Stochastic immersed boundary method incorporating thermal fluctuations

We give a brief introduction to the stochastic immersed boundary method which allows for simulation of small length-scale physical systems in which elastic structures interact with a fluid flow in the presence of thermal fluctuations. The conventional immersed boundary method is extended to account for thermal fluctuations by introducing stochastic forcing terms in the fluid equations. This gives a system of stiff SPDE's for which standard numerical approaches perform poorly. We discuss a numerical method derived using stochastic calculus to overcome the stiff features of the equations. We then discuss results which indicate that the method captures physical features predicted by statistical mechanics for small length-scale systems. The stochastic immersed boundary method holds promise as a numerical approach in simulating microscopic mechanical systems in which thermal fluctuations play a fundamental role. A more detailed discussion of this work is given in [1, 2, 3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A hydrodynamic model of human cochlea

A two-compartment model of the human cochlea is proposed. When stretched out, the bony spiral tube looks like two chambers separated by a membrane. Both chambers are filled with viscous fluid called perilymph; they communicate with one another via a canal. Sound vibrations enter the cochlea through the oval window and cause periodic change of pressure in the perilymph, which, in turn, causes the membrane to vibrate. The motion of the fluid is described by hydrodynamic equations, which are supplemented with the membrane vibration equation. The equations are linearized in the amplitude of the vibrations, and their solution is sought in the form of Fourier harmonics with a given frequency. To determine the harmonics, a system of linear boundary value problems for ordinary differential equations with variable coefficients is obtained. The numerical solution of this system using finite difference method fails because it involves a large parameter and the problem is close to a singular one. We propose a novel numerical method without saturation that enables us to obtain solutions in a wide range of frequencies up to an arbitrary and controllable accuracy. The computations confirm the Bekesy theory stating that high-frequency sounds cause the membrane to bend near the apex of the cochlea, and low-frequency sounds cause it to bend near the base of the cochlea.     

On a model of a flexural prestressed shell

E. Sanchez‐Palencia We derive a linearized prestressed elastic shell model from a nonlinear Kirchhoff model of elastic plates. The model is given in terms of displacement and micro‐rotation of the cross‐sections. In addition to the standard membrane, transverse shear, and flexural terms, the model also contains a nonstandard prestress term. The prestress is of the same order as flexural effects; hence, the model is appropriate when flexural effects dominate over membrane ones. We prove the existence and uniqueness of the solutions by Lax–Milgram theorem and compare solution with the solution of the standard shell model via numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

The coupling system of Navier–Stokes equations and elastic Navier–Lame equations in a blood vessel

In this paper, the blood flow problem is considered in a blood vessel, and a coupling system of Navier–Stokes equations and linear elastic equations, Navier–Lame equations, in a cylinder with cylindrical elastic shell is given as the governing equations of the problem. We provide two finite element models to simulating the three-dimensional Navier–Stokes equations in the cylinder while the asymptotic expansion method is used to solving the linearly elastic shell equations. Specifically, in order to discrete the Navier–Stokes equations, the dimensional splitting strategy is constructed under the cylinder coordinate system. The spectral method is adopted along the rotation direction while the finite element method is used along the other directions. By using the above strategy, we get a series of two-dimensional-three-components (2D-3C) fluid problems. By introduce the S-coordinate system in E³ and employ the thickness of blood vessel wall as the expanding parameter, the asymptotic expansion method can be established to approximate the solution of the 3D elastic problem. The interface contact conditions can be treated exactly based on the knowledge of tensor analysis. Finally, numerical test shows that our method is reasonable.     

Numerical stability for a dynamic Naghdi shell

We consider an elastic model for a shell incorporating shear, membrane, bending and dynamic effects. We make use of the theory proposed by Arnold and Brezzi [1] based on a locking free non-standard mixed variational formulation. This method is implemented in terms of the displacement and rotation variables as the minimization of an altered energy functional. We extend this theory to the shell vibrations problem and establish optimal error estimates independent of the thickness, thereby proving that shear and membrane locking is avoided. We study the numerical stability both in static and dynamic regimes. The approximation schemes are tested on specific examples and the numerical results confirm the estimates obtained from theory.     

