Noname manuscript No. (will be inserte by the eitor) Effective Rheological Properties in Semiilute Bacterial Suspensions Mykhailo Potomkin Shawn D. Ryan Leoni Berlyan Receive: ate / Accepte: ate Abstract Interactions between swimming bacteria have le to remarkable experimentally observable macroscopic properties such as the reuction of the effective viscosity, enhance mixing, an iffusion. In this work, we stuy an iniviual base moel for a suspension of interacting point ipoles representing bacteria in orer to gain greater insight into the physical mechanisms responsible for the rastic reuction in the effective viscosity. In particular, asymptotic analysis is carrie out on the corresponing kinetic equation governing the istribution of bacteria orientations. This allows one to erive an explicit asymptotic formula for the effective viscosity of the bacterial suspension in the limit of bacterium non-sphericity. The results show goo qualitative agreement with numerical simulations an previous experimental observations. Finally, we justify our approach by proving existence, uniqueness, an regularity properties for this kinetic PDE moel. Keywors effective viscosity kinetic moels bacterial suspension asymptotic analysis Contents 1 Introuction...................................... 2 2 Moel for Semiilute Bacterial Suspensions.................. 4 M. Potomkin Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 E-mail: mup20@math.psu.eu S. D. Ryan Department of Mathematical Sciences an Liqui Crystal Institute, Kent State University, Kent, OH 44240 E-mail: sryan18@kent.eu L. Berlyan Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 E-mail: berlyan@math.psu.eu
2 Mykhailo Potomkin et al. 2.1 Definition of the effective viscosity for a suspension of point force ipoles....................................... 7 3 Conitions impose to erive an explicit formula for the effective viscosity 9 3.1 Separation of variables............................. 9 3.2 Existence of a steay state P ()...................... 9 3.3 P x(x) is constant in the z-irection..................... 10 4 Derivation of asymptotic expression for P for small B........... 10 4.1 Evaluation of Fourier transforms...................... 12 4.2 The form of asymptotic expansion in B.................. 13 4.3 Contribution at O(B).............................. 14 4.4 Contribution at O(B 2 )............................. 15 5 Explicit formula for the effective viscosity................... 17 5.1 Mechanisms require for the ecrease in the effective viscosity... 18 5.2 Effective noise conjecture........................... 18 5.3 Comparison to numerical simulations................... 19 6 Global solvability of the kinetic equation.................... 21 7 Conclusions....................................... 28 A Appenix: Explicit form of integral terms I i from (39)............ 30 B Appenix: Justification of the representation formula (17)......... 33 1 Introuction Bacterial suspensions exhibit remarkable macroscopic properties ue to the emergence of self-organization among its components. In particular, interesting effective properties such as enhance iffusivity, the formation of sustaine whorls an jets, an the ability to extract useful work among other results have been recently observe for suspensions of bacteria, such as Bacillus subtilis [40, 37, 22, 34, 8]. The striking experimental observations on the effective viscosity provie the motivation for stuying a suspension s effective properties; namely, the observation of a seven-fol reuction in the effective viscosity of a suspension of swimming B. subtilis [35]. This reuction is observe below 2% volume fraction typically referre to as the ilute regime where bacteria are far apart an essentially interact with the backgroun flui only. With the assumption of no interbacterial interactions, this regime has been stuie analytically in recent works (e.g., [32, 17, 15, 16]). There bacterial tumbling was introuce in orer for the formula to preict a ecrease in the effective viscosity [16]. However, in the absence of tumbling (e.g., for anaerobic bacteria) the ecrease is still observe experimentally [35]. It was shown recently in [29] that interbacterial interactions substantially contribute to effective viscosity an an estimate for this contribution was given. Rigorous analysis of this contribution an its corresponing effect on the effective viscosity of the suspension is the main component of this paper. We begin with an iniviual base moel (IBM) previously introuce in [29,28], which has been successfully use to capture the ecrease in the effective viscosity an other collective phenomena. Such suspensions, where interbacterial interactions play an important role an are moele as a sum of pairwise interactions, are referre to as semi-ilute. Our goal is to ientify the unerlying mechanisms that contribute to the ecrease of the effective
Effective Rheological Properties in Semiilute Bacterial Suspensions 3 viscosity in this concentration regime. The main tool we employ is a kinetic theory erive from this iniviual base moel. The purpose for employing a kinetic approach is to replace a large system of couple ifferential equations by a single continuum partial ifferential equation with respect to a probability istribution of bacteria positions an orientations. Note that it is natural to consier probabilistic quantities since the main focus of this work is the stuy of the effective properties. The main computational avantage of the kinetic approach is that the number of bacteria N oes not increase the complexity of the problem [39,5]. Namely, the PDE coul be solve numerically with a fixe spatial or temporal gri inepenent of N. In aition to the ability to consier many ifferent initial conitions at once, another avantage to introucing this probabilistic framework is to consier the limiting regime as N, the so-calle mean fiel limit. More information on kinetic equations can be foun in the seminal works of the 1970 s [24,6,10] or more contemporary reviews [7,19,25,9]. Significant ifficulty in the analysis come from the incorporation of interactions. First, they appear in the kinetic equation as a non-local term ue to the fact that the suspension of interacting bacteria is generally escribe analytically by configurations of all bacteria. Secon, the main interactions that are taken into account are hyroynamic, which iverge as bacteria approach one another as the square of their istance. This results in a singular kernel in this non-local term. Thus, the kinetic equation consists of a nonlocal, nonlinear PDE ue to the presence of interactions. Using a kinetic approach, the main result of this paper is an explicit asymptotic formula for the effective viscosity with interbacterial interactions taken into account. The formula reveals the physical mechanisms necessary for the ecrease in effective viscosity observe experimentally. To achieve this result we first fin the steay state solution of the kinetic equation an then use this solution to compute the effective viscosity. For completeness, we also establish the well-poseness of the kinetic equation. This paper is organize as follows. Section 2 begins by introucing the iniviual base moel uner consieration for a semi-ilute bacterial suspension. From this, the kinetic equation for the orientation istribution is formally erive. The reason we begin with the IBM is that the effective properties of a suspension are erive from knowlege of microscopic configurations, which is transferre from the IBM to the kinetic moel. In Section 3 we introuce the main conitions uner which we erive the asymptotic formula for the effective viscosity an iscuss their physical significance. Section 4 contains the erivation of the asymptotic steay state solution to the kinetic equation for the orientation istribution in the limit of small non-sphericity. The effective viscosity from the asymptotic formula is then compare to the same quantity compute from irect simulations of the iniviual base moel in Section 5. The important physical mechanisms for the ecrease in viscosity are ientifie an the orientation istribution is compare to the results of previous works in the ilute case. In aition, the normal stress ifferences an relaxation time are consiere. The existence, uniqueness an regularity properties of a
