Stability of downflowing gyrotactic microorganism suspensions in a two-dimensional vertical channel

Transcription

1 Under consideration for publication in J. Fluid Mech. 1 Stabilit of downflowing grotactic microorganism suspensions in a two-dimensional vertical channel Y O N G Y U N H W A N G 1 A N D T. J. P E D L E Y 2 1 Department of Civil and Environmental Engineering, Imperial College London, South Kensington, London SW7 2AZ, UK 2 Department of Applied Mathematics and Theoretical Phsics, Universit of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 WA, UK (Received 29 April 214 and in revised form??) Hdrodnamic focusing of cells along the region of the most rapid flow is a robust feature in downflowing suspensions of swimming grotactic microorganisms. Experiments performed in a downward pipe flow have reported that the focussed beam-like structure of the cells is often unstable and results in the formation of regular-spaced axismmetric blips, but the mechanism b which the are formed has not been well understood. To elucidate this mechanism, in this stud, we perform a linear stabilit analsis of a downflowing suspension of randoml swimming grotactic cells in a two-dimensional vertical channel. On increasing the flow rate, the basic state exhibits a focussed beam-like structure. It is found that this focussed beam is unstable with the varicose mode, the spatial structure, wavelength, phase speed, and behaviour with the flow rate of which are remarkabl similar to those of the blip instabilit in the pipe flow experiment. To understand the phsical mechanism of the varicose mode, we perform an analsis which calculates the term-b-term contribution to the instabilit. It is shown that the leading phsical mechanism in generating the varicose instabilit originates from the horizontal gradient in the cell-swimming-vector field formed b the non-uniform shear in the base flow. This mechanism is found to be supplemented b cooperation with the grotactic instabilit mechanism observed in uniform suspensions. 1. Introduction Grotaxis is the term that describes the biased swimming of microorganisms such as Chlamdomonas, Dunaliella, and Volvox in the presence of shear in the surrounding fluid (Kessler 1984). These microorganisms are structurall bottom-heav: i.e. their centre of mass is located behind the centre of buoanc. Therefore, when a microorganism of this tpe is not aligned with the vertical, the bottom-heaviness generates a gravitational torque which reorients the cell to the vertical. This mechanism leads the microorganism to be naturall gravitactic: it swims against gravit in the absence of shear. However, when a shear is imposed in the surrounding fluid, the microorganism also experiences a viscous torque due to the shear. Therefore, the orientation of the cell is determined b the balance between the gravitational and the viscous torques, and the term grotaxis refers to this process (Kessler 1984, 1985a,b, 1986). In addition to determining the swimming orientation of a single cell, grotaxis plas address for correspondence:.hwang@imperial.ac.uk

2 2 Y. Hwang and T. J. Pedle a crucial role in generating instabilities in suspensions of bottom-heav microorganisms (Pedle & Kessler 1992; Hill & Pedle 25; Pedle 21b). The stationar bottomstanding streak plumes observed in a deep and uniform suspension (d > 1cm where d is the depth of the suspension) are an example of the grotactic instabilit (see e.g. Kessler 1986; Pedle & Kessler 1992). We suppose a small blob of higher concentration of cells in an otherwise uniform suspension. As the blob is heavier than the surroundings, it will sink while generating a downward shear flow in its wake. Due to the grotactic nature of the bottom-heav cell, the downward shear flow leads the other cells to swim toward the blob and its wake. The blob therefore becomes denser and creates even stronger downflow, causing an instabilit in the suspension (Pedle et al. 1988; Pedle & Kessler 199; Pedle 21a,b). Another example of the crucial role plaed b grotaxis in generating instabilit is bioconvection which appears in a shallow laer (d < 1cm) of the suspension (Childress et al. 1975; Kessler 1985a; Hill et al. 1989; Bees & Hill 1998; Hwang & Pedle 214). Here, the gravitactic up-swimming cells quickl accumulate at the top end of the suspension and form a laer of cells heavier than the surroundings. This leads the suspension to experience a gravitational instabilit and generates a flow pattern similar to that of Raleigh-Bénard convection (Pedle & Kessler 1992; Bees & Hill 1997). However, contrar to the classical Raleigh-Bénard convection, the pattern in bioconvection is not a simple consequence of the gravitational instabilit. As recentl shown b the authors (Hwang & Pedle 214), the grotactic instabilit mechanism also plas an important cooperative role in generating the instabilit (particularl at high wavenumbers). Furthermore, under an unstabl stratified background cell concentration, the grotaxis ields non-negligible amounts of cross-diffusion flux to locall denser regions of the cells, which also significantl contributes to the bioconvective instabilit (Hwang & Pedle 214). The instabilities observed in the suspensions of grotactic cells are often a consequence of cooperation and/or competition of multiple phsical processes, and this feature often hampers precise understanding of the origin of the instabilities. An example of this is the blip instabilit observed in the earl experiment of Kessler (1984, 1985a,b, 1986) which demonstrated grotaxis of the bottom-heav cells. The experiment is composed of a vertical pipe in which a downward through flow of the cell suspension is applied. Kessler then demonstrated the grotaxis b showing the formation of a vertical beam-like structure along the pipe axis as a consequence of the accumulation of the bottom-heav cells in the region of most rapid downward flow. In this experiment, interestingl, he reported that the vertical beam exhibits an instabilit in the form of regularl-spaced axismmetric blips, each of which eventuall develops into a vortex ring further downstream (Kessler 1985a, 1986). The tpical spacing between the blips was O(1cm) and each of them was found to fall faster than the bulk fluid. He also reported that the blips predominantl appear at relativel low flow rates and that the graduall disappear on increasing the flow rate. This experiment was recentl repeated b Denissenko & Lukaschuk (27) and the reported that the blips are also suppressed at ver low flow rates. Despite the relativel well-documented features of the blip instabilit, its origin has remained poorl understood. The earl review b Pedle & Kessler (1992) speculated that at least two mechanisms would be involved, one of which is from the gravit-driven instabilit observed in a dense falling plume through a surrounding fluid (Lister 1987; Smith 1989), and the other is associated with the inflection points in the velocit profile of the downflowing suspension caused b the strong gravitational force along the beam-like structure (see also a further discussion in 4.2). Recentl, Denissenko & Lukaschuk (27) performed a simple analsis on the blip instabilit b deriving a sstem of equations for the width and the densit of the axial beam. However, their analsis is too simple and

