Linear Momentum Equations Research Articles

An inverse problem for the system modeling nonhomogeneous asymmetric fluids

In this paper, we prove the local uniqueness of an inverse problem arising in the nonstationary flow of a nonhomogeneous incompressible asymmetric fluid in a bounded domain with a smooth boundary. The direct problem is an initial boundary value problem for a system where the unknowns are the velocity field of the fluid particles, the angular velocity of rotation of the fluid particles, the mass density of the fluid, and the pressure distribution. The inverse problem consists of recovering external forces to the linear and angular momentum equations by assuming a set of measurements as an integral overspecified condition. We introduce and prove several a priori estimates. We characterize the inverse problem solutions using an operator equation of a second kind, deduced from applying the Helmholtz decomposition. We prove several properties of the associated operator, which implies that the Tikhonov fixed point theorem hypothesis is valid. Then, we deduce the local unique solvability of the inverse problem.

An efficient numerical solver for highly compliant slender structures in waves: Application to marine vegetation

This paper presents a fully explicit coupled wave–vegetation interaction model capable of efficiently solving the coupled wave dynamics and flexible vegetation motion with large deflections. The flow model is formulated using the continuity equation and linearized momentum equations of an incompressible fluid, with additional terms within the canopy region accounting for the presence of vegetation. This linearized flow solver is unconditionally stable and second-order accurate. The flow model is validated and verified against experimental measurements and analytical solutions for waves over a rigid canopy, demonstrating its capability to accurately capture the wave dissipation and flow velocity profiles, even with a relatively coarse grid. A truss-spring model is proposed to capture vegetation motion with substantial deflections, and is proven to be mathematically consistent with the governing equation for the flexible vegetation motion. It allows for explicit time integration with large time steps when dealing with highly compliant vegetation. The truss-spring model is validated and verified by experimental and numerical results for large-amplitude motions of a single elastic blade subjected to waves and sinusoidal oscillatory flows. The coupled model, combining the linearized flow solver and the truss-spring model, is applied to investigate wave propagating over a heterogeneous, suspended, and flexible canopy, showing high efficiency and good agreement with the experiments concerning wave attenuation and the hydrodynamic loads on the vegetation.

Comparative characterization of heat transfer and entropy generation of micropolar‐Jeffrey, micropolar‐Oldroyd‐B, and micropolar‐Second grade binary nanofluids in PTSCs settings

AbstractIn this paper, we delve into the behavior of binary micropolar nanofluids, specifically micropolar‐Jeffrey, micropolar‐Oldroyd‐B, and micropolar‐Second grade, within the parabolic trough solar collector (PTSC) configurations. The primary objective is to enhance the collective efficiency of this device by means of a comprehensive comparison amongst the three aforementioned nanofluids. The governing equations, including continuity, linear momentum, angular momentum, and energy equations, were systematically formulated. Subsequently, the introduction of suitable similarity variables facilitated the transformation of the intricate partial differential equations into manageable ordinary differential equations. These resultant equations were then tackled utilizing the shooting method via the bvp4c numerical package in MATLAB. The study critically examines the influence of diverse parameters that dictate the flow dynamics of the nanofluids. This encompasses nanofluid velocity, angular velocity, temperature distribution, entropy generation, skin friction coefficient, and local Nusselt number. Remarkably, the research uncovers that the maximum temperature levels experienced enhancements of 12.1134%, 12.0616%, and 11.0645% for the micropolar‐Jeffrey, micropolar‐Oldroyd‐B, and micropolar‐Second grade nanofluids, respectively. These results imply that the introduction of these binary micropolar nanofluids leads to notable thermal enhancements in the PTSCs settings.

