Unsteady Equations Research Articles

Determination of the Thermal Conductivity and Volumetric Heat Capacity of Substance from Heat Flux

The study of nonlinear problems related to heat transfer in a substance is of great practical important. Earlier, this paper’s authors proposed an effective algorithm for determining the volumetric heat capacity and thermal conductivity of a substance based on experimental observations of the dynamics of the temperature field in the object. In this paper, the problem of simultaneous identification of temperature-dependent volumetric heat capacity and thermal conductivity of the substance under study from the heat flux at the boundary of the domain is investigated. The consideration is based on the first boundary value problem for a one-dimensional unsteady heat equation. The coefficient inverse problem under consideration is reduced to a variational problem, which is solved by gradient methods based on the application of fast automatic differentiation. The uniqueness of the solution of the inverse problem is investigated.

A reduced-dimension method of Crank-Nicolson finite element solution coefficient vectors for the unsteady Burgers equation

This paper primarily focuses on the dimensionality reduction of finite element (FE) solution coefficient vectors for the unsteady Burgers equation, solved using the Crank-Nicolson FE (CNFE) method. The proper orthogonal decomposition (POD) basis is constructed from the snapshot matrix, which is formed using the first L solutions, where L is significantly smaller than the total number of time steps N of the CNFE method. By reconstructing the matrix form of the CNFE method, a reduced-dimension Crank-Nicolson finite element (RDCNFE) method is proposed and stability analysis and error estimates are discussed. Numerical tests are implemented to verify the theoretical results and demonstrate the high efficiency of the RDCNFE method.

Numerical Solution of Burgers Equation Using Finite Difference Methods: Analysis of Shock Waves in Aircraft Dynamics

In this research, the Lax, the Upwind, and the MacCormack finite difference methods are applied to the experimental solving of the one-dimensional (1D) unsteady Burger's Equation, a Hyperbolic Partial Differential Equation. These three numerical analysis-solving methods are implemented for accurate modeling of shock wave behavior high-speed flows that are necessary for aerospace engineering design. This research analysis proves that the MacCormack technique is the one that treats the differential equations with second-order accuracy. This method is quite preferred when it comes to numerical simulations because of its advanced level of accuracy. Although the Upwind and Lax methods are slightly less accurate, they show the development of shock waves that give visualizations to better understand the flow dynamics. Also, in this study, the impact of varying viscosity coefficients on fluid flow characteristics by using the lax (a numerical method for solving the viscous Burgers equation) is investigated. This identification of the phenomenon sheds light on the behavior of boundary layers, which, in turn, can be used to improve the design of high-speed vehicles and lead to a greater understanding of the area of ​​fluid dynamics.

Nonlinear evolution of vortical disturbances entrained in the entrance region of a circular pipe

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a circular pipe is investigated using asymptotic and numerical methods. Attention is focused on the low-frequency disturbances that induce streamwise elongated structures. A pair of vortical modes with opposite azimuthal wavenumbers is used to model the free-stream disturbances. Their amplitude is assumed to be intense enough for nonlinear interactions to occur inside the pipe. The formation and evolution of the perturbation flow are described by the nonlinear unsteady boundary-region equations in the cylindrical coordinate system, derived and solved herein for the first time. Matched asymptotic expansions are employed to construct appropriate initial conditions and the initial–boundary value problem is solved numerically by a marching procedure in the streamwise direction. Numerical results show the stabilising effect of nonlinearity on the intense algebraic growth of the disturbances and an increase of the wall-shear stress due to the nonlinear interactions. A parametric study is carried out to evince the effect of the Reynolds number, the streamwise and azimuthal wavelengths, and the radial length scale of the inlet disturbance on the nonlinear flow evolution. Elongated pipe-entrance nonlinear structures (EPENS) occupying the whole pipe cross-section are discovered. EPENS with $h$ -fold rotational symmetry comprise $h$ high-speed streaks positioned near the wall, and $h$ low-speed streaks centred around the pipe core. These distinct structures display a striking resemblance to nonlinear travelling waves found numerically and observed experimentally in fully developed pipe flow. Good agreement of our mean-flow and root mean square data with experimental measurements is obtained.

