Advances in modelling of biomimetic fluid flow at different scales
© Saha and Celata; licensee Springer. 2011
Received: 26 November 2010
Accepted: 15 April 2011
Published: 15 April 2011
The biomimetic flow at different scales has been discussed at length. The need of looking into the biological surfaces and morphologies and both geometrical and physical similarities to imitate the technological products and processes has been emphasized. The complex fluid flow and heat transfer problems, the fluid-interface and the physics involved at multiscale and macro-, meso-, micro- and nano-scales have been discussed. The flow and heat transfer simulation is done by various CFD solvers including Navier-Stokes and energy equations, lattice Boltzmann method and molecular dynamics method. Combined continuum-molecular dynamics method is also reviewed.
Replicating natural manufacturing methods as in the production of chemical compounds by plants and animals.
Mimicking mechanisms found in nature such as Velcro and Gecko tape.
Imitating organizational principles from social behaviour of organisms like ants, bees and microorganisms.
Russia has developed a systematic means for integrating the natural knowledge into humankind's technology using 'Teoriya Resheniya Izobretatelskikh Zadatch (TRIZ)', i.e. the theory of inventive problem solving, which provides an objective framework based on functionality for accessing solutions from other technologies and sciences. TRIZ also prevents waste of time trying to find a solution where none exists. The four main tools of TRIZ are a knowledge database arranged by function, analysis of the technical barriers to progress (contradictions), the way technology develops (ideality) and the maximization of resource usage. The biology-based technology 'Biomimetics' suggests new approaches resulting in patents and some into production:
Strain gauging based on receptors in insects ,
Deployable structures based on flowers and leaves ,
Tough ceramics based on mother-of-pearl ,
Drag reduction based on dermal riblets on shark skin ,
Tough composites based on fibre orientations in wood ,
Underwater glues based on mussel adhesive ,
Flight mechanisms based on insect flight ,
Extrusion technology based on the spinneret of the spider ,
Self-cleaning surfaces based on the surface of the lotus leaf .
The importance of Biomimetics will increase as the incidence of genetic manipulation increases and the genetic manufacturing is developed. In the result, the area between living and non-living materials, where biology interacts with engineering, e.g. bioengineering and biomechatronics, is benefited.
There are innumerable examples of interactions with the environment and balanced and efficient heat, mass, momentum and species transfer through the microstructures in the fluid flow in the manifested living world of plants, animals and other living creatures. Biomimetics involve mimicking these interactions across the functional surfaces with the surrounding environments in the technological design. The physical nature is numerically modelled and simulated using computational fluid dynamics (CFD).
Geometrical analogy as well as physical similarity is to be studied to design technological functional surfaces imitating microstructural and biological functional surface morphologies. CFD at micro- or meso-scales and other numerical methodologies are necessary for this [18–24].
The meso- and micro-scale methods are also being developed in parallel with the continuum theory-based conventional CFD techniques-using finite volume method (FVM) and finite element method (FEM). In the mesoscopic lattice Boltzmann method (LBM), fluid flow is simulated by tracking the development of distribution functions of assemblies of molecules. It is difficult to capture the interfacial dynamics, which is essential for multiphase flow, at the macroscopic level. LBM captures the interaction of fluid particles and is, therefore, helpful for multiphase flow with phase segregation and surface tension. Also, the LBM is computationally more efficient than molecular dynamics (MD) method since it does not track individual molecules; the solution algorithm is explicit, easy to implement and parallel computation can be done. Micro/nano-scale simulations in micro/nano-scale geometries and micro time scales are done in MD method and direct simulation of Monte Carlo (DSMD) method. Coupled macro-scale simulation is being done using high performance computer (HPC). This article makes a review of the advances in multiscale biomimetic fluid flow modelling and simulation of difficult physics problems with complex biological interfaces.
Macroscopic biomimetic flow modelling
where x and u are Cartesian coordinates and velocities, respectively, and t is time. Velocity u, density ρ, viscosity μ and other solution variables represent ensemble-averaged (or time-averaged) values. Reynolds stress, is modelled and related to the mean velocity gradients by Boussinesq hypothesis. k is the turbulence kinetic energy, ε the kinetic energy dissipation rate and μt the turbulent viscosity. C is constant, σ the Prandtl number. Gk represents the generation of turbulence kinetic energy due to the mean velocity gradients. μt is the turbulent viscosity.
