skip to main content


Title: Passive concentration dynamics incorporated into the library IB2d, a two-dimensional implementation of the immersed boundary method
Abstract In this paper, we present an open-source software library that can be used to numerically simulate the advection and diffusion of a chemical concentration or heat density in a viscous fluid where a moving, elastic boundary drives the fluid and acts as a source or sink. The fully-coupled fluid-structure interaction problem of an elastic boundary in a viscous fluid is solved using Peskin’s immersed boundary method. The addition or removal of the concentration or heat density from the boundary is solved using an immersed boundary-like approach in which the concentration is spread from the immersed boundary to the fluid using a regularized delta function. The concentration or density over time is then described by the advection-diffusion equation and numerically solved. This functionality has been added to our software library, IB2d , which provides an easy-to-use immersed boundary method in two dimensions with full implementations in MATLAB and Python. We provide four examples that illustrate the usefulness of the method. A simple rubber band that resists stretching and absorbs and releases a chemical concentration is simulated as a first example. Complete convergence results are presented for this benchmark case. Three more biological examples are presented: (1) an oscillating row of cylinders, representative of an idealized appendage used for filter-feeding or sniffing, (2) an oscillating plate in a background flow is considered to study the case of heat dissipation in a vibrating leaf, and (3) a simplified model of a pulsing soft coral where carbon dioxide is taken up and oxygen is released as a byproduct from the moving tentacles. This method is applicable to a broad range of problems in the life sciences, including chemical sensing by antennae, heat dissipation in plants and other structures, the advection-diffusion of morphogens during development, filter-feeding by marine organisms, and the release of waste products from organisms in flows.  more » « less
Award ID(s):
2111765 1853608
NSF-PAR ID:
10321896
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
Bioinspiration & Biomimetics
Volume:
17
Issue:
3
ISSN:
1748-3182
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Interactions between an evolving solid and inviscid flow can result in substantial computational complexity, particularly in circumstances involving varied boundary conditions between the solid and fluid phases. Examples of such interactions include melting, sublimation, and deflagration, all of which exhibit bidirectional coupling, mass/heat transfer, and topological change of the solid–fluid interface. The diffuse interface method is a powerful technique that has been used to describe a wide range of solid-phase interface-driven phenomena. The implicit treatment of the interface eliminates the need for cumbersome interface tracking, and advances in adaptive mesh refinement have provided a way to sufficiently resolve diffuse interfaces without excessive computational cost. However, the general scale-invariant coupling of these techniques to flow solvers has been relatively unexplored. In this work, a robust method is presented for treating diffuse solid–fluid interfaces with arbitrary boundary conditions. Source terms defined over the diffuse region mimic boundary conditions at the solid–fluid interface, and it is demonstrated that the diffuse length scale has no adverse effects. To show the efficacy of the method, a one-dimensional implementation is introduced and tested for three types of boundaries: mass flux through the boundary, a moving boundary, and passive interaction of the boundary with an incident acoustic wave. Two-dimensional results are presented as well these demonstrate expected behavior in all cases. Convergence analysis is also performed and compared against the sharp-interface solution, and linear convergence is observed. This method lays the groundwork for the extension to viscous flow and the solution of problems involving time-varying mass-flux boundaries.

     
    more » « less
  2. Abstract

    In this article we study the stability of explicit finite difference discretization of advection–diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations that is obtained by discretizing the ADE in space and then extends to fully discretized methods in combination with explicit Runge–Kutta methods. In particular, we prove that all stable semi‐discretization of the ADE leads to a conditionally stable fully discretized method as long as the time‐integrator is at least first‐order accurate, whereas high‐order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of the article, the analysis and the stability results are extended to a partially dissipative wave system, which serves as a model for common practice in many fluid mechanics applications that incorporate a viscous stress in the momentum equation but no heat dissipation in the energy equation. Finally, the major theoretical predictions are verified by numerical examples.

