Browsing by Author "Lai, Ming-Chih"

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A conservative scheme for solving coupled surface-bulk convection-diffusion equations with an application to interfacial flows with soluble surfactant
(2014-01-15)
Chen, Kuan-Yu
;
Lai, Ming-Chih
;
應用數學系
;
數學建模與科學計算所(含中心)
;
Department of Applied Mathematics
Many physical problems arising in biological or material sciences involve solving partial differential equations in deformable interfaces or complex domains. For instance, the surfactant (an amphiphilic molecular) which usually favors the presence in the fluid interface may couple with the surfactant soluble in one of bulk domains through adsorption and desorption processes. Thus, it is important to accurately solve coupled surface-bulk convection-diffusion equations especially when the interface is moving. In this paper, we first rewrite the original bulk concentration equation in an irregular domain (soluble region) into a regular computational domain via the usage of the indicator function so that the concentration flux across the interface due to adsorption and desorption processes can be termed as a singular source in the modified equation. Based on the immersed boundary formulation, we then develop a new conservative scheme for solving this coupled surface-bulk concentration equations which the total surfactant mass is conserved in discrete sense. A series of numerical tests has been conducted to validate the present scheme. As an application, we extend our previous work [M.-C. Lai, Y.-H. Tseng, H. Huang, An immersed boundary method for interfacial flows with insoluble surfactant, J. Comput. Phys. 227 (2008) 7279-7293] to the soluble case. The effects of solubility of surfactant on drop deformations in a quiescent and shear flow are investigated in detail. (C) 2013 Elsevier Inc. All rights reserved.
A coupled grid based particle and implicit boundary integral method for two-phase flows with insoluble surfactant
(2019-10-15)
Hsu, Shih-Hsuan
;
Chu, Jay
;
Lai, Ming-Chih
;
Tsai, Richard
;
應用數學系
We develop a coupled grid based particle and implicit boundary integral method for simulation of three-dimensional interfacial flows with the presence of insoluble surfactant. The grid based particle method (GBPM, Leung and Zhao [20]) tracks the interface by the projection of the neighboring Eulerian grid points and does not require stitching of parameterizations nor body fitted moving meshes. Using this GBPM to represent the interface, the surfactant equation defined on the interface is discretized naturally following a new volumetric constant-along-surface-normal extension approach (Chu and Tsai [4]). We first examine the proposed scheme to solve the convection-diffusion equation for the problems with available analytical solutions. The numerical results demonstrate secondorder accuracy of the scheme. We then perform a series of simulations for interfacial flows with insoluble surfactant. The numerical results agree well with the theory, and are comparable with other numerical works in literature. (C) 2019 Elsevier Inc. All rights reserved.
A coupled immersed boundary and immersed interface method for interfacial flows with soluble surfactant
(2018-05-30)
Hu, Wei-Fan
;
Lai, Ming-Chih
;
Misbah, Chaouqi
;
應用數學系
;
Department of Applied Mathematics
In this paper, we develop a coupled immersed boundary (IB) and immersed interface method (IIM) to simulate the interfacial flow problems with soluble surfactant. That is, a coupled system of surface-bulk convection-diffusion equations must be solved not only on a moving interface but also in an evolving irregular domain. Based on the immersed interface framework, we first begin with the numerical development for the diffusion equation in a fixed irregular domain and then extend the scheme to solve the convection-diffusion equation in an evolving domain. The fluid motion governed by the incompressible Navier-Stokes equations is solved by using traditional immersed boundary method while the bulk surfactant is solved by the proposed immersed interface method. A series of numerical tests for the present scheme have been conducted to illustrate the accuracy and applicability of the method. We first perform the accuracy and efficiency tests for the present IIM solver. We then check the convergence of the surface and bulk surfactant and the fluid variables in the interfacial flow problems. We further run a series of numerical simulations for a suspended droplet under shear flow with presence of soluble surfactant to study the effects of the dimensionless Biot number and the bulk Peclet number on the droplet deformation in details. (C) 2018 Elsevier Ltd. All rights reserved.