4 Mykhailo Potomkin et al. solution to the kinetic PDE are proven in Section 6. Finally, we formulate our conclusions an outline potential future investigations in Section 7. 2 Moel for Semiilute Bacterial Suspensions We begin by introucing the couple PDE/ODE system governing the flui an bacteria ynamics respectively. Each bacterium is represente as a point force ipole. One force represents the bacterium s propulsion mechanism (e.g., flagellar motion) an the other is the opposing viscous rag exerte by the bacterium s boy on the flui. This approximation has been experimentally verifie by observing the flow ue to a bacterium (e.g., Bacillus subtilis) in a flui an comparing it to that of a force ipole [11]. As a bacterium swims through the flui its trajectory may be altere through interactions with other bacteria an the backgroun flow. At every moment in time a bacterium propels itself in the irection in which it is oriente. If one bacterium comes into close contact with another, then a collision can occur altering the bacterium s position. This is moele by an exclue volume potential. Finally, the flow itself has an impact on a bacterium trajectory through the ambient backgroun flow an the sum of flows inuce from the propulsion of all the other bacteria on its position. To make these ieas more concrete we now introuce an iniviual base moel (IBM), which governs a bacterium s position an orientation. We consier N bacteria with the position of the center of mass of the ith bacterium x i = (x i, y i, z i ) an orientation i = ( i 1, i 2, i 3). A bacterium s translational velocity is erive from a balance of forces ue to self-propulsion, collisions, an the flow fiel acting on the position of the bacterium. A bacterium s orientation velocity is erive from a balance of torques in the form of Jeffery s equation for an ellipsoi in a linear flow with aitional terms ue to the flows generate by the other bacteria in the suspension [20]. Thus, the equations of motion for bacterial positions x an orientations originally introuce from first principles in [29] are ẋ i = V 0 i + ( u j (x i, j ) + F j (x i ) ) + u BG (x i ), (1) j i ḋ i = 1 2 i ω BG i B i 0 (x i ) + ω j (x i, j ) j i E BG 0 (x i ) + E j (x i, j ) i + 2DẆ, (2) j i where V 0 is an iniviual bacterium s swimming spee an B is the Bretherton constant which takes into account the geometry of the bacterium s boy (B 1: near spherical, B 1: neele-like). The externally-impose planar shear flow contributes to each bacterium s motion through the flui velocity,
Effective Rheological Properties in Semiilute Bacterial Suspensions 5 u BG = (0, γx, 0) T, as well as its effect on a bacterium s( orientation through the vorticity ω BG 0 = x u BG an rate of strain E BG 0 = 1 2 x u BG + ( x u BG ) ) T. Here W is a white noise an we let D B 2 be the iffusion coefficient. This orer of D will be use throughout this work an represents the iea that the ranom motion present in the system has a greater effect the more elongate a particle is. The aitional terms in Jeffrey s equation (2) beyon the contribution from the backgroun flow are ue to the vorticity ω j an rate of strain E j generate by the j-th ipole on position of the i-th ipole ω j = x u j, E j = 1 2 ( x u j + ( x u j ) T ). Each of these terms epens on the flui velocity u j, which is governe by Stokes equation an will be escribe in greater etail below. Remark 1 The equations of motion (1)-(2) are a 5N couple system of orinary ifferential equations in comparison to the ilute case stuie in [16] where there were only two ODEs governing the evolution of a single bacterium in an infinite meium (only epening on a single bacterium s orientation). Thus, the semi-ilute system of equations as a greater complexity than the ilute case previously stuie. The use of Stokes equation to moel the flui is justifie by estimating the Reynol s number. Base on the typical size l 0 1 µm an swimming spee V 0 20 µm/s of a bacterium, in aition to the typical ynamic viscosity η 0 10 3 Pa s an ensity ρ 10 3 kg/m 3 of the suspening flui, the flow has a Reynols number Re aroun 2 10 5 1. Thus, inertial effects can be neglecte. Also, it is assume that a steay-state flow is establishe on a timescale much smaller than the characteristic timescale, which is the time for a bacterium to swim its length. The flow at the position of bacterium i ue to bacterium j is given by u j (x i, j ) = u(x j x i, j ) where u(x, ) is a solution of the Stokes problem η 0 x u(x, ) x p(x, ) = x [D()δ(x) ], x R 3, x u(x, ) = 0, x R 3, (3) u(x, ) 0, x, where η 0 is the ambient flui viscosity an p is the pressure. The ipole tensor D = {D lm } is given by D lm () := U 0 ( l m 1 ) 3 δ lm, (4) where U 0 is the strength of the ipole referre to as the ipole moment. For pushers, bacteria that propel themselves from behin such as B. subtilis, U 0 < 0. Equation (3) has an explicit solution: u k (x, ) := 1 8πη 0 3 l=1 m=1 3 D lm ()G kl,m (x), (5)
6 Mykhailo Potomkin et al. ( ) where G kl (x) = 1 δkl 8πη 0 x + x kx l x is the Oseen tensor. 3 Remark 2 In orer to stuy the role of interactions in semi-ilute suspensions it is natural to eal with a point representation of swimmers such that the whole suspension is moele by points interacting in the flui. In our paper, a swimmer is represente by a point force ipole with the ipole tensor (4). In general, for a given moel of a swimmer, such a point representation can by foun as the secon orer term in the multipole expansion, see [21]. We note that all results of this paper such as the asymptotic formula for orientation istribution an effective viscosity can be easily moifie to semi-ilute suspensions with swimmers whose ipole tensor is ifferent from (4). In orer to analyze the system (1)-(2), the associate kinetic theory for the probability ensity of bacterial configurations (positions an orientations of each bacterium) is stuie. In general, to erive the corresponing kinetic equation one assumes that initial conitions are ranom. Then each sum in the equations of motion is a sum of ientically istribute ranom variables. The key step in the formal erivation of the kinetic equation is replacing all sums in the equations of motion by their expectations [26,39,18]. This allows one to replace all the sums representing interactions by integrals with respect to a probability ensity function P (t, x, ) of fining a given bacterium at position x with orientation. By replacing the sums with integrals in the system (1)-(2) an enforcing conservation of probability, a stanar Fokker-Planck equation escribing the evolution of the ensity P is obtaine t P + x (VP ) + (ΩP ) D P = 0, (6) where the translational an orientation fluxes are efine by V(x, ) := V 0 + 1 up (x, )x S + u BG (x), (7) V L S 2 V L Ω(x, ) := 1 ω + BE, P (x, ) x S + ω BG () + BE BG (). (8) V L S 2 Here <, > enotes the uality with respect to the L 2 -norm, V L := [ L, L] 3, an we neglect the Lennar Jones term F ue to the fact that collisions only play a small role at low concentrations. The functions u, ω, an E uner the integral sign epen on x x, an, an they are efine as follows u(x, ) := U0 8πη 0 x [( I/3)G(x)], ω(x,, ) := 1 2 [ x u(x, )], (9) E(x,, ) := [ D x (u(x, ))], where D x (u) := 1 2 ( xu + [ x u] T ) represents the symmetric graient an I is the ientity matrix. Also, ω BG () an E BG () are efine in the same way as (9), but with the flui velocity u replace with the backgroun flow u BG.
