Convective Equation Research Articles

Diffusion - Convection Equation of Galactic Cosmic Rays (GCR) in the Atmosphere and Its Analytical, Numerical Solutions by Using Finite Elements Method Using Parker Transport Equation

This study depicts that to find the Analytical and numerical solutions of Diffusion – Convections Equations of Galactic Cosmic Rays (GCRs) by Finite Element method (FEM) and also to find the Energy Equation of GCRs by using a part of Parker’s transport equation. This considers moreover centres on the exactness and acknowledgment of the FEM strategy by utilizing dissemination blunder, scattering mistake and add up to blunder investigation. The comes about are depicted both graphically and in a unthinkable frame, which essentially guarantees the method’s legitimacy and the algorithm’s proficiency to maintain the exactness, effortlessness, and nonlinear Convection-Diffusion Equation conditions. The proposed method may be connected for tackling any nonlinear convection diffusion equation. We concentrate on assaying the confluence and stability of the nonlinear parabolic partial differential equation. This study focuses on the delicacy and acceptance of FEM method by exercising dispersion error dissipation error, and total error analysis.

A novel physics-informed neural networks approach (PINN-MT) to solve mass transfer in plant cells during drying

Predicting microscale mechanisms of plant-based food materials has been an enduring challenge due to the inherent complexity of involved physics and prohibitively-high computational costs. As an alternative to conventional methods of solving corresponding Partial Differential Equations (PDEs), Physics-Informed Neural Networks (PINNs) have recently received significant attention from the research community. This article investigates the feasibility of PINNs' Automatic Differentiation (AD) to solve mass transfer (PINN-MT) and effectively predict cellular-level mass loss and subsequent moisture variations during low-temperature drying. PINN-MT has been developed by incorporating the residuals of convective mass transfer equation, Fick's law of diffusion and relevant initial conditions into the loss function. The integration of Adaptive Activation (AA) and Transfer Learning (TL) leads to improved computational efficiency, prediction accuracy and consistency. To the best of the authors' knowledge, this study is the first of its kind to apply PINNs' AD to solve mass transfer for food-drying. The PDE-solving capacity of the proposed approach has been compared with that of the state-of-the-art Finite Element Analysis (FEA) techniques. Compared to benchmark FEA results, percentage errors for PINN-MT-based mass loss and moisture distribution predictions remain below 0.23% and 0.33%, respectively. In addition, TL enhances the solution-based prediction efficiency of PINN-MT by more than 80%. Further, a novel approach has been proposed to achieve time-efficient predictions by feeding mass transfer coefficient-hm as an independent input to the proposed predictive framework, which has the potential to tackle complex and highly non-linear physics-based models, where conventional computational approaches usually struggle.

Parameter uniform numerical method for a system of singularly perturbed parabolic convection–diffusion equations

This article presents a numerical study of the initial boundary value problem for a singularly perturbed system of two equations of convection–diffusion type. The perturbation parameter in both equations leads to the boundary layer in both solution components. The sign of the convection coefficient decides the position of the boundary layer at the right end of the spatial domain. We suggest a numerical method composed of a spline-based scheme with a Shishkin mesh for solving the proposed system. Convergence analysis shows that the numerical technique is nearly second-order uniformly convergent concerning the perturbation parameter. The numerical illustration is delivered to support the theoretical results.

Existence and upper semicontinuity of random pullback attractors for 2D and 3D non-autonomous stochastic convective Brinkman-Forchheimer equations on whole space