A coupled immersed interface and grid based particle method for three-dimensional electrohydrodynamic simulations
(2019-12-01)
Hsu, Shih-Hsuan
;
Hu, Wei-Fan
;
Lai, Ming-Chih
;
應用數學系
;
Department of Applied Mathematics
In the present work, we propose a coupled immersed interface and grid based particle method to solve two-phase electrohydrodynamic problems in three dimensions. The problem considers a leaky dielectric (weakly conducting) droplet immersed in another leaky dielectric fluid under electric field where the non-homogeneous droplet surface charge effect is taken into account. Due to the mismatch of electrical properties between two fluids, the electric potential satisfying Laplace equation with jump conditions across the droplet surface is coupled with the conservation equation for the surface charge density. Consequently, we first develop a three-dimensional augmented immersed interface method (IIM) which incorporates some known jump conditions naturally along the normal direction and check the desired accuracy. Here, the grid based particle method (GBPM) is used to track the interface by the projection of the neighboring Eulerian grid points so no requirement for stitching of parameterizations nor body fitted moving meshes. Within the leaky dielectric framework, the electric stress can be treated as an interfacial force so that both the surface tension and electric force can be formulated in a unified continuum force in the Navier-Stokes equations. A series of numerical tests have been carefully conducted to illustrate the accuracy and applicability of the present method to simulate droplet electrohydrodynamics. In particular, we investigate the droplet equilibrium dynamics under weak and strong electric fields in detail. It is interesting to find out a chaotic tumbling motion with irregular rotating modes which we believe that is the first numerical verification to the recent experiments. (C) 2019 Elsevier Inc. All rights reserved.
A Coupled Immersed Interface and Level Set Method for Three-Dimensional Interfacial Flows with Insoluble Surfactant
(2014-02-01)
Xu, Jian-Jun
;
Huang, Yunqing
;
Lai, Ming-Chih
;
Li, Zhilin
;
應用數學系
In this paper, a numerical method is presented for simulating the 3D interfacial flows with insoluble surfactant. The numerical scheme consists of a 3D immersed interface method (IIM) for solving Stokes equations with jumps across the interface and a 3D level-set method for solving the surfactant convection-diffusion equation along a moving and deforming interface. The 3D IIM Poisson solver modifies the one in the literature by assuming that the jump conditions of the solution and the flux are implicitly given at the grid points in a small neighborhood of the interface. This assumption is convenient in conjunction with the level-set techniques. It allows standard Lagrangian interpolation for quantities at the projection points on the interface. The interface jump relations are re-derived accordingly. A novel rotational procedure is given to generate smooth local coordinate systems and make effective interpolation. Numerical examples demonstrate that the IIM Poisson solver and the Stokes solver achieve second-order accuracy. A 3D drop with insoluble surfactant under shear flow is investigated numerically by studying the influences of different physical parameters on the drop deformation.
A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical coordinates
(2007-04-01)
Lai, Ming-Chih
;
Tseng, Jui-Ming
;
應用數學系
;
Department of Applied Mathematics
In this paper, we extend our previous work (M.-C. Lai, A simple compact fourth-order Poisson solver on polar geometry, J. Comput. Phys. 182 (2002) 337-345) to 3D cases. More precisely, we present a spectral/finite difference scheme for Poisson equation in cylindrical coordinates. The scheme relies on the truncated Fourier series expansion, where the partial differential equations of Fourier coefficients are solved by a formally fourth-order accurate compact difference discretization. Here the formal fourth-order accuracy means that the scheme is exactly fourth-order accurate while the poles are excluded and is third-order accurate otherwise. Despite the degradation of one order of accuracy due to the presence of poles, the scheme handles the poles naturally; thus, no pole condition is needed. The resulting linear system is then solved by the Bi-CGSTAB method with the preconditioner arising, from the second-order discretization which shows the scalability with the problem size. (c) 2006 Elsevier B.V. All tights reserved.