3 Stabilit of downflowing grotactic microorganism suspensions 3 x Flow g Figure 1. Schematic diagram of flow configuration in the present stud. is based on a number of assumptions that are not full justified: for example, timedependence of the momentum equation of the fluid is ignored, and translational and rotational diffusion of the cells are completel neglected. Furthermore, the analsis does not provide an information on the spatial spacing of the blips as it assumes that the solution varies slowl along the vertical direction. The goal of the present stud is to gain fundamental understanding of the origin of the blip instabilit within a theoretical framework. We start with a more complete form of mathematical model developed b Pedle (21a) where the distribution of the randoml swimming cells in time, space, and the cell swimming orientation-space is described with a probabilit densit function satisfing a Smoluchowski equation. We then follow the approximations given in Hwang & Pedle (214) so that the computational cost of the present stud is manageable. For simplicit, the flow configuration is set as a twodimensional vertical channel instead of the circular pipe (see figure 1). Although this configuration is not exactl the same as that of the experiment b Kessler (1985a) and Kessler (1986), we will see that man features of the instabilit found in the present stud are remarkabl similar to the blips observed in the pipe experiment. Finall, we should emphasise that the present stud should be distinguished from previous work b Ghorai & Hill (1999, 2) who performed a linear stabilit analsis of naturall arising standing grotactic plumes. In the present stud, the basic state, obtained in the form of the beam-like structure as in the experiment, is essentiall caused b the imposed flow. Therefore, the basic state given here is continuousl connected to the uniform suspension of the cells in the absence of the imposed flow (see 3.1). In contrast, the basic state in Ghorai & Hill (1999, 2) was given in the form of a verticall uniform stationar plume, generated as a consequence of the grotactic instabilit in a stationar suspension, the top and bottom ends of which are closed b the no-fluid- and no-cell-flux conditions. In this respect, their analsis is precisel a secondar stabilit analsis of a uniform suspension in the given flow configuration (i.e. stabilit of the state bifurcated via a primar instabilit), which clearl differs from the present analsis. It should finall be mentioned that this interpretation is consistent with Bees & Hill (1999) where the emergence and stabilit of stationar grotactic plumes in an unbounded and uniform suspension are analsed. This paper is organised as follows: In 2, the equations of motion are introduced and formulated for a linear stabilit analsis. Parameters and numerical methods used in the present stud are then given. In 3, the basic states are presented, and their linear stabilit is examined. In 4, we compare the present results with the previous laborator

4 4 Y. Hwang and T. J. Pedle experiments, and provide a detailed discussion on the origin of the blip instabilit using a budget analsis. 2. Problem formulation 2.1. Equation of motion We consider a downward fluid flow in a vertical two-dimensional channel in which swimming grotactic microorganisms are suspended (see figure 1). We denote x and as the vertical and horizontal directions, respectivel, and t is the time (the superscript indicates dimensional variables). The suspension is assumed to be dilute (the average volume fraction of cells is tpicall less than 1%), and is bounded b the two walls located at = ±h, respectivel. The fluid in the suspension has densit ρ and kinematic viscosit ν, and gravit heads downward in the x direction. The suspended microorganisms are assumed to be spherical puller -tpe swimmers, which generate thrust b pulling fluid from the front to the back (e.g. C. nivalis, Dunaliella, and Volvox) as in the experiment of Kessler (1986). Their swimming speed is Vc and the are assumed to perform random walks as a result of repeated reorientations. Following Hwang & Pedle (214), where a more detailed discussion of the present model is given, the equations of motion are then written as u =, (2.1a) u t + (u )u = 1 ρ p + ν 2 u n υg i, (2.1b) n t + [n (u + Vc e )] = (D T n ), with boundar conditions u =±h = (, ), (2.1c) (2.1d) [n (u + Vc e ) D T n ] =±h j =. (2.1e) Here, u is the velocit, p the pressure, n the cell number densit, g = g ρ/ρ the reduced gravit (g and ρ are the gravitational acceleration and the densit difference between cell and fluid, respectivel), υ the volume of a single cell, e the unit vector indicating the swimming direction, D T the diffusivit tensor for the cell number densit, and i and j are respectivel the unit vectors in the vertical and wall-normal directions. The no-slip condition (2.1d) is imposed at the wall for the fluid velocit, and the no-flux condition (2.1e) is applied for the cell-number densit so that total number of the cells is preserved in time. In (2.1c), indicates the local ensemble average at given x and time t to take the random walk of the cell into account: for example, the mean swimming vector e implies e (x, t ) e =1 ef(x, e, t ) d 2 e, (2.2a) where f(x, e, t ) is the probabilit densit function (p.d.f.) for the cell orientation, satisfing f(x, e, t ) d 2 e = 1. (2.2b) e =1 Under the assumption that the random walk is Brownian, the p.d.f obes the following