In this work, we analyze the long time behavior of 2D as well as 3D convective Brinkman-Forchheimer (CBF) equations and its stochastic counterpart with non-autonomous deterministic forcing term in $\mathbb{R}^d$ $ (d=2, 3)$: $$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u} + (\boldsymbol{u}\cdot\nabla)\boldsymbol{u} + \alpha\boldsymbol{u} + \beta|\boldsymbol{u}|^{r-1} \boldsymbol{u} + \nabla p=\boldsymbol{f},\quad \nabla\cdot\boldsymbol{u}=0, $$ where $r\geq1$. We prove the existence of a unique global pullback attractor for non-autonomous CBF equations, for $d=2$ with $r\geq1$ and $d=3$ with $r > 3$ (for any $\beta,\mu > 0$), and $d=r=3$ with $2\beta\mu\geq1$. For the same cases, we show the existence of a unique random pullback attractor for non-autonomous stochastic CBF equations with linear multiplicative white noise. In addition, a remark on the existence of a unique weak pullback mean random attractor for the stochastic CBF equations is also provided. Finally, we establish the upper semicontinuity of the random pullback attractors, that is, the random pullback attractors converge towards the global pullback attractor when the noise intensity approaches to zero. Since we do not have compact Sobolev embeddings on unbounded domains, the pullback asymptotic compactness of the random dynamical system is proved by the method of energy equations given by Ball. For the case of Navier-Stokes equations defined on $\mathbb{R}^d$, such results are not available and the presence of Darcy term $\alpha\boldsymbol{u}$ helps us to establish the above mentioned results for CBF equations on whole space.

Inclusion of anisotropy in understanding rock deformation and inter-well fracture growth in layered formation through CZM based XFEM

In layered formations such as the Wufeng-Longmaxi marine shale, mechanical rock properties may be uniform horizontally within a layer, but vary vertically, and from layer to layer. In this study, we implement Newton-Raphson method within a rigorous two-dimensional scheme for computing fluid-solid coupling at every time increment in an anisotropic rock frame. The investigated domain consists of intercalations of soft and hard rock layers with a consideration of a cluster of three horizontal wells. The linear convection equation describing the fluid flow is discretized using upwind scheme whose integration points are determined by the intersection between the crack geometry and the underlying discretization of the extended finite element method (XFEM). A normalized slurry concentration averaged over the fracture width is used to establish a basis for analysing fluid viscosity effects with subsequent XFEM computation indicating that an increased slurry concentration leads to increasing fracture aperture. We further conduct a thorough investigation of an inter-well fracture development with the analyses revealing that the fractures kink at the interfaces of the intercalations of soft and hard layers. It is also observed that, when the elastic contrast between the soft and hard strata decreases, fracture kinking reduces and the inter-well fracture interference mechanism switches to linking or coalescence. Before the linking, the fluid pressure fields inside the separate fractures are independent, whereas an equilibrated pressure field develops inside the inter-linked fracture system. Oriented initial perforations show no link-growth of fractures. However, we observe that the highest inclination angle, which is 75° in our case, yields the highest treatment pressure of 62.8 MPa, whereas the lowest inclination angle of 15° yields the lowest treatment pressure of 56.3 MPa. We validate our results through comparison with a triaxial fracturing cell experiment, and Kristianovich-Geertsma-de Klerk (KGD) fracture problem. In all cases, the findings agree with the numerical results.

Validity of the boundary layer assumptions for natural convection around a cylinder in a porous medium

In this research, it has been investigated that under what conditions it is correct to use the boundary layer equations for natural convection over an embedded horizontal cylinder in an infinitely porous medium, assuming Local Thermal Non-Equilibrium. For this purpose, the results obtained for the average Nusselt values using boundary layer assumptions have been compared with the results obtained from COMSOL software (complete two-dimensional equations). For the boundary layer results, the physical coordinate of the problem has been converted to a computational Cartesian system, and then the parabolic equations of the boundary layer have been solved numerically by coding in Fortran 90 using the implicit finite difference method of Keller box. Removing some terms in the boundary layer equations in order to make the equations parabolic causes errors in the results. The effect of Rayleigh number, ratio of solid-to-fluid conduction coefficient, Bio number, porosity, and diameter ratio parameter on the amount of this deviation has been shown. Difference between the consequences of the two methods increases by increasing the contribution of the solid matrix in heat transfer. In high porosity, high Bio number, low conductivity ratio, and the low ratio diameter of cylinder to particle, the results of the two solution methods approach each other. So, in this condition, the boundary layer equations for the free convection on the embedded horizontal cylinder in the porous medium can be used and other than these conditions, the assumption of the boundary layer equations has a significant error.

