The Squirmer model and the Boundary Element Method

Transcription

1 Hauptseminar : Active Matter The Squirmer model and the Boundary Element Method Miru Lee April 6, 017 Physics epartment, University of Stuttgart 1 Introduction A micro scale swimmer exhibits directional motion in the fluid without any external force and torque. Our main goal is to understand and explain mathematically how it moves at very low Reynolds number in a Newtonian fluid. Micro-organisms such as Opalina moving via mechanical motions of densely packed cilia covering the whole surface can be modeled as if they swim via a motion of an "envelope" covering the surface as shown in figures 1a and 1b. The motion of the envelope, which gives rise to local fluid motion on the surface (i.e., the boundary conditions of the flow field), is called squirming motion [1,, and the swimmers that can be understood by means of the squirming motion are called squirmers. We will show an analytic solution to the squirmer model problem under the assumptions that the flow field due to the squirmer is axisymmetric and the shape of the body is a sphere. Later, we will introduce the rigid body and the far-field approximations to gain intuition about the flow field. This will allow us to identify four distinct types of squirmer. On the other hand, a chemically active particle can also be a micro-swimmer. Basic mechanism of which a chemically active particle may take advantage is the so-called phoretic mechanism; a local flow field in the vicinity of the particle s surface, which is created by the short-range interaction between the surface and local field gradients around the particle (e.g., solute density or temperature field), drives the motion of the particle. If the particle is able to induce the local chemical gradients around its surface itself, it can move autonomously [3, 4. We will focus on self-diffusiophoresis to explain the directional motion of a chemically active particle, and we will show that the mathematical structure of a simplified model for self-diffusiophoresis is very similar with that of the squirmer even though the basic mechanisms seem very different from each other. Therefore, this model can be regarded as a chemical version of the squirmer. If a particle, however, does not possess a particularly simple shape (i.e., spherical shape, axial symmetry, etc.), analytic solutions cannot be found, and thus numerical approaches have to be considered. There are several numerical methods to solve partial differential equations, and what we will present here is the boundary element method (BEM). The main idea of the method is that; 1) find a fundamental solution of the relevant equation, describing a field due to a point source, ) express the general solution in terms of boundary distributions of the fundamental solutions, and 3) compute the densities of the distributions to satisfy the given boundary conditions [5. We will use Laplace s equation as an example to explain the methodology of BEM because Laplace s equation is not only relevant to the chemically active particle but also less cumbersome than the Stokes equation, though the letter can also be solved using the same methodology. Squirmer model The squirmer at low Reynolds number in the incompressible flow of the unbounded Newtonian fluid is assumed to induce the axisymmetric flow and to have a sphere shape in order to simplify the 1

2 (a) Opalina: a micro-organism having cilia around the J. Fluid Mech. (1971) body. (b) The squirming motion: the cilia motions are assumed to be a periodic wave motion [1 Figure 1: Opalina (a) and the squirming motion of Opalina (b). mathematical structure. Then, the motion of the envelope is replaced by the flow field on the surface, that is, a boundary condition [. Before dealing with the problem, we have to choose a frame. Figure shows the two possible options: the fixed laboratory frame and the so-moving frame, attached to the moving swimmer. We will work in the co-moving frame, and employ spherical coordinates..1 Stream function At low Reynolds number, the Stokes equation for the incompressible flow of the Newtonian fluid is given by: u 1 p = 0 (1) µ where u and p are the flow field and the hydrostatic pressure, and µ is the viscosity of the fluid. For an axisymmetric flow field with the axis of symmetry being the z axis, one can introduce the stream function ψ(r, θ) in terms of which the flow fields is [6: u r (r, θ) = 1 r ψ (cos θ) The flow field does not depend on the azimuthal angle φ. Taking the curl on Eq.(1) and using p = 0 leads to Substituting Eq.() into Eq.(3) gives where, in spherical coordinates, the operator E is given by The general solution of Eqs.(4) and (5) is [6: Figure : The laboratory frame and the comoving frame. u θ (r, θ) = 1 ψ r r. () u = 0. (3) E (E ψ) = 0, (4) E = r + 1 cos θ r (cos θ). (5) ψ(r, θ) = (α n r n + β n r n+1 + γ n r n+ + δ n r n+3 )G n (cos θ), n=0 + (α nr n + β nr n+1 + γ nr n+ + δ nr n+3 )H n (cos θ) n= (6)