A FRACTIONAL STEP IMMERSED BOUNDARY METHOD FOR STOKES FLOW WITH AN INEXTENSIBLE INTERFACE ENCLOSING A SOLID PARTICLE
(2012)
Lai, Ming-Chih
;
Hu, Wei-Fan
;
Lin, Wen-Wei
;
應用數學系
;
數學建模與科學計算所(含中心)
In this paper, we develop a fractional step method based on the immersed boundary (IB) formulation for Stokes flow with an inextensible (incompressible) interface enclosing a solid particle. In addition to solving for the fluid variables such as the velocity and pressure, the present problem involves finding an extra unknown elastic tension such that the surface divergence of the velocity is zero along the interface, and an extra unknown particle surface force such that the velocity satisfies the no-slip boundary condition along the particle surface. While the interface moves with local fluid velocity, the enclosed particle hereby undergoes a rigid body motion, and the system is closed by the force-free and torque-free conditions along the particle surface. The equations are then discretized by standard centered difference schemes on a staggered grid, and the interactions between the interface and particle with the fluid are discretized using a discrete delta function as in the IB method. The resultant linear system of equations is symmetric and can be solved by fractional steps so that only fast Poisson solvers are involved. The present method can be extended to Navier-Stokes flow with moderate Reynolds number by treating the nonlinear advection terms explicitly for the time integration. The convergent tests for a Stokes solver with or without an inextensible interface are performed and confirm the desired accuracy. The tank-treading to tumbling motion for an inextensible interface enclosing a solid particle with different filling fractions under a simple shear flow has been studied extensively, and the results here are in good agreement with those obtained in literature.
A Front-Tracking Method for Motion by Mean Curvature with Surfactant
(2009-04-01)
Lai, Ming-Chih
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Hsu, Che-Wei
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Huang, Huaxiong
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應用數學系
;
Department of Applied Mathematics
In this paper, we present a finite difference method to track a network of curves whose motion is determined by mean curvature. To study the effect of inhomogeneous surface tension on the evolution of the network of curves, we include surfactant which can diffuse along the curves. The governing equations consist of one parabolic equation for the curve motion coupled with a convection-diffusion equation for the surfactant concentration along each curve. Our numerical method is based on a direct discretization of the governing equations which conserves the total surfactant mass in the curve network. Numerical experiments are carried out to examine the effects of inhomogeneous surface tension on the motion of the network, including the von Neumann law for cell growth in two space dimensions.
A hybrid immersed boundary and immersed interface method for electrohydrodynamic simulations
(2015-02-01)
Hu, Wei-Fan
;
Lai, Ming-Chih
;
Young, Yuan-Nan
;
應用數學系
;
數學建模與科學計算所(含中心)
In this paper, we develop a hybrid immersed boundary (IB) and immersed interface method (IIM) to simulate the dynamics of a drop under an electric field in Navier-Stokes flows. Within the leaky dielectric framework with piecewise constant electric properties in each fluid, the electric stress can be treated as an interfacial force on the drop interface. Thus, both the electric and capillary forces can be formulated in a unified immersed boundary framework. The electric potential satisfies a Laplace equation which is solved numerically by an augmented immersed interface method which incorporates the jump conditions naturally along the normal direction. The incompressible Navier-Stokes equations for the fluids are solved using a projection method on a staggered MAC grid and the potential is solved at the cell center. The interface is tracked in a Lagrangian manner with mesh control by adding an artificial tangential velocity to transport the Lagrangian markers to ensure that the spacing between markers is uniform throughout the computations. A series of numerical tests for the present scheme have been conducted to illustrate the accuracy and applicability of the method. We first compute the potential and its gradient (electric field) to perform the accuracy check for the present augmented IIM. We then check the convergence of the interfacial electric force and the fluid variables. We further run a series of simulations with different permittivity and conductivity ratios and compare with the results obtained by the small deformation theory and other numerical results in literature. In addition, we also study the electric effect for a drop under shear flow. (C) 2014 Elsevier Inc. All rights reserved.