Use of high-speed angiography HSA-derived boundary conditions and Physics Informed Neural Networks (PINNs) for comprehensive estimation of neurovascular hemodynamics.

Physics-informed neural networks (PINNs) and computational fluid dynamics (CFD) have both demonstrated an ability to derive accurate hemodynamics if boundary conditions (BCs) are known. Unfortunately, patient-specific BCs are often unknown, and assumptions based upon previous investigations are used instead. High speed angiography (HSA) may allow extraction of these BCs due to the high temporal fidelity of the modality. We propose to investigate whether PINNs using convection and Navier-Stokes equations with BCs derived from HSA data may allow for extraction of accurate hemodynamics in the vasculature. Imaging data generated from in vitro 1000 fps HSA, as well as simulated 1000 fps angiograms generated using CFD were utilized for this study. Calculations were performed on a 3D lattice comprised of 2D projections temporally stacked over the angiographic sequence. A PINN based on an objective function comprised of the Navier-Stokes equation, the convection equation, and angiography-based BCs was used for estimation of velocity, pressure and contrast flow at every point in the lattice. Imaging-based PINNs show an ability to capture such hemodynamic phenomena as vortices in aneurysms and regions of rapid transience, such as outlet vessel blood flow within a carotid artery bifurcation phantom. These networks work best with small solution spaces and high temporal resolution of the input angiographic data, meaning HSA image sequences represent an ideal medium for such solution spaces. The study shows the feasibility of obtaining patient-specific velocity and pressure fields using an assumption-free data driven approach based purely on governing physical equations and imaging data.

Non-overlapping High-accuracy Parallel Closure for Compact Schemes: Application in Multiphysics and Complex Geometry

Compact schemes are often preferred in performing scientific computing for their superior spectral resolution. Error-free parallelization of a compact scheme is a challenging task due to the requirement of additional closures at the inter-processor boundaries. Here, sources of the error due to sub-domain boundary closures for the compact schemes are analyzed with global spectral analysis. A high-accuracy parallel computing strategy devised in “ A high-accuracy preserving parallel algorithm for compact schemes for DNS. ACM Trans. Parallel Comput. 7, 4, 1-32 (2020)” systematically eliminates error due to parallelization and does not require overlapping points at the sub-domain boundaries. This closure is applicable for any compact scheme and is termed here as non-overlapping high-accuracy parallel (NOHAP) sub-domain boundary closure. In the present work, the advantages of the NOHAP closure are shown with the model convection equation and by solving the compressible Navier–Stokes equation for three-dimensional Rayleigh–Taylor instability simulations involving multiphysics dynamics and high Reynolds number flow past a natural laminar flow airfoil using a body-conforming curvilinear coordinate system. Linear scalability of the NOHAP closure is shown for the large-scale simulations using up to 19,200 processors.

Control of Chemoconvection in a Rectangular Slot by Changing Its Spatial Orientation

Recently, we found that a two-layer miscible system placed in a vertical slab reactor shows an occurrence of a density shock-wave-like pattern. This wave resembles a turbulent bore separating immobile fluid and an area of intense mixing. It travels away from the convective core of the system and is highly dependent on the intensity of a gravity-dependent chemoconvection in the cocurrent flow. The novelty of this work is that we demonstrate that the change in angle between gravity and wave direction allows controlling the chemoconvection intensity and, consequently, the rate of a spatially-extended reaction. We study both experimentally and numerically the effect of the spatial orientation of a slab reactor to a gravity field on a flow structure induced by a neutralization reaction. In experiments, we use aqueous mixtures of nitric acid and sodium hydroxide. We apply the Fizeau interferometry to visualize the flow and use the PIV method to measure the fluid velocity. The mathematical model includes reaction–diffusion–convection equations that describe 3D flows. We study the flow modifications with a change in the inclination angle from 0 to 90 degrees. At small angles (up to 30), the cocurrent flow becomes spatially heterogeneous, and the fields of salt and acid are separated. If the inclination exceeds 50 degrees, the wavefront is deformed, and the wave breaks up, resulting in a sharp decrease in the reaction rate.