The turbulent flow induced by the fish-tail oscillation is characterized by fluctuating velocity fields. The instantaneous governing equations are time averaged to reduce the computational time and complexity which is done in the form of turbulence models like the semi-empirical k-ε work-horse turbulence model for practical engineering flow calculations.
where is the flow velocity vector, is the grid velocity of the moving mesh, Γ is the diffusion coefficient, Sφ is the source term of φ and ∂V is the boundary of the control volume V.
Other example of using CFD to study biomimetic fluid flow problems include simulation of air flow around flapping insect wings, numerical simulation of electro-osmotic flow near earthworm surface and simulation of explosive discharge of the bombardier beetle.
The dynamic mesh CFD model is used to examine critical flight simulations of normal aircraft, like the undercarriage lowering at low air speed, or the movement of sweep wings of fighter jets at high air speed. Next to flight applications, the dynamic mesh model can also simulate moving heart valves in the biomedical area, or small flapping membrane valves in micro-fluidics or the flow around any arbitrary moving part in other industry or sports applications.
Hybrid molecular-continuum fluid dynamics simulation
Nanoscale systems such as GaAsMESFETs and SiMOSFETs semiconductor devices, ultra-fast (picoseconds or femtoseconds) pulsed lasers do not conform to the classical Fourier heat diffusion theory in which the mean free path of the energy carriers becomes comparable to or larger than the characteristic length scale of the particle device/system or the time scale of the processes becomes comparable to or smaller than the relaxation time of the energy carriers. Although numerical techniques like Boltzmann transport equation (BTE) or atomic-level simulation (MD) and Monte Carlo simulation (MCS) can capture the physics in this regime, they require large computational resources. The C-V hyperbolic equation, which is not subject to the Fourier law assumption of infinite thermal propagation speed, is also not free from anomalies.
Limitations of continuum description of a system
Finite difference and finite element methods serve well for continuum description of a system governed by a set of differential equations and boundary conditions. However, the problem arises when the system has atomic fabric of matter such as in the case of friction problems and phase-change problems of fluid freezing into a solid or dynamic transition such as intermittent stick-slip motion .
The molecular dynamics (MD) method
When a system is modelled on the atomic level such as in case of MD, the motion of individual atoms or molecules is approximated. The particle motion is controlled by interaction potentials and equations of motion. MD is used for systems on the nanometre scale.
Coupling two very different descriptions of fluids at MD-continuum interface is a serious issue. The overlapping region of two descriptions must be coupled over space as well as time giving consistent physical quantities like density, momentum and energy and their fluxes must be continuous. Quantities of particles may be averaged locally and temporally to obtain boundary conditions of continuum equations. Getting microscopic quantities from macroscopic non-unique ensembles is, however, difficult.
Several coupling schemes [40–44] have been developed and the two solutions relax in a finite overlap region before they are coupled. Equations of motion are the language of particles and these are coupled with the continuum language, i.e. the differential equations. The coupling mechanism transmits mass flux, momentum flux and energy flux across the domain boundary. If the remaining boundaries are sealed, i.e. the simulated system is closed; the coupling ensures conservation of mass, momentum and energy.
The two domains are coupled to each other by ensuring that the flux components normal to the domain boundary match. If particles flow towards the boundary, a corresponding amount of mass, momentum and energy must be fed into the continuum. Conversely, any transport in the vicinity of the boundary on the part of the continuum must provide a boundary condition for transport on the part of the particles.
Smoothed particle hydrodynamics
Sousa  presented a scientific smoothed particle hydrodynamic (SPH) multiphysics simulation tool applicable from macro to nanoscale heat transfer. SPH  is a meshless particle based Lagrangian fluid dynamic simulation technique; the fluid flow is represented by a collection of discrete elements or pseudo particles. These particles are initially distributed with a specified density distribution and evolve in time according to the fluid heat, mass, species and momentum conservation equations. Flow properties are determined by an interpolation or smoothing of the nearby particle distribution with the help of a weighting function called the smoothing kernel. SPH is advantageous in (1) tracking problems dealing with multiphysics, (2) handling complex free surface and material interface, (3) parallel computing with relatively simple computer codes, (4) dealing with transient fluid and heat transport.
Following the original approach of Olfe  and Modest  in case of radiative heat transfer, Sousa  made the SPH numerical modelling for the ballistic-diffusive heat conduction equation. In this method, the heat carriers inside the medium are split into two components: ballistic and diffusive. The ballistic component is determined from the prescribed boundary condition and/or nanoscale heat sources and it experiences only outscattering; the transport of the scattered and excited heat carriers inside the medium is treated as diffusive component.