5 Stabilit of downflowing grotactic microorganism suspensions 5 equation (Hwang & Pedle 214): f t + (u )f + e [ 1 2B [i (i e)e]f Ω ef] = DR 2 e f, (2.2c) where B = να /2gl is the grotactic time scale (l is the centre of gravit offset of the cell and α (=.6) is a geometrical constant for the spherical cell), Ω the vorticit, and DR the rotational diffusivit. We note that equations (2.1c) and (2.2c) are derived b approximating the Smoluchoski equation equation which gives the probabilit densit function for distribution of the cell number densit and the swimming orientation (Hwang & Pedle 214). The approximation is valid if the length scale and the shear rate (or vorticit) of the sstem are sufficientl large: i.e. h O(.1cm) and O(u /h) O(B 1 ). However, this does not limit the present stud as the experiments of Kessler (1985a, 1986) were performed under such conditions. Finall, in the present stud, the cell swimming in (2.2c) is full three dimensional (i.e. e = (e 1, e 2, e 3 )) although the flow field is considered to be two-dimensional. In this respect, the present stud is precisel an analsis of twodimensional dnamics in a three-dimensional vertical channel. For the diffusivit tensor in (2.1c), we use the simplified expression given in Pedle & Kessler (199): D T = Vc 2 τ( ee e e ), (2.3) where τ is the correlation time scale for the cell orientation. The assumption of a constant τ ma not be a reasonable description particularl if the local vorticit (or the flow rate) is large (see also e.g. Bearon et al. 212; Croze et al. 213). It can be improved b the generalised Talor dispersion theor as recentl addressed (Hill & Bees 22; Malena & Frankel 23; Bearon et al. 211). However, it should also be pointed out that it is not evident et whether the generalised Talor dispersion theor itself would provide a good quantitative prediction for D T in real cell suspensions. An important assumption in the generalised Talor dispersion theor, as also in the present stud, is that the rotational diffusivit does not change with the local vorticit (see e.g. Hill & Bees 22; Malena & Frankel 23). However, in realit, this does not seem to be true: the rotational diffusivit does depend on the vorticit (S. Furlan, T. J. Pedle & R. E. Goldstein 214, unpublished observation). The underling difficult caused b the lack of an experimentall verified expression for D T leads us still to rel on the simple expression (2.3) and to perform a parametric stud with respect to the correlation time τ and the rotational diffusivit DR instead (see 3.3) Non-dimensionalization The governing equation (2.1) is made dimensionless b introducing x = x h, = h, t = t V c h, u = u V c, p = p ρv c 2, n = n N, (2.4) where N = 1/V Ω n dv and Ω is the domain of interest, with volume V. We note that the velocit scale is chosen to be Vc instead of the one associated with the translational diffusion flux (Vc 2 τ/h) in our previous stud (Hwang & Pedle 214). The dimensionless form of (2.1) is then given as u =, u t + (u )u = p + 1 Re 2 u Ri ni, n t + [n(u + e )] = 1 ScRe (D T n), (2.5a) (2.5b) (2.5c)

6 6 Y. Hwang and T. J. Pedle D 1 f R t + D 1 R (u )f + e [λ[i (i e)e]f + 1 Ω ef] = 2 e f, 2D R with boundar conditions where c h Re = V ν (2.5d) u =±1 = (, ), (2.5e) [n(u + V c e ) 1 ScRe D T n] =±1 j =,, Ri = Nυg h Vc 2, Sc = ν V c 2 τ, D T = (2.5f) D T Vc 2 τ, λ = 1 2BDR, D R = D R h Vc. (2.5g) Here, Re is the Renolds number based on the cell swimming velocit, Ri the Richardson number indicating the ratio of the potential energ of the suspension to the kinetic energ caused b the swimming motion, Sc the Schmidt number representing the ratio of the kinematic viscosit to the translational diffusivit of the cell number densit, λ the inverse of dimensionless grotactic time scale, and D R the dimensionless rotational diffusivit. We note that the Péclet number based on the cell-swimming velocit Vc is then defined as Pe = ScRe Basic state We first calculate the basic state of (2.5) (i.e. the stead solution), to which a small perturbation will be added for a linear stabilit analsis. Since the flow configuration is homogeneous along the vertical direction, the velocit and the cell number densit for the basic state satisf ielding x =, u = u (= (U (), )), n = n (), (2.6) P x + 1 d 2 U Re d 2 Ri n =, (2.7a) P =, (2.7b) d [ n e dn ] d ScRe D22 T =, d (2.7c) e [λ[i (i e)e]f + 1 Ω ef ] = 2 e f, 2D R with boundar conditions (2.7d) U =±1 =, (2.7e) n e 2 =±1 + 1 dn ScRe D22 T =. (2.7f) d =±1 Here, P is the basic-state pressure and Ω is the basic state vorticit, and the subscript in e 2 and DT 22 implies the statistical properties obtained with f. From (2.7a) and (2.7b), P / x is constant in t and. Therefore, assuming even smmetr of the basic state (i.e. U () = U ( ) and n () = n ( ); see also figure 4), integration of (2.7a) over [ 1, ] ields dp dx = 1 du + Ri. (2.8) Re d = 1

7 Stabilit of downflowing grotactic microorganism suspensions 7 (a) (b).6.4 e D T.2 e D T -.2 e D T 12 D T Ω / z D r Ωz Figure 2. (a) Mean cell swimming vector and (b) diffusivit tensor (λ = 2.2). Here, the dashed vertical lines indicate Ω z = ±4.4D r at which the deterministic swimmer begins to tumble. / D r This equation indicates that the vertical pressure gradient dp /dx is composed of two parts: one is the driving pressure gradient to generate a downward flow (the first term on the right-hand side) and the other is the hdrostatic pressure required to balance the gravitational force caused b the negativel buoant cells (the second term on the righthand side). It is important to note that the hdrostatic part of the pressure gradient is independent of the flow rate as Ri is given b the properties of the cells and the flow geometr. Therefore, for a given flow configuration, the hdrostatic part dp h /dx = Ri can be separated from dp /dx, leading to dp d dx + 1 d 2 U Re d 2 Ri (n 1) =, (2.9) where dp d /dx(= 1/Re du /d = 1 ) is the pressure gradient for driving the flow. We also note that (2.9) provides a phsicall consistent solution of (2.7) for the stationar case: i.e. U () = and n () = 1 (uniform suspension) if dp d /dx =. In obtaining the solution of (2.7), it is also important to note that (2.7d) does not contain an partial derivative in x. This implies that f (x, e) is not explicitl dependent upon the spatial location x, and onl depends on the local vorticit Ω (= Ω z k where Ω z = du /d). Therefore, once e and D T are obtained in terms of Ω b solving (2.7d), the can be used as functions of Ω for (2.7c). Numerical solution of (2.7d) for a given local vorticit (or shear rate) has been obtained in detail in our previous stud (Hwang & Pedle 214). Therefore, in the present stud, we briefl report how e and D T behave with the local vorticit Ω z (see figure 2). In the absence of the vorticit (Ω z = ), the mean swimming orientation e heads upward (figure 2a) and the diffusivit tensor D T becomes highl anisotropic (figure 2b) due to the gravitaxis. As Ω z is increased, but while it is small (i.e. < Ω z /D R < 4 5), the mean swimming vector is graduall tilted along the rotational direction of the given vorticit, and the cross component of D T (i.e. DT 12 ) is also generated. However, if the strength of the vorticit is sufficientl large ( Ω z /D R > 4 5), this tendenc is changed. In this case, the size of the mean swimming vector ( e ) decas and the diffusivit tensor tends towards isotropicit as the vorticit strength is increased. It is worth noting that this change is seen to appear from Ω z /D R 4.4, the value at which a deterministic swimmer would begin to tumble. In general, the nonlinear dependence of e and D T on Ω z and the two-wa coupling between (2.7c) and (2.9) ield a difficult in seeking a general analtic solution of (2.7),

