some information on the boundary from the given data. In the present paper we prove that, under some assumptions on the coefficients of the method which are mainly always satisfied, no numerical differentiation is required to approximate that information in order to achieve order 4 for parabolic problems with Dirichlet boundary conditions. With Robin/Neumann ones, just numerical differentiation in time may be necessary for order 4, but none for order ≤3.Furthermore, as with this technique it is not necessary to impose any stiff order conditions, in search of efficiency, we recommend some methods of classical orders 2, 3 and 4 and we give some comparisons with several methods in the literature, with the corresponding stiff order.

Generalized magneto-thermoelasticity problem in an orthotropic hollow sphere under thermal shock

This article addresses the magneto-thermoelasticity problem within an orthotropic hollow sphere. The governing equations are formulated based on the Lord-Shulman coupled theory of thermoelasticity. Time-dependent thermal and mechanical boundary conditions are imposed on the inner and outer surfaces of the hollow sphere. The analytical solution is obtained using the finite Hankel transform. A thermal shock in the form of heat flux is prescribed on the sphere’s inner surface, and subsequently, the transient thermal response of the sphere, along with radial displacement and radial, tangential, and circumferential stresses, are derived. The influence of different magnetic field values, orthotropic material properties, and relaxation time is investigated and presented graphically. The obtained results exhibit excellent agreement with existing literature, validating the robustness of the proposed model. The findings not only contribute to the fundamental understanding of magneto-thermoelasticity but also offer practical insights for engineering applications involving orthotropic materials in complex thermal and magnetic environments. The availability of an exact solution facilitates the validation of numerical models, ensuring their accuracy and reliability in capturing the intricate behavior of orthotropic materials under coupled fields.

Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

In this paper, two-dimensional (2D) heat equations on disjoint rectangles are considered. The solutions are connected by interface Robin’s-type internal conditions. The problem has external Dirichlet boundary conditions that, in the forward (direct) formulation, are given functions. In the inverse problem formulation, the Dirichlet conditions are unknown functions, and the aim is to be reconstructed upon integral observations. Well-posedness both for direct and inverse problems is established. Using the given 2D integrals of the unknown solution on each of the domains and the specific interface boundary conditions, we reduce the 2D inverse problem to a forward heat 1D one. The resulting 1D problem is solved using the explicit Saul’yev finite difference method. Numerical test examples are discussed to illustrate the efficiency of the approach.

Solving reaction-diffusion problems with explicit Runge–Kutta exponential methods without order reduction

In this paper a technique is given to recover the classical order of the method when explicit exponential Runge–Kutta methods integrate reaction-diffusion problems. In the literature, methods of high enough stiff order for problems with vanishing boundary conditions have been constructed, but that implies restricting the coefficients and thus, the number of stages and the computational cost may significantly increase with respect to other methods without those restrictions. In contrast, the technique which is suggested here is cheaper because it just needs, for any method, to add some terms with information only on the boundaries. Moreover, time-dependent boundary conditions are directly tackled here.

Numerical modelling of thermal hysteresis in melting and solidification of phase change materials

This research focuses on exploring the impact of thermal hysteresis effects on the performance of Phase Change Materials (PCMs) used in thermal storage or buffering applications. Computational Fluid Dynamics modeling in ANSYS Fluent is employed to investigate thermal hysteresis phenomena during solid-liquid phase change. The traditional enthalpy-porosity method is extended by introducing an alternative relationship between the liquid fraction and PCM temperature, implemented in Fluent through User Defined Functions. The study demonstrates the developed model by simulating a rectangular PCM enclosure with a heated wall featuring a time-dependent temperature boundary condition. Considering convective heat transfer, a comparison is made between the newly developed hysteresis model and the default solidification-melting model of ANSYS Fluent. The new model facilitates numerical analysis of thermal hysteresis during partial and complete melting and solidification processes, enabling the study of hysteresis effects on the part-load operation of PCM systems.

Formation Rate and Energy Efficiency of Ice Plug in Pipelines Driven by the Cascade Utilization of Cold Energy

Ice plug technology is an effective method for isolating the pipeline system, which are promising methods utilized in the nuclear, chemical, and power industries. To reduce the cold energy consumption and temperature stress, the multi-stage (1–10) of time-dependent thermal boundary conditions was proposed for the formation of ice plug, while the gradient cooling wall temperature of multi-stage was applied. A numerical model considering the liquid–solid phase change, heat transfer, and time-dependent thermal boundary condition has been established. The effects of the ratio of length and diameter of the cooling wall lc/d (1–9) on the formation rate and heat flux of ice plug in the pipe has been investigated. The fastest formation rate of ice plug with 800 mm in the axial direction (7.47 cm3/s) was observed in the pipe with the lc/d of 5. The formation rate of ice plug and the ice formation volume under unit energy consumption VE under various stages (1–10) of cooling wall temperature have been compared. The VE of eight temperature stages (1.45 cm3/kJ) was 1.16 times than the VE of one temperature stage, which satisfied the freezing rate at the same time. This investigation provides insight for proposing an energy-saving system for the formation of ice plug.