A least squares augmented immersed interface method for solving Navier-Stokes and Darcy coupling equations
(2018-05-15)
Li, Zhilin
;
Lai, Ming-Chih
;
Peng, Xiaofei
;
Zhang, Zhiyue
;
應用數學系
A new finite difference method based on Cartesian meshes and fast Poisson/Helmholtz solvers is proposed to solve the coupling of a fluid flow modeled by the incompressible Navier-Stokes equations and a porous media modeled by the Darcy's law. The finite difference discretization in time is based on the pressure Poisson equation formulation. At each time step, several augmented variables along the interface between the fluid flow and the porous media are introduced so that the coupled equations can be decoupled into several Poisson/Helmholtz equations with those augmented variables acting as jumps of the unknown solution and some directional derivatives. The augmented variables should be chosen so that the Beavers-Joseph-Saffman (BJS) or Beavers-Joseph (BJ)and other interface conditions are satisfied. It has been tested that a direct extension of the augmented idea in [27] does not work well when the fluid flow is modeled by the Navier-Stokes equations. One of the new ideas of this paper is to enforce the divergence free condition at the interface from the fluid side. In this way, the Schur complement matrix for the augmented variables is over-determined and the least squares solution is used for the coupling problem. The new augmented approach enables us to solve the Navier-Stokes and Darcy coupling efficiently with second order accurate velocity and pressure in the L-infinity norm for tested problems. The proposed new idea in enforcing the divergence free condition at the interface from the fluid side has also been utilized to solve the Stokes and Darcy coupling equations and shown to outperform the original method in [27]. In additional to the detailed accuracy check for the present method, some interesting numerical simulations for Navier-Stokes and Darcy coupling have been conducted in this paper as well. (C) 2018 Elsevier Ltd. All rights reserved.
A level-set method for two-phase flows with soluble surfactant
(2018-01-15)
Xu, Jian-Jun
;
Shi, Weidong
;
Lai, Ming-Chih
;
應用數學系
;
Department of Applied Mathematics
A level-set method is presented for solving two-phase flows with soluble surfactant. The Navier-Stokes equations are solved along with the bulk surfactant and the interfacial surfactant equations. In particular, the convection-diffusion equation for the bulk surfactant on the irregular moving domain is solved by using a level-set based diffusive-domain method. A conservation law for the total surfactant mass is derived, and a re-scaling procedure for the surfactant concentrations is proposed to compensate for the surfactant mass loss due to numerical diffusion. The whole numerical algorithm is easy for implementation. Several numerical simulations in 2D and 3D show the effects of surfactant solubility on drop dynamics under shear flow. (C) 2017 Elsevier Inc. All rights reserved.
A note on pressure accuracy in immersed boundary method for Stokes flow
(2011-06-01)
Chen, Kuan-Yu
;
Feng, Ko-An
;
Kim, Yongsam
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Lai, Ming-Chih
;
應用數學系
In this short note, we provide a simplified one-dimensional analysis and two-dimensional numerical experiments to predict that the overall accuracy for the pressure or indicator function in immersed boundary calculations is first-order accurate in L(1) norm, half-order accurate in L(2) norm, but has O(1) error in L(infinity) Despite the pressure has O(1) error near the interface, the velocity field still has the first-order convergence in immersed boundary calculations. (C) 2011 Elsevier Inc. All rights reserved.
A PARAMETRIC DERIVATION OF THE SURFACTANT TRANSPORT EQUATION ALONG A DEFORMING FLUID INTERFACE
(2008-01-01)
Huang, Huaxng
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Lai, Ming-Chih
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Tseng, Hsiao-Chieh
;
應用數學系
;
Department of Applied Mathematics
A parametric derivation of the surfactant transport equation along a deforming fluid interface is presented in this paper. The derivation is based on the Lagrangian formulation of the interface with a parametric representation. Comparisons with some of the existing derivations are also given.