8 8 Y. Hwang and T. J. Pedle except for Ri = which admits the following solution: U () = Re dp d 2 dx (2 1), (2.1a) n () = N e ScRe e 2 D T 22 d, (2.1b) where N is a normalization constant giving 1/V Ω n () dv = 1. We note that this is simpl a plane Poiseulle flow, and it can occur if the cells are neutrall buoant. For Ri, the solution of (2.7) is obtained numericall as detailed in 2.6. In the present stud, we perform the entire analsis b prescribing the flow rate, = 1 1 U () d, (2.11) with a given value rather than giving a fixed pressure gradient dp d /dx (see 3.1 for a detailed discussion). Using the flow rate, the Renolds number based on the bulk flow velocit U b (= /2) is obtained as Re b = Re 2, (2.12) where the factor 2 in the denominator originates from the channel width (2h) Linear stabilit analsis Now, we consider a small perturbation about the basic state: u = u (x) + ϵu (x, t) + O(ϵ 2 ), p = P (x) + ϵp (x, t) + O(ϵ 2 ), (2.13) n = n (x) + ϵn (x, t) + O(ϵ 2 ), f = f (e) + ϵf (x, e, t) + O(ϵ 2 ), where u = (u, v ) and ϵ 1. Here, we note that the perturbation of the vertical velocit and the cell number densit over the given control volume should satisf u dv = and n dv =, (2.14) Ω as the flow rate is fixed and the total number of the cells over the entire domain is preserved in time. The linearised equations for the small perturbation are then given as Ω u x + v =, (2.15a) u t + U u U + v x = p x + 1 Re 2 u Ri n, v t + U v x = p + 1 Re 2 v, (2.15b) (2.15c) n t + (U + e 1 ) n x + e n 2 + d e 2 n + v dn d d + e 2 dn d + n = 1 [ D 11 2 n T ScRe x 2 + 2D12 T + D12 T x e 1 x + n 2 n x + dd21 T d e 2 dn d + D22 T dn d + D22 T n x + 2 n D22 T 2 + dd22 T n d d 2 n d 2 ], (2.15d)

9 Stabilit of downflowing grotactic microorganism suspensions 9 (a) (b) ζ 1,ζ ζ Ω / z D r ζ 2 ζ 3,ζ ζ Ωz / D Figure 3. Dependence of (a) ζ 1, ζ 2 and (b) ζ 3, ζ 4 on Ω z (λ = 2.2). Here, the dashed vertical lines indicate Ω z = ±4.4D r at which the deterministic swimmer begins to tumble. D 1 f R t + D 1 R U f x + e [λ[j (j e)e]f + 1 Ω ef ] 2 e f 2D R = D 1 R e [ 1 2 Ω ef ], with boundar conditions r ζ 4 (2.15e) u =±1 = v =±1 =, (2.15f) [ e 2 n + e 2 n ] 1 [ D 12 n T ScRe x + n D22 T + D22 dn ] T =, (2.15g) d =±1 where the superscript for e i and D ij T (i, j = 1, 2) indicates the statistical properties obtained from f. As discussed in Hwang & Pedle (214), performing a linear stabilit analsis directl with (2.15) would not be computationall feasible in practice due to the large dimension of (2.15) and the number of parameters to be studied. Therefore, here we approximate (2.15e) as in Hwang & Pedle (214) where f is assumed to be quasi stead and quasi uniform: i.e. e [λ[j (j e)e]f + 1 2D R Ω ef ] e 2 f = 1 2D R e [Ω ef ]. (2.16) It is evident that the approximation is valid onl if f varies slowl in time and space: f (t, x, e) = f (T, X, e) where T = δt and X = δx with δ 1. We note that the instabilities in the present stud indeed appear in such situations (see 3.2), indicating that relaxing the approximation would not significantl modif the parameter range over which the instabilities appear. The approximated equation (2.16) allows us to write its solution as f (t, x, e) = D 1 R ω z(t, x)f ω z (e) where ω z is the perturbed spanwise vorticit and f ω z (e) is the solution of (2.16) if ω z moments in (2.15d) such as e 1, e 2, DT 12 = 1 and D R = 1. Therefore, the statistical, and D 22 T are given in the form of e 1 = 1 ω D zζ 1, e 1 = 1 ω R D zζ 2, D 12 1 T = ω R D zζ 3, D 22 1 T = ω R D zζ 4, (2.17) R where ζ i for i = 1, 2, 3, 4 are obtained from the first- and the second-order moments of f ω z (e) (for further details, see also Hwang & Pedle 214). The computed ζ i using the numericall obtained f ω z (e) in Hwang & Pedle (214) are shown in figure 3. In practice, the use of (2.16) instead of (2.15e) makes the present approach practicall identical to that in Pedle & Kessler (199) where e and D T are generall obtained

