Sheared bioconvection in a horizontal tube

The recent interest in using microorganisms for biofuels is motivation enough to study bioconvection and cell dispersion in tubes subject to imposed flow. To optimize light and nutrient uptake, many microorganisms swim in directions biased by environmental cues (e.g. phototaxis in algae and chemotaxis in bacteria). Such taxes inevitably lead to accumulations of cells, which, as many microorganisms have a density different to the fluid, can induce hydrodynamic instabilites. The large-scale fluid flow and spectacular patterns that arise are termed bioconvection. However, the extent to which bioconvection is affected or suppressed by an imposed fluid flow and how bioconvection influences the mean flow profile and cell transport are open questions. This experimental study is the first to address these issues by quantifying the patterns due to suspensions of the gravitactic and gyrotactic green biflagellate alga Chlamydomonas in horizontal tubes subject to an imposed flow. With no flow, the dependence of the dominant pattern wavelength at pattern onset on cell concentration is established for three different tube diameters. For small imposed flows, the vertical plumes of cells are observed merely to bow in the direction of flow. For sufficiently high flow rates, the plumes progressively fragment into piecewise linear diagonal plumes, unexpectedly inclined at constant angles and translating at fixed speeds. The pattern wavelength generally grows with flow rate, with transitions at critical rates that depend on concentration. Even at high imposed flow rates, bioconvection is not wholly suppressed and perturbs the flow field.     

The Effects of Thermal Radiation on an Unsteady MHD Axisymmetric Stagnation-Point Flow over a Shrinking Sheet in Presence of Temperature Dependent Thermal Conductivity with Navier Slip

In this paper, the magnetohydrodynamic (MHD) axisymmetric stagnation-point flow of an unsteady and electrically conducting incompressible viscous fluid in with temperature dependent thermal conductivity, thermal radiation and Navier slip is investigated. The flow is due to a shrinking surface that is shrunk axisymmetrically in its own plane with a linear velocity. The magnetic field is imposed normally to the sheet. The model equations that describe this fluid flow are solved by using the spectral relaxation method. Here, heat transfer processes are discussed for two different types of wall heating; (a) a prescribed surface temperature and (b) a prescribed surface heat flux. We discuss and evaluate how the various parameters affect the fluid flow, heat transfer and the temperature field with the aid of different graphical presentations and tabulated results.     

Inflation waves induced by axial acceleration of the aorta

The phenomenon of high-amplitude inflation waves resulting from a sharp axial acceleration of the aorta, as may occur in road accidents, is investigated theoretically. The aorta is modeled as an axisymmetric tapered membranic shell (tube) made of an incompressible, nonlinear viscoelastic material with cylindrical orthotropy. It is filled with an inviscid, incompressible fluid whose flow is considered as quasi-one dimensional along the tube axis. The equations of motion of the tube and of the fluid are solved numerically, by using a two-step explicit scheme, for several axial acceleration profiles. The solutions shows that an inflation wave is generated and it propagates in opposite direction to that of the acceleration. The wall stresses, deformations and their time derivatives as well as fluid velocity and pressure are determined along the tube at different time intervals. Peak axial and circumferential stresses are high, with the latter far exceeding the former. These stresses may cause rupture of the aorta.     

Static equilibrium configurations of a model red blood cell

Summary The membrane of the red blood cell is modeled as a fluid shell which resists bending and changes in area. The differential equations governing the mechanical equilibrium of such a membrane are derived and axisymmetric solutions are obtained numerically.     

On the paths of fluid particles in an axisymmetrical aneurysm

The aim of this study is the characterization of the pulsatile flow field by demonstration of the paths of single particles in a model of an axisymmetric aneurysm. The detailed analysis of the flow field can give additional information on the flow pattern and the time of transition of blood particles in the segment. The basis of the calculations is the system of the Navier-Stokes equations for incompressible Newtonian fluid flow. To solve these equations numerically the finite element method was used. The trajectory equations for a fluid particle were solved by use of a predictor-corrector procedure. The results of the computer simulation demonstrate the development, shift and disappearance of vortices in the excavation and give references to zones of stasis. This behavior can be an important factor in thrombogenesis.     