A penalty immersed boundary method for viscoelastic particulate flows
(2018-08-01)
Kim, Yongsam
;
Lai, Ming-Chih
;
Seol, Yunchang
;
應用數學系
;
丘成桐中心
We extend the penalty immersed boundary (pIB) method to investigate the interaction between circular rigid particles and a surrounding viscoelastic Oldroyd-B fluid. The basic idea of the pIB method is the splitting of an immersed boundary, which here is a rigid body, notionally into two Lagrangian components: one is a massive component carrying all mass of the rigid body, and the other is massless. These two components are connected by a system of stiff springs with zero rest length. The massive component has no direct interaction with the surrounding fluid and behaves as though in a vacuum, following the dynamics of a rigid body, in which the acting forces and torques are generated from the system of stiff springs that connects the two Lagrangian components. The massless component interacts with the surrounding Oldroyd-B fluid: it moves at the local fluid velocity and exerts force locally on the fluid. We verify the pIB method combined with Oldroyd-B fluid model by investigating the effects of the wall and elasticity of the fluid on the lateral position of a circular particle falling under the influence of gravity and by studying convergence of the numerical solutions. We also simulate the interaction between multiple circular particles and the surrounding Oldroyd-B fluid and compare the dynamics of the particles in various flow conditions.
A short note on Navier-Stokes flows with an incompressible interface and its approximations
(2017-03)
Lai, Ming-Chih
;
Seol, Yunchang
;
應用數學系
;
Department of Applied Mathematics
In biological applications, a cell membrane consisting of a lipid bilayer usually behaves as fluid-like interface with surface incompressibility. Here we consider a mathematical formulation for an incompressible interface immersed in Navier-Stokes flows and study the mathematical and physical features for this incompressible interface. The model formulation introduces an unknown tension which acts as a Lagrange\'s multiplier to enforce such surface incompressibility. In this note, we show that the spreading operator of the tension and the surface divergence operator of the velocity are skew-adjoint with each other which indicates physically that the tension does not do extra work to the fluid under the condition of surface incompressibility. In order to avoid solving the unknown tension to enforce the surface incompressibility, we adopt a nearly surface incompressible approach (or penalty approach) by introducing two different modified elastic tensions which can be used efficiently in practical numerical simulations. Furthermore, we show that the resultant modified elastic forces have the same mathematical form as the original one derived from the unknown tension. (C) 2016 Elsevier Ltd. All rights reserved.
A simple implementation of the immersed interface methods for Stokes flows with singular forces
(2008-02-01)
Lai, Ming-Chih
;
Tseng, Hsiao-Chieh
;
應用數學系
;
Department of Applied Mathematics
In this paper, we introduce a simple version of the immersed interface method (IIM) for Stokes flows with singular forces along an interface. The numerical method is based on applying the Taylor's expansions along the normal direction and incorporating the solution and its normal derivative jumps into the finite difference approximations. The fluid variables are solved in a staggered grid, and a new accurate interpolating scheme for the non-smooth velocity has been developed. The numerical results show that the scheme is second-order accurate. (c) 2007 Elsevier Ltd. All rights reserved.
A simple projection method for the coupled Navier-Stokes and Darcy flows
(2019-02-01)
Lai, Ming-Chih
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Shiue, Ming-Cheng
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Ong, Kian Chuan
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應用數學系
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Department of Applied Mathematics
The flow through both an incompressible fluid region and a porous medium occurs in a wide range of applications encompassing industrial processes and geological phenomena. Recently, the fluid transport phenomena across the interface between the distinct regions have received increasing attention both from the mathematical and the numerical points of view. The primary objective of the present study lies in the development of a simple and efficient method for the computations of coupled Navier-Stokes and Darcy flows with complex interface conditions. A finite difference projection method is developed within a staggered grid framework to solve the coupled system in a segregated manner using primitive variables. Numerical simulations are carried out to demonstrate the order of convergence and its capability. The proposed method renders the versatility in solving the coupled system, and it is readily extendible to multi-physics fluid flows and turbulent flows for a broad range of applications.