Numerical estimation of wave breaking forces on seawalls

ABSTRACT This paper presents a computationally efficient numerical scheme to estimate wave forces from plunging breakers on vertical rigid seawalls. The approximated velocity equation, an asymptotical solution of the Green-Naghdi equation in the Camassa-Holm scaling, is utilized to obtain the evolution of the free surface and the depth-averaged horizontal velocity over time, leading to the formation of a plunging breaker. The uniform dynamic pressure along the depth is obtained by utilizing the unsteady Bernoulli equation. The wave forces on seawalls are then determined using Froude-Krylov theory, with adjustments for diffraction effects. The results, while being comparable to some existing formulations, notably differ with some others. This highlights the limitations imposed by the inherent assumptions in the existing empirical formulations and underlines the importance of the proposed methodology that is founded on an accurate modelling of the wave breaking process to estimate wave breaking forces on seawalls.

Source term estimation of a time-varying source around a building based on Bayesian inference and unsteady adjoint equations

In actual pollutant dispersion accidents, the location of the source is typically concealed and the intensity of the source varies with time. It is important to accurately estimate source parameters based on limited sensor data. However, previous studies were based on the assumption of stabilized sources and concentration fields, and ignored the process of sensor concentration changes over time, which affects the accuracy of the estimation. Therefore, this study applied a source term estimation (STE) method which combines the Bayesian inference method with unsteady adjoint equations to a time-varying source around building. The influences of the release forms, locations, and heights of the source were analyzed from the flow field and transient stage perspectives. We found that the estimation of the time-varying source performed worse than that of the constant source assumed in existing studies. The uncertainty of the estimated results increased with the complexity of the release forms of the source. In particular, the estimation of the location and strength of the period source had a wider probability distribution, higher uncertainty, and was more susceptible to changes in source location and height. The results showed that for time-varying sources, the estimated results fluctuated strongly over time with the pre-developmental and stabilization phases, and it was critical to estimate the source term based on sensor data at various time points.

Studies on drug delivery from dissolving microneedle and its uptake in tissue

Abstract In this study, a mathematical model for transdermal drug administration using a dissolving conical microneedle (MN) is developed. The drug is absorbed in the blood compartment after passing through the skin’s viable epidermis. Reversible reaction kinetics then allow the drug to enter the tissue compartment. An unsteady reaction equation is used to predict the drug’s distribution into the skin following MN’s dissolution. An unsteady reaction mechanism is used to determine the concentrations in the blood and tissue compartments. Using the fourth-order Runge–Kutta method, the governing equations with their initial conditions are solved numerically. The MN is expected to dissolve completely in 108.1 s, according to the predicted results, while the concentration in the skin peaks at 94.4 s. The concentrations in the tissue and blood compartments, however, reach their respective maximal amounts later on. In this investigation, the impacts of the drug’s mass fraction, volume fraction, and penetration rate cannot be completely ruled out. A thorough sensitivity analysis has been carried out for some of the parameters involved. Our findings closely align with the findings reported in the literature.

A generalized mathematical model for the damped free motion of a liquid column in a vertical U-tube