Effective Rheological Properties in Semiilute Bacterial Suspensions 7 1 Remark 3 Since ω, E x x, the integrals with respect to the spatial variables must be consiere in the istributional or principal value sense (which 3 are equivalent here). Namely, < u i x j, ϕ >= C ij ()ϕ(0) + ui x j (ϕ(x) ϕ(0))x, where C ij () = lim ε 0 x =ε u i n j s x. The orientation vector S 2 can be represente by two inepenent angles in spherical coorinates := (cos α sin β, sin α sin β, cos β) = ( 1, 2, 3 ), (10) for azimuthal angle α [0, 2π) an polar angle β [0, π) with unit basis vectors ˆα := ( sin α, cos α, 0) an ˆβ := (cos α cos β, sin α cos β, sin β) respectively. Here one must be careful to note that the ivergence an the Laplacian in orientations (the Laplace-Beltrami operator) in (6) are taken over the unit sphere. In particular, for any fiel A = A() the following efinition hols A := 1 sin β [ α(a α ) + β (sin βa β )] = A { 2 (A ) } =1, (11) where A α = A ˆα, A β = A ˆβ, an is the classical graient. 2.1 Definition of the effective viscosity for a suspension of point force ipoles To efine the effective viscosity consier the contributions to stress: (i) ue to ipolar hyroynamic interactions Σ lm() := N i=1 U 0 V L ( l m δ lm /3), l, m = 1, 2, 3, epening only on each particle s orientation [3] an (ii) ue to soft collisions (the exclue volume constraints) Σ LJ lm (x) := N i=1 j i F l (x i x j )(x i m x j m), l, m = 1, 2, 3, V L epening only on the relative positions of each bacterium [41]. Both are combine to form the total stress ue to interactions first use in [29,28].
8 Mykhailo Potomkin et al. We assume that all bacteria are in the volume V L at any instant of time. The bacterial configurations are enote by x := (x 1,..., x N ) an := ( 1,..., N ). The ultimate goal is to compute the effective viscosity ue to hyroynamic interactions at low concentrations for comparison with experimental observation [35] an numerical simulations. At lower concentrations φ, where the striking experimental ecrease in the effective viscosity was observe, the contribution ue to collisions is relatively small an for the proceeing analysis will be neglecte Σ(x, ) = Σ () + Σ LJ (x) Σ (), for φ small. (12) The exact concentration interval where the formula (12) works well will be etermine later by comparison with irect numerical simulations of the suspension. Thus, it is sufficient to restrict attention to the ensity of orientations enote P () efine as P () := 1 P (x, )x, where P () = 1. (13) N V L S 2 For comparison with experiment, the main quantity of interest is the shear viscosity or component η 1212 of the fourth orer viscosity tensor relating the stress to the strain, henceforth, enote as ˆη. We efine the effective viscosity as the average ratio of the corresponing components of the stress an strain tensors ˆη η 0 := 1 Σ xy η 0 V L V L S γ P (x, )xs = ρ Σ γ xy()p ()S, (14) 2 S 2 as in [29,28]. Here ρ = N/ V L is the mean concentration or number ensity. The following nonlinear, nonlocal integro-ifferential equation escribes the evolution of the orientation ensity P (t, ) where < Ω > x = interaction terms t P (t, ) = (< Ω > x P (t, )), (15) 1 N V L ΩP x (t, x)x, Ω contains the backgroun flow an Ω(t, x, ) = ω BG + E BG + 1 N V L S 2 V L ω + BE, P (t, x, ) x. Equation (15) is obtaine by integrating (6) in x an iviing by N. Remark 4 In this work, lower concentrations of bacteria are consiere where the primary contribution to the effective viscosity from interactions is the ipolar component of the stress, Σ, which only epens on the set of bacterium orientations. Thus, the ẋ equation will not factor into the final formula; however, F is the force associate to a truncate Lennar-Jones type potential imposing exclue volume constraints. For more information on its efinition an why it is neee for global solvability see [28]. This quantity still remains in the original couple ODE system use for simulations to ensure that particles remain a finite istance apart avoiing an artificial ivergence in the flui velocity u 1/ x i x j (see section 5.3).