Convergence proof of the velocity field for a stokes flow immersed boundary method

The immersed boundary (IB) method is a computational framework for problems involving the interaction of a fluid and immersed elastic structures. It is characterized by the use of a uniform Cartesian mesh for the fluid, a Lagrangian curvilinear mesh on the elastic material, and discrete delta functions for communication between the two grids. We consider a simple IB problem in a two‐dimensional periodic fluid domain with a one‐dimensional force generator. We obtain error estimates for the velocity field of the IB solution for the stationary Stokes problem. We use this result to prove convergence of a simple small‐amplitude dynamic problem. We test our error estimates against computational experiments. © 2007 Wiley Periodicals, Inc.     

Fluid dynamics of an underwater electric discharge on the axis of a shell immersed in a fluid

We study the motion of a fluid during an electrical discharge on the axis of an elastic cylindrical shell immersed in the fluid. We estimate the influence of the parameters of the shell on the dynamics of a gasvapor chamber. We point out several properties of the computation of the pulsed flows in an infinite volume of fluid (the choice of the method of constructing the grid, the limitation of the computational region). We obtain the pressure distribution over space at large distances from the shell. We analyze the influence of the shell on the parameters of the flow in the interior and exterior regions. Four figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 120–126.     

Coupled differential algebraic equations in the simulation of flexible multibody systems with hydrodynamic force elements

The mathematical modelling of elastohydrodynamic bearings in combustion engines leads to a coupled system of (partial) differential algebraic equations, which is represented by a flexible multibody system model of the engine including crankshaft and bearing and by the Reynolds equation that describes the non-linear effects in the fluid film. The hydrodynamic forces depend strongly on the position and the elastic deformation of crankshaft and bearing shell. We discuss the influence of the spatial discretization on accuracy and numerical effort. Since a fine spatial discretization substantially slows down the numerical solution, we propose a semi-analytical method based on singular perturbation theory to speed-up time integration. Numerical tests for a simplified benchmark problem illustrate the advantages of this approach. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Oscillation of the basilar membrane in the mammalian cochlea

In the model of the cochlea considered it is interpreted as a hydrodynamical wave-guide with properties that vary smoothly along its axis. The basilar membrane itself is treated as an anisotropic membrane. An analogue of the WKB method is used to compute the displacements of the basilar membrane. The results of the computations are illustrated with a number of graphs.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 54–62, 1979.     

Incompressible flows in elastic domains: an immersed boundary method approach

This study illustrates how the immersed boundary method may be applied to perform the numerical simulation of incompressible flows in two-dimensional domains bounded by elastic boundaries. It presents the basic intermediate steps involved in the derivation of a solution methodology, from a scientific motivation to the numerical results, which can be applied for both steady and transient problems, even when the boundaries have an arbitrary shape. Its motivation, briefly presented, was borne in a bioengineering problem: the numerical simulation of the performance of ventricular assist devices. The mathematical model is composed by the Navier–Stokes equations, where the forcing term contains singular forces which arise from the elastic stresses acting on the boundaries. The incompressibility constraint is modified to introduce the inflow and outflow conditions into the problem through the use of sources and sinks. The methodology is applied to simulate two problems: the steady flow between two parallel plates, for which the exact solution is known and can be used to validate the approach, and the periodic flow in a winding channel, a transient problem in a non-trivial domain.     

Direct numerical simulation of particles in a fluid interacting with a wall

We develop a code to be applied in the context of the cleaning of wafer surfaces by hydrodynamic forces. Our goal is to study the detachment of (submicron) particles, exposed to a shear flow, from a wall by means of direct numerical simulation. The particles are treated as rigid bodies fully interacting with the fluid. To simulate moving particles in the fluid we implement an immersed boundary method with direct forcing into OpenFOAM. The particle-wall interaction is treated with a soft contact model. As first simple examples we study the elastic normal impact of a cylinder onto a wall as well as the onset of sliding of a cylinder pressed to a horizontal wall by gravity under a time-depended drag force. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transversally inhomogeneous sharply tuned linear cochlear model

A linear mechanism of sharp frequency selectivity in the inner ear is developed. The cochlea is assumed to be a slightly varying wave guide with inhomogeneous cross section. The tectorial membrane is considered as an additional mass loading the narrow strip of the basilar membrane that underlies the rows of outer hair cells. A high concentration of mass along the middle line of the cochlea partition provides the sharpening of the tuning curve without significantly altering the phase. The dissipation of energy is assumed to hold in fluid boundary layers near the cochlea partition. The responses of the model with one and the same set of input parameters are compared with different experimental data obtained during the last decade, in the basal and apical parts of the cochlea. This paper demonstrates that a system such as cochlea is capable of performing sharply tuned linear frequency analysis without adding any outside energy to the input waveform to be analyzed. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 213–240. Translated by S. M. Novoselova.     