A one-dimensional mathematical model is developed to analyze the unsteady characteristics of arbitrary damped free motion, denoted as Z(τ), in a rigid, constant-diameter vertical column of a U-tube filled with an incompressible Newtonian liquid, where τ represents time. This model utilizes a simplified unsteady momentum equation derived from the Navier–Stokes equations in a circular coordinate system. Moreover, it incorporates assumptions about the periodicity of arbitrary damped free oscillations and employs the Fourier series representation to characterize the damped free motion. The combination of assumptions made for periodicity, the simplified momentum equation, and the Fourier series representation makes the current mathematical model unique and novel compared to prevailing models in the literature. In this model, the governing partial differential equation contains two dependent variables: Z(τ) is the known variable, as one can measure from experiments, and the instantaneous velocity uz is the unknown variable. Fitting the experimental data into the Fourier series provides the Fourier coefficients associated with the specific experiment. The Laplace transform method is used to determine the analytical solution for uz corresponding to the known Z(τ). The analytical expressions for instantaneous flow characteristics of practical importance, including area-averaged velocity, wall shear stress, and acceleration/deceleration, are deduced from uz. The analytical solutions presented are valid for generalized unsteady motions, including underdamped oscillations with varying amplitudes and periods, underdamped oscillations with varying amplitudes and constant period, and overdamped motion that does not exhibit a single oscillation. The findings from the present model offer insights for formulating a new friction model.

The continuous adjoint to the incompressible (D)DES Spalart-Allmaras turbulence models

This article formulates the continuous adjoint method for the gradient-based shape optimization of fluid flows governed by the incompressible Detached Eddy Simulation (DES) and Delayed-DES (DDES) models, based on the Spalart-Allmaras turbulence model. As both flow models are inherently unsteady, challenges arise regarding the availability of flow fields during the backward in time integration of the unsteady adjoint equations. To minimize both the computational cost and the memory demands, the computed flow fields are compressed using the iPGDZ+ lossy compression technique, recently developed by the authors. Its application in the context of turbulence-resolving flows, where the compression of flow fields poses an intricate challenge, is a second original contribution of this article. Everything is implemented as an extension to the publicly available adjointOptimisation library of OpenFOAM, which is used to solve the flow and adjoint equations and conduct the optimization. Using two shape optimization problems in external aerodynamics, it is demonstrated that including the adjoint to the turbulence model equation is crucial for the computation of accurate sensitivity derivatives. In contrast to sensitivities computed under the “frozen turbulence” assumption, which neglects variations in turbulent viscosity due to changes in the design variables, the proposed adjoint method yields sensitivities that align with those obtained using Finite Differences. This is due to the Think-Discrete Do-Continuous adjoint method which, inspired by hand-differentiated discrete adjoint, gives rise to consistent discretization schemes of the terms involved in the equations derived by continuous adjoint. Furthermore, it is demonstrated that the proposed adjoint method can significantly benefit from the iPGDZ+ algorithm, by reducing memory requirements by more than two orders of magnitude, eliminating the need for flow recomputations, while maintaining the accuracy of the computed derivatives. Ways to handle large integration windows of the objective function with this type of flow models are beyond the scope of this article.

The electrokinetic energy conversion analysis of viscoelastic Maxwell nanofluids with couple stress in circular microchannels

The present study focuses on the unsteady flow of a viscoelastic Maxwell nanofluid with couple stress in a circular microchannel under the combined action of periodic pressure and magnetic field. The Green's function method is applied to the unsteady Cauchy momentum equation to derive the velocity field. We strive to check out the analytical solutions of the current model by validating them with existing results. In addition, the effects of several dimensionless parameters (such as the couple stress parameter γ, the Deborah number De, and the dimensionless frequency ω) on the streaming potential and the electrokinetic energy conversion (EKEC) efficiency of the three waveforms (cosine, square, and triangular) are portrayed via graphical illustrations. Within the range of parameters chosen in this study, the results demonstrate that increasing the value of the Deborah number or decreasing the dimensionless frequency can effectively enhance the streaming potential. The velocity field and EKEC efficiency are improved with increasing couple stress parameters. Furthermore, our investigation is extended to compare the EKEC efficiency for square and triangular waveforms when the couple stress parameters are set to 20 and 40, respectively. The results also indicate that increasing the couple stress parameter significantly improves the EKEC efficiency, particularly in the context of the square waveform. These findings will provide valuable assistance in the design of periodic pressure-driven microfluidic devices.

