A high-resolution finite-difference method for simulating two-fluid, viscoelastic gel dynamics

Wright, Grady B.; Guy, Robert D.; Du, Jian; Fogelson, Aaron L.

doi:10.1016/j.jnnfm.2011.07.002

Cited by 8 publications

(9 citation statements)

References 35 publications

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“…Our development of a two-material immersed boundary method draws on our previous work on numerical methods for two-material mixtures [24] and on the immersed boundary methodology for a single fluid [1], so we begin this section by introducing the two-material model and reviewing the classical immersed boundary method.…”

Section: Model Equations and Numerical Methodsmentioning

confidence: 99%

“…We are modeling the network as an Oldroyd-B fluid with spatially and temporally varying elastic modulus. See [24] for a more detailed discussion of the model.…”

Section: Model Equations and Numerical Methodsmentioning

confidence: 99%

“…(3), (16) and (17) as described in detail in [24]. We write the equations in matrix-vector form as:

\nabla \cdot (θ^{normaln} τ) + \nabla \cdot (θ^{normaln} σ^{normaln}) - ξ θ^{normaln} θ^{normals} ({boldu}^{normaln} - {boldu}^{normals}) - θ^{normaln} \nabla p + θ^{normaln} {boldf}_{o}^{normaln} + θ^{normaln} θ^{normals} {boldf}_{p}^{normaln} = 0,

where

\begin{matrix} right L_{normaln, normals} = & left [\begin{matrix} α_{n, s} \partial_{x} (θ^{n, s} \partial_{x}) + μ_{n, s} \partial_{y} (θ^{n, s} \partial_{y}) & μ_{n, s} \partial_{y} (θ^{n, s} \partial_{x}) + λ_{n, s} \partial_{x} (θ^{n, s} \partial_{y}) \\ μ_{n, s} \partial_{x} (θ^{n, s} \partial_{y}) + λ_{n, s} \partial_{y} (θ^{n, s} \partial_{x}) & α_{n, s} \partial_{y} (θ^{n, s} \partial_{y}) + μ_{n, s} \partial_{x} (θ^{n, s} \partial_{<...>} \end{matrix} \end{matrix}

…”

Section: Model Equations and Numerical Methodsmentioning

confidence: 99%

“…When discretized, these equations lead to a large, sparse linear system of saddle point type. A multigrid preconditioned GMRES solver is used to solve the system of equations [33, 24]. The transport equation (1) is then solved by the second-order corner transport upwind (CTU) scheme as described in [27].…”

Section: Model Equations and Numerical Methodsmentioning

confidence: 99%

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An immersed boundary method for two-fluid mixtures

Guy

Fogelson

2014

Journal of Computational Physics

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We present an Immersed Boundary method for interactions between elastic boundaries and mixtures of two fluids. Each fluid has its own velocity field and volume-fraction. A penalty method is used to enforce the condition that both fluids’ velocities agree with that of the elastic boundaries. The method is applied to several problems: Taylor’s swimming sheet problem for a mixture of two viscous fluids, peristaltic pumping of a mixture of two viscous fluids, with and without immersed particles, and peristaltic pumping of a mixture of a viscous fluid and a viscoelastic fluid. The swimming sheet and peristalsis problems have received much attention recently in the context of a single viscoelastic fluid. Numerical results demonstrate that the method converges and show its capability to handle a number of flow problems of substantial current interest. They illustrate that for each of these problems, the relative motion between the two fluids changes the observed behaviors profoundly compared to the single fluid case.

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Section: Model Equations and Numerical Methodsmentioning

confidence: 99%

“…We are modeling the network as an Oldroyd-B fluid with spatially and temporally varying elastic modulus. See [24] for a more detailed discussion of the model.…”

Section: Model Equations and Numerical Methodsmentioning

confidence: 99%

“…(3), (16) and (17) as described in detail in [24]. We write the equations in matrix-vector form as:

\nabla \cdot (θ^{normaln} τ) + \nabla \cdot (θ^{normaln} σ^{normaln}) - ξ θ^{normaln} θ^{normals} ({boldu}^{normaln} - {boldu}^{normals}) - θ^{normaln} \nabla p + θ^{normaln} {boldf}_{o}^{normaln} + θ^{normaln} θ^{normals} {boldf}_{p}^{normaln} = 0,

where

\begin{matrix} right L_{normaln, normals} = & left [\begin{matrix} α_{n, s} \partial_{x} (θ^{n, s} \partial_{x}) + μ_{n, s} \partial_{y} (θ^{n, s} \partial_{y}) & μ_{n, s} \partial_{y} (θ^{n, s} \partial_{x}) + λ_{n, s} \partial_{x} (θ^{n, s} \partial_{y}) \\ μ_{n, s} \partial_{x} (θ^{n, s} \partial_{y}) + λ_{n, s} \partial_{y} (θ^{n, s} \partial_{x}) & α_{n, s} \partial_{y} (θ^{n, s} \partial_{y}) + μ_{n, s} \partial_{x} (θ^{n, s} \partial_{<...>} \end{matrix} \end{matrix}

…”

Section: Model Equations and Numerical Methodsmentioning

confidence: 99%

Section: Model Equations and Numerical Methodsmentioning

confidence: 99%

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An immersed boundary method for two-fluid mixtures

Guy

Fogelson

2014

Journal of Computational Physics

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“…The term θ n ψ(θ n ) is an additional pressure which can be used to represent osmotic effects [17,18] that can cause the mixture to swell or deswell, or to represent active-contractile processes such as those in the cell cytoplasm [19]. We have developed numerical algorithms for solving these equations in the cases that the solvent phase is a viscous fluid and the network phase is either a viscous fluid [20] or an Oldroyd-B viscoelastic fluid [21]. Many biological problems involve free boundaries whose dynamics are determined as part of the solution.…”

Section: Introductionmentioning

confidence: 99%

An Interface-Capturing Regularization Method for Solving the Equations for Two-Fluid Mixtures

Guy

Fogelson

et al. 2013

Commun. comput. phys.

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Abstract. Many problems in biology involve gels which are mixtures composed of a polymer network permeated by a fluid solvent (water). The two-fluid model is a widely used approach to described gel mechanics, in which both network and solvent coexist at each point of space and their relative abundance is described by their volume fractions. Each phase is modeled as a continuum with its own velocity and constitutive law. In some biological applications, free boundaries separate regions of gel and regions of pure solvent, resulting in a degenerate network momentum equation where the network volume fraction vanishes. To overcome this difficulty, we develop a regularization method to solve the two-phase gel equations when the volume fraction of one phase goes to zero in part of the computational domain. A small and constant network volume fraction is temporarily added throughout the domain in setting up the discrete linear equations and the same set of equation is solved everywhere. These equations are very poorly conditioned for small values of the regularization parameter, but the multigrid-preconditioned GMRES method we use to solve them is efficient and produces an accurate solution of these equations for the full range of relevant regularization parameter values.

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Low Reynolds Number Swimming with Slip Boundary Conditions

Alshehri

Althobaiti

2020

Lecture Notes in Computer Science

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We investigate the classical Taylor's swimming sheet problem in a viscoelastic fluid, as well as in a mixture of a viscous fluid and a viscoelastic fluid. Extensions of the standard Immersed Boundary (IB) Method are proposed so that the fluid media may satisfy partial slip or free-slip conditions on the moving boundary. Our numerical results indicate that slip may lead to substantial speed enhancement for swimmers in a viscoelastic fluid and in a viscoelastic two-fluid mixture. Under the slip conditions, the speed of locomotion is dependent in a nontrivial way on both the viscosity and elasticity of the fluid media. In a two-fluid mixture with free-slip network, the swimming speed is also significantly affected by the drag coefficient and the network volume fraction.

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A high-resolution finite-difference method for simulating two-fluid, viscoelastic gel dynamics

Cited by 8 publications

References 35 publications

An immersed boundary method for two-fluid mixtures

An immersed boundary method for two-fluid mixtures

An Interface-Capturing Regularization Method for Solving the Equations for Two-Fluid Mixtures

Low Reynolds Number Swimming with Slip Boundary Conditions

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