A new analytical method for transient heat conduction in composite disks applied to thermal plug and abandonment of oil wells

A new hybrid method for transient heat conduction problems is developed and applied to simulate a Plug and Abandonment (P&A) operation of oil wells. An oil well is approximated by concentric disks in a one-dimensional configuration, which allows for use of polar coordinates. The application of the Separation of Variables Method (SVM) is used as the analytical framework for the solution of the conductive heat transfer arising from a volumetric heat source located in the center of the disks. The SVM is able to solve only time-independent boundary conditions. However, using the Duhamel's theorem, the solution determined with the SVM can be used to achieve the solution when both time-dependent boundary conditions and internal heat generation are prescribed. Lastly, for verification purposes, a commercial software that solves the transient temperature field by means of numerical procedures, providing reliability to the analytical method proposed in this work.

Convective thermal losses of long-term underground hot water storage

A case of underground long-term hot water storage is investigated numerically. The study is based on the unsteady two-dimensional Navier-Stokes equations in Boussinesq approximation applied to a closed cavern with time-dependent temperature boundary conditions on the walls. The problem formulated in a vorticity-stream function statement is solved by finite difference method (FDM) for high values of the Rayleigh number and for the Prandtl number of water. Streamlines, velocity and temperature fields are presented graphically for given moments of time. The evolution of the thermocline thickness in the mid-section of the cavern is slow and illustrates that the hot water zone occupies more than the half of the cavern even after 6 months period. The Nusselt number on the walls shows that the convective thermal losses are small and after certain period of time tend to decrease due to the diminished temperature difference at the walls. The influence of the fluid convection on the thermal losses is evaluated quantitatively, showing high seasonal thermal efficiency of the insulated hot water storage.

Topography optimisation using a reduced-dimensional model for transient conjugate heat transfer between fluid channels and solid plates with volumetric heat source

Consideration of transient effects is important for industrial applications of heat transfer structure optimisation studies; however, the huge computational cost associated with transient problems is a pressing concern. This paper proposes an extension of a previous reduced-dimensional model to transient conjugate heat transfer between a fluid flow and solid-heated plates in a plate heat exchanger. The extended reduced-dimensional model introduces the temperature field of the plate governed by the heat conduction equation, which is coupled to the temperature field of the fluid, governed by the convection-diffusion equation, through the heat flux balance equation at the contact surface. The model is based on assumptions of fully developed flow and constant temperature profile, reducing the three-dimensional problem to a planar problem and significantly reducing computational costs. The accuracy of the model for the simulation of transient heat transfer is verified by comparison with a three-dimensional model. In this paper, the topography of the heat exchanger plate is optimised for both steady-state and transient conditions by applying the reduced-dimensional model. The effectiveness of the optimised design was demonstrated by the cross-check of both the reduced-dimensional and full three-dimensional models. Furthermore, this work considers the effect of time-independent boundary conditions and time-dependent boundary conditions on transient optimisation. The transient and steady-state optimised designs are analysed and compared for both conditions, and the necessity of transient optimisation is discussed.

CFD Analysis of Transient Conjugated Forced Convection in Minipipes with Time-Dependent Periodical Convection Boundary Conditions

CFD Analysis of Transient Conjugated Forced Convection in Minipipes with Time-Dependent Periodical Convection Boundary Conditions

Perfect plasticity versus damage: an unstable interaction between irreversibility and Γ-convergence through variational evolutions

This paper addresses the question of the interplay between relaxation and irreversibility through quasi-static evolutions in damage mechanics, by inquiring the following question: can the quasistatic evolution of an elastic material undergoing a rate-independent process of plastic deformation be derived as the limit model of a sequence of quasi-static brittle damage evolutions? This question is motivated by the static analysis performed in [J.-F. Babadjian et al. Commun. Pure Appl. Math. 74 (2021) 1803–1854], where the authors have shown how the brittle damage model introduced by Francfort and Marigo (see [G.A. Francfort and J.-J. Marigo, Eur. J. Mech. A Solids 12 (1993) 149–189, G.A. Francfort and J.-J. Marigo, J. Mech. Phys. Solids 46 (1998) 1319–1342]) can lead to a model of Hencky perfect plasticity. Problems of damage mechanics being rather described through evolution processes, it is natural to extend this analysis to quasi-static evolutions, where the inertia is neglected.We consider the case where the medium is subjected to time-dependent boundary conditions, in the one-dimensional setting. The idea is to combine the scaling law considered in [J.-F. Babadjian, et al. Commun. Pure Appl. Math. 74 (2021) 1803–1854] with the quasi-static brittle damage evolution introduced in [G.A. Francfort and A. Garroni, Arch. Rational Mech. Anal. 182 (2006) 125–152] by Francfort and Garroni, and try to understand how the irreversibility of the damage process will be expressed in the limit evolution. Surprisingly, the interplay between relaxation and irreversibility is not stable through time evolutions. Indeed, depending on the choice of the prescribed Dirichlet boundary condition, the effective quasi-static damage evolution obtained may not be of perfect plasticity type.