Intrinsic complex issues in hybrid method
MD model and the Maxwell-Boltzmann velocity distribution
The MD atomistic model in the micro-scale framework is a deterministic method. In this model, the evolution of the molecular system is obtained by computing the trajectories of the particles based on the classical molecular model. The continuum conditions can be applied to molecular domain either by the method based on continuous rescaling of atomic velocities or by the periodic resampling method of atomistic velocities that employs velocity distribution functions such as Maxwell-Boltzmann or Chapman-Enskog distribution for non-equilibrium situations of hybrid simulations in dilute gases employing geometrical decomposition and state coupling. The Maxwell-Boltzmann velocity distribution is the natural velocity distribution of an atomic or molecular system in an equilibrium state defining the probability of one-dimensional velocity components of an atom assuming a specific value based on temperature and the atomic mass. The reflective plane placed at the upper boundary of the boundary condition transfer region maintains every particle inside the molecular domain. This scheme is simpler than the velocity reversing scheme, but this can be applied only to incompressible flows because the normal pressure is a result of the reflected atoms.
In the rescaling techniques, in addition to the velocity restrictions, the continuum pressure applies to the atomistic region. The normal pressure is applied through external forces generating a potential energy field. Energy is decreased because of the reduction of potential energy of the atoms moving towards the continuum boundary. The resulting energy oscillations in the molecular system are reduced by velocity reversing of the outermost atoms. This scheme is simple and robust because of uncontrolled transfer of energy. The continuum temperature to the molecular system is accomplished by an energy transfer scheme. The energy is added or removed from the microscopic system to parallel the macroscopic temperature without modifying the mean velocity of the particles. The energy transfer takes place independent of each dimension and is accomplished by the velocity vectors of the atoms [42, 61–68].
Issues related to boundary conditions in hybrid multiscaling modelling
Drikakis and Asproulis  applied macroscopic boundary conditions in hybrid multiscale modelling. MD microscopic simulation was employed. They employed the methods for various liquid and gas flows with heat transfer and identified specific parameters for accuracy and efficiency. Their work has shown that knowledge about boundary conditions development and application is needed in multiscale computational frameworks. Continuum temperature and velocity as well as macroscopic pressure constrain molecular domain. Inconsistent pressure can shrink the simulation domain and the particles may drift away generating errors and instabilities in the hybrid procedure. Also, the size of the regions for the application of velocity constrains is important to avoid unrealistic heat transfer across the computational domain and inconsistencies between the molecular and continuum state. Resampling frequency and the termination of the atomistic region have significant impact in the resampling techniques and these can influence trapping of particles in the constrained region and may cause deviations between the macroscopic and microscopic velocities. The domain termination needs correct continuum pressure application.
Challenge in biomimetic flow simulation
The task of imitating biological functional surfaces with variety of complex three-dimensional micro- and nano-structures is very challenging in biomimetic flow simulation. The transfer of biological morphologies of plants and animals by imitating both geometrical and physical similarity to technological applications is to be identified [70–127]. Studies on micro surface structures of different species are to be made by scanning electron microscope (SEM) and atomic force microscope (AFM) to imitate engineering functional surfaces. The mesoscopic LBM has been applied in studying electro-osmotic driving flow within the micro thin liquid layer near an earthworm body surface . The moving vortices give the effect of anti soil adhesion. Few multiphase LBM models are the pseudo-potential model, the free energy model and the index-function model [129–132]. In LBM, effective interaction potential describes the fluid-fluid interaction. Interface is introduced by modelling the Boltzmann collision operator imposing phase separation. Also, the fluid-fluid interactions are represented by a body force term in Boltzmann equation. In this case, second-order terms in the pressure tensor are removed and more realistic interfacial interactions are produced.
Hard spheres fluids, square well fluids and Lennard-Jones fluids are model fluids in MD. The fluid flow and heat transfer in micro-scale and nano-scale systems get microscopic and nanoscopic insight from MD .
A comprehensive and state-of-the-art review of CFD techniques for numerical modelling of some biomimetic flows at different scales has been done. Fluid-fluid interfaces contacting with functional solid surfaces have been discussed. The multiphysics modelling at different scales by Navier-Stokes and energy equations, mesoscopic LBM, MD method and combined continuum-MD method with appropriate coupling schemes have been dealt with in detail.
atomic force microscope
Boltzmann transport equation
computational fluid dynamics
direct simulation of Monte Carlo
finite element method
finite volume method
high performance computer
lattice Boltzmann method
lattice Poisson method
Monte Carlo simulation
scanning electron microscope
smoothed particle hydrodynamic
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