A first‐order hyperbolic arbitrary Lagrangian Eulerian conservation formulation for non‐linear solid dynamics

SummaryThe paper introduces a computational framework using a novel Arbitrary Lagrangian Eulerian (ALE) formalism in the form of a system of first‐order conservation laws. In addition to the usual material and spatial configurations, an additional referential (intrinsic) configuration is introduced in order to disassociate material particles from mesh positions. Using isothermal hyperelasticity as a starting point, mass, linear momentum and total energy conservation equations are written and solved with respect to the reference configuration. In addition, with the purpose of guaranteeing equal order of convergence of strains/stresses and velocities/displacements, the computation of the standard deformation gradient tensor (measured from material to spatial configuration) is obtained via its multiplicative decomposition into two auxiliary deformation gradient tensors, both computed via additional first‐order conservation laws. Crucially, the new ALE conservative formulation will be shown to degenerate elegantly into alternative mixed systems of conservation laws such as Total Lagrangian, Eulerian and Updated Reference Lagrangian. Hyperbolicity of the system of conservation laws will be shown and the accurate wave speed bounds will be presented, the latter critical to ensure stability of explicit time integrators. For spatial discretisation, a vertex‐based Finite Volume method is employed and suitably adapted. To guarantee stability from both the continuum and the semi‐discretisation standpoints, an appropriate numerical interface flux (by means of the Rankine–Hugoniot jump conditions) is carefully designed and presented. Stability is demonstrated via the use of the time variation of the Hamiltonian of the system, seeking to ensure the positive production of numerical entropy. A range of three dimensional benchmark problems will be presented in order to demonstrate the robustness and reliability of the framework. Examples will be restricted to the case of isothermal reversible elasticity to demonstrate the potential of the new formulation.

Mixing-phase model for shear-induced contractive/dilative effects in unsteady water-sediment mixture flows

Among the geophysical surface processes, mud and debris flows show one of the most complex and challenging behaviour for scientists and modellers. These flows consist of highly-unsteady gravity-driven movements of water-sediment mixtures with non-Newtonian rheology where the solid concentration could be about 40%–80% of the flow volume and which occur along steep and irregular terrains. Furthermore, the appearance of dynamic pressures in the fluid filling the intergranular pores increases the complexity and dominates the behaviour of the fluidized water-sediment material, leading to the appearance of significant density gradients during the movement. The dynamic pressure in the pore-fluid changes the effective normal stress within the mobilized material, affecting the frictional shear stress between grains and leading to the solid phase dilation/contraction. This must be properly accounted for when developing realistic models for water-sediment surface flows. In this work, a novel physically-based approach for modelling multi-grain dense-packed water-sediment flows is presented. A novel closure formulation for the pressure distribution within the pore-fluid during the movement of dense-packed water-sediment materials has been derived. This closure allows to relate the appearance of shear-induced dynamic pore pressures to the contractive/dilative behaviour of the solid aggregate. The resultant system of depth-averaged conservation laws includes continuity of the density-variable water-sediment material and the different solid classes transported in the flow, as well as the linear momentum equation for the fluidized bulk material, and it is solved using a well-balanced fully-coupled Finite Volume (FV) method. The resultant simulation tool is faced to synthetic, laboratory and real-scale benchmark cases to test its robustness and accuracy. The presence of dynamic pore pressures within the pore-fluid leads to the appearance of a deviatoric contribution to the solid flux, which causes the shear-induced separation of the solid and liquid phases and sustains the flow mobility for long distances, as it has been observed in real mud and debris events.

A thermodynamically consistent phase field model for brittle fracture in graded coatings under thermo-mechanical loading

This paper presents a thermodynamically consistent phase field formulation designed for investigating quasi-static brittle fracture under thermo-mechanical loading conditions in functionally graded materials. The governing equations for linear momentum and crack evolution are derived by minimizing the free energy functional, which includes elastic strain energy and surface energy, with respect to displacement field and the phase field variable, respectively. A thermodynamically consistent formulation is employed to establish the governing equation for heat conduction. The proposed formulation is converted into a set of finite element equations using the Ritz method and implemented in COMSOL Multiphysics, a commercial finite element software which allows for writing governing equations in their strong form. A segregated solver is utilized to solve iteratively the coupled problem. The proposed model is verified against existing mechanical and thermo-mechanical phase field models, demonstrating good agreement with the literature. To explore the model’s capabilities, a graded layer composed of Zirconia and a Titanium alloy with a surface crack undergoing thermo-mechanical loading is considered as a case study for this investigation. Additionally, the model incorporates a thermal conductivity degradation proposed. A parametric study is performed to investigate the effect of varying parameters like volume fraction, thermal loading rate, and the thermal conductivity degradation on the load–displacement curve of the graded layer. These findings contribute to the understanding of thermo-mechanical behavior of surface graded layers and serve as a foundation for addressing more complex thermo-mechanical problems.