Flow of water out of a funnel

The unsteady Bernoulli equation is used to numerically determine the surface height and velocity distribution of water flowing out of a conical tube as a function of time. The speed is found to interpolate between freefall for a cylindrical pipe of constant radius and Torricelli’s law for a funnel having a small exit hole. In addition, the applied force needed to hold the conical vessel in place is calculated using Newton’s second law including the rocket thrust due to the water flowing out of the funnel. A comparison is made with the analogous expressions for the flow through and holding force on a right cylindrical tank having a hole in its bottom face. The level of presentation is appropriate for an undergraduate calculus-based physics course in mechanics that includes a module on fluid dynamics.

Steady and transient analysis on thermal hydraulic characteristic of helical coiled once-through steam generator of liquid metal fast reactor

A transient model for thermal–hydraulic characteristics of helical coiled once-through steam generator (HCOTSG) of liquid metal fast reactor was established based on global discrete grid method. The model meshed the liquid metal circuit and water-steam circuit. Meanwhile, unsteady heat conduction equations were built to accurately simulate coupled heat transfer between two sides. Four-equation drift-flux method was adopt for water-steam flow. Taking lead–bismuth fast reactor as example, thermal–hydraulic characteristics of HCOTSG were first calculated under steady conditions. It was found that heat flux distribution along steam generator is extremely uneven, and the difference between maximum and minimum wall heat flux reaches hundreds of times. Then, the transient response was simulated when inlet conditions of primary side were perturbed by step changes, and it was found that maximum wall heat flux increases from 1175.5 kW/m2 to 1365 kW/m2, increasing by nearly 16 %, when inlet lead–bismuth temperature step increases from 450°C to 480 °C.

Interfacial physico-chemical equilibrium control of lotus-type pore formation in solid

This study presents a challenging analysis of interfacial equilibrium conditions that control the evolution of lotus-type pores in both metals and nonmetals during solidification. It incorporates Henry’s or Sieverts’ law, affecting solute transfer at the cap and top free surface, and pore evolution. The significance of the directional and lightweight characteristics of lotus-type porous materials makes them vitally important in functional heat sinks, energy absorption, biomedical devices, and other applications. The study extends previous solute transfer models based on solute concentration deviations in the liquid from the top surface and convection-affected segregation at the advancing liquid–solid interface by further considering the effects of interfacial equilibrium conditions on pore development. Typical data selected for the dimensionless Henry’s law constant at the cap and top free surface is 0.175, while the Sieverts’ law constant at the cap and top free surface is 0.03. MATLAB Simulink and Simscape (version R2020b) with the solver ode113 are utilized to solve the resulting simultaneous system of unsteady first-order differential equations. The results show that the size of lotus-type pores increases as the Henry’s law constant at the cap decreases while the Henry’s law constant at the top free surface increases. Similar results are observed for Sieverts’ law. Lotus-type pores readily form as the Henry’s law constant at the cap increases while that at the top free surface decreases. The lotus pore length can also be determined and interpreted algebraically using solute content conservation. The model’s predictions closely match analytical findings previously validated by experimental data

Similarity solutions of a class of unsteady laminar boundary layer

The study of laminar unsteady boundary layer flows is essential for understanding the transition from laminar to turbulent flow, as well as the origins of turbulence. However, finding solutions to this phenomenon poses significant challenges. In this study, we introduce a novel method that employs a similarity transformation to convert the two-dimensional unsteady laminar boundary layer equations into a single partial differential equation with constant coefficients. By applying this transformation, we successfully derive similarity solutions for flat plate boundary layer flow, expressed in terms of Kummer functions. For convergent boundary layer flow, we derive an approximate analytical solution that includes both shock wave and soliton wave solutions. The superposition of these solutions provides evidence for the existence of solitons or soliton-like coherent structures (SCS) within boundary layers. Additionally, this paper explores two- and three-dimensional laminar flows, as well as three-dimensional turbulent flow equations, revealing that they all incorporate third-order derivatives with respect to spatial coordinates. This finding suggests that all viscous fluid motions have the potential to exhibit solitons/like coherent structures (SCS).

