Squirming Sphere in an Ideal Fluid

Transcription

1 Squirming Sphere in an Ideal Fluid D Rufat June 8, 9 Introduction The purpose of this wor is to derive abalytically and with an explicit formula the motion due to the variation of the surface of a squirming body in a fluid. Such a derivation has been done before, but only in two dimensions, and there is no literature on any attempts to generalize to three dimensions, which is cosiderably more involved algebraically. But utilizing software such as Mathematica, the derivation is quite straightforward. The primary motivation behind this wor is the search for innovative propulsion mechanisms. Surface variations may not lead to a very high speed, but they provide higher maneuverability and efficiency and are the primary source of motion of aquatic animals. Two types of fluid are studied - an Eulerian (potential fluid where all viscous effects are ignored and inertia is the only factor, and a Stoesian (creeping fluid where all inertial effects are ignored and viscous friction is the only factor. We consider only those cases because they linearize the equations governing the fluid motion, and permit analytic solutions. Hydromechanical Connections Consider a body immersed in a fluid. We regard its shape as a shape manifold M. And the configuration manifold Q = M G represents the complete position and orientation of the body in space. Q = M G can be thought of as a trivial bundle with a group structure G, and an action of G on Q given by Φ h (r, g = (r, hg for (r, g Q and h G. An element of the tangent space is given by q = (ṙ, ġ T Q. We are intererested in computing a connection on the configuration bundle Q that will allow us to compute the fiberwise translations resulting from cyclical changes in shape.

2 A connection is a vector valued one-form Γ : T Q g on Q. We follow (Kelly, 998 in defining two types of connections: Mechanical connection Γ mech : (q, q I (qj(q, q where I : g g is the loced inertia tensor such that I(qξ, η = ξ Q (q, η Q (q KE and I(r, g = Ad g I loc(rad g and J : T Q g is the momentum map. Stoes connection Γ Stoes : (q, q V (qk(q, q where K : T Q g is a momentum map K(q, q, ξ = F (q, q, ξ Q (q associated with a dissipative force F, and V : g g is a viscosity tensor such that V(qξ, η = F (ξ Q (q, η Q (q and V(r, g = Ad g V loc(rad g Kelly uses the following local expressions Γ mech = Ad g (g ġ + A mech ṙ Γ mech = Ad g (g ġ + A Stoes ṙ and for zero initial momentum g ġ = A mech ṙ g ġ = A Stoes ṙ

3 3 z F (θ, ϕ θ y ϕ x Figure : Coordinate system. which completely describe the evolution of a Potential and Stoesian swimmers respecitlvey. 3 Squirming Sphere Consider a three dimensional nearly spherical body whose surface is described by the following equation: r = F (θ, ϕ; s,..., s n Such a body has already been studied previously numerically (Rufat, 5, and the current method that allows us to obtain explicit formulas for the displacement as will be later seen, are valuable tools for validating numerical simulations. As an example we consider the propulsive movement of a roughly spherical device whose boundary shape is modulated by a small amount. We parametrize the boundary in terms of the shape variables s i (t, and express it as a sum of an unperturbed base shape and a perturbed part: The configuration manifold is F (θ; s = + ɛ (s cos θ + s cos 3θ Q = R SE(3

4 4 z F(θ, s s s Figure : Boundary perturbations. 3. Potential Flow Irrotational potential flow is best described in terms of the velocity potential, which satisfies with the Neumann boundary condition φ = where u is the prescribed boundary motion. φ n = u n ( For an axisymmetric problem, such as this one, the solution of the Laplace equation can be expressed as a series φ(r, θ = = ( A r + B r + P (cos θ where P (x are the Legendre polynomials satisfying the orthogonality condition: ˆ + P m (xp n (xdx = n + δ mn