Non-linear bioconvection in a deep suspension of gyrotactic swimming micro-organisms

 The non-linear structure of deep, stochastic, gyrotactic bioconvection is explored. A linear analysis is reviewed and a weakly non-linear analysis justifies its application by revealing the supercritical nature of the bifurcation. An asymptotic expansion is used to derive systems of partial differential equations for long plume structures which vary slowly with depth. Steady state and travelling wave solutions are found for the first order system of partial differential equations and the second order system is manipulated to calculate the speed of vertically travelling pulses. Implications of the results and possibilities of experimental validation are discussed. Received: 26 May 1997 / Revised version: 10 May 1998     

Fluid–structure interaction analysis of turbulent pulsatile flow within a layered aortic wall as related to aortic dissection

Turbulent pulsatile flow and wall mechanics were studied numerically in an axisymmetric three-layered wall model of a descending aorta. The transport equations were solved using the finite element formulation based on the Galerkin method of weighted residuals. A fully-coupled fluid–structure interaction (FSI) analysis was utilized in this investigation. We calculated Von Mises wall stress, streamlines and fluid pressure contours. The findings of this study show that peak wall stress and maximum shear stress are highest in the media layer. The difference in the elastic properties of contiguous layers of the wall of the aorta probably determines the occurrence of dissection in the media layer. Moreover, the presence of aortic intramural hematoma is found to have a significant effect on the peak wall stress acting on the inner layer.     

An integral method for the analysis of blood flow

The existing methods to solve the problems of pulsatile flow in the cardiovascular system are based on either linear axisymmetric equations or non-linear one-dimensional equations. The solutions thus obtained give only a mediocre comparison with measurements. In this paper, a non-linear axisymmetric theory is proposed. The starting point of the present theory is a third degree polynomial representation of the velocity profile. Integral methods are then applied to obtain the governing equations. To ascertain the accuracy of the theory proposed above, the calculations for a simple case involving pulsatile flow in a long rigid tube were performed. The results are: (a) the average velocities compare very well with exact solutions and (b) the velocity profiles for a given frequency agree very well with exact solutions for flow in small tubes, but tend to differ as tube size is increased.     

Subcritical flutter in collapsible tube flow: a model of expiratory flow in the trachea

Using an axisymmetric geometry that retains certain qualitative features of the trachea, we extend one-dimensional modeling of flow in collapsible tubes to include both curved shell effects and, for untethered tubes, wall inertia. A systematic scaling of the finite deformation membrane equations leads to an approximate set which is consistent with the one-dimensional fluid model; axial and normal wall variables are coupled elastically, but only axial inertia is retained. Transverse curvature causes elastic coupling that can give rise to axial wall motion and a flutter instability. The source of instability is the product of a nonzero reference axial curvature with axial tension variation due to axial stretching. The numerical results suggest that this mechanism may be significant even in processes which cannot be assumed one-dimensional.     

The Orientation of Swimming Biflagellates in Shear Flows

Biflagellated algae swim in mean directions that are governed by their environments. For example, many algae can swim upward on average (gravitaxis) and toward downwelling fluid (gyrotaxis) via a variety of mechanisms. Accumulations of cells within the fluid can induce hydrodynamic instabilities leading to patterns and flow, termed bioconvection, which may be of particular relevance to algal bioreactors and plankton dynamics. Furthermore, knowledge of the behavior of an individual swimming cell subject to imposed flow is prerequisite to a full understanding of the scaled-up bulk behavior and population dynamics of cells in oceans and lakes; swimming behavior and patchiness will impact opportunities for interactions, which are at the heart of population models. Hence, better estimates of population level parameters necessitate a detailed understanding of cell swimming bias. Using the method of regularized Stokeslets, numerical computations are developed to investigate the swimming behavior of and fluid flow around gyrotactic prolate spheroidal biflagellates with five distinct flagellar beats. In particular, we explore cell reorientation mechanisms associated with bottom-heaviness and sedimentation and find that they are commensurate and complementary. Furthermore, using an experimentally measured flagellar beat for Chlamydomonas reinhardtii, we reveal that the effective cell eccentricity of the swimming cell is much smaller than for the inanimate body alone, suggesting that the cells may be modeled satisfactorily as self-propelled spheres. Finally, we propose a method to estimate the effective cell eccentricity of any biflagellate when flagellar beat images are obtained haphazardly.     