Entropy generation analysis of a micropolar fluid in a corrugated channel with convective and slip conditions

This study focuses on examining the entropy generation in a corrugated channel, considering convective boundary conditions and slip flow, with a micropolar fluid. The governing equations for the micropolar fluid flow, including linear and angular momentum equations, as well as the energy equation, are solved using the perturbation technique. The effects of corrugations, slipping flow, and convective boundary conditions on the entropy generation and the flow behavior are analyzed. The results show that the entropy generation enhanced with the corrugation amplitude, while it reduced for the slip flow. Moreover, entropy generation is affected by the convective boundary conditions, and it is an increasing function with the convective heat transfer coefficient. Additionally, the study demonstrates that micropolar fluid exhibits distinct flow characteristics in comparison to classical Newtonian fluids. These findings have practical implications for the design and optimization of microfluidic devices, as well as for gaining insights into the behavior of micropolar fluids in various engineering applications.

A thermo-chemo-mechanically coupled peridynamics for investigating crack behavior in solids

In engineering applications, the phenomenon of cracking is often accompanied by a coupled multiphysics effect. Peridynamics (PD) is an effective approach for solving cracking problems, but currently, no general PD model accounts for the coupling of multiple physical fields. In this work, we develop a PD model of coupled deformation, heat conduction, species diffusion, and chemical reactions. First, we establish the equations for mass, linear momentum, and energy. Then we establish fully coupled constitutive laws that interpret the interactions between the various fields and formulate evolution equations that govern the flux of species and heat. These laws and equations are developed based on the inequality of energy dissipation and the principles of chemical kinetics. Species diffusion and chemical reactions are treated as separate processes to study their effects on the Helmholtz free energy density of solids and the subsequent formation and propagation of cracks. In addition, certain coupling coefficients are calibrated by equating the corresponding physical quantities in the PD model with those in continuum mechanics. Four specific cases are simulated to validate the model, including redistribution of vacancies in ceramics, hydrogen traps, and embrittlement in metals.

Creep failure of hierarchical materials

Creep failure of hierarchical materials is investigated by simulation of beam network models. Such models are idealizations of hierarchical fibrous materials where bundles of load-carrying fibers are held together by multi-level (hierarchical) cross-links. Failure of individual beams is assumed to be governed by stress-assisted thermal activation over local barriers, and beam stresses are computed by solving the global balance equations of linear and angular momentum across the network. Disorder is mimicked by a statistical distribution of barrier heights. Both initially intact samples and samples containing side notches of various length are considered. Samples with hierarchical cross-link patterns are simulated alongside reference samples where cross-links are placed randomly without hierarchical organization. The results demonstrate that hierarchical patterning may strongly increase creep strain and creep lifetime while reducing the lifetime variation. This is due to the fact that hierarchical patterning induces a failure mode that differs significantly from the standard scenario of failure by nucleation and growth of a critical crack. Characterization of this failure mode demonstrates good agreement between the present simulations and experimental findings on hierarchically patterned paper sheets.

