Aplicação da álgebra geométrica na hidrodinâmica
Data
2022-12-19
Autores
Belançon, Emerson Dionísio
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Editor
Universidade Estadual Paulista (Unesp)
Resumo
Na resolução de problemas envolvendo a dinâmica de fluidos é comum aparecer singularidades. Tais singularidades podem ser impulsionadas pelo movimento da fronteira em conjunto com a viscosidade do fluido ou singularidades de superfície livre associadas à tensão superficial e viscosidade ou ainda singularidades de pontos interiores da vorticidade associadas ao intenso alongamento do vórtice. Diante disto, alguns métodos são utilizados para tentar contornar esta situação uma vez que as singularidades dificultam uma boa aproximação para a solução das equações de movimento. Alguns métodos são, os Quatérnions, Simetria de Lie ou aproximações numéricas como elementos finitos os quais permitem ajustar a solução. No final do século XIX, William Kingdon Clifford desenvolveu uma estrutura matemática conhecida como Álgebra Geométrica de Clifford ou simplesmente Álgebra Geométrica. Esta estrutura matemática não se baseia em conceito de vetores mas em subespaços capaz de gerar objetos matemáticos de dimensões variadas, conhecidos como multivetores. A estrutura dessa álgebra generaliza e integra naturalmente conceitos, como matrizes, coordenadas de Plücker, números complexos, quatérnions, álgebra de Lie. Assim, este trabalho propõe uma maneira simples para compactar as equações usadas para descrever o escoamento de um fluido ideal e incompressível em uma única equação utilizando a estrutura da álgebra geométrica. Depois da compactação, resolver dois problemas clássicos da hidrodinâmica: o escoamento bidimensional em torno de um cilindro circular. O primeiro é manter o cilindro fixo e o segundo é acrescentar um vórtice simples no centro do cilindro. O primeiro problema será resolvido de duas formas diferentes: a primeira é aplicar um método semelhante a segunda fórmula (ou identidade) de Green, a qual é uma das ferramentas fundamentais no desenvolvimento deste trabalho. A segunda é compactar as equação clássicas da hidrodinâmica com a estrutura da álgebra geométrica e utilizar as funções de Green para resolver essa equação. O segundo problema será resolvido utilizando a compactação das equações clássicas que é o método proposto neste trabalho.
In solving problems involving fluid dynamics, it is common for singularities to appear. Such singularities can be driven by the movement of the boundary together with the fluid viscosity or free surface singularities associated with surface tension and viscosity or even singularities at interior points of vorticity associated with the intense elongation of the vortex. In view of this, some methods are used to try to circumvent this situation since the singularities make it difficult to get a good approximation for the solution of the equations of motion. Some methods are, the Quaternions, Lie Symmetry or numerical approximations as finite elements which allow to adjust the solution. In the late 19th century, William Kingdon Clifford developed a mathematical framework known as Clifford’s Geometric Algebra or simply Geometric Algebra. This mathematical structure is not based on the concept of vectors but on subspaces capable of generating mathematical objects of varied dimensions, known as multivectors. The structure of this algebra naturally generalizes and integrates concepts such as matrices, Plücker coordinates, complex numbers, quaternions, Lie algebra. Thus, this work proposes a simple way to compress the equations used to describe the flow of an ideal and incompressible fluid into a single equation using the structure of geometric algebra. After compaction, solve two classic hydrodynamics problems: he two-dimensional flow around a circular cylinder. The first is to keep the cylinder fixed and the second is to add a simple vortex in the center of the cylinder. The first problem will be solved in two different ways: the first is to apply a method similar to Green’s second formula (or identity), which is one of the fundamental tools in the development of this work. The second is to compact the classical equations of hydrodynamics with the structure of geometric algebra and use Green’s functions to solve this equation. The second problem will be solved using the compression of classical equations, which is the method proposed in this work.
In solving problems involving fluid dynamics, it is common for singularities to appear. Such singularities can be driven by the movement of the boundary together with the fluid viscosity or free surface singularities associated with surface tension and viscosity or even singularities at interior points of vorticity associated with the intense elongation of the vortex. In view of this, some methods are used to try to circumvent this situation since the singularities make it difficult to get a good approximation for the solution of the equations of motion. Some methods are, the Quaternions, Lie Symmetry or numerical approximations as finite elements which allow to adjust the solution. In the late 19th century, William Kingdon Clifford developed a mathematical framework known as Clifford’s Geometric Algebra or simply Geometric Algebra. This mathematical structure is not based on the concept of vectors but on subspaces capable of generating mathematical objects of varied dimensions, known as multivectors. The structure of this algebra naturally generalizes and integrates concepts such as matrices, Plücker coordinates, complex numbers, quaternions, Lie algebra. Thus, this work proposes a simple way to compress the equations used to describe the flow of an ideal and incompressible fluid into a single equation using the structure of geometric algebra. After compaction, solve two classic hydrodynamics problems: he two-dimensional flow around a circular cylinder. The first is to keep the cylinder fixed and the second is to add a simple vortex in the center of the cylinder. The first problem will be solved in two different ways: the first is to apply a method similar to Green’s second formula (or identity), which is one of the fundamental tools in the development of this work. The second is to compact the classical equations of hydrodynamics with the structure of geometric algebra and use Green’s functions to solve this equation. The second problem will be solved using the compression of classical equations, which is the method proposed in this work.
Descrição
Palavras-chave
Álgebra geométrica, Multivetores, Hidrodinâmica, Funções de Green, Geometric algebra, Multivector, Hydrodynamics, Green’s Functions