10 1 Y. Hwang and T. J. Pedle from the quasi-stead and quasi-uniform Fokker-Planck equation: i.e. (2.2c) without the convective derivative terms. Within this framework, e and D T on Ω z become identical to e and D T on Ω z given in figure 1. For two-dimensional fluid flows, as in the present stud, this implies that e and D T are also obtained from a Talor-series expansion of e and D T around e and D T : i.e. e = e + ϵ d e ω z + O(ϵ 2 ), D R dω Ω=Ω D T = D T + ϵ dd T D R dω Ω=Ω ω z + O(ϵ 2 ), (2.18) where e = e Ωz =Ω z and D T = D T Ωz =Ω z. Therefore, ζ i are simpl given as ζ 1 = d e 1 dω z Ωz =Ω z, ζ 2 = d e 2 dω z Ωz =Ω z, ζ 3 = dd12 T Ωz, ζ 4 = dd22 T Ωz, (2.19) dω z =Ω z dω z =Ω z and are identical to those obtained using f ω z (e) in Hwang & Pedle (214). We now eliminate p in (2.15) following the standard procedure and rewrite (2.15a), (2.15b), and (2.15c) into a single equation in terms of v. We then consider a normal mode solution, v (x,, t) = ˆv()e i(αx ωt), n (x,, t) = ˆn()e i(αx ωt), (2.2) where α is the streamwise wavenumber and ω is the frequenc. This leads to the following equations for linear stabilit: ( ) ( ) ( ) ( ) M LOS iαrid iω + 1 ˆvˆn L v =, C L C ˆvˆn where M = α 2 D 2, (2.21a) (2.21b) L OS = iαu M + iαd 2 U + 1 Re M2, (2.21c) L v C = Dn + G 1 [ ζ 1 n M + i α D(ζ 2n M)] G 2 [ ζ 3 Dn M + i α D(ζ 4Dn M)] (2.21d) L C = iαu + iα e 1 + e 2 D + D e [ ] α 2 DT 11 2iαDT 12 ScRe D DT 22 D 2 iαddt 12 DDT 22 D, (2.21e) with boundar conditions ˆv =±1 = Dˆv =±1 =, (2.21f) [ e 2 ˆn 1 ( ) iαdt 12 + DT 22 ScRe D ˆn + i ] α (G 1ζ 2 n G 2 ζ 4 Dn )Mˆv =. =±1 (2.21g) Here, D d/d, L OS is the Orr-Sommerfeld operator, L v C the coupling operator between ˆv and ˆn in the equation for ˆn, and L C the homogeneous part of the equation for ˆn,

11 Stabilit of downflowing grotactic microorganism suspensions 11 respectivel. The parameters G 1 and G 2 in (2.21d) are defined as G 1 = 1 = V c D R DR h, G 1 2 = = V c 2 τ ScReD R DR. h2 (2.21h) As discussed in Hwang & Pedle (214), G 1 indicates the importance of swimming relative to rotational diffusion and G 2 represents the importance of translational diffusion relative to rotational diffusion. Finall, it should be pointed out that L C is not a simple advection-diffusion operator because of the term D e 2 in the top line of (2.21e). This term appears to represent the production caused b the gradient in the mean swimming vector field D e 2, and it is found to pla a crucial role in generating the instabilities in the present stud as we shall see in Numerical methods The equations for the basic state (2.7) are solved numericall. The horizontal direction is discretized using a Chebshev-collocation method (Weideman & Redd 2), and the boundar conditions are implicitl imposed on the discretized operator so that its inversion is numericall possible. The stead solution of (2.7) is then obtained using the Newton-Raphson iteration with a prescribed flow rate. For the smallest considered, the iteration is started with a guess U () = 3/4( 2 1) and n () = 1. On graduall increasing, the initial guess is obtained using a numerical continuation: the solution obtained for the previous is imposed as the initial guess for the new. The numerical solution is obtained with N = 151, and it is found to be identical to that with N = 251 within the residual of the iteration. The solution is also validated b an independentl written numerical solver of (2.7). In this solver, the horizontal direction is discretized using a second-order finite volume method, and the solution is obtained b solving the unstead (2.7) for which the time integration is conducted semi-implicitl: the diffusion terms are marched using the second-order Crank-Nicolson method, and the rest of the terms are advanced using a third-order Runge-Kutta method. The solution is obtained with N = 51 to capture the spectral accurac of the Newton-Raphson solver, and shows excellent agreement with that of the Newton-Raphson solver. The equations for linear stabilit (2.21) are solved b modifing the solver validated in our previous stud (Hwang & Pedle 214). The horizontal direction is discretised using the same Chebshev-collocation method. The discretized eigenvalue problem is solved using the function eig in Matlab. The computation is performed with N = 151, showing no difference from the results with N = Parameters Table 1 summarises a list of the parameters and their values in the present stud. The values of the parameters are taken from those for C. nivalis as in previous studies (e.g. Pedle & Kessler 199; Bees & Hill 1998; Pedle 21b; Hwang & Pedle 214). The width of the channel is chosen to be close to the scale of the laborator experiments (Kessler 1985a, 1986; Denissenko & Lukaschuk 27). It should be pointed out that the mean cell number densities N considered is one order of magnitude lower than that in the tpical experimental condition. However, the onset of the instabilit is, in fact, found under such conditions at least within the framework of the present model. We note that this does not ield an inconsistenc with the existing experiments as the critical cell number densit for the blip instabilit has not been discussed at all. Furthermore, performing a linear stabilit analsis with the N used in the experiment would not be phsicall so meaningful from a theoretical perspective as this regime is quite far from

12 12 Y. Hwang and T. J. Pedle Parameter Description Value (Ref. value) Units ρ Fluid densit 1 g/cm 3 g Gravitational acceleration 98 cm/s 2 ν Kinematic viscosit.1 cm 2 /s d(= 2h) Channel width.4.8 (.4) cm N Cell mean number densit cells/cm 3 ρ/ρ Relative cell densit.5 υ Cell volume cm 3 g (= g ρ/ρ) Relative gravit 49 cm/s 2 B grotactic time scale 3.4 sec Vc Swimming speed cm/s τ Correlation time scale 2 5 (5) s DV (= Vc 2 τ) Nominal translation cell diffusivit cm 2 /s DR Rotational diffusivit (.67) 1/s Table 1. Parameters and their values in the present stud (the ones in parentheses indicate the reference values). The parameters for the cell properties are taken from the data for C. nivalis (Pedle & Kessler 199; Bees & Hill 1998; Pedle 21b; Hwang & Pedle 214). Parameter Description Value (Ref. value) Sc Schmidt number (5) Ri Richardson number 2 Dimensionless flow rate 8 Re Renolds number based on Vc (.13) Re b Renolds number based on the bulk flow velocit 1 G 1 (= 1/D R ) see (2.21h) (.47) G 2 see (2.21h) (.74) λ inverse of grotactic time scale normalised b DR (2.2) D R Rotational diffusivit normalised b Vc /h (2.1) Table 2. Dimensionless parameters and their values in the present stud (the ones of parentheses indicate the reference values). the onset of instabilit. Based on the parameter values given in table 1, the dimensionless parameters and their values are obtained as in table Results 3.1. Basic state The basic state is first calculated b solving (2.7), and is reported in figure 4 for a sufficientl large Ri(= 9). For zero flow rate ( = ), U () = and n () = 1 as discussed. These solutions continuousl deform as the flow rate is graduall increased. For small flow rates ( < 2), the velocit profile exhibits upward backflow near the wall ( >.6) though the bulk fluid flows downward (figure 4a). This backflow tpicall appears at large Ri as will be discussed with figure 5. The cell-number densit profile n () shows its maximum at the channel centre = (figure 4b), and it is clearl a consequence of the grotaxis of the cells as in the experiment of Kessler (1985a): the downward flow ields the cells to swim towards the centreline (i.e. e 1 > at < with Ω z and e 1 < at > with Ω z ) while the swim upward ( e 2 > ). In this respect, it is interesting to note that n () at the walls ( = ±1) in this case is