On the stability of strong-stability-preserving modified Patankar–Runge–Kutta schemes

In this paper, we perform a stability analysis for classes of second and third order accurate strong-stability-preserving modified Patankar–Runge–Kutta (SSPMPRK) schemes, which were introduced in Huang and Shu [J. Sci. Comput. 78 (2019) 1811–1839] and Huang et al. [J. Sci. Comput. 79 (2019) 1015–1056] and can be used to solve convection equations with stiff source terms, such as reactive Euler equations, with guaranteed positivity under the standard CFL condition due to the convection terms only. The analysis allows us to identify the range of free parameters in these SSPMPRK schemes in order to ensure stability. Numerical experiments are provided to demonstrate the validity of the analysis.

Nonlinear development of convective patterns driven by a neutralization reaction in immiscible two-layer systems.

This article provides the results of a theoretical and experimental study of buoyancy-driven instabilities triggered by a neutralization reaction in an immiscible two-layer system placed in a vertical Hele-Shaw cell. Flow patterns are predicted by a reaction-induced buoyancy number [Formula: see text], which we define as the ratio of densities of the reaction zone and the lower layer. In experiments, we observed the development of cellular convection ([Formula: see text]), the fingering process with an aligned line of fingertips at a slightly denser reaction zone ([Formula: see text]) and the typical Rayleigh-Taylor convection for [Formula: see text]. A mathematical model includes a set of reaction-diffusion-convection equations written in the Hele-Shaw approximation. The model's novelty is that it accounts for the water produced during the reaction, a commonly neglected effect. The persisting regularity of the fingering during the collapse of the reaction zone is explained by the dynamic release of water, which compensates for the heavy fluid falling and stabilizes the pattern. Finally, we present a stability map on the plane of the initial concentrations of solutions. Good agreement between the experimental data and theoretical results is observed. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Statistical Image Analysis on Liquid-Liquid Mixing Uniformity of Micro-Scale Pipeline with Chaotic Structure

The aim of this work is to introduce a novel statistical technique for quantifying the concentration field uniformity of the liquid-liquid mixing process within a micro-scale chaotic pipeline. For illustration, the microscale liquid-liquid mixer in which the inlet direction is parallel to the mixing unit is designed by using the chaotic pipeline with Baker map. Meanwhile, the non-uniformity coefficient method is adopted quantificationally instead of qualitatively estimating the concentration field uniformity of the chaotic micromixer based on uniform design theory and image analysis. Results show that the concentration distribution of the chaotic mixing process of liquid-liquid under various working conditions is obtained by solving the steady-state Navier–Stokes and diffusion convection equations. The average contribution ratio of the three basic mixing units of the chaotic Baker pipeline to the concentration field uniformity is approximately 6:3:1, which is calculated aligned with the fluid flow direction successively. The optimal mixing uniformity can be obtained as the initial velocity is 0.05 m/s and the diffusion coefficient is 5 × 10−9 m2/s, respectively. The reliability of the new method for estimating the concentration field uniformity parameters is explained from three dimensions. The statistical image analysis technique is illustrated to be reliable and effective in yielding accurate concentration field information of the simulated chaotic mixer. Furthermore, it can be adapted to examine a variety of concentration distribution issues in which concentrations are evaluated under distinct scales.

Finite difference scheme for a non-linear subdiffusion problem with a fractional derivative along the trajectory of motion

Abstract The article is devoted to the construction and study of a finite-difference scheme for a one-dimensional diffusion–convection equation with a fractional derivative with respect to the characteristic of the convection operator. It develops the previous results of the authors from [5, 6] in the following ways: the differential equation contains a fractional derivative of variable order along the characteristics of the convection operator and a quasi-linear diffusion operator; a new accuracy estimate is proved, which singles out the dependence of the accuracy of mesh scheme on the curvature of the characteristics.

