Field theoretic formulation of fluid mechanics according to the geometric algebra

Abstract

A simple and coherent approach to fluid mechanics is presented using a proper formalism of geometric algebra. The analogy between the equations of electromagnetism and fluid mechanics provides reinterpretation of the equations for two constituent (vorticity and the Lamb vector) fields. Identifying certain quantities as the source fields, the guiding Navier–Stokes (NS) equations of fluid mechanics can be formulated as a set of four geometrically distinct field equations, resembling exactly the Maxwell equations for the constituent magnetic and electric fields. The same set of equations works for all the cases of compressible, incompressible, viscous and the inviscid fluid motions with appropriately modified source terms. The analogy is completed by defining the combined ‘fluidomechanic’ bivector field in space–time algebra and further extended to the fluidic analogue of the Poynting theorem, Poynting vector and Lorentz force.

Appendix

Using eqs (5) and (7), the evolution equation of the Lamb vector field ${\textbf{l}}$ can be written as

$$\begin{aligned} \partial _{t} {\textbf{l}}= & {} \partial _{t}({\textbf{v}} \cdot {\textbf{W}}-{\textbf{k}})=\partial _{t}{\textbf{v}} \cdot {\textbf{W}}+{\textbf{v}}\cdot \partial _{t}{\textbf{W}} -\partial _{t}{\textbf{k}}\nonumber \\= & {} (-{\textbf{l}}-\nabla \Phi )\cdot {\textbf{W}} +{\textbf{v}}\cdot (-\nabla \wedge {\textbf{l}})\nonumber \\{} & {} -\eta \{\nabla ^{2}+(3^{-1}\nabla )\nabla \cdot \} \partial _{t} {\textbf{v}}\nonumber \\= & {} -{\textbf{l}}\cdot {\textbf{W}}-(\nabla \Phi ) \cdot {\textbf{W}}-{\textbf{v}}\cdot (\nabla \wedge {\textbf{l}})\nonumber \\{} & {} -\eta \{\nabla ^{2}+(3^{-1}\nabla ) \nabla \cdot \}(-{\textbf{l}}-\nabla \Phi ). \end{aligned}$$

(A.1)

With two vector fields ${\textbf{l}}$ and ${\textbf{v}}$, a known identity of geometric calculus reads as

$$\begin{aligned} \nabla ({\textbf{v}}\cdot {\textbf{l}})&=({\textbf{v}}\cdot \nabla ) {\textbf{l}}{+}({\textbf{l}}\cdot \nabla ) {\textbf{v}} {+}(\nabla \wedge {\textbf{v}})\cdot {\textbf{l}} {+}(\nabla \wedge {\textbf{l}})\cdot {\textbf{v}}\nonumber \\&\Rightarrow (\nabla \wedge {\textbf{l}}) \cdot {\textbf{v}} \equiv -{\textbf{v}} \cdot (\nabla \wedge {\textbf{l}})=-\nabla ({\textbf{v}} \cdot {\textbf{k}})\nonumber \\&\quad -({\textbf{v}}\cdot \nabla ){\textbf{l}} -({\textbf{l}} \cdot \nabla ){\textbf{v}}+{\textbf{l}}\cdot {\textbf{W}} \end{aligned}$$

(A.2)

since ${\textbf{v}}\cdot ({\textbf{v}}\cdot {\textbf{W}})$ vanishes identically. Now, substitution of (A.2) in (A.1) gives

$$\begin{aligned} \partial _{t} {\textbf{l}}= & {} -\nabla ({\textbf{v}}\cdot {\textbf{k}}) -({\textbf{v}}\cdot \nabla ){\textbf{l}}-({\textbf{l}}\cdot \nabla ) {\textbf{v}}+{\textbf{W}}\cdot \nabla \Phi \\{} & {} +\eta \{\nabla ^{2} +(3^{-1}\nabla )\nabla \cdot \} ({\textbf{l}}+\nabla \Phi ). \end{aligned}$$

