Incompressible Mixture Research Articles

Wong–Zakai approximation for a stochastic 2D Cahn–Hilliard–Navier–Stokes model

AbstractIn this paper, we demonstrate the Wong–Zakai approximation results for two dimensional stochastic Cahn–Hilliard–Navier–Stokes model. The model consists of a Navier–Stokes system coupled with convective Cahn–Hilliard equations. It describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids under the influence of multiplicative noise. Our main result describes the support of the distribution of solutions. As in [2], both inclusions are proved by means of a general Wong–Zakai type result of convergence in probability for nonlinear stochastic PDEs driven by a Hilbert‐valued Brownian motion and some adapted finite‐dimensional approximation of this process. Note that the coupling between the Navier–Stokes system and the Cahn–Hilliard equations makes the analysis more involved.

Thermodynamically consistent diffuse-interface mixture models of incompressible multicomponent fluids

The prototypical diffuse-interface model for incompressible fluid mixtures is the Navier–Stokes Cahn–Hilliard (Allen–Cahn) model. Despite its foundation in continuum mixture theory, it is not fully compatible with this theory due to the diffusive flux approximation. This paper introduces a class of thermodynamically consistent diffuse-interface incompressible fluid mixture models that is fully compatible with the continuum theory of mixtures. The proposed models can be formulated in either constituent or mixture quantities, enabling a direct comparison with the Navier–Stokes Cahn–Hilliard (Allen–Cahn) model with non-matching densities. This comparison reveals the key modelling simplifications employed in the latter.

Computational workflow to monitor the electroosmosis of nanofluidic flow in the vicinity of a bounding surface

This investigation studies the flow of a boundary layer with electroosmotic forces on a nanofluid with gyrotactic microorganisms along the vertical Riga plate. This sort of fluid movement necessitates specific mathematical techniques and numerical simulations. In addition, the boundary-layer flow is induced by the mass and heat transfer, Joule heating, and viscous dissipation. A mathematical model is simulated by non-linear partial differential equations (PDEs). The combination of PDEs is turned into a set of non-linear ordinary differential equations using proper transformations. Some analysis tools are used to investigate the morphological characteristics of the problem while applying suitable boundary conditions. The influence of parameters on the derived solutions is numerically and visually explained through sets of figures. It is elucidated that the concentration of the microorganisms reduces due to an increase in Lewis number which leads to a decrease in the motile microorganism’s density profile. It is seen that the temperature distribution is improved when the chemical reaction and Casson parameter increase. In addition, we find that the application of electro-osmotic forces applied to the surfaces helps to dewater and separate microorganisms from incompressible solid and liquid mixtures.

Comparative analysis of entropy generation and heat transfer in a tilted partially heated square enclosure using the finite difference method

PurposeIn recent times, there has been a growing interest in buoyancy-induced heat transfer within confined enclosures due to its frequent occurrence in heat transfer processes across diverse engineering disciplines, including electronic cooling, solar technologies, nuclear reactor systems, heat exchangers and energy storage systems. Moreover, the reduction of entropy generation holds significant importance in engineering applications, as it contributes to enhancing thermal system performance. This study, a numerical investigation, aims to analyze entropy generation and natural convection flow in an inclined square enclosure filled with Ag–MgO/water and Ag–TiO2/water hybrid nanofluids under the influence of a magnetic field. The enclosure features heated slits along its bottom and left walls. Following the Boussinesq approximation, the convective flow arises from a horizontal temperature difference between the partially heated walls and the cold right wall.Design/methodology/approachThe governing equations for laminar unsteady natural convection flow in a Newtonian, incompressible mixture is solved using a Marker-and-Cell-based finite difference method within a customized MATLAB code. The hybrid nanofluid’s effective thermal conductivity and viscosity are determined using spherical nanoparticle correlations.FindingsThe numerical investigations cover various parameters, including nanoparticle volume concentration, Hartmann number, Rayleigh number, heat source/sink effects and inclination angle. As the Hartmann and Rayleigh numbers increase, there is a significant enhancement in entropy generation. The average Nusselt number experiences a substantial increase at extremely high values of the Rayleigh number and inclination.Practical implicationsThis numerical investigation explores advanced applications involving various combinations of influential parameters, different nanoparticles, enclosure inclinations and improved designs. The goal is to control fluid flow and enhance heat transfer rates to meet the demands of the Fourth Industrial Revolution.Originality/valueIn a 90° tilted enclosure, the addition of 5% hybrid nanoparticles to the base fluid resulted in a 17.139% increase in the heat transfer rate for Ag–MgO nanoparticles and a 16.4185% increase for Ag–TiO2 nanoparticles compared to the base fluid. It is observed that a 5% nanoparticle volume fraction results in an increased heat transfer rate, influenced by variations in both the Darcy and Rayleigh numbers. The study demonstrates that the Ag–MgO hybrid nanofluid exhibits superior heat transfer and fluid transport performance compared to the Ag–TiO2 hybrid nanofluid. The simulations pertain to the use of hybrid magnetic nanofluids in fuel cells, solar cavity receivers and the processing of electromagnetic nanomaterials in enclosed environments.