Numerical study on bacterial flagellar bundling and tumbling in a viscous fluid using an immersed boundary method

The bundling and tumbling behavior of bacterial flagella in a viscous fluid has got immense significance in the field of biological fluid dynamics. In this paper we investigate the hydrodynamic interaction among two and more than two flagella in a viscous fluid based on an immersed boundary method. We model each helical flagellum by a number of triangular cross-sections with three immersed boundary (IB) points on each cross-section. Three types of elastic links are generated from each IB point to create an elastic network model of the flagellum and the first cross-section is modeled as the flagellar motor. The elastic forces are computed based on the elastic energy approach and the motor forces are obtained from the applied angular frequency of rotation of the motor. The Stokes equations governing the flow are solved on a staggered Cartesian grid system using a fractional-step based finite-volume method. It is observed that when two left-handed helical flagella rotate in the counter-clockwise direction, the resulting hydrodynamic interaction leads to bundling. When one of the flagella reverses the direction of rotation to clockwise the hydrodynamic interaction results in tumbling. During the bundling, the flagella wrap and intertwine each other, whereas during the tumbling they separate in an erratic way. There exists an exact combination of the handedness and rotational direction of the flagella to achieve the bundling. The bundling-to-tumbling behavior of the flagella is studied and it is concluded that the tumbling occurs faster than the bundling. Further, the hydrodynamic interaction among three flagella in a viscous fluid is studied for the cases of rotation in the same direction and in different directions. The bundling and tumbling behavior is well captured even for the case of multiple (more than two) flagella using the developed model.     

A fluid–structure interaction model with interior damping and delay in the structure

A coupled system of partial differential equations modeling the interaction of a fluid and a structure with delay in the feedback is studied. The model describes the dynamics of an elastic body immersed in a fluid that is contained in a vessel, whose boundary is made of a solid wall. The fluid component is modeled by the linearized Navier-Stokes equation, while the solid component is given by the wave equation neglecting transverse elastic force. Spectral properties and exponential or strong stability of the interaction model under appropriate conditions on the damping factor, delay factor and the delay parameter are established using a generalized Lax-Milgram method.     

Fluid-structure interaction of an elastic pipe immersed in a coaxial rigid pipe: a model for the Tullio Phenomenon

An axisymmetric, elastic pipe is filled with an incompressible fluid and is immersed in a second, coaxial rigid pipe which contains the same fluid. A pressure pulse in the outer fluid annulus deforms the elastic pipe which invokes a fluid motion in the fluid core. It is the aim of this study to investigate streaming phenomena in the core which may originate from such a fluid-structure interaction. This work presents a numerical solver for such a configuration. It was developed in the OpenFOAM software environment and is based on the Arbitrary Lagrangian Eulerian (ALE) approach for moving meshes. The solver features a monolithic integration of the one-dimensional, coupled system between the elastic structure and the outer fluid annulus into a dynamic boundary condition for the moving surface of the fluid core. Results indicate that our configuration may serve as a mechanical model of the Tullio Phenomenon (sound-induced vertigo). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure

We develop a loosely coupled fluid‐structure interaction finite element solver based on the Lie operator splitting scheme. The scheme is applied to the interaction between an incompressible, viscous, Newtonian fluid, and a multilayered structure, which consists of a thin elastic layer and a thick poroelastic material. The thin layer is modeled using the linearly elastic Koiter membrane model, while the thick poroelastic layer is modeled as a Biot system. We prove a conditional stability of the scheme and derive error estimates. Theoretical results are supported with numerical examples. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1054–1100, 2015     

Unsteady motion of a spherical bubble in a complex fluid: Mathematical modelling and simulation