3 where G n and H n are Gegenbauer functions of the first kind and the second kind, respectively. 1 The two functions can also be re-presented in terms of Legendre polynomials of the first and the second kinds, P n and Q n : G n (cos θ) = P n (cos θ) P n (cos θ), H n (cos θ) = Q n (cos θ) Q n (cos θ) n 1 n 1 for n. (7) For n, G n and H n are defined as. Flow field Using G 0 (cos θ) = H 1 (cos θ) = 1, G 1 (cos θ) = H 0 (cos θ) = cos θ. (8) P n 1 (cos θ) = dg n(cos θ) d(cos θ), Q n 1(cos θ) = dh n(cos θ) d(cos θ) and substituting Eq.(6) in to Eq.() leads to u r (r, θ) = (α n r n + β n r n 1 + γ n r n + δ n r n+1 )P n 1 (cos θ) n=1 (α nr n + β nr n 1 + γ nr n + δ nr n+1 )Q n 1 (cos θ), n= u θ (r, θ) = [nα n r n (n 1)β n r n 1 + (n + )γ n r n (n 3)δ n r n+1 G n(cos θ) n=0 + [nα nr n (n 1)β nr n 1 + (n + )γ nr n (n 3)δ nr n+1 H n(cos θ). n= (9) For our squirmer, the flow field should be a bounded function at every point. Legendre functions of the second kind and Gegenbauer functions of the second kind diverge at cos θ = ±1. In addition, G 0 (cos θ) and G 1(cos θ) become infinite at = 0. Hence, in order for u(r, θ) to be finite for all θ, the related coefficients must vanish [6, i.e., α n = β n = γ n = δ n = 0 for all n, β 0 = γ 0 = δ 0 = 0, α 1 = γ 1 = δ 1 = 0. (10).3 Boundary conditions The system is a particle in an unbounded fluid. Thus, we have boundary conditions at infinity and on the surface of the particle. We begin with the boundary conditions at infinity. The fluid at infinity in the co-moving frame is moving in the negative z-direction with the velocity U. Consequently, the corresponding boundary condition is [6 which implies u(r ) Uê z, (11) { U for n =, α n = 0 otherwise, and γ n = 0 for all n. (1) 1 Note that the summation of the second line on Eq.(6) starts from n =. This is because of the definitions of G n and H n at n = 0 and n = 1. Gn(cos θ) For n, G n(cos θ) always contains term, so is well behaved. 3

4 As a result, the flow field becomes ( U + β u r (r, θ) = β 1 r P 0(cos θ) r 3 + δ r ( u θ (r, θ) = U β r 3 + δ ) G (cos θ) r n= ) P 1 (cos θ) ( βn+1 n= ( nβn+1 r n+ + (n )δ n+1 r n ) P n (cos θ), r n+ + δ n+1 r n ) Gn+1 (cos θ). The squirmer model is defined by the velocities on the surface as functions of θ, i.e., u r (θ) = f(θ) and u θ (θ) = g(θ), with f and g known functions. Since it is possible to expand any well behaving function of θ in terms of P n (cos θ) and G n (cos θ)/, the flow on the surface can be written as [ u r (a, θ) = f(θ) = A n P n (cos θ), u θ (a, θ) = g(θ) = B n V n (cos θ) (14) n=0 n=1 where V n (cos θ) = G n+1(cos θ), and {A n 0 } and {B n 1 } are known quantities. Evaluating Eq.(13) at r = a and matching with Eq.(14) provides the rest of the coefficients {β n } and {δ n }: [( ) n β 1 = a A 0, β n+1 = a n+ 1 A n B n, δ n+1 = an ( β = a3 3 A 1 + B 1 + δ a ), U = 1 3 A B 1 δ 3a. (na n B n ), However, β and δ are still unknown because U is undetermined yet. Thus, we need an additional condition..4 Force-free condition Let us consider the net force and the net torque acting on the swimmer. Given that the swimmer moves with the constant velocity, the net force and the net torque on it must vanish [7, 8. There are two forces and torques acting on the body: the external force and torque (F ext and T ext ) and the hydrodynamic force and torque (F hydro and T hydro ), hence F net = F ext + F hydro = 0, T net = T ext + T hydro = 0. Since the external force and torque are zero, the hydrodynamic force and torque, which are given by [6 F hydro = Π ˆndS, T hydro = r (Π ˆn)dS, (17) have to be zero. Using (13), one can find that the first term of Eq.(17) reduces to At the same time, due to the axial symmetry, (13) (15) (16) F z = 4πµδ. (18) T hydro 0. The force-free condition thus implies δ = 0. Putting everything together, one arrives at: [ u r (r, θ) = a r A a 3 0P 0 (cos θ) + 3 r 3 (A 1 + B 1 ) U P 1 (cos θ) [( n a n + r n n a n+ ) ( a n+ an r n+ A n + )B rn+ r n n P n (cos θ), [ 1 a 3 u θ (r, θ) = 3 r 3 (A 1 + B 1 ) + U V 1 (cos θ) + 1 ) ( [(n an+ n(n ) a n (n )an rn+ r n B n + )A r n an+ r n+ n V n (cos θ), U = 1 3 (B 1 A 1 ). (19) 4