An Experimental Investigation of Bioconvection in Three Species of Microorganisms*

SYNOPSIS. A series of experiments was carried out on bioconvection using cultures of Polytomella agilis, Tetrahymena pyriformis and Chlamydomonas moewusii. During convection these microorganisms disperse into more and less dense regions causing visible patterns. These patterns, called the cell plan forms, were found to change with increasing Rayleigh numbers above Rc, the lowest concentration gradient at which bioconvection occurs. Hexagonal convection cells were observed near the critical Rayleigh number, Rc. As the Rayleigh number was increased the cell plan form changed from hexagons to 2-dimensional rolls and then to square convection cells. The size of the observed convection cells was less than expected for such a physical phenomenon. This appears to result from an increasing tendency of the microorganisms to circulate near the upper surface at higher Rayleigh numbers. The square convection cells appear to be unique to bioconvection. The inhibition of bioconvection is directly linked to an increased mortality rate. Observations on the nature of negative geotaxis were made which tend to support the statocyst hypothesis.     

Leukocyte deformability: finite element modeling of large viscoelastic deformation.

An axisymmetric deformation of a viscoelastic sphere bounded by a prestressed elastic thin shell in response to external pressure is studied by a finite element method. The research is motivated by the need for understanding the passive behavior of human leukocytes (white blood cells) and interpreting extensive experimental data in terms of the mechanical properties. The cell at rest is modeled as a sphere consisting of a cortical prestressed shell with incompressible Maxwell fluid interior. A large-strain deformation theory is developed based on the proposed model. General non-linear, large strain constitutive relations for the cortical shell are derived by neglecting the bending stiffness. A representation of the constitutive equations in the form of an integral of strain history for the incompressible Maxwell interior is used in the formulation of numerical scheme. A finite element program is developed, in which a sliding boundary condition is imposed on all contact surfaces. The mathematical model developed is applied to evaluate experimental data of pipette tests and observations of blood flow.     

The distribution of pressure behind a soft contact lens

We determine the pressure distribution behind a soft contact lens that is necessary to keep the lens in conformity with an axisymmetric substrate. The substrate consists of two regions: a central portion, the cornea, supposed to be an ellipsoid; and a peripheral region, the sclera, taken to be a sphere. The pressure is obtained as part of a numerical solution of the axisymmetric equilibrium equations for an initially curved, linearly elastic membrane. The relaxed shape of the lens is assumed to be an axisymmetric ellipsoid with a central curvature and a shape factor different from those of the cornea. The variation in the thickness of the lens from its center to edge is approximated by a polynomial. Pressure distributions are obtained for several typical soft contact lens fittings.     

Effect of initial stresses on the wave propagation in arteries

A theoretical analysis for the problem of wave propagation in arteries is presented. Blood is treated as a Newtonian, viscous incompressible fluid. On the basis of information derived from experimental investigations on the mechanical properties of wall tissues, the arterial wall is considered to be nonlinearly viscoelastic and orthotropic. The analysis is carried out for a cylindrical artery, under the purview of membrane theory, by taking account the effect of initial stresses. The motion of the wall and that of the fluid are assumed to be axisymmetric. Particular emphasis has been paid to the propagation of small amplitude harmonic waves whose wavelength is large compared to the radius of the vessel. By employing the equations of motion of the fluid and those for the wall, together with the equations of continuity, a frequency equation is derived by exploiting the conditions of continuity of the velocity of the arterial wall and that of blood on the endosteal surface of the wall. In order to illustrate the validity of the derived analytical expressions a quantitative analysis is made for the variations of the phase velocities as well as the transmission coefficient with frequency corresponding to different transmural pressures which relate to different initial stresses. Computational results indicate that phase velocities increase with the increase of transmural pressures.     