Unifying temperature definition in atomistic and field representations of conservation laws

In this work, a field representation of the conservation law of linear momentum is derived from the atomistic, using the theory of distributions as the mathematical tool, and expressed in terms of temperature field by defining temperature as a derived quantity as that in molecular kinetic theory and atomistic simulations. The formulation leads to a unified atomistic and continuum description of temperature and a new linear momentum equation that, supplemented by an interatomic potential, completely governs thermal and mechanical processes across scales from the atomic to the continuum. The conservation equation can be used to solve atomistic trajectories for systems at finite temperatures, as well as the evolution of field quantities in space and time, with atomic or multiscale resolution. Four sets of numerical examples are presented to demonstrate the efficacy of the formulation in capturing the effect of temperature and thermal fluctuations, including phonon density of states, thermally activated dislocation motion, dislocation formation during epitaxial processes, and attenuation of longitudinal acoustic waves as a result of their interaction with thermal phonons.

A complete physical 3D model from first principles of vibrational-powered electromagnetic generators

The dynamic behavior of a vibrational electromagnetic generator using a magnetic levitation architecture was theoretically and experimentally studied in great detail, when operating under a wide range of three-dimensional excitations. We developed a complete rigorous physical model from first principles based on the theory of electrodynamics of continua, centered on the laws of electrodynamics and balance of mass, linear momentum, angular momentum, energy and entropy. Local electromagnetic and gravitational body forces, couples and powers were considered, and the surface tractions were divided into constraint and friction components, as well as those due to external mechanical energy sources. The balance of linear momentum, angular momentum and circuit equations resulted in up to 13 non-linear differential equations describing the dynamics of the levitating-magnet and container, relating input forces and torques with output displacement, constraint forces and voltage. The balance of energy yielded a consistent equivalence between the time rate of change of the internal kinetic and potential energies of the generator and the output power, associated with the external circuit, Ohmic losses and friction losses, as well as the input mechanical power being supplied to the system by the environment. Both the input and output powers were proven to tend to increase equally when operating the generator under resonant conditions. The levitating generator was shown to be sensitive to axial translational and centrifugal inertial forces, each one effectively resulting in a uni-stable or bi-stable system. The dynamical response yielded multiple initial conditions dependent steady-states, hysteretic frequency output and chaotic characteristics. Relevant guidelines to optimize the energy conversion efficiency of energy harvesters are provided. This model was validated by experimental tests, including general 3D motions combining translations and rotations: cross-correlations exceeding 90% were achieved. Such Newtonian and Langrangian modelling approaches hold great potential to be easily adapted to a wide range of other electromagnetic generators, with multiple degrees-of-freedom and operating under various environments, such that significant advances in energy technologies can be supported.

Irreversibility mixed convection Jeffrey fluid flow analysis in a vertical channel with an inclined magnetic field and Soret effects under slip conditions

Jeffrey fluid flow is found to have extensive applications in many fields of engineering. There are numerous applications of Jeffrey fluid in polymer industries and industrial fluids such as paints, paper, toothpaste, ketchup, etc. The present work explains the irreversibility of Jeffrey fluid in mixed convection Navier–slip flow through a vertical channel having an inclined magnetic field. The model equations of linear momentum, energy, and concentration are formulated and numerically solved using the spectral quasi-linearization method. The outcomes are depicted graphically and quantitatively discussed for active parameters that appeared in mathematical formulations. Moreover, flow and cross-flow velocity increase with an enhancement in Soret number and magnetic parameter, whereas heat transfer rate decreases with these parameters. Temperature, cross-flow velocity, and heat transfer decrease as the inclination angle increases. Flow velocity, concentration, and mass transfer rise as the inclination angle rises. The results demonstrate that as the magnetic parameter, hall parameter, angle of inclination, and slip condition at the right plate increase, the entropy generation number also increases. Conversely, it decreases as the slip condition at the left plate and Jeffrey fluid parameter increase.