Transient Coupled Flow Model for Matrix-Fracture in High-Pressure Gas Reservoirs Opened by Gas Drilling

Gas drilling technology is an effective method to protect low permeability tight sandstone reservoirs from damage. However, using this technology to drill high-pressure fractured reservoirs, the fluid flow characteristics are extremely complex, leading to drilling safety challenges. Therefore, based on the conservation equations of mass, momentum, and energy, a new transient coupled flow model in matrix-fracture is established for high-pressure gas reservoirs opened by gas drilling, considering convection term and energy equation, to research the pressure and flow velocity transient behaviors. This model is solved by Shaher Momani’s algorithm and CyabJiBe format, and the type curves of the pressure and flow velocity are plotted. The three flow stages are developed, and the impact of parameters and equation terms on the transient flow behavior is analyzed and discussed, such as the energy equation, the matrix permeability ( k ), the fracture aperture ( L f ), and the fracture outlet pressure. It is found that the energy equation term of the model has a great influence on the transient characteristics such as velocity and pressure, which cannot be ignored. In the first flow stage, the flow velocity characteristics are not affected by the matrix parameters being extremely unstable, but in the second and third stages, the influence of the gas supplied to fracture cannot be ignored. The proposed transient coupled flow model provides a new understanding of the transient flow behavior in matrix-fracture for high-pressure reservoirs during gas drilling, which can be used to interpret the characteristics of pressure and velocity more realistically, to provide support for safe drilling of gas drilling.

The Study of Three-Dimensional Fluid Flow Regimes on the Basis of the Exact Solution to the Convection Equations

We consider the evaporation/condensation process of diffusion type in two-layer fluid flows under the assumption that both viscous heat-conducting incompressible fluids, a liquid and a gas-vapor mixture, have a common interface and occupy an infinite threedimensional rectangular channel. Based on the exact solution to the convection equations, we study the fluid flows arising in the case of a linearly distributed thermal load on the substrate and thermal insulation of the upper and lateral walls of the channel. It is observed that some complex vortex structures are formed under the action of the longitudinal temperature gradient and transversely directed gravity field. For the ethanol–nitrogen system we present the characteristics of hydrodynamic, thermal, and concentration fields, in particular, the profiles of the evaporative mass flow rates and interface tangential stresses, the topology of flows, and the temperature and vapor concentration distributions.

Effect of U-shape external wall on heat transfer coefficients and building performances

We keep the focus in this paper on the heat transfer coefficients affecting the performance of building for different climatic zones of Morocco. In this respect, the building model is developed using COMSOL Multiphysics based on the convective and radiative equations. Furthermore, the U-shape of the south external wall of the building is modified and tested according to three scenarios (A 0.25 m, B 0.75 m, and C 1.25 m). The results showed a significant increase of radiative heat transfer coefficient (HTC) by 33% and 64.62% by increasing the U-shape from 0.25 to 0.75 m and from 0.75 m to 1.25 m respectively. In fact, the convective coefficient decreases progressively in the U-shape wall. Hence, a maximum reduction of total energy consumption is observed as well as the HTC decreases.

New traveling wave solutions of a nonlinear diffusion–convection equation by using standard tanh method

Exact solutions of traveling wave are acquired by employ a relatively new technique which is called standard tanh method for a nonlinear diffusion–convection equation. The tanh method is applied for the first time for finding travelling wave solutions for the nonlinear diffusion–convection equation , where n and m, are integers, and , of order (m=4, n=7) and (m=5, n=9). Analytical solutions of a nonlinear diffusion–convection equations are obtained as a polynomial in tanh(x), and the plots for exact solutions are given. The obtained results are compared with F-Expansion method to validate the proposed approach.

