Bitensorial formulation of the singularity method for Stokes flows

Giuseppe Procopio; Massimiliano Giona; Giuseppe Procopio; Massimiliano Giona

doi:10.3934/mine.2023046

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-34. doi: 10.3934/mine.2023046

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Research article

Bitensorial formulation of the singularity method for Stokes flows

Giuseppe Procopio ,
Massimiliano Giona ^,

DICMA, La Sapienza Università di Roma, via Eudossiana 18, Rome 00184, Italy

Received: 08 June 2022 Revised: 14 July 2022 Accepted: 15 July 2022 Published: 04 August 2022

This paper develops the bitensorial formulation of the system of singularities associated with unbounded and bounded Stokes flows. The motivation for this extension is that Stokesian singularities and hydrodynamic fundamental solutions are multi-point functions, and bitensor calculus provides either the proper geometrical setting, in order to avoid inconsistencies and misunderstandings on the role of the different tensorial indices, or a way for compactly deriving hydrodynamic properties. A first relevant result is to provide a clear definition of the singularities (both bounded and unbounded) in Stokes flow, specifying the associated differential equations and boundary conditions. Using this formalism for bounded flows, we show the existence of an integro-differential operator providing the whole system of hydrodynamic singularities by acting on the unbounded Green function (Stokeslet) at its pole and we derive its explicit representation in terms of moments. In the case of an immersed body in a unbounded fluid, we show that, the operator furnishing the disturbance field of a purely $ n $-th order ambient flow, is a generalized $ n $-th order Faxén operator, i.e., it yields the $ n $-th moment on the body if applied to a generic ambient flow, and that a generic disturbance field can be expressed by a summation of the generalized $ n $-th order Faxén operators. Furthermore, we find that the operator providing the disturbance of an ambient flow coincides with the reflection operator for the Stokes solutions in the same flow geometry. We apply this result to the paradigmatic case of fundamental singularities for the Stokes flow bounded by a plane. In this way, we obtain in an alternative and easy way the image system for the Sourcelet and the Rotlet (already derived in the literature) and for the Source Doublet and the Strainlet (presented here for the first time).
- Stokes flows,
- Bitensor calculus,
- singularity method,
- Green functions,
- generalized functions,
- hydrodynamics
Citation: Giuseppe Procopio, Massimiliano Giona. Bitensorial formulation of the singularity method for Stokes flows[J]. Mathematics in Engineering, 2023, 5(2): 1-34. doi: 10.3934/mine.2023046

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Abstract

This paper develops the bitensorial formulation of the system of singularities associated with unbounded and bounded Stokes flows. The motivation for this extension is that Stokesian singularities and hydrodynamic fundamental solutions are multi-point functions, and bitensor calculus provides either the proper geometrical setting, in order to avoid inconsistencies and misunderstandings on the role of the different tensorial indices, or a way for compactly deriving hydrodynamic properties. A first relevant result is to provide a clear definition of the singularities (both bounded and unbounded) in Stokes flow, specifying the associated differential equations and boundary conditions. Using this formalism for bounded flows, we show the existence of an integro-differential operator providing the whole system of hydrodynamic singularities by acting on the unbounded Green function (Stokeslet) at its pole and we derive its explicit representation in terms of moments. In the case of an immersed body in a unbounded fluid, we show that, the operator furnishing the disturbance field of a purely $ n $-th order ambient flow, is a generalized $ n $-th order Faxén operator, i.e., it yields the $ n $-th moment on the body if applied to a generic ambient flow, and that a generic disturbance field can be expressed by a summation of the generalized $ n $-th order Faxén operators. Furthermore, we find that the operator providing the disturbance of an ambient flow coincides with the reflection operator for the Stokes solutions in the same flow geometry. We apply this result to the paradigmatic case of fundamental singularities for the Stokes flow bounded by a plane. In this way, we obtain in an alternative and easy way the image system for the Sourcelet and the Rotlet (already derived in the literature) and for the Source Doublet and the Strainlet (presented here for the first time).

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