An evaluation of three-dimensional diarthrodial joint contact using penetration data and the finite element method

We have developed an approximate method for simulating the three-dimensional contact of soft biphasic tissues in diarthrodial joints under physiological loading. Input to the method includes: (i) kinematic information describing an in vitro joint articulation, measured while the cartilage is deformed under physiological loads, (ii) geometric properties for the relaxed (undeformed) cartilage layers, obtained for the analyses in this study via stereophotogrammetry, and (iii) material parameters for the biphasic constitutive relations used to represent cartilage. Solid models of the relaxed tissue layers are assembled in physiological positions, resulting in a mathematical overlap of the cartilage layers. The overlap distribution is quantified and converted via the biphasic governing equations into applied traction boundary conditions for both the solid and fluid phases for each of the contacting layers. Linear, biphasic, three-dimensional, finite element analysis is performed using the contact boundary conditions derived for each of the contacting layers. The method is found to produce results consistent with the continuity requirements of biphasic contact. Comparison with results from independent, biphasic contact analyses of axisymmetric problems shows that the method slightly underestimates the contact area, leading to an overestimation of the total traction, but yields a good approximation to elastic stress and solid phase displacement.     

Effects of transmural pressure and wall shear stress on LDL accumulation in the arterial wall: a numerical study using a multilayered model

The accumulation of low-density lipoprotein (LDL) is recognized as one of the main contributors in atherogenesis. Mathematical models have been constructed to simulate mass transport in large arteries and the consequent lipid accumulation in the arterial wall. The objective of this study was to investigate the influences of wall shear stress and transmural pressure on LDL accumulation in the arterial wall by a multilayered, coupled lumen-wall model. The model employs the Navier-Stokes equations and Darcy's Law for fluid dynamics, convection-diffusion-reaction equations for mass balance, and Kedem-Katchalsky equations for interfacial coupling. To determine physiologically realistic model parameters, an optimization approach that searches optimal parameters based on experimental data was developed. Two sets of model parameters corresponding to different transmural pressures were found by the optimization approach using experimental data in the literature. Furthermore, a shear-dependent hydraulic conductivity relation reported previously was adopted. The integrated multilayered model was applied to an axisymmetric stenosis simulating an idealized, mildly stenosed coronary artery. The results show that low wall shear stress leads to focal LDL accumulation by weakening the convective clearance effect of transmural flow, whereas high transmural pressure, associated with hypertension, leads to global elevation of LDL concentration in the arterial wall by facilitating the passage of LDL through wall layers.     

Effects of friction on the unconfined compressive response of articular cartilage: a finite element analysis

A finite element analysis is used to study a previously unresolved issue of the effects of platen-specimen friction on the response of the unconfined compression test; effects of platen permeability are also determined. The finite element formulation is based on the linear KLM biphasic model for articular cartilage and other hydrated soft tissues. A Galerkin weighted residual method is applied to both the solid phase and the fluid phase, and the continuity equation for the intrinsically incompressible binary mixture is introduced via a penalty method. The solid phase displacements and fluid phase velocities are interpolated for each element in terms of unknown nodal values, producing a system of first order differential equations which are solved using a standard numerical finite difference technique. An axisymmetric element of quadrilateral cross-section is developed and applied to the mechanical test problem of a cylindrical specimen of soft tissue in unconfined compression. These studies show that interfacial friction plays a major role in the unconfined compression response of articular cartilage specimens with small thickness to diameter ratios.     

Modeling pulsatile flow in aortic aneurysms: effect of non-Newtonian properties of blood

Pulsatile flow in an axisymmetric rigid-walled model of an abdominal aorta aneurysm was analyzed numerically for various aneurysm dilations using physiologically realistic resting waveform at time-averaged Reynolds number of 300 and peak Reynolds number of 1607. Discretization of the governing equations was achieved using a finite element scheme based on the Galerkin method of weighted residuals. Comparisons with previously published work on the basis of special cases were performed and found to be in excellent agreement. Our findings indicate that the velocity fields are significantly affected by non-Newtonian properties in pathologically altered configurations. Non-Newtonian fluid shear stress is found to be greater than Newtonian fluid shear stress during peak systole. Further, the maximum shear stress is found to occur near the distal end of AAA during peak systole. The impact of non-Newtonian blood flow characteristics on pressure compared to Newtonian model is found insignificant under resting conditions. Viscous and inertial forces associated with blood flow are responsible for the changes in the wall that result in thrombus deposition and dilation while rupture of AAA is more likely determined by much larger mechanical stresses imposed by pulsatile pressure on the wall of AAA.