Abstract.
In a previous paper, we presented a (noncanonical) Hamiltonian model for the dynamic interaction of a neutrally buoyant rigid body of arbitrary smooth shape with N closed vortex filaments of arbitrary smooth shape, modeled as curves, in an infinite ideal fluid in \(\mathbb{R}^3\). The setting of that paper was quite general, and the model abstract enough to make explicit conclusions regarding the dynamic behavior of such systems difficult to draw. In the present paper, we examine a restricted class of such systems for which the governing equations can be realized concretely and the dynamics examined computationally. We focus, in particular, on the case in which the body is a smooth sphere. The equations of motion and Hamiltonian structure of this dynamic system, which follow from the general model, are presented. Following this, we impose the constraint of axisymmetry on the entire system and look at the case in which the rings are all circles perpendicular to a common axis of symmetry passing through the center of the sphere. This axisymmetric model, in our idealized framework, is governed by ordinary differential equations and is, relatively speaking, easily integrated numerically. Finally, we present some plots of dynamic orbits of the axisymmetric system.
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Abbreviations
- a:
-
radius vector of sphere
- a :
-
=| a |
- Γ i :
-
strength of i th ring
- U :
-
translation velocity vector of sphere= Ub in the axisymmetric model
- L :
-
‘linear momentum’ of body+fluid system= Lb in the axisymmetric model
- M :
-
mass plus added mass of sphere
- C i :
-
arc-length parameterized curve representing i th ring
- s i :
-
arc-length parameter of i th ring
- R i :
-
radius of i th ring in the axisymmetric model
- z i :
-
position of (center of) i th ring along axis of symmetry measured from origin of body-fixed frame
- u R,i :
-
velocity field due to the i th ring in unbounded flow
- u I,i :
-
velocity field of image of i th ring
- u SI,i :
-
self-induced velocity of i th ring, for the axisymmetric model u SI,i = u LI,i b where LI stands for the local induction approximation
- ΦI,i:
-
velocity potential of u I,i
- Φ B :
-
velocity potential of the Kirchhoff velocity field
- ∇Φ B :
-
Kirchhoff velocity field
- n i :
-
principal unit normal field on i th ring
- b :
-
unit binormal field — parallel to axis of symmetry for all rings
- t i :
-
unit tangent field on i th ring
- N ij :
-
n i component of \({\mathbf{u}}_{R,j|_{C_i}}\)
- B ij :
-
b component of \({\mathbf{u}}_{R,j|_{C_i}}\)
- \({\mathcal{N}}_{ij}\) :
-
n i component of \({\mathbf{u}}_{I,j|_{C_i}}\)
- \({\mathcal{B}}_{ij}\) :
-
b component of \({\mathbf{u}}_{I,j|_{C_i}}\)
- \(\mathfrak{N}_{i}\) :
-
n i component of \(\nabla\Phi_{B|_{C_i}}\)
- \(\mathfrak{B}_{i}\) :
-
b component of \(\nabla\Phi_{B|_{C_i}}\)
- ∂B:
-
surface of sphere
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Communicated by D. H. Sattinger
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Shashikanth, B.N., Sheshmani, A., Kelly, S.D. et al. Hamiltonian Structure and Dynamics of a Neutrally Buoyant Rigid Sphere Interacting with Thin Vortex Rings. J. Math. Fluid Mech. 12, 335–353 (2010). https://doi.org/10.1007/s00021-008-0291-0
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DOI: https://doi.org/10.1007/s00021-008-0291-0