Semi-analytical models for two-dimensional multispecies transport of sequentially degradation products influenced by rate-limited sorption subject to arbitrary time-dependent inlet boundary conditions

Most of the semi-analytical and analytical models employed to depict multidimensional, multispecies transport of sequentially degrading reaction products are built upon solving a set of coupled advection-dispersion equations (ADEs). Within these equations, sorption is considered as being equilibrium-controlled. However, it has been demonstrated that more realistic predictions of the transport of contaminants in the groundwater could be obtained by the use of a rate-limited sorption process instead of an equilibrium sorption assumption. This study is thus designed to devise semi-analytical models for the two-dimensional multispecies transport of a chemical mixture comprised of a parent compound and its degradation-daughter products which is influenced by rate-limited sorption subject to arbitrary time-dependent inlet boundary conditions. Three integral transforms are applied to generate a set of linear algebraic equations (AEs) resulting from the reductions of the ADEs. The contaminant concentration of each species is calculated by solving these AEs and then retransforming the solutions back to the original time-space domain. The simulation results obtained with our newly developed semi-analytical model are nearly identical to those generated using a numerical model based on the Laplace transform finite difference (LTFD) method. This confirms the validity of the new semi-analytical model. An investigation of the effect of rate-limited sorption on the migration of the contaminant plume is carried out using various time-dependent inlet boundary conditions. The sorption rate values vary from low to high, specifically, 0.05, 0.5, 5, and 50 years−1. The results show that the predicted concentrations of all contaminants within in the decay chain decrease as the sorption rate constant increases, for both constant and exponentially time-dependent boundary sources. However, under pulse loading boundary conditions, the concentrations of the later degradation products tend to increase with the sorption rate constant. The semi-analytical models created in this study, allow for different inlet boundary conditions, so can be utilized to simulate the transport of sequentially degrading reaction products. This greater accuracy makes them useful for many applications.

Analysis of Multilayer Cylindrical Thermal Conduction With a Time-Varying Convective Boundary Condition

Abstract Heat transfer in a multilayer body plays a key role in design and optimization of several engineering systems. While the analysis of simple multilayer problems is quite straightforward, realistic scenarios such as time-dependent boundary conditions result in significant complications in analysis. This work presents thermal analysis of a one-dimensional heat-generating multilayer cylinder with time-varying convective heat transfer at the boundary. Such a scenario may occur in applications such as nuclear reactors, jet impingement cooling, turbine blade heat transfer, as well as casting and related manufacturing processes. Analysis is presented for both annular and solid cylinders. A derivation for the temperature distribution is carried out, using a shifting function to split the time-dependent boundary condition into two parts, followed by appropriate mathematical substitution. For particular special cases, the analytical results derived here are shown to reduce exactly to results from past work. Good agreement of the theoretical results with numerical simulations is also demonstrated. Thermal response to various realistic time-dependent boundary conditions is analyzed. This work contributes towards the design of realistic multilayer problems and may facilitate the optimization of engineering systems where multilayer thermal conduction plays a key role.

Temperature Response of Double-Layer Steel Truss Bridge Girders

Double-layer steel truss continuous girders are prone to significant temperature stress, deviation, torsion, and warping, thus causing adverse temperature structural responses, and also affecting the safety and durability of bridge structures. This paper presents an investigation on time-dependent characteristics in the temperature field and temperature response of double-layer steel truss continuous bridge girders, fully considering the shielding effect subjected to different solar radiation angles during the high-temperature season. The time-dependent thermal boundary conditions and support conditions provided for the steel truss bridge structure were determined. Subsequently, a thermal analysis model for the entire structure of double-layer steel truss continuous girders was established to attain the temperature distribution law. The research results show that significant differences occur in the position and temperature difference of temperature gradients exhibited in the vertical, horizontal, and longitudinal directions in the double-layer steel truss bridge structure. The temperature distribution pattern within the chord section is mainly influenced by the environmental temperature and solar radiation intensity, along with the heat exchange between different panels. Thereafter, a validated temperature gradient formula for the component section has been proposed. The time-dependent laws in structural displacement, stress, and rotation angle under daily temperature cycling conditions have been revealed, thereby providing a theoretical basis for the life cycle construction and safety maintenance of double-layer steel truss structure bridges.