Effective Rheological Properties in Semiilute Bacterial Suspensions 9 3 Conitions impose to erive an explicit formula for the effective viscosity To calculate the effective viscosity we impose three conitions to make the system more amenable to mathematical analysis. 3.1 Separation of variables In this paper only small concentrations are consiere where collisions are not important, yet the flow of each bacterium affects all others. The bacteria are at large istances apart an, thus, since the backgroun flow provies the major contribution to bacterial motion, then istributions of positions an orientations become essentially inepenent of one another. This can be justifie from the experimental work of Aranson et al. (e.g., see [37, 2]). Henceforth, it is assume that the positions an orientations are ecouple. Conition (C1): The ensity P (x, ) can be written as P (x, ) = P x (x)p () (separation of variables), (16) where P () = 1 N V L P (x, )x an P S 2 ()S = 1. Here N is the number of bacteria, supp(p x (x)) V L, where the spatial ensity P x (x) can be foun by P x (x) = P (x, )S S 2. This conition is use twice. First, the effective viscosity at low concentration only epens on the orientation (see Remark 4). Thus, using conition (C1) an explicit equation for the evolution of the orientation istribution can be erive from (15). Secon, V formally contains iverging integrals (e.g., FxS since F x 12 ), which will no longer be present in the equation for the orientation istribution P () allowing for further mathematical analysis. It will be observe at the en of this work that the asymptotic expansion for P () epens on P x (x) through the coefficients, thus all the information about spatial patterns is preserve. 3.2 Existence of a steay state P () A steay state solution to (15) is efine as follows Definition 1 ˆP () is calle a steay state solution to (15) if it solves ( ) 0 = < Ω > x ˆP (). To compute time inepenent effective viscosity we impose the following conition. Conition (C2): There exists a nontrivial steay state solution to (15).
10 Mykhailo Potomkin et al. First, note that there is no trivial steay state unless B = 0 in which case we fin the uniform orientation istribution P () = 1 4π. This can be obtaine both in the limit as B 0 in the asymptotic results erive herein for P () an from observing that the trivial steay state woul be a constant satisfying the constraint P S 2 ()S = 1. One still nees to prove the existence of a steay state in the general case B 0. The conition (C2) can be formulate as a theorem an its proof may be the topic of a future work. Here we remain focuse on the stuy of the effective viscosity. 3.3 P x (x) is constant in the z-irection We assume that P x (x) is constant in z for the case of the planar shear backgroun flow uner consieration in this work. This is consistent with past numerical observations by Ryan et al. [29] an experimental observation in [38] since the suspension remains below any critical concentration for threeimensional collective motion. Also, collective motion even in full 3D experiments an simulations in planar shear flow has been observe to be essentially 2D in the shearing plane [38]. Thus, following experimental observation, we assume the same. Conition (C3): The ensity P x (x) is constant in z. The conition (C3) essentially follows from the physical setup of the quasi- 2D thin film suspension. In Appenix B we show that the conition (C3) leas to the following representation formula for the Fourier transform of the spatial istribution F [P x ]: (F [P x ]) 2 = δ(k 3 ) ˆP 2 12(k 1, k 2 ). (17) Here k = (k 1, k 2, k 3 ) is the Fourier variable, an ˆP 12 (k 1, k 2 ) is a smooth function efine in k-space inepenent of k 3. 4 Derivation of asymptotic expression for P for small B In this section, an expression for the orientation istribution P () is erive. Since (15) is a nonlinear integro-ifferential equation it is challenging, in general, to fin an analytical solution. Thus, we look for P () by asymptotic expansion in the limit of small non-sphericity (B 1). This will allow us to apply analytical techniques an erive an expression, which will provie physical insight into the mechanisms contributing to the ecrease in the effective viscosity. Rewrite the equation for the orientation ensity P () (15) as (the argument t is suppresse for simpler notation) t P + [(ω BG + BE BG ] 1 )P + N V L ( ˆΩP (x, ))x = 0, (18)
Effective Rheological Properties in Semiilute Bacterial Suspensions 11 where ˆΩ(x, ) := 1 ω + BE, P x (x ) x P ( ). (19) V L S 2 Herein ˆΩ will enote the component of the orientational flux Ω ue to interactions. Observe that the ω an E are functions of x x,, an. Using Conition (C1) efine in (16) we obtain a close form equation for a steay state P () (provie that P x is given): 0 = [(ω BG + BE BG )P () ] + 1 ( ˆΩ(x,, )P x (x)p ()) S x. (20) N V L V L The first term in (20) is the contribution ue to the backgroun planar shear flow: [(ω BG () + BE BG ())P () ] = 3γB 2 sin2 β sin 2αP () + γ 2 (1 + B cos 2α){ αp ()} (21) + γb 4 sin 2α sin 2β{ βp ()}. The secon term in (20) is the contribution of hyroynamic interactions between bacteria. Notice the convolution form of the nonlocal terms in the spatial variable. In the next section, the Fourier transform will be utilize to compute quantities necessary to erive the formula for the effective viscosity. Specifically, using tools such as Parseval s Theorem, one can take the spatial integrals an consier them in Fourier space where they will prove easier to analyze. After using the separation of variables (16), the ensity will be expresse in terms of the Fourier frequencies k. The main goal for the remainer of this section is to write the system in a convenient form for using the Fourier transform. This iea follows naturally from the aforementione observation that all the interactions terms take the form of a convolution. Introuce the Fourier transform C(k) := F [P x ](k): P x (x) = 1 (2π) 3 e ik x C(k)k. (22) Define H(x x,, ) := ω(x x,, )+BE(x x,, ), then the following equalities hol < H P x, P x > x =< F [H P x ], F [P x ] > k =< F [H], (F [P x ]) 2 > k, (23) where an F stan for convolution an Fourier transform, respectively. The first equality is Parseval s ientity an the secon is the fact that the Fourier transform of a convolution is the prouct of Fourier transforms. Thus, one can rewrite equation (20) in the following form [(ω BG + BE BG )P () ] + {P ()P ( ) < F [H](F [P x ]) 2 } > k S = 0. (24) S 2