5 Note that the force and torque free condition results in vanishing 1/r term, making 1/r term be the leading order..5 The laboratory frame The flow field in the lab frame is given by: v(r) = u(r) + U + R Ω (0) where R r + r c (see figure 3). In Eq.(0), Ω is zero due to the axisymmetric flow field, leaving v(r) = u(r) + U. (1) Accordingly, the flow field in the lab frame is given by: Figure 3: The relation between the lab frame and the co-moving frame. v r (R) = a r A 0P 0 (cos θ) + 3 r 3 (A 1 + B 1 )P 1 (cos θ) [( n a n + r n n a n+ ) ( a n+ r n+ A n + a 3 a 3 v θ (R) = 1 3 r 3 (A 1 + B 1 )V 1 (cos θ) + 1 [(n an+ U = 1 3 (B 1 A 1 ). (n )an rn+ r n ) B n + an rn+ r n )B n P n (cos θ) ( n(n ) a n )A r n an+ r n+ n V n (cos θ) () Note that R r, unless one chooses the origin of the lab frame to coincide with the current center of the squirmer..6 Reduced models We will consider examples of which there is only a tangential velocity at the surface of the particle, which would cover the case of an impermeable, rigid sphere with just tangential motion, that is, slip. This leads to u r (a, θ) = n=0 A n P n (cos θ) = 0 A n = 0 for all n. (3) Additionally, since each mode B n contributes to the flow field with the order of 1/r n and 1/r n+, the first and the second term (i.e., B 1, B modes) dominates the flow field at a point far away from the body; B 1 mode corresponds to a uniform flow, while B mode accounts for a disturbance. Thus, for the sake of gaining physical intuition, we will only examine B 1 and B modes. Accordingly, the flow field in the co-moving frame is approximately given by ( a 3 u r (r, θ) = 3 r 3 P 1(cos θ) ) ( a 4 ) 3 cos θ B 1 + r 4 a r B P (cos θ), ( 1 a 3 u θ (r, θ) = 3 r 3 V 1(cos θ) + ) 3 B 1 + a4 r 4 B V (cos θ), U = 3 B 1. Since B 1 is directly connected to U, it is clear that B determines the appearance of the flow field. Hence, by introducing the dimensionless parameter ξ := B /B 1, one can evaluate a configuration of the flow field; given ξ, the appearance of the flow field will be the same regardless of the value of B 1. (4) 5