5 5 We require that u as r, and therefore we must tae A = for. φ(r, θ = = a P (cos θ r + a Φ The problem of solving for φ is reduced to finding the coefficients a. First, in order to enforce the boundary conditions on the perturbed surface, we need to compute the normal n at each point. The two tangential components on the surface are = t = F (F ˆr = θ θ ˆr + F ˆθ t = F (F ˆr = ˆr + F sin θ ˆϕ ϕ ϕ and therefore the normal component can be evaluated as the cross product of the two n = t t = F sin θ ˆr F F θ sin θ ˆθ F F ϕ ˆϕ Note that we need not normalize the expression for n since it occurs on both sides of the boundary condition equation (. Expanding the spherical coordinates (r, θ, ϕ of n in terms of ɛ: n = n ( +ɛn ( +ɛ n ( (c[θ]s + c[3θ]s = + ɛ s[θ]s + 3s[3θ]s + ɛ (c[θ]s + c[3θ]s (c[θ]s + c[3θ]s (s[θ]s + 3s[3θ]s where c[θ] = cos θ and s[θ] = sin θ. Next, we expand the potential in a power series in terms of the small perturbation paramater ɛ. φ = φ ( + ɛφ ( + ɛ φ ( ɛ i φ (i +... where φ (i = a (i Φ = Plugging bac in, we expand both sides in terms of ɛ φ n = u n = i= ɛ i = ɛ i u (i i= a (i f (i

6 6 and equate terms of equal power = a (i f (i = u (i where the first few terms are = = η = 3 ( 3η 3 = ( 3η 5η 3 4 = 5 ( 3 3η + 35η = 3 ( 5η 7η η = 7 ( 5 5η + 35η 4 3η = ( 35η 35η η 5 49η 7 where η = cos θ By the linearity of Laplace s equation (using superposition one can write following Kirchoff, Then, up to second order in ɛ φ = φ z ż + φ s ṡ + +φ s ṡ φ z = P ( r + ɛ 9P s r + 9P s 7r 3 6P 3s 5r 4 48P 4s 35r ( 5 + ɛ (P 3339s 499s 45r 78P s s 735r 3 + 6P ( 3 38s 45s 95r P 4s s 3475r 5 6P ( 5 9s 9s 637r 6 64P 6s s r 7 5P 7s 43r 8 ( P φ s = ɛ 3r 4P ( 9r 3 + ɛ 4P s 3P s 5r 35r + 8P s r 3 + P 3s 5r 4 5P 4s 55r 5 64P 5s 63r 6 ( 3P φ s = ɛ r P ( 3 5r 4 + ɛ 34P s 39P s 35r 35r 3P s 35r 3 + P 3s 5r 4 + 8P 4s 95r 5 8P 5s 89r 6 856P 6s 67r 7 The total inetic energy is given by

7 7 L = = π ˆ F ˆ π ˆ dv φ dφ = π 5 ż ɛ π 5 +ɛ π 5 ˆ π ˆ ˆ sin θdθ r dr φ dη r dr φ ( 49 5 s ż + 6 ( 5ṡż 3 89ṡ + 4 5ṡ + ( s ṡ s ṡ ( 576 ż + 75 s s ż + O(ɛ 3 This Lagrangian is invariant under translation along z, so its momentum map is J = π 5 ż ɛπ 5 ( 49 5 s ż + 3 +ɛ π 5ṡ 5 ( s ṡ 68 ( s ṡ + 75 s s ż +O(ɛ 3 The loced inertia tensor is given by Therefore, I(s, s : ξ π ( 5 ξ ɛ 49 ( s + ɛ 75 s s + O(ɛ 3 Γ mech = I J = dz ɛ 3 ( 69 5 ds + ɛ 55 s ṡ s ṡ 49 5 s 3 + O(ɛ 3 5ṡ = dz ɛ 3 ( 8 5 ds + ɛ 5 s ṡ s ṡ + O(ɛ 3 Self-propulsion of the sphere requires that its motion remain horizontal with respet to this connection, therefore 3. Stoes Flow ż = ɛ 3 ( 68 + ɛ 5ṡ 45 s ṡ 8 5 s ṡ Thes solution of creeping-motion can be reduce in two dimensions to the biharmonic equation in terms of the stream function 4 ψ = which satisfies the no-slip and no-penetration boundary condition on the surface