Exponential stability and stabilization of a stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes equations with multiplicative noise

In this article, we study the asymptotic stability of the unique strong solution to a stochastic version of a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. The model consists of the 2D Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. We establish the existence, uniqueness and regularity results for the stationary solution of the stochastic 2D nonlocal Cahn-Hilliard Navier-Stokes equations. Furthermore, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solution. Finally, we also prove a result related to the stabilization of the stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes equations.

Well‐posedness and decay for 3D non‐isothermal Navier–Stokes–Allen–Cahn system in ℝ3

In this paper, we investigate the local existence, global existence and time decay estimates for the nonisentropic Navier–Stokes–Allen–Cahn system in . This system describes the flow of a two‐phase immiscible heat‐conducting viscous incompressible mixture. In order to overcome the difficulties caused by the derivatives of double‐well potential and the nonlinear terms, we rewrite the Cahn–Hilliard part of the system as a new equivalent equation with “good” linear principle parts. Then, on the basis of the higher order norm estimates of solutions and the mollifier technique, we obtain the local well‐posedness of strong solutions. Moreover, by using pure energy method and standard continuity argument together with negative Sobolev norm estimates, one proves the global well‐posedness and time decay estimates provided that the initial data are sufficiently small.

A thermodynamic framework for heat‐conducting flows of mixtures of two interacting fluids

AbstractWithin the theory of interacting continua, we develop a model for a heat conducting mixture of two interacting fluids described in terms of the densities and the velocities for each fluid and the temperature field for the mixture as a whole. We use a general thermodynamic framework that determines the response of the material from the knowledge of two pieces of information, namely how the material stores the energy and how the entropy is produced. This information is expressed in the form of the constitutive equations for two scalars: the Helmholtz free energy and the entropy production. Additionally, we follow the goal to determine the response of a mixture from a small (minimal) set of material parameters, including shear viscosity, bulk viscosity and heat conductivity associated with the mixture as a whole and the drag coefficient connected with the interaction force between the constituents. The same thermodynamic approach is used to obtain the model when the mixture as a whole responses as an incompressible material. For both the compressible and incompressible mixtures, we investigate three variants stemming from different definitions of the (averaged) velocity associated with the mixture as a whole. We also address the issue of identification of boundary conditions for the individual constituents from the standard boundary conditions formulated in terms of the quantities associated with the mixture as a whole.

Strong solution to stochastic 2D nonlocal Cahn-Hilliard-Oldroyd model of order one: Existence and uniqueness

We consider a stochastic model for the motion of an incompressible isothermal mixture of two immiscible non-Newtonian fluids perturbed by a multiplicative noise of Gaussian and Levy type. The model consists of the stochastic Oldroyd model of order one, coupled with a stochastic nonlocal Cahn-Hilliard model. Global existence and uniqueness of strong probabilistic solution is established. Moreover, we show that the sequence of Galerkin approximation converges in mean square to the exact strong probabilistic solution of the problem.

Nonlocal Cahn–Hilliard–Hele–Shaw Systems with Singular Potential and Degenerate Mobility

We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for an incompressible mixture of two fluids. The relative concentration difference $\varphi$ is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity $\mathbf{u}$ obeys a Darcy's law depending on the so-called Korteweg force $\mu\nabla \varphi$, where $\mu$ is the nonlocal chemical potential. In addition, the kinematic viscosity $\eta$ may depend on $\varphi$. We establish first the existence of a global weak solution which satisfies the energy identity. Then we prove the existence of a strong solution. Further regularity results on the pressure and on $\mathbf{u}$ are also obtained. Weak-strong uniqueness is demonstrated in the two dimensional case. In the three-dimensional case, uniqueness of weak solutions holds if $\eta$ is constant. Otherwise, weak-strong uniqueness is shown by assuming that the pressure of the strong solution is $\alpha$-H\"{o}lder continuous in space for $\alpha\in (1/5,1)$.