A well-conditioned augmented system for solving Navier-Stokes equations in irregular domains
(2009-04-20)
Ito, Kazufumi
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Lai, Ming-Chih
;
Li, Zhilin
;
應用數學系
;
Department of Applied Mathematics
An augmented method based on a Cartesian grid is proposed for the incompressible Navier-Stokes equations in irregular domains. The irregular domain is embedded into a rectangular one so that a fast Poisson solver can be utilized in the projection method. Unlike several methods suggested in the literature that set the force strengths as unknowns, which often results in an ill-conditioned linear system, we set the jump in the normal derivative of the velocity as the augmented variable. The new approach improves the condition number of the system for the augmented variable significantly. Using the immersed interface method, we are able to achieve second order accuracy for the velocity. Numerical results and comparisons to benchmark tests are given to validate the new method. A lid-driven cavity flow with multiple obstacles and different geometries are also presented. (C) 2008 Elsevier Inc. All rights reserved.
ADAPTIVE MESH REFINEMENT FOR ELLIPTIC INTERFACE PROBLEMS USING THE NON-CONFORMING IMMERSED FINITE ELEMENT METHOD
(2011)
Wu, Chin-Tien
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Li, Zhilin
;
Lai, Ming-Chih
;
應用數學系
;
數學建模與科學計算所(含中心)
In this paper, an adaptive mesh refinement technique is developed and analyzed for the non-conforming immersed finite element (IFE) method proposed in [27]. The IFE method was developed for solving the second order elliptic boundary value problem with interfaces across which the coefficient may be discontinuous. The IFE method was based on a triangulation that does not need to fit the interface. One of the key ideas of IFE method is to modify the basis functions so that the natural jump conditions are satisfied across the interface. The IFE method has shown to be order of O(h(2)) and O(h) in L(2) norm and H(1) norm, respectively. In order to develop the adaptive mesh refinement technique, additional priori and posterior error estimations are derived in this paper. Our new a-priori error estimation shows that the generic constant is only linearly proportional to ratio of the diffusion coefficient beta(-) and beta(+), which improves the corresponding result in [27]. We also show that a-posteriori error estimate similar to the one obtained by Bernardi and Verfurth [4] holds for the IFE solutions. Numerical examples support our theoretical results and show that the adaptive mesh refinement strategy is effective for the IFE approximation.
Amoeboid swimming in a channel
(2016)
Wu, Hao
;
Farutin, Alexander
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Hu, Wei-Fan
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Thiebaud, Marine
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Rafai, Salima
Several micro-organisms, such as bacteria, algae, or spermatozoa, use flagellar or ciliary activity to swim in a fluid, while many other micro-organisms instead use ample shape deformation, described as amoeboid, to propel themselves either by crawling on a substrate or swimming. Many eukaryotic cells were believed to require an underlying substratum to migrate (crawl) by using membrane deformation (like blebbing or generation of lamellipodia) but there is now increasing evidence that a large variety of cells (including those of the immune system) can migrate without the assistance of focal adhesion, allowing them to swim as efficiently as they can crawl. This paper details the analysis of amoeboid swimming in a confined fluid by modeling the swimmer as an inextensible membrane deploying local active forces (with zero total force and torque). The swimmer displays a rich behavior: it may settle into a straight trajectory in the channel or navigate from one wall to the other depending on its confinement. The nature of the swimmer is also found to be affected by confinement: the swimmer can behave, on average over one swimming cycle, as a pusher at low confinement, and becomes a puller at higher confinement, or vice versa. The swimmer\'s nature is thus not an intrinsic property. The scaling of the swimmer velocity V with the force amplitude A is analyzed in detail showing that at small enough A, V similar to A(2)/eta(2) (where eta is the viscosity of the ambient fluid), whereas at large enough A, V is independent of the force and is determined solely by the stroke cycle frequency and the swimmer size. This finding starkly contrasts with models where motion is based on ciliary and flagellar activity, where V similar to A/eta. To conclude, two definitions of efficiency as put forward in the literature are analyzed with distinct outcomes. We find that one type of efficiency has an optimum as a function of confinement while the other does not. Future perspectives are outlined.