8 8 u = (u r, u θ = ( F, Following (Leal, 7 p.46 the general solution is ψ = (A r +3 + B r + + C r + D r Q (η = where Q (x are polynomial functions defined in terms of the Legendre polynomials Q (x = and satisfiying the orthogonality condition ˆ x P (xdx ˆ + Q m (xq n (x x dx = n(n + (n + δ mn Again we want the velocity to vanish at infinity, so we will neglect the first two terms ψ = (a r + b r Q (η = u r = r ψ η u θ η = r ψ r Again we will exploit the linearity of the equation by using Kirchoff s method to express teh streamfunction as ψ = ψ z ż + ψ s ṡ + +ψ s ṡ ψ z = ( r 3r [ ( s Q + ɛ Q 3 [ ( +ɛ Q 3r ( 448s + 45s 5 ( +Q 3 6 ( 38s 35s 95r ( 48s +Q 7 43r 7 536s 43r 5 ( + Q 9s r + 9rs + Q 4 ( 4s 5r 3 s 5r 7r 4 4s 7r s + 85s ( + Q 87 45r 49 s s + 97s s 49r 4 ( 38s + 45s ( 95r 3 + Q 4 764s s 695r 4 836s s 695r ( ( 48 9s + Q 54s 5 637r 5 3 ( 9s 8s 637r 3 + Q ( 7s ( 4s s + Q 6 r 6 6s s r 4 4 7s ] 4r ] Due to the lac of time, I am unable to continue with the calculations. The Mathematica code that I use seems to have trouble computing the ψ s and ψ s components. This could be related

9 9 the 3D equivalent of the Stoe s paradox, or could simply be a bug in the code. In any case, it would be important to carry this calculation to completion at some point in the future in order to compare the displacement resulting from creeping flow to that in potential flow. 4 Discussion A calculation similar to the one done above was carried by Kelly for a two dimensional circle where the algebra is less involved. Figure 3 compares the results. In all cases the shape trajectory is a unit circle (s, s shape space. potential 3d potential d z stoes d.3.. s s Figure 3: Geometric phase for a particular gate. TODO: stoes 3d. Explicitly, we quote the results. For the the potential flow case, Kelly obtained: and for Stoe s case ż = ɛ (s ṡ s ṡ + O(ɛ 3 ż = ɛ ( 4 s ṡ + s ṡ + O(ɛ If we compare those, to our result

10 Figure 4: Ilustration of axisymmetroc squirming. ż = ɛ 3 ( 68 + ɛ 5ṡ 45 s ṡ 8 5 s ṡ ( The first thing that we notice is the presence of O(ɛ in the 3d case (our solution. However, this term has no effect on the net displacement and it only results in a more pronounced oscillation of the center of mass during each cycle. Also we notice that the body achieves higher displacementin 3d. This simplicity with which such explicit equations such as can be obtained is a testament to the power of the geometric approach to mechanics. References Bloch A, 7 Nonholonomic Mechanics and Control (Springer Holm D D, 8 GEOMETRIC MECHANICS: Dynamics and Symmetry illustrated edition edition (Imperial College Press Kelly S D, 998 The Mechanics and Control of Robotic Locomotion with Appliations to Aquatic Vehicles Ph.D. thesis California Institute of Technology Kohr M, Pop I, 4 Viscous Incompressible Flow for Low Reynolds Numbers (Advances in Boundary Elements illustrated edition edition (WIT Press (UK Leal L G, 7 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes st edition (Cambridge University Press Marsden J E, Ratiu T S, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems nd edition (Springer Mason R, Burdic J, 999 Propulsion and control of deformable bodies in an ideal fluid in 999 IEEE International Conference on Robotics and Automation, 999. Proceedings, volume

11 Melli J B, Rowley C W, Rufat D S, 6 Motion planning for an articulated body in a perfect planar fluid SIAM Journal on Applied Dynamical Systems 5 65 Nair S, Kanso E, 7 Hydrodynamically coupled rigid bodies Journal of Fluid Mechanics Rufat D S, 5 Locomotion of a Three Dimensional Variable Shape Body in Inviscid Fluid Undergraduate independent wor Princeton University