Moreover, from the following identity of GC:

$$\begin{aligned} \nabla \cdot ({\textbf{v}} \wedge {\textbf{l}}){} & {} ={\textbf{l}}(\nabla \cdot {\textbf{v}})+({\textbf{v}} \cdot \nabla ){\textbf{l}}-{\textbf{v}}(\nabla \cdot {\textbf{l}}) -({\textbf{l}}\cdot \nabla ){\textbf{v}}\nonumber \\{} & {} \Rightarrow ({\textbf{v}} \cdot \nabla ){\textbf{l}}=\nabla \cdot ({\textbf{v}} \wedge {\textbf{l}}) -{\textbf{l}}(\nabla \cdot {\textbf{v}})\nonumber \\{} & {} \qquad +{\textbf{v}} (\nabla \cdot {\textbf{l}})+({\textbf{l}}\cdot \nabla ){\textbf{v}} \end{aligned}$$

and we can write:

$$\begin{aligned} \partial _{t} {\textbf{l}}= & {} -\nabla ({\textbf{v}}\cdot {\textbf{k}}) -\nabla \cdot ({\textbf{v}} \wedge {\textbf{l}})+{\textbf{l}}(\nabla \cdot {\textbf{v}}) -{\textbf{v}}(\nabla \cdot {\textbf{l}})\nonumber \\{} & {} -2({\textbf{l}} \cdot \nabla ){\textbf{v}}+{\textbf{W}} \cdot \nabla \Phi \nonumber \\{} & {} +\eta \{\nabla ^{2}+(3^{-1}\nabla )\nabla \cdot \} ({\textbf{l}}+\nabla \Phi ). \end{aligned}$$

(A.3)

Expressing the second term on the right-hand side of the above equation as

$$\begin{aligned}{} & {} -\nabla \cdot ({\textbf{v}} \wedge ({\textbf{v}}\cdot {\textbf{W}})) +\nabla \cdot ({\textbf{v}} \wedge {\textbf{k}})\\{} & {} \quad =-\nabla \cdot \{v^{2}{\textbf{W}}-({\textbf{v}} \wedge {\textbf{W}}) \cdot {\textbf{v}}\}+\nabla \cdot ({\textbf{v}} \wedge {\textbf{k}})\\{} & {} \quad =-\nabla v^{2}\cdot {\textbf{W}}-v^{2}\nabla \cdot {\textbf{W}} +\{\nabla \cdot ({\textbf{v}} \wedge {\textbf{W}})\}\cdot {\textbf{v}}\\{} & {} \qquad +({\textbf{v}} \wedge {\textbf{W}}):\nabla \wedge {\textbf{v}}+\nabla \cdot ({\textbf{v}} \wedge {\textbf{k}}), \end{aligned}$$

eq. (A.3) can be written as

$$\begin{aligned} \partial _{t} {\textbf{l}}= & {} -\nabla ({\textbf{v}}\cdot {\textbf{k}})-v^{2} \,\nabla \cdot {\textbf{W}}+ \{\nabla \cdot ({\textbf{v}} \wedge {\textbf{W}})\}\cdot {\textbf{v}}\\{} & {} +({\textbf{v}} \wedge {\textbf{W}}): {\textbf{W}} +\nabla \cdot ({\textbf{v}} \wedge {\textbf{k}})+{\textbf{l}} (\nabla \cdot {\textbf{v}})-{\textbf{v}} (\nabla \cdot {\textbf{l}})\\{} & {} -2\,({\textbf{l}}\cdot \nabla ){\textbf{v}}+{\textbf{W}} \cdot \nabla (\Phi +v^{2})+\eta \nabla ^{2} ({\textbf{l}}+\nabla \Phi )\\{} & {} +3^{-1} \eta \nabla \{\nabla \cdot ({\textbf{l}}+\nabla \Phi )\}. \end{aligned}$$