Decoupling Method for the Convective-Dominated Leaching Process of Ion-Adsorption-Type Rare-Earth Ores

Ion-adsorption-type rare-earth ores (IATREOs) that have experienced granite weathering have good permeability, and their leaching process involves the solute transport problem, which is dominated by convection. Because of the oscillation and dispersion errors of existing numerical methods for solving the convective-dominated solute transport equation, the results have low precision in the twin leaching process. In this paper, the convection–dispersion equation is decoupled into the dispersion equation, the convection equation, and the source-sink equation; the Crank–Nicolson and implicit difference methods are used to solve the dispersion equation and the source-sink equation, respectively. The solution of the convection equation is achieved on the basis of its physical interpretation. Therefore, a decoupling method for the convective-dominated solute transport equation is established. In comparison to the two examples with analytical solutions, the calculation errors of the method established in this paper are less than 2.00%, and it can solve the oscillation and dispersion problems. The rationality of the method is further demonstrated through the column leaching experiment of IATREOs. In comparison to the test results, the coefficients of determination of the breakthrough curves of rare-earth ions and ammonium ions calculated by the proposed method are all greater than 0.850, and the peak concentration error of rare-earth ions is less than 7.00%. This indicates that the proposed method can simulate the leaching process well. Furthermore, by combining the multiple/half method and the dichotomy method, an optimization method for determining the leaching agent amount was established to analyze the relationship between the leaching agent and the ratio of dispersion to pillar length. The results can provide a solution that can be used to mine IATREOs from experience to theory.

A STUDY OF MIXED CONVECTIVE TERNARY HYBRID NANOFLUID FLOW OVER A STRETCHING SHEET WITH RADIATION AND TRANSPIRATION

This study explains the motion of a mixed convective ternary hybrid nanofluid flow with radiation and transpiration. Ternary hybrid nanofluid is the colloidal mixture of three district types of particles, namely, Al<sub>2</sub>O<sub>3</sub>, single-wall CNT, and graphene. The ordinary differential equations (ODEs) of the equation is yielded by converting partial differential equations (PDEs) with the help of suitable variables for analyzing the spherical, cylindrical, and platelet nanoparticles. Then the analytical solution can be conducted by using different shapes of the nanoparticles, namely, spherical, cylindrical, and platelet, to get the solution domain. Main results on the applications of suction and results of four roots are investigated. Also by using this domain fluctuation of friction, heat and mass transfer across the flow can be analyzed with various physical parameters. Increasing knowledge of these ternary hybrid nanofluids helps to improve the heat transfer rate. This is useful in dynamics of fuel and coolant in automobiles. Furthermore, the goal and novelty of the present work mainly explains the ternary hybrid nanofluid flow with three different shapes of nanoparticles in the presence of mixed convection. The given mixed convective Navier-Stokes equations given in PDE form are converted into ODE form using suitable transformation. The analytical results are expressed in terms of Prandtl number and in terms of four roots. Finally, the impact of different shapes of nanoparticles and parameters can be discussed using a graphical scenario. The main finding is that the Dufour number is greater for more values of thermal radiation. Also, transverse and tangential velocities as well as temperature and concentration profiles are greater for more values of solution domain and Darcy number.

Numerical Study of Natural Solutal Convection in an Isoscele Trapezoidal Cavity

A numerical study of the natural heat and mass transfer in a cavity with a straight isoscele cross-section containing air is made in this paper. The two inclined side walls are kept in natural convection with the surrounding environment. The upper horizontal wall is subjected to a heat flux of constant density, while the lower one is adiabatic. Under the Boussinesq assumption, the thermodynamic conditions are numerically studied using the unsteady convection equations formulated as a secondary variable of vorticity and a stream function, energy and moisture. The system of equations discretised by the implicit finite difference method is solved by the Thomas algorithm. The results show a flow structure, isotherms and isohumidities dependent on the study parameters. Thus, on one hand, an increase in the inclination angle of the walls is accompanied by an increase in the velocity of the fluid. On the other hand, an increase in the aspect ratio or Lewis number leads to a decrease in the fluid’s velocity. The average Nusselt number, which is independent of the Rayleigh number, increases slightly as the inclination angle of the walls decreases. The increase of the Lewis number results in the decrease of the flow velocity components values. It is observed that the maxima values of velocity components were reached for Rayleigh number equal to 7.103.