An accurate theoretical formula for linear momentum, force and kinetic energy

The paper demonstrates that the existing mathematical formulas of linear momentum, force and kinetic energy in physics are incomplete, since such formulas have been formulated without incorporation of mass-energy equivalence relation E = mc2. Therefore, new reformulations of the main equations of linear momentum, force and kinetic energy in the realm of special relativity are proposed. The proposed formulas provide same mathematical outcomes as the old formulas, displaying same behavior of the system when velocity approaches to speed of light, but, most importantly, comprise only velocity of the light and mass of object to provide well-defined expressions. [Formulae available with the full text]. These formulas vividly reveal that every physical variable depends solely on relativistic mass. Therefore, it modifies Newton’s second law of motion and states that the force depends on rate of change of relativistic mass of object rather than its velocity. In this highly interesting topic, primary purpose here has been to present a succinct and the carefully reasoned account of a new aspect of the Newton’s second law of motion which properly allows to derive the new mathematical formulas of linear momentum, force and kinetic energy.

A velocity-coefficient independent pressure correction equation for accelerated convergence of linear systems in incompressible flows

A velocity-coefficient independent pressure correction equation is proposed based on a modified SIMPLE algorithm for the solution of incompressible Navier-Stokes equations. In the conventional approach of SIMPLE, the central velocity coefficient of linearized momentum equation updates with every non-linear iteration within a time step. Since the pressure correction equation is coupled to the velocity coefficient term the pressure matrix coefficients obtained from the discretized pressure Poisson’s equation also get updated. As a consequence, the preconditioning step used to accelerate the convergence of pressure linear system becomes computationally expensive as the preconditioner is computed at every SIMPLE iteration. In the proposed approach this difficulty is circumvented wherein the velocity coefficient term in the pressure correction equation is repositioned with the velocity term such that the pressure matrix coefficients no longer depend on the same. Hence the preconditioner is computed only once at the first non-linear iteration that is stored and re-used for the subsequent iterations thereby reducing the overall CPU time of simulation.

Motion of Air Bubbles in a Cement Slurry.

The dynamics of air (gas) bubbles in a column of cement slurry is examined numerically. The air injected at the bottom of a laboratory-scale column through a porous distributor plate spatially distributes and migrates as a swarm of bubbles throughout the slurry toward the freeboard. The two-phase system of the cement slurry and the air bubbles is modeled using the conservation equations of mass and linear momentum in the framework of the volume-of-fluid (VOF) approach. The cement slurry is modeled using the Herschel-Bulkley and Bingham fluid models. Results show that the mean Sauter diameter and the mean rise velocity of the bubbles decrease with the gas flow rate. Meanwhile, it is found that the rising of the bubbles is controlled by breakup events, along with relatively weak path instabilities of the bubbles resulting in relatively straight trajectories, independent of the gas flow rate. The extent of the yielded region appears larger for the Herschel-Bulkley model compared to the Bingham fluid model (by approximately 10%).

Arbitrary mesh‐moving velocity‐based space‐time finite element method for large deformation analysis of solids

SummaryThis paper presents a new velocity‐based space‐time finite element method for computing the large deformation of solids over arbitrary Lagrange/Eulerian moving meshes. The proposed method is oriented toward quasi‐static deformation problems with hypoelastic and neo‐Hookean hyperelastic constitutive models. The Cauchy stress in the weak form of the linear momentum equation is explicitly described in terms of deformation velocity through consistent time integration of objective stress rates transferred in the arbitrary moving mesh. Consequently, the resultant system equation only contains the space‐time nodal velocity as the primary unknown. An iterative algorithm for solving the discretized system equation is implemented, whereby the nonlinearities induced by geometrical change in the computational domain as well as nonlinear constitutive equations are computed in an iterative manner updating the stress and the strain over the moving mesh of the deforming domain. Three benchmark problems, such as uniform beam compression, thick‐cylinder compression with superimposed rigid body rotation and extrusion of a column involving material loss, have been solved in order to examine the accuracy of the proposed numerical scheme. The numerical results have revealed the proper computation of the stress and the displacement of the solids undergoing the large deformation and rotation over the arbitrary Lagrange/Eulerian moving meshes, and have shown the validity and applicability of the proposed method with linear convergence of the iterative algorithm.