6 (a) Puller (b) Pusher (c) Neutral (d) Shaker Figure 4: Flow field due to a squirmer for various values of ξ. The top row: in the co-moving frame. The bottom row : in the lab frame. [( a 3 u r (r, θ) = B 1 3 r 3 P 1(cos θ) ) ( a 4 ) 3 cos θ + ξ r 4 a r P (cos θ), [( 1 a 3 u θ (r, θ) = B 1 3 r 3 V 1(cos θ) + ) 3 B 1 + ξ a4 r 4 V (cos θ). By varying ξ, four distinct flow configurations are established (see figure 4) [7: (5) ξ > 0 puller, ξ < 0 pusher, ξ = 0 neutral, ξ shaker. puller: it pulls the fluid to move forward (panel (a)). pusher: it pushes the fluid to move forward (panel (b)). neutral: it neither pushes nor pulls. It moves by sucking and releasing the fluid (panel (c)). shaker: it actuates the fluid without directional motion (panel (d))..7 Self-diffusiophoretic particle Self-diffusiophoresis of a Janus particle involves local fluid motion around the surface of the particle (i.e., the boundary condition of the flow field on the surface), and the fluid motion is generated by the chemical interaction between an anisotropically distributed solute and the surface, which the so-called phoretic mechanism. The gradients of the solute density are induced by the particle, e.g., releasing a solute from a certain fraction of its surface (as shown in figure 5) [4. For this model, we assume that the solute density c(r) reaches the steady-state in a very short time scale compared to the motion of the particle. Therefore, the solute density is governed by Laplace s equation [3: c(r) = 0. (6) Figure 5: Self-diffusiophoretic Janus particle releases a solute from parts of the surface to induce chemical composition gradients. [4 6

7 The boundary condition for the solute density c(r) on the surface (r = a) of the particle is given: { k on the active surface, ˆn c(a, θ) = 0 on the inert surface. where and k is the diffusion coefficient of the solute and the solute flux constant. normal vector of the surface. At infinity, the boundary condition is (7) ˆn denotes a The slip velocity at the surface of the particle is given by c(r ) = c. (8) v s = b c(r) r=a (9) where is the projection of the gradient onto the particle surface, leaving only the tangential component. Eq.(9) involves the effect of the chemical interaction between the particle and the anisotropically diffused solute in the vicinity of it. The surface mobility b, which accounts for the strength and the type (i.e., repulsive or attractive) of the interaction, is assumed to be constant on the surface [3, 4. We further assume that the self-diffusiophoresis is spherical, axisymmetric, and thus the corresponding flow field is also axisymmetric. Consequently, the particle moves along the z direction without rotation, that is, U = Uê z and Ω = 0. The Stokes equation is the same as that of the squirmer model, that is, Eq.(13) with the slip velocity Eq.(9) [4. Therefore, one already has the solution, from section.4. 3 Boundary Element Method If the particle has an arbitrary shape, one has to approach the problem numerically. The Boundary Elements Method (BEM) is a tool to solve partial differential equations. The working mechanism of BEM will be illustrated for the case of Laplace s equation. That of the Stokes equation is similar with Laplace s equation, but the details are more complex due to the vectorial character of the field. Nevertheless, one can use the same mathematical structure to solve the Stokes equation. 3.1 Green s function for Laplace s equation Since the Laplacian is a linear operator, the solution for Laplace s equation can be expressed in terms of a superposition of fundamental solutions; the fundamental solutions are Green s functions and Green s function dipoles. Therefore, understanding Green s function is of importance. A Green s function G(r, r 0 ) is defined as satisfying Laplace s equation except at a single point r 0, namely a singular point or a pole [5: G(r, r 0 ) + δ(r r 0 ) = 0. (30) Eq.(30) is Poisson s equation, and the Green s function describes a field at a point r due to a point source located at r 0. In three dimensional free space, it has the explicit form: 1 G(r, r 0 ) = 4π r r 0. (31) Figure 6: A field due to a pair of a source and a sink of identical strength; the white circle and the black circle denote the source and the sink respectively. The colormap indicates the strength of the field. 7