13 (a) 1 Stabilit of downflowing grotactic microorganism suspensions 13 (b) 4 3 ) U ( -1 ) n ( (c) (d) ) U ( -5-1 ) n ( Figure 4. Profiles of the basic state respectivel for (a, b) low ( =,.4,.8, 1.2, 1.6) and (c, d) high flow rates ( =, 5, 1, 15, 2) at Ri = 9: (a, c) vertical velocit U (); (b, d) cell-number densit n (). slightl larger than it is close b due to the upward backflow. As the flow rate is further increased, the upward backflow in U () diminishes due to the strong pressure gradient applied downward (figure 1c). Also, U () becomes closer to the parabolic profile given in (2.1a) because the forcing b the pressure gradient dominates over the gravitational one. The peak value of the cell-number densit profile n () at the channel centre is found to increase with the flow rate (figure 4d), indicating that the beam-like structure along the channel centre is more focused on increasing at least for the values considered here. However, it should be mentioned that this behaviour ma not persist at ver large flow rates. If the flow rate is ver large, local Ω z would also be ver large. In this case, e 2 decreases with the increase of while DT 22 approaches a constant value D22 T.33 (see figure 2). Since n () is given b (2.1b) in the limit of, the width of the focussing in n () at the channel centre (i.e. l D 22 T / e 2 ) would become thicker with an increase of. However, we should note that this could be an artifact of the present model as the expression for the diffusivit tensor (2.3) would not be reliable at ver large Ω z (or ). For further detailed discussion on this issue, the reader ma refer to appendix in Hwang & Pedle (214) where the relevance of the expression (2.3) is analsed in comparison to the generalised Talor dispersion theor. The range of the flow rate and the Richardson number Ri leading to the backflow in U () is also identified b further inspecting (2.9). Integrating (2.9) over [ 1, ]

14 14 Y. Hwang and T. J. Pedle 6 dp d dx 4 2 Ri Figure 5. Dependence of the relation between the driving pressure and the flow rate on the Richardson number (Ri =, 1, 2, 3,..., 9). gives dp d dx = 1 Re du d. (3.1) = 1 If the backflow in U () appears near the wall, du /d = 1 >, ielding dp d /dx <. Figure 5 shows the relation between dp d /dx and obtained from the computed basic state for a number of Ri. The region for dp d /dx < appears at relativel small (< 3) and large Ri(> 7), indicating that the the backflow appears if the gravitational forcing dominates over the pressure gradient in the flow-field formation. Furthermore, dp d /dx < implies that the driving pressure is applied upward, suggesting that the backflow is caused b an upward pressure gradient. Therefore, it appears that the downward flow in this case is generated essentiall b the gravitational force, and the role of the imposed pressure gradient is limited to simpl balancing with the gravitational force so that the given flow rate is maintained. It should be pointed out that this feature would make it difficult to interpret an analsis like the present one but with dp d /dx prescribed instead of flow rate. As shown in figure 5, for a given dp d /dx, two stead solutions emerge at sufficientl large Ri. A time dependent simulation using the numerical solver used for validation shows that the one with smaller is an unstable stead solution. This suggests that if dp d /dx is chosen as the control parameter for the flow rate, the stead solutions exhibit a saddle-node bifurcation on increasing dp d /dx, creating an additional complexit in analzing the flow field Linear stabilit analsis Now, we perform a linear stabilit analsis with the reference parameters given in tables 1 and 2. We first consider a uniform suspension without an flow (i.e. n () = 1 and = ). Figure 6 shows the growth rate of the two most unstable modes with respect to the vertical wavenumber α for several Ri near the onset of the instabilit, Ri c 66 (note that Ri = 5 corresponds to N cells/cm 3 for the reference parameters). It is found that one branch depends on Ri (figure 6a), whereas the other remains unchanged regardless of the value of Ri (figure 6b). While the former gives a non-trivial eigenfunction, the latter is numericall found to ield a simple eigenfunction, ˆv = and ˆn = const. This feature in the latter enables us to analticall obtain its dispersion relation as ω = α e 1 idt 11 α2 /(ScRe), and this is indeed identical to the numerical one given in figure 6 (b). We note that this mode gives zero growth rate at α = (i.e. ω i = ). However, the constraint given in (2.14) would exclude an initial condition for (2.15) that would give rise to this mode, so it can be safel neglected.