     
    more » « less
  3. We present a multi-scale mathematical model and a novel numerical solver to study blood plasma flow and oxygen concentration in a prototype model of an implantable Bioartificial Pancreas (iBAP) that operates under arteriovenous pressure differential without the need for immunosuppressive therapy. The iBAP design consists of a poroelastic cell scaffold containing the healthy transplanted cells, encapsulated between two semi-permeable nano-pore size membranes to prevent the patient’s own immune cells from attacking the transplant. The device is connected to the patient’s vascular system via an anastomosis graft bringing oxygen and nutrients to the transplanted cells of which oxygen is the limiting factor for long-term viability. Mathematically, we propose a (nolinear) fluid–poroelastic structure interaction model to describe the flow of blood plasma through the scaffold containing the cells, and a set of (nonlinear) advection–reaction–diffusion equations defined on moving domains to study oxygen supply to the cells. These macro-scale models are solved using finite element method based solvers. One of the novelties of this work is the design of a novel second-order accurate fluid–poroelastic structure interaction solver, for which we prove that it is unconditionally stable. At the micro/nano-scale, Smoothed Particle Hydrodynamics (SPH) simulations are used to capture the micro/nano-structure (architecture) of cell scaffolds and obtain macro-scale parameters, such as hydraulic conductivity/permeability, from the micro-scale scaffold-specific architecture. To avoid expensive micro-scale simulations based on SPH simulations for every new scaffold architecture, we use Encoder–Decoder Convolution Neural Networks. Based on our numerical simulations, we propose improvements in the current prototype design. For example, we show that highly elastic scaffolds have a higher capacity for oxygen transfer, which is an important finding considering that scaffold elasticity can be controlled during their fabrication, and that elastic scaffolds improve cell viability. The mathematical and computational approaches developed in this work provide a benchmark tool for computational analysis of not only iBAP, but also, more generally, of cell encapsulation strategies used in the design of devices for cell therapy and bio-artificial organs. 
    more » « less
  4. Abstract

    Efficient and accurate modeling of the coupled thermal‐hydraulic‐mechanical‐chemical (THMC) processes in various rock formations is indispensable for designing energy geo‐structures such as underground repositories for high‐level nuclear wastes. This work focuses on developing and verifying an implicit finite element solver for generic coupled THMC problems in geological settings. Starting from the mass, momentum, and energy balance laws, a specialized set of governing equations and a thermoporoelastic constitutive model is derived. This system is then solved by an implicit finite element (FE) scheme. Specifically, the residuals and the Jacobians are scripted in a user‐defined element (UEL) subroutine which is then combined with the general‐purpose FE software Abaqus Standard to solve initial‐boundary value problems. Considering the complexity of the system, the UEL development follows a stepwise manner by first solving the coupled hydraulic‐mechanical (HM) and thermal‐hydraulic‐mechanical (THM) equations before moving on to the full THMC problem. Each implementation step consists of at least one verification test by comparing computed results with closed‐form analytical solutions to ensure that the various coupling effects are correctly realized. To demonstrate the robustness of the algorithm and to validate the UEL, a three‐dimensional case study is performed with reference to the in‐situ heating test of ATLAS at Belgium in 1980s. A hypothetical radionuclide leakage event is then simulated by activating the chemical‐concentration degree of freedom and prescribing a constant high concentration at the heater's surface. The model predicts a limited contaminated regime after six years considering both diffusion and advection effects on species transport.

     
    more » « less
  5. We perform direct numerical simulations of a gas bubble dissolving in a surrounding liquid. The bubble volume is reduced due to dissolution of the gas, with the numerical implementation of an immersed boundary method, coupling the gas diffusion and the Navier–Stokes equations. The methods are validated against planar and spherical geometries’ analytical moving boundary problems, including the classic Epstein–Plesset problem. Considering a bubble rising in a quiescent liquid, we show that the mass transfer coefficient $k_L$ can be described by the classic Levich formula $k_L = (2/\sqrt {{\rm \pi} })\sqrt {\mathscr {D}_l\,U(t)/d(t)}$ , with $d(t)$ and $U(t)$ the time-varying bubble size and rise velocity, and $\mathscr {D}_l$ the gas diffusivity in the liquid. Next, we investigate the dissolution and gas transfer of a bubble in homogeneous and isotropic turbulence flow, extending Farsoiya et al. ( J. Fluid Mech. , vol. 920, 2021, A34). We show that with a bubble size initially within the turbulent inertial subrange, the mass transfer coefficient in turbulence $k_L$ is controlled by the smallest scales of the flow, the Kolmogorov $\eta$ and Batchelor $\eta _B$ microscales, and is independent of the bubble size. This leads to the non-dimensional transfer rate ${Sh}=k_L L^\star /\mathscr {D}_l$ scaling as ${Sh}/{Sc}^{1/2} \propto {Re}^{3/4}$ , where ${Re}$ is the macroscale Reynolds number ${Re} = u_{rms}L^\star /\nu _l$ , with $u_{rms}$ the velocity fluctuations, $L^*$ the integral length scale, $\nu _l$ the liquid viscosity, and ${Sc}=\nu _l/\mathscr {D}_l$ the Schmidt number. This scaling can be expressed in terms of the turbulence dissipation rate $\epsilon$ as ${k_L}\propto {Sc}^{-1/2} (\epsilon \nu _l)^{1/4}$ , in agreement with the model proposed by Lamont & Scott ( AIChE J. , vol. 16, issue 4, 1970, pp. 513–519) and corresponding to the high $Re$ regime from Theofanous et al. ( Intl J. Heat Mass Transfer , vol. 19, issue 6, 1976, pp. 613–624). 
    more » « less