12 Mykhailo Potomkin et al. In orer to compute F [H] one must first unerstan how the Fourier transform acts on the flui velocity u an its erivatives. 4.1 Evaluation of Fourier transforms In orer to analyze (24), an analytical expression for the Fourier transform F [H] = F [ω] + BF [E] is neee. Both terms epen on the flui velocity u efine by (3). Recall the ipolar stress Σ(x, ) = D()δ(x) = U 0 ( I/3)δ(x). (25) Then the Stokes equation in (3) can be written as η 0 x u + x p = x Σ(x, ), x u = 0. (26) Denote the Fourier transform of a function f(x) as f(k) = F [f] (k) = e i(k x) f(x)x, an compute the Fourier transform of u an the symmetric graient D x (u). Proposition 1 Let u be a solution of (3) an let Σ be efine by (25). Then (i) Σ( ) = U 0 ( I/3), (ii) ũ(k) = i ) (I kk Σ(k) k η 0 k k 2 k, (27) (iii) F [D x (u)] = 1 ( ) 2η 0 k 4 k 2 Σkk 2kk Σkk + k 2 kk Σ. (28) Here enotes the transpose. Proof The part (i) follows from the fact that the Fourier transform of δ- function is 1. We split the proof of (ii) into two steps: First, we fin the Fourier transform of the pressure p, then by using the first equation in (3) we fin ũ. Step 1: Evaluation of p = F [p]. By taking the ivergence of (26) in x we obtain x p = x ( x Σ). (29) Observe that F [ x p] = k 2 p(k), F [ x ( x Σ)] = Σ : 2 xe ik x x = Σ(k) : kk. Substituting these formulas into (29) we obtain k 2 p(k) = Σ(k) : kk, an, thus, we fin an expression for the Fourier transform of the pressure p: p(k) = 1 k 2 Σ(k) : kk. (30)
Effective Rheological Properties in Semiilute Bacterial Suspensions 13 Step 2: Evaluation of ũ = F [u]. Return to Stokes equation (26) an observe that η 0 F [ x u] = η 0 k 2 ũ(k), F [ x Σ] = i Σ(k)k. F [ x p] = ik p(k), Using these relations one fins that η 0 k 2 ũ(k) + ik p(k) = i Σ(k)k. After rearranging the terms an using (30) we complete the proof of (ii). To prove (iii) we first observe that F [D x (u)] = i 2 (ũk + kũ ). Plug the Fourier transform of u from (ii) into this expression to fin F [D x (u)] = i 2 (ũk + kũ ) = 1 2η 0 k 2 ((I kk k 2 ) Σ(k)kk + kk Σ(k)(I kk k 2 ) Use the fact that Σ is symmetric ( Σ = Σ ) to complete the proof of (iii). Remark 5 It is easily seen that F [D x (u)] oes not epen on k, since F [D x (u)] can be rewritten as F [D x (u)] = 1 k Σ η 0 k k 2 k η 0 k k Σ k k k + k η 0 k k Σ. k This subsection is conclue by summarizing the analytical expressions for the two main components of F [H] = F [ω] + BF [E]: F [E] = ( F [D x (u)] ) = F [D x (u)] F [D x (u)] (31) F [ω] = 1 2 F [ x u] = 1 [ ik F [u]], (32) 2 where F [u] an F [D x (u)] are given by Proposition 1. k k k ). 4.2 The form of asymptotic expansion in B Recall the steay-state Liouville equation (24) with the backgroun terms substitute in: 0 = 3γB 2 sin2 β sin 2αP () + γ 2 (1 + B cos 2α) αp () + γb 4 sin 2α sin 2β βp () + 1 {P ()P ( ) < F [H], (F [P x ]) 2 } > k S. (33) N V L S 2 We consier the asymptotic expansion in the Bretherton constant, B 1, for the orientation istribution, P (), up to the secon orer: P (α, β) = P (0) (1) (2) (α, β) + P (α, β)b + P (α, β)b2 + O(B 3 ). (34)
14 Mykhailo Potomkin et al. Substituting (34) into (33) we get ifferent equations at ifferent orers of B. It is straightforwar that P (0) 1 (α, β) = 4π (surface area of the unit sphere is 4π) solves the equation at orer O(1). We want to consier the asymptotic expansion about the uniform istribution because it has been extensively ocumente in theory an experiment that as the bacterium boies become or spherical (B 0), then the istribution in angles is uniform [29,16]. In the next two subsections, the linear orer term P (1) (α, β) an quaratic orer term P (2) (α, β) are compute. 4.3 Contribution at O(B) First, notice that ω(x x,, ) = 0. Inee, this follows from (11) since ω = 0 an the classical ivergence of ω with respect to is zero (note that ω = A, where A = x u oes not epen on ). This observation implies F [H] = B F [E]. Using this equality an expaning the ivergence uner the integral sign we rewrite (33) as follows: 0 = γ 2 [B sin(2α) sin β (cos β βp 3 sin βp ) + (1 + B cos(2α)) α P ] + B P ( )P () (F [E()])(F [P x ]) 2 k S (35) N V L S 2 + 1 [P ()]P ( ) F [H()](F [P x ]) 2 k S. N V L S 2 The first integral at O(B) is 1 16π 2 (F [E()])(F [P x ]) 2 k S, (36) N V L S 2 By switching the orer of integration an noting ΣS S 2 = U S 2 0 [ ( ) I/3]S = 0 we obtain that (36) is zero using (31) an (28). Since both [P ()] an BE are of the orer O(B), the secon integral in (35) at O(B) is 1 4πN V L S 2 P (1) () F [ω](f [P x]) 2 k S which is also zero ue to S 2 U 0 ( I/3)S = 0. Thus, the integral terms o not contribute to equation (35) at orer O(B), an it has the following form: 0 = γ 2 After substituting P (0) = 1 4π [ ] 3P (0) sin(2α) sin2 β + α P (1). (37) an solving (37), one fins that P (1) (α, β) = 3 8π sin2 β cos(2α). (38)
