A 3-dimensional model of flagellar swimming in a Brinkman fluid

We investigate 3-dimensional flagellar swimming in a fluid with a sparse network of stationary obstacles or fibers. The Brinkman equation is used to model the average fluid flow where a resistance term is inversely proportional to the permeability
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  This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics  1 A 3-dimensional model of flagellar swimmingin a Brinkman fluid NguyenHo Ho 1 , Karin Leiderman 2 , and Sarah Olson 3 † , 1 Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way,Cincinnati, Ohio 45221, USA. 2 Applied Mathematics and Statistics, Colorado School of Mines, 1500 Illinois Street, Golden,Colorado 80401 USA. 3 Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road,Worcester, Massachusetts 01609, USA.(Received xx; revised xx; accepted xx) We investigate 3-dimensional flagellar swimming in a fluid with a sparse network of stationary obstacles or fibers. The Brinkman equation is used to model the average fluidflow where a resistance term is inversely proportional to the permeability and representsthe effect due to the presence of fibers. The flagellum is represented as a Kirchhoff rod thatcan exhibit propagating planar or spiral bending. To solve for the local fluid velocity andangular velocity, we use the method of regularized Brinkmanlets and extend it to the casefor a Kirchhoff rod that is discretized as point forces and torques along the centerline. Thenew numerical method is validated by comparing to asymptotic swimming speeds derivedfor an infinite-length cylinder propagating lateral or spiral waves in a Brinkman fluid.Similar to the asymptotics, we observe that in the case of small amplitude, swimmingspeed is enhanced relative to the Stokes case as the resistance is increased. For largeramplitude bending, the simulations show a non-monotonic change in swimming speed asthe resistance is varied, with a peak when the resistance parameter is near one. This isdue to the emergent amplitude and wavelengths; as the resistance is increased (or as thenumber of stationary fibers is increased), the emergent amplitude of the swimmer has atendency to decrease. Key words:  Authors should not enter keywords on the manuscript, as these mustbe chosen by the author during the online submission process and will then be addedduring the typesetting process (see for the full list) 1. Introduction Microorganisms such as spermatozoa make forward progression by propagating bend-ing along their flagellum. The emergent flagellar curvature and beat frequency depends onthe fluid properties as well as the chemical concentrations within the flagellum (Gaffney et al.  2011; Miki 2007; Smith  et al.  2009; Suarez & Pacey 2006; Woolley & Vernon 2001).The fluid environment experienced by mammalian sperm includes complex geometriesand background flows due to interactions with other sperm, cilia, and walls (Fauci& Dillon 2006; Ho & Suarez 2001; Suarez & Pacey 2006; Suarez 2010). As a sperm †  Email address for correspondence:  2  N. Ho, K. Leiderman, and S. Olson  progresses towards the egg, the fluid could contain differing amounts of uterine cells,sulfomucins, protein networks and other macromolecules, especially at different times inthe menstrual cycle (Katz & Berger 1980; Katz  et al.  1989; Mattner 1968; Suarez 2010).Many experiments have examined sperm motility in gels such as methylcellulose (MC)or polyacrylamide (PA), which may be more representative of the  in vivo  environmentthan sperm motility in a culture medium. In experiments, the emergent beat frequencyand wavelength varied with viscosity in MC gels (Smith  et al.  2009) and the swimmingspeed of mouse sperm decreased in both MC and PA gels (relative to the culture medium)(Suarez & Dai 1992). This motivates the development of a 3-dimensional (3D) frameworkto study emergent properties (e.g., waveform or swimming speed) of a sperm whencoupled with this protein network.Previous computational studies of finite-length swimmers in a Newtonian fluid withpreferred bending kinematics have identified that there is a non-monotonic relationshipbetween emergent swimming