15 .1.5 Stabilit of downflowing grotactic microorganism suspensions 15 (a) (b) Ri.1.5 ω i ω i α Figure 6. Growth rate of the first two most unstable modes in stationar suspensions ( = ) with the vertical wavenumber α near the critical Ri c 66 (Ri = 5, 6, 7, 8, 9): (a) the mode depending on Ri and (b) the one independent of Ri. α (a).2 (b) ω i -.1 ω i (c) α (d) α ωi ωi α Figure 7. Growth rate with the vertical wavenumber α for (a, b) small ( =,.4,.8, 1.2, 1.6) and (c, d) large ( =, 4, 8, 12, 16) flow rates (Ri = 9): (a, c) sinuous and (b, d) varicose modes. In (a, b),, = ;, =.4;, =.8;, = 1.2;, = 1.6, and, in (c, d),, = ;, = 4;, = 8;, = 12;, = 16. α On increasing the flow rate, the branches of the two modes continuousl change as shown in figure 7. The mode depending on Ri at = develops into the sinuous mode (see also figure 8a), while the one independent of Ri at = smoothl changes into the varicose mode (see also figure 8b). For small flow rates ( < 1), the sinuous mode at low vertical wavenumbers (α < 1) is slightl destabilised with a small increase of

16 16 Y. Hwang and T. J. Pedle (a) (b) x x Figure 8. Spatial structure of the (a) sinuous and (b) varicose modes (Ri = 9, = 4, and α = 1.3). Here, the contours indicate n () + γn (, x) with an arbitrar value of γ for visualization, and the vectors represent the vertical and horizontal perturbation velocit field. (figure 7a) whereas at high wavenumbers (α > 1.5) it is rapidl stabilised. However, as is further increased, the sinuous mode for all α is stabilised, and this tendenc continues even at ver large flow rates (figure 7c). On the other hand, the varicose mode is significantl destabilised with an increase of except at ver small (<.4) (figure 7b). It becomes the most unstable mode for most of the flow rates considered ( > 2 in this case). The destabilization with the increase of continues until 4 5, after which the varicose mode turns out to be slowl restabilised (figure 7d) on increasing. The varicose mode also becomes stable at ver large flow rates ( > 16). The tpical spatial structure of the sinuous and the varicose modes is visualised in figure 8. The sinuous mode is characterised b a meandering motion of the focused beam along with an alternating column of vortices, spanning the whole width of the channel (figure 8a). The rotational direction of the vortex alternates along the vertical direction, and the focussed beam of the cell concentration meanders along the downwash side of each vortex. On the other hand, the beam-like structure in the varicose mode exhibits a periodic thickness variation along the vertical direction and a pair of counter-rotating vortices are aligned along the beam (figure 8b). This structure is clearl reminiscent of the blip instabilit observed in the downward pipe flow experiment of Kessler (1985a, 1986), and man other features of the varicose mode are also found to be strikingl similar to those of the blip instabilit as we shall see further in 4.1. In figure 9, we show contours of the largest growth rate ω i,max and the corresponding α max in the -Ri plane. Here, the values of α used to search α max are chosen to var from α =.1 so that the branch giving ω i = at α = is not numericall included (figure 6b). However, even in this case, drawing a smooth neutral stabilit curve (ω i = ) on the contour has been found to be difficult because this branch still gives ω i of the most unstable mode to be close to zero at the smallest α(=.1). For this reason, we simpl provide a guide isoline ω i =.1 (the solid line in figure 9a), which appears to be a little smoother than the precise neutral line ω i =. For =, the instabilit appears at Ri Ri c ( 66). As discussed above, the mode associated with this instabilit transits into the sinuous mode on increasing, and is rapidl stabilised for >.4 (see also figures

17 Stabilit of downflowing grotactic microorganism suspensions 17 (a) 9.2 (b) Ri Figure 9. (a) The maximum growth rate ω i,max and (b) the corresponding α max in the Ri- plane. Here, α is chosen from α =.1 to α = 3.1 with an increment α =.2. The solid line in (a) is ω i =.1. 7a and c). On the other hand, the varicose mode is significantl destabilised on increasing, ielding a branch switching behaviour with a kink near 1.5 in the contour of ω i,max (figure 9a). The varicose mode gives the largest growth rate ω i,max for 1.5. For sufficientl large flow rates ( 5), the varicose mode is stabilised as the flow rate is increased. The most unstable varicose mode involves a non-zero α max (.5 1.2) as alread discussed (figure 9b). The vertical wavenumber giving the largest growth rate is found to increase on increasing for 8, and it then decreases with a further increase of ( 8) Other parameters In this section, we test the robustness of the present results b examining a few different sets of the parameters. As discussed previousl, a limitation of the present model is the lack of an experimentall justified expression for the translational diffusivit tensor. We therefore focus on investigating how the results with the reference parameters would change as the diffusion characteristics of the sstem are changed. The following three parameters are considered to var: 1) τ the correlation time scale in the translational diffusivit model (2.3); 2) h the half width of the channel; 3) DR the rotational diffusivit. First, varing τ onl changes the Schmid number Sc while the rest of the dimensionless parameters remain the same. This therefore allows us to examine solel the role of the translational diffusivit. Second, varing the half width of the channel h, which also enables us to test the robustness to the flow geometr, changes Re, Ri, and D R in (2.5). However, the change in D R does not ield a change in the solution of (2.5d) because Ω in it is also made dimensionless with the length scale h. Since Ri is also a control parameter for the instabilities, varing h would onl change Re. Finall, varing the rotational diffusivit ields a change of λ(= 1/(2BDR )) in (2.5d). This therefore results in a change in the statistical properties such as e and D T, leading the entire sstem to experience an intricate change. However, it is worth pointing out that the increase of DR generall weakens the role of the swimming in the sstem: for example, in the limit of DR, e = and D T = 1/3I in (2.5c), indicating that the effect of the grotaxis would diminish. Figure 1 shows the contours of ω i,max and α max in the -Ri plane for τ = 2 s (Sc = 126). The overall tendenc of the contours is qualitativel similar to that for the reference value, and onl a quantitative change appears. For all flow rates, the instabilit appears at much lower Ri (figure 1a) compared to that with τ = 5s as