Effective Rheological Properties in Semiilute Bacterial Suspensions 15 Since the integral terms are zeros at orer O(B), the contribution ue to interactions oes not appear at orer O(B) an thus the only contribution is ue to the backgroun flow. It will be shown later that up to O(B) the contribution to the effective viscosity by the bacteria is zero. This will she light on the fact that interactions are necessary to see the ecrease in the effective viscosity an the backgroun flow alone is insufficient. Note that even though this is the contribution ue to the backgroun flow the strain rate γ is not present. Therefore, the magnitue of the flow will not have an effect on the longtime limit of the effective viscosity at O(B). However, once the terms at the next orer are compute one observes a competition evelop between the backgroun flow an the flow ue to inter-bacterial interactions. In this case the magnitue of the shear γ becomes important. 4.4 Contribution at O(B 2 ) Consier terms in (35) of orer O(B 2 ): 0 = γ 2 sin(2α) sin β cos β βp (1) () 3γ 2 sin(2α) sin2 (β)p (1) () + γ 2 αp (2) () + γ 2 cos(2α) αp (1) () 1 + 4πN V L F [E]F [P x ] 2 k P (1) ( )S S 2 1 + [P (2) 4πN V L ()] F [ω](f [P x]) 2 k S (39) S 2 1 + [P (1) 4πN V L ()] F [E](F [P x]) 2 k S S 2 + 1 [P (1) (1) ()]P N V L ( ) F [ω](f [P x ]) 2 k S. S 2 Denote the four integral terms in equation (39) by I 1, I 2, I 3 an I 4, respectively. The following equalities hol: U 0 ( I 1 = A sin 2 β cos(2α) + C sin 2 β sin(2α) ), 40πη 0 N V L I 2 = I 3 = 0, 3U 0 I 4 = 10πη 0 N V L D sin(2α) sin2 β, where constants A, C, an D are efine as follows A := 1 2 sin 2 (2θ) ˆP 12k 2 2 kθ, C := 1 2 sin(4θ) 2 ˆP 12 k 2 kθ, D := cos(θ) sin(θ) ˆP 12k 2 2 kθ. (40)
16 Mykhailo Potomkin et al. Here ˆP 12 is from (17), an we use spherical coorinates in the Fourier space (k = k, θ, φ). The calculations of I i can be foun in Appenix A. After substitution of the expressions for each I i, we get the following equation for P (2) (): 0 = γ 2 sin(2α) sin β cos β βp (1) () 3γ 2 sin(2α) sin2 (β)p (1) () + γ 2 αp (2) () + γ 2 cos(2α) αp (1) () (41) U 0 ( + A sin 2 β cos(2α) + C sin 2 β sin(2α) ) 40πη 0 N V L 3U 0 + 10πη 0 N V L D sin2 β sin(2α). Base on the form of the equation (41), the following representation is use to fin P (2) (): P (2) () = C 1 sin 4 β cos(4α) + C 2 sin 2 β cos(2α) + C 3 sin 2 β sin(2α). (42) In orer to fin each C i substitute (42) into (41): [ ] [ 3γ 0 = 8π 2γC 1 sin(4α) sin 4 U 0 A β + γc 3 + 40πη 0 N V L [ + γc 2 + U 0 C 40πη 0 N V L + 3U 0 D 10πη 0 N V L ] sin 2 β sin(2α). ] sin 2 β cos(2α) Since the factors are linearly inepenent, each coefficient is zero an, thus, we fin the C i s: C 1 = 3 16π, C 2 = U0(C+12D) 40γπη 0N V L, C 3 = U 0A 40γπη 0N V L. Using these coefficients one obtains an explicit formula for the orientation istribution up to O(B 3 ): P (α, β) = 1 4π 3 [ 3 8π sin2 β cos(2α)b + 16π sin4 β cos(4α) C + 12D U 0 40γπη 0 N V L sin2 β cos(2α) (43) ] U 0 A 40γπη 0 N V L sin2 β sin(2α) B 2 + O(B 3 ). Formula (43) is the main result of Section 4. Since A, C, an D contain ˆP 12, all the spatial information is embee in these coefficients. In particular, we foun the lowest orer (in B) contribution of hyroynamic interactions to the P () occurs at O(B 2 ). In the following section, the contribution of hyroynamic interactions to the effective viscosity is compute as well as the change in the effective normal stress coefficients. The combination of these two quantities will escribe the total effect of hyroynamic interactions on the rheological behavior of the bacterial suspension.
Effective Rheological Properties in Semiilute Bacterial Suspensions 17 5 Explicit formula for the effective viscosity Using the expression for the orientation istribution, P () efine in (43), an the formula for the effective viscosity for ipoles in a suspension (14), we compute the contribution to the effective viscosity ue to interactions: η int := η η 0 η 0 = U 2 0 B 2 ρ 2  75γ 2 πη 0 < 0. (44) where  = 1 N A O(1) an the equality hols up to orer O(B 3 ). The 2 quantity η int behaves like ρ 2 in concentration (cf. [4] where an expansion for the effective viscosity to orer two in concentration is erive for passive spheres corresponing to pairwise interactions). As an aitional check of consistency, consier the imensions of the final quantity. The ipole moment [U 0 ] = kg m2 s, both the Bretherton constant B an  are imensionless, the 2 concentration/number ensity [ρ] = 1 m, the ambient viscosity [η 3 0 ] = kg m s, an the strain rate [γ] = 1 s resulting in ηint being imensionless. In aition, the orientation istribution P () from (43) can be use to compute the effective first an secon ipolar normal stress coefficients N 12 = Σ 11 Σ 22 γ an N 2 23 = Σ 22 Σ 33 γ to investigate the effect of hyroynamic interactions. The main avantage of the mathematical moel is that 2 the computation of the effective normal stress coefficients is straightforwar in contrast to experiment where its measurement can be quite complicate [13]. These coefficients can provie important information about the suspension. For example, the ratio of the first normal stress to the viscosity etermines the effective relaxation time [13]. Also, phenomena such as extruate swelling [1] an seconary flow [27] are important in many technological applications. A simple calculation shows that N 12 = Σ 11 Σ 22 γ 2 N 13 = Σ 22 Σ 33 γ 2 = U 0ρ γ 2 = U 0ρ γ 2 [ 2 5 2U 0ρ(C + 12D) ] B 2 75γπη 0 [ 1 5 + U 0ρ(C + 12D) 75γπη 0 B 2 (45) ]. (46) The approximations are vali for B 1, so for pushers (U 0 < 0) N 12 > 0 an N 23 < 0 where as for pullers (U 0 > 0) N 12 < 0 an N 23 > 0. Both results are consistent with the preictions in [16, 31] while proviing aitional information about the concentration epenence. The effective normal stress coefficients grow linearly with concentration in the presence of interacting bacteria; however, the fact that the normal stresses of active suspensions are non-zero in the case of a planar shear flow inicate the emergence of non- Newtonian behavior. One sees in (45)-(46) that as the shear rate γ the normal stresses approach zero inicating the ominance of the backgroun flow on the suspension overwhelming any contribution from interactions.