The nonlinear response of an oscillatory bubble in a complex fluid is studied. The bubble is immersed in a Newtonian liquid, which may have a dilute volume fraction of anisotropic additives such as fibers or few ppm of macromolecules. The constitutive equation for the fluid is based on a Maxwell model with an extensional viscosity for the viscous contribution. The model is considered new in the study of bubble dynamics in complex fluids. The numerical computation solves a system of three first order ordinary differential equations, including the one associated with the solution of the convolution integral, using a fifth order Runge–Kutta scheme with appropriated time steps. Asymptotic solutions of governing equation are developed for small values of the pressure forcing amplitude and for small values of the elastic parameter. A study of the bubble collapse radius is also presented. We compare the results predicted by our model with other model in the literature and a good agreement is observed. The calculated asymptotic solutions are also used to test the results of the numerical simulations. In addition, the orientation of the additives is considered. The angular probability density function is assumed to be a normal distribution. The results show that the model based on the fully aligned additives with the radial direction overestimates the tendency of the additives to stabilize the bubble motion, since the effect of extensional viscosity occurs due to the particle resistance to the movement throughout its longitudinal direction.     

Computing Viscous Flow in an Elastic Tube

We have developed a numerical method for simulating viscous flow through a compliant closed tube, driven by a pair of fluid source and sink. As is natural for tubular flow simulations, the problem is formulated in axisymmetric cylindrical coordinates, with fluid flow described by the Navier-Stokes equations. Because the tubular walls are assumed to be elastic, when stretched or compressed they exert forces on the fluid. Since these forces are singularly supported along the boundaries, the fluid velocity and pressure fields become unsmooth. To accurately compute the solution, we use the velocity decomposition approach, according to which pressure and velocity are decomposed into a singular part and a remainder part. The singular part satisfies the Stokes equations with singular boundary forces. Because the Stokes solution is unsmooth, it is computed to second-order accuracy using the immersed interface method, which incorporates known jump discontinuities in the solution and derivatives into the finite difference stencils. The remainder part, which satisfies the Navier-Stokes equations with a continuous body force, is regular. The equations describing the remainder part are discretized in time using the semi-Lagrangian approach, and then solved using a pressure-free projection method. Numerical results indicate that the computed overall solution is second-order accurate in space, and the velocity is second-order accurate in time.     

USTIFICATION OF A TWO-DIMENSIONAL NONLINEAR SHELL MODEL OF KOITER'S TYPE

A two-dimensional nonlinear shell model"of Koiter's type"has recently been proposed by the first author. It is shown here that, according to two mutually exclusive sets of assumptions bearing on the associated manifold of admissible inextensional displacements, the leading term of a formal asymptotic expansion of the solution of this two-dimensional model, with the thickness as the"small" parameter, satisfies either the two-dimensional equations of a nonlinearly elastic "membrane" shell or those of a nonlinearly elastic "flexural" shell. These conclusions being identical to those recently drawn by B. Miara, then by V. Lods and B. Miara, for the leading term of a formal asymptotic expansion of the solution of the equations of three-dimensional nonlinear elasticity, again with the thickness as the "small" parameter, the nonlinear shell model of Koiter's type considered here is thus justified, at least formally.     

An immersed boundary–lattice Boltzmann approach to study the dynamics of elastic membranes in viscous shear flows

A combined immersed boundary–lattice Boltzmann approach is used to simulate the dynamics of elastic membrane immersed in a viscous incompressible flow. The lattice Boltzmann method is utilized to solve the flow field on a regular Eulerian grid, while the immersed boundary method is employed to incorporate the fluid–membrane interaction with a Lagrangian representation of the deformable immersed boundary. The distinct feature of the method used here is to employ the combination of simple Peskin's IBM and standard LBM. In order to obtain more accurate and truthful solutions, however, a non-uniform distribution of Lagrangian points and a modified Dirac delta function are used. Two test cases are presented. In the first case, we consider a vesicle suspended in a simple shear flow commonly known as tank-treading motion. The computed results were compared with experiments, which showed reasonably good agreement. For the second test case, we consider individual healthy (soft) and sick (stiff) RBCs suspended in a shear flow. The simulation results demonstrated that elastic deformation plays an important role in overall RBC motions characterized as tank-treading and tumbling motions, in which the natural state of the elastic membrane is an essential consideration. In addition, the results confirm that the combination of the immersed boundary and lattice Boltzmann methods permits the simulation of the complex biological phenomena.