Characterisation of damage by means of electrical measurements: Numerical predictions

AbstractUnderstanding damage mechanisms and quantifying damage is important in order to optimise structures and to increase their reliability. To achieve this goal, experimental‐ and simulation‐based techniques are to be combined. Different methods exist for the analysis of damage phenomena such as fracture mechanics, phase field models, cohesive zone formulations and continuum damage modelling. Assuming a typical ‐type damage formulation, the governing equations of continua that account for gradient‐enhanced ductile damage under mechanical and electrical loads are derived. The mechanical and electrical sub‐problems give rise to the local form of the balance equation of linear momentum, the micromorphic balance relation and the continuity equation for the electric charge, respectively. Experimental investigations indicate that changes in electrical conductivity arise due to the evolution of the underlying microstructure, for example, of cracks and dislocations. Therefore, motivated by deformation‐induced property changes, the effective electrical conductivity is assumed to be a function of the damage variable. This eventually allows the prediction of experimentally recorded changes in the electrical resistance due to mechanically‐induced damage processes. Interpreting the resistivity as a fingerprint of the material microstructure, the simulation approach proposed in the present work contributes to the development of non‐destructive electrical‐resistance‐based characterisation methods. To demonstrate the applicability of the proposed framework, different representative simulations are studied.

Current density profiles at thin layer electrochemical cells under laminar and turbulent regimes

The transition of laminar to turbulent regimes on smooth electrocatalysts in thin-film electrochemical cells is a subject of study with the aim of predicting current density profiles. Analytical solutions are obtained for velocity profiles under steady state conditions for both hydrodynamic conditions applying a proper velocity contour condition on the catalyst surface. This state is necessary to describe electrochemical cell performances since in these cases, electrode reactions occur at the surface and not in the bulk of the electrolyte. The obtained equations for linear momentum under asymptotic conditions reduce to the solutions obtained by classical Power Series methods years ago. Concentration and current density profiles are also solved accordingly and compared with experimental data of fast redox and dissolved gas reactions on smooth noble metal electrocatalysts.

Local gradient theory of dielectrics incorporating polarization inertia and flexodynamic effect

A higher-grade theory of non-ferromagnetic thermo-elastic dielectrics which incorporates the local mass displacement, the heat flux gradient, polarization inertia, and flexodynamic effects is developed. The process of local mass displacement is associated with changes in material microstructure. Using the fundamental principles of continuum mechanics, electrodynamics, and non-equilibrium thermodynamics, the gradient-type constitutive equations are derived. Due to accounting for the polarization inertia, the rheological constitutive equation for the polarization vector is obtained. In the balance equation of linear momentum, an additional term with the second time derivative of the polarization vector appears in comparison with the classical theory. This term controls the influence of the dynamic flexoelectric effect on the mechanical motion of dielectric solids. The propagation of a plane harmonic wave is analyzed within the context of the developed theory. It is shown that the theory allows for capturing the experimentally observed phenomenon of high-frequency dispersion of a longitudinal elastic wave. The theory may be useful for modeling coupled processes in nanodielectrics and heterogeneous polarized systems.

Effective properties of fractional viscoelastic composites via two‐scale asymptotic homogenization

Driven by the growing interest in fractional constitutive modeling, we adopt a two‐scale asymptotic homogenization approach to study the effective properties of fractional viscoelastic composites. We focus on a purely mechanical setting and derive the cell and homogenized problems corresponding to the balance of linear momentum equation in the absence of body forces and inertial terms. In doing this, we reformulate the original framework in the Laplace–Carson domain and discuss how to obtain the effective coefficients in the time domain. We particularize the general setting of our work by considering memory functions that describe special types of fractional linear viscoelastic behaviors, and after presenting the limit cases of our selections, we framed the homogenization results to account for benchmark problems with different combinations of constitutive models. Specifically, these latter involve elastic, fractional Kelvin–Voigt, fractional Zener and fractional Maxwell constituents. Our results permit us to reinforce the interpretation of the theoretical findings and to elucidate the role of the fractional constitutive models on the effective properties of the composites under investigation.