speeds and bending amplitude (Dillon  et al.  2006; Elgeti et al.  2010; Fauci & McDonald 1995; Olson & Fauci 2015; Yang  et al.  2008). Onthe other hand, for infinite-length flagella with prescribed bending, the asymptoticswimming speeds are an increasing function with respect to the bending amplitude(Taylor 1951, 1952). Since most gels and biological fluids contain proteins and othermacromolecules, recent studies have focused on swimmers in complex fluids. In fluids thatexhibit contributions from viscous and elastic effects, the swimming speeds of infinite-length flagella with prescribed kinematics decrease in comparison to the Newtonian case(Fu  et al.  2009; Lauga 2007). In contrast, swimming speeds increase for certain parameterchoices for a finite-length swimmer in a nonlinear viscoelastic fluid and a Carreau fluid(Newtonian fluid with shear-dependent viscosity) (Montenegro-Johnson  et al.  2012; Teran et al.  2010; Thomases & Guy 2014). An enhancement in swimming speed relative to aNewtonian fluid has also been observed in a model of a two-phase fluid for a gel whenthe elastic network is stationary (Fu  et al.  2010).The average fluid flow through an array of sparse, spherical particles can be modeledvia the Brinkman equation (Auriault 2009; Brinkman 1947; Durlofsky & Brady 1987;Howells 1974; Spielman & Goren 1968). A flow dependent resistance term with resistanceparameter accounts for the presence of the particles in the fluid. In the case of an infinite-length flagellum in a Brinkman fluid with prescribed bending, in both 2D and 3D, theswimming speed increases as the resistance increases (Ho  et al.  2016; Leshansky 2009).This increase in swimming speed is an enhancement relative to the Newtonian case; thepresence of particles or fibers actually aids in forward progression. In contrast, for a finite-length swimmer with preferred planar bending, there was a non-monotonic relationshipbetween swimming speed and the resistance parameter (Cortez  et al.  2010; Olson &Leiderman 2015; Leiderman & Olson 2016).To explore emergent properties of flagellar swimming in a fluid with a sparse andstationary protein network, we use the incompressible Brinkman equations to govern thefluid motion (Brinkman 1947; Howells 1974): −∇  p + µ∆ u − µα 2 u + f  b =  0 ,  (1.1) ∇· u  = 0 .  (1.2)Here,  p  is the average fluid pressure (force per area),  u  is the average fluid velocity (lengthper time),  f  b represents the body force (force per unit volume) applied on the fluid bythe immersed structure,  µ  is the viscosity (force time per area), and  α  = 1 / √  γ   is theresistance parameter (inverse length), which is inversely proportional to the square rootof the permeability  γ  . One can think of the Brinkman equation as the addition of a lower-  Regularized Kirchhoff Rod model   3 0 0.05 0.1 0.15 0.200.0050.010.0150.020.025 a f  / √  γ      ϕ   Volume FractionCollagen GelCervical Mucus Figure 1.  The volume fraction   ϕ  solved for in Eq. (1.3) is plotted as a function of   a f  / √  γ   where  a f   is fiber radius and   γ   is permeability. The green and red markers represent the volume fraction  for a collagen gel (  γ   = 8 . 6 ) and cervical mucus (  γ   = 0 . 0085 ), respectively. order resistance term to the Stokes equations for low Reynolds number flow (since thelength scale of these swimmers is small, they live in a viscosity dominated environmentwhere inertia can be neglected). As  α  →  0 (no resistance), the incompressible Stokesequations are recovered and as α →∞ (high resistance), the term µ∆ u becomes negligibleand Eq. (1.1) behaves like Darcy’s law. An important characteristic of a Brinkman fluidis the Brinkman screening length,  √  γ  , which marks the approximate length over whicha disturbance to the velocity would decay.To consider a microorganism swimming in this environment, we assume that theobstacles are at a low enough volume fraction that the distance between fibers islarger than the diameter of the microorganism. In the case of randomly oriented fibers,Spielman & Goren (1968) have derived a relationship between the volume fraction  ϕ , thepermeability  γ  , and the radius of the fiber  a f   as a 2 f  γ   = 4 ϕ  13 a 2 f  γ   + 56 a f  √  γ K  1 ( a f  / √  γ  ) K  0 ( a f  / √  γ  )  .  (1.3)Here,  K  0 ( · ) and  K  1 ( · ) are the zeroth and first order modified Bessel functions of thesecond kind. Figure 1 shows the volume fraction  ϕ  as a function of the ratio  a f  / √  γ  ,indicating a relevant biological range for  ϕ . For reference, we also indicate the volumefraction of cervical mucus and a collagen gel based on experimental values for  ϕ  and  a f  (Saltzman  et al.  1994). Fibers in vaginal fluid have been reported to be 1–20 micronsapart, whereas the flagellar diameter for many species of sperm is in the range of 0.2–0.6 microns (Bahr & Zeitler 1964; Bloodgood 1990; Hafez & Kenemans 2012; Rutllant et al.  2001, 2005). The interfiber spacing (or distance between the fibers) can also beapproximated as (Leshansky 2009) D ≈ 2 a f   12   3 πϕ  − 1  ,  (1.4)based on a known volume fraction  ϕ  and fiber radius  a f  . In the case where the ratio D/a f   ≫ 1, there are little or no interactions between a stationary network of fibers andthe swimmers. Thus, it is assumed that the fibers do not impart any additional stressonto the filament.A fundamental solution of the incompressible Brinkman equations given in (1.1)–(1.2),is well-known (Durlofsky & Brady 1987; Pozrikidis 1989). It represents the velocity due toa concentrated external force acting on the fluid at a single point. However, the velocity  4  N. Ho, K. Leiderman, and S. Olson  becomes singular when the point forces are concentrated along curves in 3D. To eliminatethese singular solutions in Stokes flow, the method of regularized Stokeslets (Cortez 2001;Cortez  et al.  2005) is employed while the method of regularized Brinkmanlets (Cortez et al.  2010) is introduced to deal with these situations in a Brinkman fluid.In order to model emergent waveforms of swimmers that can be either planar orhelical, as observed in experiments (Woolley & Vernon 2001), we use a Kirchhoff rodmodel to represent the elastic flagellum. The propagation of bending along the filamentis given as a time-dependent preferred curvature function, where deviations from thispreferred configuration lead to forces and moments (body forces). An immersed boundaryformulation of the Kirchhoff rod model was first developed by Lim  et al.  (2008) and hasbeen extended to a regularized Stokes formulation (Lee  et al.  2014; Olson  et al.  2013;Olson 2014). Here, we extend the regularized method to now study flagellar swimmingin a fluid governed by the Brinkman equation where we account for the local linear andangular velocity due to point forces and torques along the length of the flagellum. Twoapproaches for the numerical method are derived in Section 3.3, where in the limit asresistance  α → 0, the solutions approach those of Stokes equations (detailed in AppendixB). We are able to match emergent swimming speeds with asymptotic swimming speedsfor both the planar and spiral bending cases where swimming speed increases withamplitude and beat frequency for a fixed resistance. The numerical results show that forthe planar and helical bending cases, there is an optimal range of resistance  α , around α  = 1, that allows the swimmer to achieve a large bending amplitude while receivingan extra boost in propulsion from the presence of the fiber network. In addition, as  α increases, the emergent waveform of the swimmer has a decreased amplitude (relative tothe preferred amplitude), resulting in a decreased swimming speed. 2. Kirchhoff Rod Model With the Kirchhoff Rod (KR) formulation, a flagellum is described by a 3D space curve X ( s ) for 0  < s < L , where  s  is a Lagrangian parameter initialized to be the arclength and L  is the length of the unstressed rod. Here, we assume the rod length is much greater thanthe radius and that the rod is isotropic and homogeneous. The associated orthonormaltriads  { D 1 ( s ) , D 2 ( s ) , D 3 ( s ) }  follow the right-handed rule. The triad  D 3 ( s ) is effectivelyin the direction of the tangent vector while  D 1 ( s ) and  D 2 ( s ) are rotations of the normaland binormal vectors, respectively, coinciding with the principle axes of the rod crosssection.Figure 2 shows the centerline of a