Computational Analysis of MHD Nanofluid Flow Across a Heated Square Cylinder with Heat Transfer and Entropy Generation

Abstract The mixed convection heat transfer of nanofluid flow in a heated square cylinder under the influence of a magnetic field is considered in this paper. ANSYS FLUENT computational fluid dynamics (CFD) software with a finite volume approach is used to solve unsteady two-dimensional Navier-Stokes and energy equations. The numerical solutions for velocity, thermal conductivity, temperature, Nusselt number and the effect of the parameters have been obtained; the intensity of the magnetic field, Richardson number, nanoparticle volume fraction, magnetic field parameter and nanoparticle diameter have also been investigated. The results indicate that as the dimensions of nanoparticles decrease, there is an observed augmentation in heat transfer rates from the square cylinder for a fixed volume concentration. This increment in heat transfer rate becomes approximately 2.5%–5% when nanoparticle size decreases from 100 nm to 30 nm for various particle volume fractions. Moreover, the magnitude of the Nusselt number enhances with the increase in magnetic field intensity and has the opposite impact on the Richardson number. The findings of the present study bear substantial implications for diverse applications, particularly in the realm of thermal management systems, where optimising heat transfer is crucial for enhancing the efficiency of electronic devices, cooling systems and other technological advancements.

Heat and mass transfer conduct in an unsteady two- dimensional stream between parallel sheets

In the present research, an effort has been made to analytically solve heat and mass linear/ nonlinear as well as steady/ unsteady equations in a viscous nanofluid squeezed between parallel sheets. Using Python and the SymPy library, the nanofluid with viscous properties between parallel sheets has been analyzed to symbolically solve flow, heat, and mass transfer effects equations through the Homotopy Perturbation Method and Akbari-Ganji Method approaches. The two nanofluids selected to conduct this study are Copper as well as Al2O3, whose sizes are 29 nm and 47 nm respectively. The provided details encompass the outcomes of active variables on flow and the transfer of heat coupled with mass. The Homotopy Perturbation and Akbari-Ganji methods have resulted in top-of-the-line consequences compared to analytical and numerical approaches. This research study highlights a faster and more accurate computation to conduct the analytic section of the study. The outcome shows that the increase of the Prandtl number and the Eckert number will increase Nusselt. However, skin friction increases with the increase in the Schmidt number. Furthermore, a rise in Schmidt number and parameters related to chemical reactions leads to an elevated Sherwood number. The outcomes of the study presented here provide a more innovative and precise insight, and the comparison with the available literature also proves there is a well-agreed numerical calculation. Microchips in engineering and medical-related industries would enjoy the outcomes obtained from this study. This study proves that the maximum and minimum amounts of heat transfer in respect occur at η=0 and η=1. Moreover, the maximum and minimum amounts of error are equal to 0.0001 and 0.00001, respectively. The maximum and minimum amounts of concentration occur at η=1 and η=0 in order. Finally, the maximum and minimum amounts of error are equal to 0.000016 and 0.000002, respectively.

Transient study on heat transfer and flow performance of alkali metal heat pipe in space nuclear powered heat pipe radiator

The non-direct contact radiator has a simple structure and high reliability, making it more suitable for long-life nuclear powered spacecraft. This article is based on Fortran language and innovatively combines molecular dynamics models to establish a two-dimensional transient analysis program for thermo-hydraulic performance of rubidium heat pipes. The unsteady Navier Stokes control equation, heat conduction equation, and continuity equation is calculated by the finite difference method. This study adopt molecular dynamics theory for the first time to deal with condensation/ evaporation at the interface of alkali metal heat pipes under the background of space nuclear power. The delay method and multi grid method are used to ensure convergence and accelerate calculation speed. The transient thermo-hydraulic performance of different positions of heat pipes in non-direct contact radiators during start-up are analyzed. Current research can provide scientific reference for the upgradation of heat pipe radiators for nuclear powered spacecraft.