18 Mykhailo Potomkin et al. 5.1 Mechanisms require for the ecrease in the effective viscosity In this subsection, the mechanisms that lea to a ecrease in the effective viscosity are investigate. These same mechanisms are shown in [30] to be responsible for collective motion an large scale structure formation in suspensions of pushers. Our mathematical analysis provies insight beyon experiment. Formula (44) reveals that elongation of bacteria, self-propulsion, an interactions are all require to observe a ecrease in the effective viscosity; namely, for spherical bacteria (B = 0) the net change in the effective viscosity is zero. In aition, active bacteria are require, since U 0 f p = 0 results in no change in the effective viscosity where f p is the propulsion force. Finally, if the spatial ensity P x (x) is near uniform, then  = 1 2N sin 2 (2θ) ˆP 12k 2 0 resulting in 2 no change in the effective viscosity. In the limit γ the contribution to motion of bacteria ue to shear ominates the contribution ue to interactions with P () maximize at α = π/2 an β = π/2 (alignment with y-axis). This is analogous to the passive case where bacteria in a planar shear flow ten to align with the irection where the flui exerts the least amount of torque on the bacterium boy. Therefore, confirming our main conclusion that in orer to exhibit a ecrease in the effective viscosity active, elongate bacteria whose interactions result in a non-uniform istribution in space are neee. 5.2 Effective noise conjecture In this subsection, the results herein involving a semi-ilute suspension of point force ipoles are compare to the previous result for a ilute suspension of prolate spherois with propulsion moele as a point force [16]. Thus, the only contribution to bacterial motion is the backgroun flow. In [16], finite size bacteria are taken as spherois with a point force (δ function) accounting for self-propulsion. In aition, each bacterium experiences a ranom reorientation referre to as tumbling. Biologically tumbling correspons to a reorientation of a bacterium in hopes of fining a more favorable (nutrient rich) environment. Typically in experiment this is observe when the concentration of oxygen is low. Thus, bacteria enter a more ormant state resulting in a lower swimming spee an an increase tumbling rate [36]. Since only the term containing  contributes to the effective viscosity, one can choose to match the coefficient of this term P int = 1 4π 3 8π B cos(2α) sin2 β + 3 16π B2 sin 4 β cos(4α) U 0 ρ C + 12D B 2 sin 2 β cos(2α) U 0ρ B 2 sin 2 β sin(2α) + O(B 3 ) 40γπη 0 40γπη 0 with the corresponing coefficient in the erivation by Haines et al. [16], which is quaratic in the iffusion strength D. To make the formulas for the effective
Effective Rheological Properties in Semiilute Bacterial Suspensions 19 viscosity ientical, the strength of the effective noise/iffusion (tumbling) is chosen to be ˆD := 15η 0γ 2 + 225η0 2γ4 Â2 B 2 γ 2 ρ 2 U0 2 12ÂBρU > 0, 0 (since U 0 < 0 for pushers). Observe that ˆD, chosen in this way, epens only on the physical parameters present in the problem an the same effective viscosity as the ilute case stuie in [16] is foun. This ˆD is referre to as the effective noise an the phenomenon where stochasticity arises from a completely eterministic system is calle self-inuce noise. A future work may seek to explain this phenomenon rigorously using mathematical analysis. One heuristic iea is that the perioic (eterministic) Jeffrey orbits are estroye by interactions resulting in stochastic behavior. Some conclusions about this effective noise can be mae that ensure its consistency with physical reality. As bacteria become spheres B 0, ˆD 0 resulting in no change in the effective viscosity consistent with [16]. Also as the strain/shear rate γ, ˆD 0. This is physically intuitive, because as the shear rate becomes large its contribution ominates that ue to hyroynamic interactions resulting in behavior that resembles that of a passive suspension. Thus, the contribution to the effective viscosity ue to hyroynamic interactions is zero. Finally, we compare our results with irect simulations for the couple PDE/ODE system compose of Stokes PDE (3) an (1)-(2). 5.3 Comparison to numerical simulations In this section, the accuracy of the erive formula is teste by comparing it to recent numerical simulations. The numerical proceure is outline in [29]. These simulations are parallel in nature allowing them to be carrie out on GPUs for greater efficiency. Figure 1 shows how both the formula an numerical computations of viscosity change with bacterium shape as all other system parameters remain fixe. Here shape is accounte for through the Bretherton constant B = b2 a 2 b 2 +a 2 where b is the length of the major axis an a is the length of the minor axis of the ellipsoi representing a bacterium. First, notice that in both the formula an numerics the contribution to the effective viscosity ue to hyroynamic interactions ecreases with B (increasing in magnitue). This is ue to the fact that as bacteria become more asymmetrical as B 1 the inter-bacterial hyroynamic interactions have a greater effect on alignment. This alignment increases the magnitue of the ipolar stress leaing to an even bigger ecrease in the effective viscosity. The agreement between the analytical formula an numerical simulations breaks own as B becomes large, but this is expecte ue to the fact that the asymptotic formula is vali in the regime where B 1 (small non-sphericity).
20 Mykhailo Potomkin et al. 0 Effective Viscosity, η η 0-0.2-0.4-0.6-0.8 Analytical Numerical -1 0 0.1 0.2 0.3 0.4 Bretherton Constant, B Fig. 1 Comparison of the formula for the effective viscosity with numerical simulations as bacterium shape changes through the Bretherton constant B for a fixe volume fraction Φ =.02 an shear rate γ =.1. The vertical bars represent the error in the numerical approximation. Error in the analytical solutions comes from the numerical estimation of Â. Figure 2 shows how both the formula an numerical computations of viscosity change with the concentration of the suspension as all other system parameters remain fixe. It is seen that as concentration increases the effective viscosity ecreases. This can easily be explaine by the fact that as the concentration increases, the motion of bacteria begins to be ominate by inter-bacterial hyroynamic interactions. This leas to collective motion of the bacteria in the suspension, which subsequently ecreases the viscosity. The two results begin to iverge near volume fraction Φ.02. The reason the numerical simulations o not ecrease as much is that collisions are taken into account. It was shown in [29] that the stress ue to collisions is a positive contribution to the effective viscosity that is not capture by the formula. This contribution begins to become important beyon the ilute regime (Φ > 2%). 0-0.25 Effective Viscosity, η η 0-0.5-0.75-1 -1.25 Analytical Numerical -1.5 0.005 0.01 0.015 0.02 0.025 0.03 Concentration, Φ Fig. 2 Comparison of the formula for the effective viscosity with numerical simulations as the volume fraction Φ changes for a fixe shape B =.2 an shear rate γ =.1.