flagellum discretized as a helix using the centerlineapproximation with the associated orthonormal triads plotted at one point on the spacecurve. In the standard KR model,  D 3 ( s ) is enforced to be the tangent vector and the rodis inextensible. We employ an unconstrained version whereby an elastic energy penaltyis used to numerically maintain the inextensibility of the rod and keep  D 3 ( s ) as a unittangent vector along the rod (Lim  et al.  2008; Olson  et al.  2013).The derivation of the internal force and torque in terms of the associated orthonormaltriads has previously been described in detail (Lim  et al.  2008; Olson  et al.  2013). Here,we summarize the main equations of the KR model which are utilized later. The balanceof force and torque on a cross section of the rod are0 =  f   +  ∂  F ∂s ,  (2.1)0 =  m +  ∂  M ∂s  +  ∂  X ∂s  × F  ,  (2.2)  Regularized Kirchhoff Rod model   5  2 3 Figure 2.  The Kirchhoff rod representation of a flagellum is discretized as a helix using acenterline approximation with orthonormal triads  { D 1 ,  D 2 ,  D 3 }  plotted at one point on thespace curve. where  f   (units of force per length) and  m  (units of force) are part of the externalforces applied on the rod. Whereas,  F  and  M  are the internal components of the forcetransmitted across each section of the rod and are given in terms of   X ( s ) and its triads.The components of   F  and  M  can be expanded in the basis of the triads: F  = 3  i =1 F  i D i ,  M  = 3  i =1 M  i D i ,  (2.3)for  i  = 1 , 2 , 3 where both  F  and  D i are 3 by 1 vectors at a given  s  along the rod centerline.The constitutive relations for the unconstrained KR are (Olson  et al.  2013; Lim  et al. 2008) M  1 =  a 1  ∂  D 2 ∂s  · D 3 − Ω  1  , M  2 =  a 2  ∂  D 3 ∂s  · D 1 − Ω  2  , M  3 =  a 3  ∂  D 1 ∂s  · D 2 − Ω  3  , (2.4) F  1 =  b 1 ∂  X ∂s  · D 1 , F  2 =  b 2 ∂  X ∂s  · D 2 , F  3 =  b 3  ∂  X ∂s  · D 3 − 1  ,  (2.5)where the material properties of the rod are characterized through the parameters  a i  and  b i for  i  = 1 , 2 , 3. The bending moduli are  a 1 ,  a 2  and  a 3  is the twisting modulus while  b 1 ,b 2  arethe shear moduli, and  b 3  is the extensional modulus. The strain-twist vector is represented by { Ω  1 ,Ω  2 ,Ω  3 } where  Ω  3  is the intrinsic twist and  Ω  1 ,Ω  2  are the geodesic and normal curvatures,respectively, associated with the intrinsic curvature  Ω   through the equation  Ω   =   Ω  21  + Ω  22 .This vector determines the preferred configuration of the rod where internal force and torqueare generated by differences from the actual and preferred configuration. The preferred strainand twist of the rod can be varied (in time  t  and with respect to arc length parameter  s ), topropagate planar or spiral bending that are representative of sperm flagellar beatforms observedin experiments (Smith  et al.  2009; Woolley & Vernon 2001). 3. Method of Regularized Brinkmanlets for the Kirchhoff Rod Given a 3D elastic structure immersed in a Brinkman fluid, the equations of motion include theexternal forces and torques of the structure on the fluid. The solutions can be calculated exactlyin terms of fundamental solutions due to the linearity of the Brinkman equation. However,singular solutions are obtained when evaluating the flow at the location of a point force ortorque on the centerline of the structure. Eliminating these singularities requires a regularizationmethod and we utilize the Method of Regularized Brinkmanlets (MRB) (Cortez  et al.  2010).The idea is to use a smooth approximation to the singular point force or point torque. Thesmooth approximation is called a “blob” or regularization function,  φ ε ( r ), and is a radiallysymmetric function whose width is determined by the regularization parameter  ε  ≪  1. In thelimit as  ε → 0, the singular solutions are recovered.Since we want to capture both the bending and twisting motions of the rod in a 3D infinitefluid, the expression of the force density  f  b at a point  x  in the fluid is a contribution of both  f  and  m , given as f  b ( x ,t ) =   Γ   − f  ( s,t ) + 12 ∇× ( − m ( s,t ))  φ ε ( r ) ds  (3.1)
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