Assessment of Models for Nonlinear Oscillatory Flow Through a Hexagonal Sphere Pack

We review models for unsteady porous media flow in the volume-averaging framework and we discuss the theoretical relations between the models and the definition of the model coefficients (and the uncertainty therein). The different models are compared against direct numerical simulations of oscillatory flow through a hexagonal sphere pack. The model constants are determined based on their definition in terms of the Stokes flow, the potential flow and steady nonlinear flow. Thus, the discrepancies between the model predictions and the simulation data can be attributed to shortcomings of the models’ parametrisation. We found that an extension of the dynamic permeability model of Pride et al. (PRB 47(9):4964–4978, 1993) with a Forchheimer-type nonlinearity performs very well for linear flow and for nonlinear flow at low and medium frequencies, but the Forchheimer term with a coefficient obtained from the steady-state overpredicts the nonlinear drag at high frequencies. The model reduces to the unsteady Forchheimer equation with an acceleration coefficient based on the static viscous tortuosity for low frequencies. The unsteady Forchheimer equation with an acceleration coefficient based on the high-frequency limit of the dynamic tortuosity has large errors for linear flow at medium and high frequencies, but low errors for nonlinear flow at all frequencies. This is explained by an error cancellation between the inertial and the nonlinear drag.

One-level and two-level operator splitting methods for the unsteady incompressible micropolar fluid equations with double diffusion convection

In this paper, a one-level operator splitting method (OOSM) and a two-level operator splitting method (TOSM) for the two-dimensional or three-dimensional (2D/3D) unsteady incompressible micropolar fluid equations with double diffusion convection (IMFDDC) are proposed and analyzed. Firstly, a OOSM is constructed, which consists of five steps. The projection strategy is adopted to get the values of linear velocity and pressure in the first two steps, then the three elliptic subproblems about the angular velocity, the temperature, and the concentration variables are solved in the last three steps. The original variables with the predefined boundary conditions need to be solved separately in each step. Next, in order to solve the problem efficiently, a TOSM is presented, which only solves the nonlinear equations in the coarse level subspaces and the Oseen type linearized equations in the fine level. The numerical analysis of stability and error estimates of the fully discrete methods show that TOSM can reach the same accuracy as the OOSM with H∼O(h2/3). Numerical examples are provided to confirm the effectiveness and reliability of the proposed methods.

Unsteady Magnetohydrodynamics PDE of Monge–Ampère Type: Symmetries, Closed-Form Solutions, and Reductions

The paper studies an unsteady equation with quadratic nonlinearity in second derivatives, that occurs in electron magnetohydrodynamics. In mathematics, such PDEs are referred to as parabolic Monge–Ampère equations. An overview of the Monge–Ampère type equations is given, in which their unusual qualitative features are noted. For the first time, the Lie group analysis of the considered highly nonlinear PDE with three independent variables is carried out. An eleven-parameter transformation is found that preserves the form of the equation. Some one-dimensional reductions allowing to obtain self-similar and other invariant solutions that satisfy ordinary differential equations are described. A large number of new additive, multiplicative, generalized, and functional separable solutions are obtained. Special attention is paid to the construction of exact closed-form solutions, including solutions in elementary functions (in total, more than 30 solutions in elementary functions were obtained). Two-dimensional symmetry and non-symmetry reductions leading to simpler partial differential equations with two independent variables are considered (including stationary Monge–Ampère type equations, linear and nonlinear heat type equations, and nonlinear filtration equations). The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial boundary value problems described by highly nonlinear partial differential equations.