Effective Rheological Properties in Semiilute Bacterial Suspensions 21 Figure 3 shows how both the formula an numerical computations of viscosity change with the shear rate of the backgroun flow in the suspension as all other system parameters remain fixe. As expecte when the shear rate is large in both the analytical formula an simulations, the ecrease in viscosity ue to hyroynamic interactions is negligible. This is ue to the fact that the backgroun flow ominates motion of bacteria wiping out the effects of inter-bacterial interactions an stopping any collective structures from forming. When the shear rate is too small the effective viscosity becomes unboune. This makes sense given that at small shear rate the system becomes almost non-issipative an thus the effective viscosity is not well-efine. This can easily be seen by noting that the viscosity is the ratio of the stress over the strain an when the strain is essentially zero the effective viscosity becomes unboune. All three plots show goo qualitative agreement with each other, experimental observation, an physical intuition. 0 Effective Viscosity, η η 0-0.2-0.4-0.6-0.8 Analytical Numerical -1 0 0.2 0.4 0.6 0.8 1 Shear Rate, γ Fig. 3 Comparison of the formula for the effective viscosity with numerical simulations as the shear rate γ changes for a fixe volume fraction Φ =.02 an shape B =.2. 6 Global solvability of the kinetic equation In this section, we stuy solvability of the main nonlinear integro-ifferential equation (15) governing the evolution of the orientation istribution. Primarily we are intereste in existence, uniqueness, an the regularity properties of solutions of (15). First, we note that (15) is an equation of the form: ([ ] ) t P = K(, )P ( )S + k() P + D P. (47) S 2 Inee, one can obtain (15) by substituting K(, ) = ω(, ) + BE(, ), k() = ω BG () + BE BG (). (48)
22 Mykhailo Potomkin et al. Both K an k from (48) are infinitely smooth functions of. Therefore, in this section we consier (47) for the general case of smooth K an k. We follow the stanar proceure for the analysis of the well-poseness of the evolution PDEs (e.g., see [12, 23, 14]). In particular, we introuce the notion of a weak solution. By H s (s R) we enote the corresponing Sobolev spaces. Definition 2 For T > 0, the function f which belongs to space H given by H = L 2 ((0, T ), H 1 (S 2 )) H 1 ((0, T ), H 1 (S 2 )) (49) is a weak solution of (47) if for almost all t [0, T ] an all h H 1 (S 2 ) [ ] t f, h = D f, h + f, K(, )fs + k() h, (50) S 2 where, is the uality prouct for istributions on the unit sphere S 2. Remark 6 Accoring to the well-known embeing (see [33]) the fact that a weak solution f belongs to H implies that it is continuous with respect to t [0, T ] with values in L 2 (S 2 ), i.e., f C([0, T ]; L 2 (S 2 )). Definition 3 A function f C([0, T ]; L 2 (S 2 )) is calle positive in istributional sense if f, h 0 (51) for all t [0, T ] an all h C(S 2 ) such that h() 0 for all S 2. The following theorem is the main result of this section. Theorem 1 Assume f 0 L 2 (S 2 ), K C 2 (S 2 S 2 ), k C 2 (S 2 ) an T > 0. Assume also that f 0 is positive in the istributional sense. Then the following statements hol: (i) There exists the unique weak solution of (47) f on interval [0, T ] such that f t=0 = f 0. The weak solution f is positive. It continuously epens on initial conitions, i.e., there exists a positive constant C > 0 such that sup f (1) f (2) L2 (S 2 ) C f (1) 0 f (2) 0 L 2 (S 2 ), (52) t [0,T ] where f (1) an f (2) are weak solutions with initial conitions f (1) t=0 = f (1) 0 an f (2) t=0 = f (2) 0, respectively. (ii) For all s 0 if f 0 H s (S 2 ), then f C([0, T ]; H s (S 2 )). If f 0 C (S 2 ), then f C([0, T ]; C (S 2 )).
Effective Rheological Properties in Semiilute Bacterial Suspensions 23 (iii) For all s 0 if f 0 H s (S 2 ), then for all m 0 an t > 0: ( f(t) 2 H s+m (S 2 ) C 1 + 1 ) t m, (53) where the constant C epens only on f 0 H s (S 2 ), s, an m. In particular, for all p Z. f C((0, ); H p (S 2 )) Proof STEP 0. (Preliminaries) Consier spaces of functions with mean zero: L 2 (S 2 ) := L 2 (S 2 ) {f : f, 1 = 0} Ḣ s (S 2 ) := H s (S 2 ) {f : f, 1 = 0}. Note that for f L 1 (S 2 ) f, 1 = fs. S 2 We use f L2 (S 2 ) as a norm in Ḣ1 (S 2 ). In this proof we assume that f S 2 0 S = 1. Consier the mean zero component of the solution f; namely, g := f 1 4π. If f is the weak solution of (47), then g satisfies t g, h = D g, h + 1 4π + g, K(, )g( )S h S 2 + 1 [ ] 4π + g, K(, )S + k() h (54) S 2 for all h H 1 (S 2 ). Existence, uniqueness, an continuous epenence on initial conitions will be proven for g, which is equivalent to the proof of the same properties for f. Below C enotes a positive constant an it may change from line to line. STEP 1. (Local existence) Let E N be the space spanne by the first N eigenvalues of the Laplace-Beltrami operator, an let Π N be the orthogonal projector on the space E N. Introuce the Galerkin approximation g N, which is the solution of the following equation: t gn, h = D g N, h + 1 4π + gn, K(, )g N ( )S h S 2 + 1 [ ] 4π + g, K(, )S + k() h, (55) S 2 for all h E N, an g N t=0 = Π N g 0, where g 0 := f 0 1 4π. In a stanar manner, the problem (55) can be interprete as a system of N ODEs, an its solution g N exists for t [0, t N ) for some t N > 0. Taking