Eulerian finite volume formulation using Lagrangian marker particles for incompressible fluid–structure interaction problems

AbstractWe propose a monolithic fluid–structure interaction (FSI) method that uses the cell‐centered finite volume formulation in the Eulerian description, Lagrangian marker particles allocated in the solid region, and the incompressible mixture formulation. In the proposed method, we compute all the basic equations and spatial derivatives, except the solid constitutive equations, on an Eulerian mesh to avoid neighboring particle search. Although full Eulerian methods that use a Cartesian mesh are attractive for FSI problems that require large‐scale computing and include complex geometries and the large deformation of the solid, they cannot avoid the numerical dissipation of the interfaces or internal variables of the solid caused by their advection. This computational problem has been a barrier to the industrial application of full Eulerian mesh methods. In the numerical examples, we confirmed that the proposed method retains sharp interfaces, such as the corners of a square solid, and yields more accurate numerical results for the deformation, energy, and incompressibility of a solid in fluid than our conventional Eulerian FSI method.

The Double Layer Capacity of Non-Ideal Electrolyte Solutions – a Numerical Study

The capacity of electrochemical interfaces is an important characteristic of the adjacent electrode and electrolyte materials [5,2]. In order to interpret experimental results of capacity measurements, mathematical models resolving the space charge layer are necessary [3,4,6]. These are usually based on Poisson-Nernst-Planck-type equations, which – together with equilibrium conditions – allow for a numerical computation of the differential capacity. In recent research, progress has been made in the modeling of space charge layers, most notably with taking the solvation effect into account [1,3], which leads to a coupled Poisson-momentum equation system [2,3,4]∇(ε0(1+χ)∇φ) = -q(φ,p)∇p = -q(φ,p)∇φwhere φ is the electrostatic potential, p the (local) material pressure, and q the charge density.The electrolyte is mainly considered as ideal mixture, which yields for the chemical potential μα ∼ log (yα), where yα is the mole fraction of constituent Xα (α = 0, 1, ..., N), which in turn yields in thermodynamic equilibrium an analytic expression yα = yα(φ,p) and thus q = q(φ,p) via the incompressibility constraint.Non-idealities, for example Debye-Hückel-like activity coefficients [7,8], yield more complex representations μα = μα(yα) or even μα = μα(y1, ..., yN), whereby the equilibrium condition ∇μα + e0zα∇φ = 0 cannot be employed anymore to derive an explicit expression for yα. Instead, it remains an implicit (differential) equation, which has to be solved (numerically) together with the Poisson and the momentum equation.For different examples of non-ideal chemical potential functions, e.g. Debye-Hückel-like acitivity coefficients [7,8], we derive the corresponding boundary value problems. Some remarks on mathematical issues of the derivation are given throughout the presentation and we describe the general strategy how to solve the corresponding coupled, non-linear differential equation system. Subsequently we provide numerical results for the differential capacity and the corresponding space charge layers and investigate the impact of the non-ideality by comparison to ideal, incompressible solvation mixtures.[1] M. Landstorfer. Electrochemistry Communications, 92:56–59 (2018).[2] M. Landstorfer, C. Guhlke, and W. Dreyer. Electrochimica Acta, 201:187–219 (2016).[3] W. Dreyer, C. Guhlke, and M. Landstorfer. Electrochemistry Communications, 43(0):75–78 (2014).[4] W. Dreyer, C. Guhlke, and R. Muller. Phys. Chem. Chem. Phys., 15:7075–7086 (2013).[5] G. Valette. Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, 122(0):285–297 (1981).[6] C.-L. Li and J.-L. Liu. SIAM Journal on Applied Mathematics, 80(5):2003–2023 (2020).[7] P. Debye and E. Hückel. Physikalische Zeitschrift, 24:185–206 (1923).[8] K. S. Pitzer. Activity coefficients in electrolyte solutions. CRC press, 2018.

Global–in–time existence for liquid mixtures subject to a generalised incompressibility constraint

We consider a system of partial differential equations describing diffusive and convective mass transport in a fluid mixture of N>1 chemical species. A weighted sum of the partial mass densities of the chemical species is assumed to be constant, which expresses the incompressibility of the fluid, while accounting for different reference sizes of the involved molecules. This condition is different from the usual assumption of a constant total mass density, and it leads in particular to a non-solenoidal velocity field in the Navier-Stokes equations. In turn, the pressure gradient occurs in the diffusion fluxes, so that the PDE–system of mass transport equations and momentum balance is fully coupled. Another striking feature of such incompressible mixtures is the algebraic formula connecting the pressure and the densities, which can be exploited to prove a pressure bound in L1. In this paper, we consider incompressible initial states with bounded energy and show the global existence of weak solutions with defect measure.