8 3. Green s function dipole Figure 6 shows a field due to a pair of two sources of identical strength but opposite sign (i.e., a source and a sink). When the distance between the source and the sink is negligible compared to the distance between the origin and at a point r, the source and the sink form the so-called "source dipole". The field due to a source dipole is shown in figure 6, and can be expressed as [5: V dip (r, r 0 ) = d G (r, r 0 ), (3) where d is the vector from the sink to the source. G (r, r 0 ) is called Green s function dipole, and mathematically can be derived by taking the gradient of the Green s function G(r, r 0 ) with respect to the coordinates the pole r 0 : G (r, r 0 ) := 0 G(r, r 0 ). (33) Note that the Green s function dipole is a vector. In free space, it is given by (form Eq.(31)): 3.3 Green s identities To understand how the general solution of Laplace s equation can be recasted into a form of a integral representation, it is helpful to understand Green s identities beforehand. Any two functions f(r) and g(r), which are at least twice differentiable, satisfy Green s first identity [5. g f = (g f) g f. (35) Changing the order of f and g leads to f g = (f g) f g. (36) By subtracting the two equations, one can obtain Green s second identity: G (r, r 0 ) = r r 0 4π r r 0 3. (34) Figure 7: An arbitrarily chosen volume V c. denotes the boundary surface. ˆn is unit normal vector pointing into V c [5. g f f g = (g f f g). (37) If the two functions satisfy Laplace s equation, the second identity becomes: (g f f g) = 0. (38) This is called the reciprocal relation. 3 Integrating the second identity over an arbitrary volume V c, as depicted in figure 7, leads to (g f f g)dv = (g f f g)dv = (g f f g) ˆndS. (39) V c V c Note that the Gauss theorem is applied in the second equality and the negative sign comes from the choice of ˆn (as shown in figure 7). 3 In the case of Stokes equation, the analogous relation is the Lorentz reciprocal theorem. 8

9 3.4 Integral representation Assuming that f(r) satisfies Laplace s equation and replacing g(r) with the Green s function G(r, r 0 ) in Eq.(37) leads to f(r) G(r, r 0 ) = (G(r, r 0 ) f(r) f(r) G(r, r 0 )). (40) Substituting Eq.(30) into Eq.(40) and integrating over the control volume results in f(r 0 )δ(r r 0 )dv = G(r, r 0 )[ˆn f(r)ds + f(r)[ˆn G(r, r 0 )ds. V c (41) If r 0 / V c, the left-hand side of Eq.(41) becomes zero, leaving 0 = G(r, r 0 )[ˆn f(r)ds + f(r)[ˆn G(r, r 0 )ds. (4) On the other hand, if r 0 V c, Eq.(41) gives f(r 0 ) = G(r, r 0 )[ˆn f(r)ds + f(r)[ˆn G(r, r 0 )ds. (43) Since the two arguments of the Green s function is interchangeable (as one can check from Eq.(31)), one can rewrite Eq.(43) as: f(r 0 ) = G(r 0, r)[ˆn f(r)ds + f(r)[ˆn G(r 0, r)ds. (44) Eq.(44) is a boundary-integral representation in terms of the boundary distributions of the Green s function and the Green s function dipole. The first term of Eq.(44) is called the single-layer potential, and the second term is called the double-layer potential [ Boundary integral equation To use Eq.(44), we need information about both sets of the boundary values, that is, f(r) and ˆn f(r) on the boundary. In practice, only one of two sets is known. For the case of the self-diffusiophoretic particle, ˆn c(r) on the boundary of the particle is given. However, obtaining the other set of the boundary values is possible by taking the limit of Eq.(44) such that r 0 approaches the boundary [5: lim f(r 0) = lim G(r 0, r)[ˆn f(r)ds + lim f(r)[ˆn G(r 0, r)ds. (45) r 0 r 0 r 0 The single-layer potential integral of Eq.(45) is continuous as the singular point approaches and crosses the boundary, but the double-layer potential integral is not [5. By separating the integral region of the double-layer potential as shown in figure 8, one arrives at: lim r 0 f(r)[ˆn G(r 0, r)ds = P V f(r)[ˆn G(r 0, r)ds + Pole f(r)[ˆn G(r 0, r)ds, (46) where P V denotes the so-called principal value integral, which integrates over the boundary surface except for the region where the pole is located as depicted in figure 8. The second term in Eq.(46), therefore, integrates the small area around the pole r 0, and it can be set to give the result [5: Pole f(r)[ˆn G(r 0, r)ds 1 f(r 0). (47) 9