18 18 Y. Hwang and T. J. Pedle (a) 4 (b) Ri Figure 1. Contours of (a) the maximum growth rate ω i,max and (b) the corresponding α max in the Ri- plane for τ = 2s (Sc = 126). Here, α is chosen from α =.1 to α = 3.1 with an increment α =.2. The solid line in (a) is ω i =.1. (a) 4.4 (b) Ri Figure 11. Contours of (a) the maximum growth rate ω i,max and (b) the corresponding α max in the Ri- plane for h =.4 cm (Re =.25). Here, α is chosen from α =.1 to α = 3.1 with an increment α =.2. The solid line in (a) is ω i =.1. expected (figure 9a). With a small increase of the flow rate ( < 5), the varicose mode is destabilised as in the case of τ = 5 s (figure 1a), and the corresponding α max is augmented. Given the fact that the overall translational diffusivit of the cell-number densit is weaker, such a change would not be surprising. It is also interesting to note that the varicose instabilit in this case is persistent even at fairl high flow rates ( = 4) although its growth rate tends to deca on increasing. The effect of an increase of the half width of the channel (h =.4 cm) is shown in figure 11. This change ields a decrease of the dimensionless fluid viscosit and the translational diffusivit of the cell-number densit as the Renolds number increases (Re =.13.25). We also note that, in this case, Ri 1 for N cells/cm 3. Not surprisingl, the overall change caused b the increase of the channel width is qualitativel similar to that for τ = 2 s (figure 1): the critical Ri for the onset of the instabilit are considerabl lowered at all, and the α max is also increased. The varicose mode in this case is even more persistent on increasing, and it is not completel restabilised even at = 8 for the considered Ri. Finall, the dependence on the rotational diffusivit is tested with DR =.147s 1 (λ = 1), a value significantl larger than the reference DR (=.67s 1 ) (λ = 2.2), and is shown in figure 12. The much higher values of Ri c (> 25) at all the flow rates than those in figure 9 suggest that the increase of DR has a stabilizing effect on the instabilit.

19 Stabilit of downflowing grotactic microorganism suspensions 19 (a) 4.2 (b) Ri Figure 12. Contours of (a) the maximum growth rate ω i,max and (b) the corresponding α max in the Ri- plane for D R =.147s 1 (λ = 1.). Here, α is chosen from α =.1 to α = 3.1 with an increment α =.2. The solid line in (a) is ω i =.1. Interestingl, the contours of ω i,max and α max in this case looks quite different from those in figures 9, 1, and 11. A closer scrutin reveals that the most unstable mode is given b the sinuous mode instead of the varicose mode in most of the regions in the Ri- plane except for Ri 3 35 and 6 8 where the varicose mode is found to be most unstable, resulting in a small island in the contours. This suggests that the increase of the rotational diffusivit stabilises the varicose mode more than the sinuous one and that the rotational diffusivit plas an important role in determining the competition dnamics between the sinuous and the varicose modes. However, the value considered in figure 12 (D R =.147s 1 ) appears to be out of a biologicall relevant range in the stationar condition: for instance, the experiment b Vladimirov et al. (24) reported that the measured value of λ for C. nivalis in a stationar medium varies in the range of λ This suggests that the sinuous-mode dominance in the competition dnamics between the two modes would probabl not be observed in realit. 4. Discussion Thus far, we have analsed linear instabilities emerging in a downflowing suspension of grotactic microrganisms through a two-dimensional vertical channel. At zero flow rate ( = ), inspection of the two most unstable modes near the critical Richardson number Ri c shows that one is destabilised b an increase of Ri while the other is found to be independent of Ri. On increasing the flow rate, the former continuousl changes into the sinuous mode (figure 8a) whereas the latter is deformed into the varicose mode (figure 8b). For a given Ri (i.e. averaged cell concentration for a given flow configuration), the sinuous mode is slightl destabilised b a small increase of the flow rate and is quickl restabilised when it is further increased. On the other hand, the varicose mode is much more significantl destabilised than the sinuous one b an increase of the flow rate, being the most unstable mode in a wide range of the flow rate. At ver large flow rates, both sinuous and varicose modes are found to be restabilised. These qualitative features are robust to changes of the parameters such as the translational diffusivit and the channel width as long as the value of the rotational diffusivit is kept in the biologicall relevant range.

20 2 Y. Hwang and T. J. Pedle cr U c, Poi U( = ), U c, Poi Figure 13. Centreline velocit U ( = ) and phase speed of the varicose mode c r (= ω r /α max ) normalised b the corresponding centreline velocit of the plane Poiseulle flow U c,p oi (= 3/4) with respect to the flow rate (Ri = 9):, c r/u c,p oi;, U ( = )/U c,p oi Comparison with experiments Let us first compare the present results with the original experiment b Kessler (1986) and the recent one b Denissenko & Lukaschuk (27). Although their flow configuration is composed of a vertical pipe rather than a channel as in the present stud, a number of the present findings show qualitativel good agreement with those in the experiments. As mentioned previousl, the spatial structure of the varicose mode is ver similar to that of the blip instabilit: for example, the pair of counter rotating vortices in figure 8 (b) is reminiscent of the vortex ring in the late stage of the blip instabilit in pipe flow (see e.g. figure 5 in Kessler 1986). The vertical wavenumber α max of the most unstable varicose mode in the present stud has been robustl found to be α max 1 2 regardless of the change of some parameters (figures 9, 1, and 11). This ields λ x,max cm, which compares fairl well with that observed in the experiments: e.g. λ x 1.5 cm in Kessler (1986) and λ x.5.7 cm in Denissenko & Lukaschuk (27). Furthermore, the experiments reported that the blips disappear at sufficientl high flow rates, and the same qualitative feature is retrieved as seen in figure 9. The phase speed of the varicose mode is also compared with the falling speed of the blips. Figure 13 shows the phase speed of the varicose mode c r (= ω r /α max ) with the centreline velocit of the base flow U ( = ). Here, both c r and U ( = ) are normalised b U c,p oi to compare them with the centreline velocit of the plane Poiseulle flow at the same flow rate U c,p oi (= 4/3). The phase speed of the varicose mode c r is found to be faster than U c,p oi, indicating that the varicose mode falls faster than the corresponding centreline velocit of the plane Poiseulle flow. Interestingl, it is almost identical to the centreline velocit of the base flow U ( = ) except for < 4 where c r is a little slower than U ( = ), indicating that the varicose mode originates from =. Furthermore, this suggests that the faster speed of the varicose mode is essentiall due to the focussed beam which increases the centreline flow speed via the gravitational force. This finding is consistent with the pipe flow experiment of Kessler (1986) where he reported that the blips fall faster than the corresponding centreline velocit of the Poiseulle flow U c,p oi. The speed of the blips in his experiment was found as 2 5U c,p oi, comparable with c r = 1 2U c,p oi in figure 3, given the flow geometr of the present stud. Despite such an encouraging comparison, care needs to be taken in interpreting the present results quantitativel. An important reason for this stems from the lack of a reliable model for the diffusivit tensor. As discussed, the rotational diffusivit is significantl