Transfer equation for the strain rate tensor and description of an incompressible dispersed mixture (incompressible fluid) by a system of equations of dynamic type

It is known that the application of the vector operation rot to the equations of hydrodynamics leads to the Helmholtz-Friedman equation for a vortex. A dispersed mixture, tensor transformations are used, in a certain sense generalizing the vector operation rot, which gives more than one, a couple of equations. One of them describes the transfer of vorticity is the well-known Helmholtz-Friedman equation. The second equation was obtained for the first time, and it describes the transfer of the strain rate tensor. Any tensor decomposes into symmetric and antisymmetric parts. By definition, the symmetric part of the tensor U is the strain rate tensor. The antisymmetric part of U is a tensor whose components are related in a known manner to the pseudovector angular velocity.

Approximation of a stochastic two-phase flow model by a splitting-up method

<p style='text-indent:20px;'>In this paper, we consider a stochastic Allen-Cahn Navier-Stokes system in a bounded domain of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^d, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ d = 2,3 $\end{document}</tex-math></inline-formula>. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on splitting-up method as well as some compactness method.

Three-dimensional numerical investigation of the displacement of shear-thinning milk protein concentrates in spacer-filled channel

Rinsing is an important part of the cleaning procedure applied to the spiral-wound membrane (SWM) modules in the dairy industry. In this work, we numerically investigated the displacement of non-Newtonian reconstituted milk protein concentrates in a feed spacer channel using OpenFOAM and compared the results with experimental data obtained in an SWM pilot plant. The model consists of the classical mass and momentum balance equations for an incompressible viscous mixture coupled with an advection-diffusion equation for the milk concentration. The power-law model was utilized to describe the shear-thinning viscosity of the milk protein concentrates. The shear-thinning behavior was further assumed to depend on the milk protein content. The calculations confirmed significantly shorter rinsing times and lower mixed-phase volumes for lower milk protein concentrations. The rinsing time can further be lowered by increasing the volumetric flow rate. Through the numerical simulations, we identified nearly inaccessible zones behind the filament crossings that prolong the rinsing process.

Strong solutions for the stochastic Cahn-Hilliard-Navier-Stokes system

A well-known diffuse interface model consists of the Navier-Stokes equations for the average velocity, nonlinearly coupled with a convective Cahn-Hilliard type equation for the order (phase) parameter. This system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors. Here we consider a stochastic version of this model forced by a multiplicative white noise on a bounded domain of Rd, d=2,3. We prove the existence and uniqueness of a local maximal strong solution when the initial data (u0,ϕ0) takes values in H1×H2. Moreover in the two-dimensional case, we prove that our solution is global.

Global Existence of Martingale Solutions and Large Time Behavior for a 3D Stochastic Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Shear Dependent Viscosity

In this paper, we consider a stochastic version of a nonlinear system which consists of the incompressible Navier–Stokes equations with shear dependent viscosity controlled by a power $$p>2$$ , coupled with a convective nonlocal Cahn–Hilliard-equations. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluid having the same density. We prove the existence of a weak martingale solutions when $$p\in [11/5,12/5)$$ , and their exponential decay when the time goes to infinity.

Existence of a solution to the stochastic nonlocal Cahn–Hilliard Navier–Stokes model via a splitting-up method

We consider a stochastic diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids under the influence of stochastic external forces in a bounded domain of , d = 2, 3. The model consists of the stochastic Navier–Stokes equations coupled with a nonlocal Cahn–Hilliard equation. We prove the existence of a global weak martingale solution via a numerical scheme based on splitting-up method.

Splitting-up scheme for the stochastic Cahn–Hilliard Navier–Stokes model

In this paper, we consider a stochastic Cahn–Hilliard Navier–Stokes system in a bounded domain of [Formula: see text] [Formula: see text]. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on the splitting-up method as well as some compactness method.

Artificial Compressibility Method for the Navier–Stokes–Maxwell–Stefan System

The Navier–Stokes–Maxwell–Stefan system describes the dynamics of an incompressible gaseous mixture in isothermal condition. In this paper we set up an artificial compressibility type approximation. In particular we focus on the existence of solution for the approximated system and the convergence to the incompressible case. The existence of the approximating system is proved by means of semidiscretization in time and by estimating the fractional time derivative.