10 Consequently, Eq.(45) turns to f(r 0 ) = + P V G(r 0, r)[ˆn f(r)ds f(r)[ˆn G(r 0, r)ds. (48) Since ˆn f(r) is known, Eq.(48) can be expressed compactly as f(r 0 ) = where P V f(r)[ˆn G(r 0, r)ds + Φ(r 0 ), Φ(r 0 ) G(r 0, r)[ˆn f(r)ds. (49) Figure 8: A schematic picture of Eq.(46). The principal value integral corresponds to integrating the boundary following the orange line (i.e., the boundary surface). Thus, it integrates over the whole boundary surface except for the area around the singular point. This equation is called a Fredholm integral equation of the second kind for the boundary values of f(r) [ How to use Eq.49 in a numerical scheme? To apply Eq.(49) in a numerical scheme, one has to implement a so-called "discretization". Figure 9 represents a discretized boundary. Each triangle is called a boundary element. The boundary elements can be realized such that the element is small enough so that the value of f(r) over that region can be approximated to be constant f(r 0 Σ n ) f n. (50) Therefore, taking Eq.(49) with r 0 Σ n for n = 1,,...N leads to Figure 9: A discretized boundary. f 1 = f =..., f N = N i=1 N i=1 N i=n f i a (1) i + Φ (1) i, f i a () i + Φ () i, f i a (N) i + Φ (N) i. (51) where a (k) i and Φ (k) i are defined as P V a (k) i = ˆn G(r 0 Σ k, r)ds, Φ (k) i Σ i = ˆn f(r) r Σi Σ i G(r 0 Σ k, r)ds. (5) Eq.(51) consists linear system of N equations for the N unknowns for f i. Accordingly, the discretized form of f(r) is determined. Knowing f(r), one has all the information needed for using Eq.(44), and therefore, f(r) at any point is computable by interpolation. 10

11 4 Summary We have explored the squirmer model, and BEM. The squirmer model is a model to explain the ciliated self-propelling micro-organisms at low Reynolds number. In order to understand such kind of swimmers, we approximated the flow produced by the cilia motion with a "squirming" fluid velocity, providing the boundary conditions on the surface. We solved the Stokes equation analytically, in the axisymmetric flow field with a spherical shaped body. Then, introducing the rigid body and the far-field approximations allows us to identify the four distinct types of squirmer. We also studied self-diffusiophoretic particles, which is driven by the phoretic mechanism, and showed that the simple basic model of self-diffusiophoresis reduces to a squirmer model. Lastly, for the case of which an analytic approach is not available, we introduced and explained the boundary element method. The first step was to re-expressing the function of our interest into boundary integral representation in terms of two sets of the boundary values of the function. In practice, only one set is given, so the other set has to be obtained. In order to find the unknown set, we established an integral equation. It can be solved by discretizing the boundary into the boundary elements. After determining the two sets of the boundary values, one could compute the function at any point via the integral representation. Even though we illustrated the methodology of Laplace s equation, the same structure can be applied to solving the Stokes equation. References [1 J. Blake, A spherical envelope approach to ciliary propulsion, Journal of Fluid Mechanics, vol. 46, pp , [ M. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers, Communications on Pure and Applied Mathematics, vol. 5, pp , 195. [3 W. Uspal, M. N. Popescu, S. ietrich, and M. Tasinkevych, Self-propulsion of a catalytically active particle near a planar wall: from reflection to sliding and hovering, Soft Matter, vol. 11, pp , 015. [4 S. Michelin and E. Lauga, Phoretic self-propulsion at finite Péclet numbers, Journal of Fluid Mechanics, vol. 747, pp , 014. [5 C. Pozrikidis, A practical guide to boundary element methods with the software library BEMLIB. CRC Press, 00. [6 J. Happel and H. Brenner, Low Reynolds number hydrodynamics: with special applications to particulate media. Springer Science & Business Media, 01, vol. 1. [7 M. T. ownton and H. Stark, Simulation of a model microswimmer, Journal of Physics: Condensed Matter, vol. 1, p , 009. [8 O. S. Pak and E. Lauga, Generalized squirming motion of a sphere, Journal of Engineering Mathematics, vol. 88, pp. 1 8,