Helmholtz minimum dissipation theorem

In fluid mechanics, Helmholtz minimum dissipation theorem (named after Hermann von Helmholtz who published it in 1868[1][2]) states that the steady Stokes flow motion of an incompressible fluid has the smallest rate of dissipation than any other incompressible motion with the same velocity on the boundary.[3][4] The theorem also has been studied by Diederik Korteweg in 1883[5] and by Lord Rayleigh in 1913.[6]

This theorem is, in fact, true for any fluid motion where the nonlinear term of the incompressible Navier-Stokes equations can be neglected or equivalently when ∇ × ∇ × ω = 0 {\displaystyle \nabla \times \nabla \times {\boldsymbol {\omega }}=0} , where ω {\displaystyle {\boldsymbol {\omega }}} is the vorticity vector. For example, the theorem also applies to unidirectional flows such as Couette flow and Hagen–Poiseuille flow, where nonlinear terms disappear automatically.

Mathematical proof

Let u ,   p {\displaystyle \mathbf {u} ,\ p} and E = 1 2 ( ∇ u + ( ∇ u ) T ) {\displaystyle E={\frac {1}{2}}(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{T})} be the velocity, pressure and strain rate tensor of the Stokes flow and u ′ ,   p ′ {\displaystyle \mathbf {u} ',\ p'} and E ′ = 1 2 ( ∇ u ′ + ( ∇ u ′ ) T ) {\displaystyle E'={\frac {1}{2}}(\nabla \mathbf {u} '+(\nabla \mathbf {u} ')^{T})} be the velocity, pressure and strain rate tensor of any other incompressible motion with u = u ′ {\displaystyle \mathbf {u} =\mathbf {u} '} on the boundary. Let u i {\displaystyle u_{i}} and e i j {\displaystyle e_{ij}} be the representation of velocity and strain tensor in index notation, where the index runs from one to three.

Consider the following integral,

∫ ( e i j ′ − e i j ) e i j   d V = ∫ ∂ ( u i ′ − u i ) ∂ x j e i j   d V {\displaystyle {\begin{aligned}\int (e_{ij}'-e_{ij})e_{ij}\ dV&=\int {\frac {\partial (u_{i}'-u_{i})}{\partial x_{j}}}e_{ij}\ dV\end{aligned}}}

where in the above integral, only symmetrical part of the deformation tensor remains, because the contraction of symmetrical and antisymmetrical tensor is identically zero. Integration by parts gives

∫ ( e i j ′ − e i j ) e i j   d V = ∫ ( u i ′ − u i ) e i j n j   d A − 1 2 ∫ ( u i ′ − u i ) ( ∇ 2 u i )   d V . {\displaystyle \int (e_{ij}'-e_{ij})e_{ij}\ dV=\int (u_{i}'-u_{i})e_{ij}n_{j}\ dA-{\frac {1}{2}}\int (u_{i}'-u_{i})(\nabla ^{2}u_{i})\ dV.}

The first integral is zero because velocity at the boundaries of the two fields are equal. Now, for the second integral, since u i {\displaystyle u_{i}} satisfies the Stokes flow equation, i.e., μ ∇ 2 u i = ∇ p {\displaystyle \mu \nabla ^{2}u_{i}=\nabla p} , we can write

∫ ( e i j ′ − e i j ) e i j   d V = − 1 2 μ ∫ ( u i ′ − u i ) ∂ p ∂ x i   d V . {\displaystyle \int (e_{ij}'-e_{ij})e_{ij}\ dV=-{\frac {1}{2\mu }}\int (u_{i}'-u_{i}){\frac {\partial p}{\partial x_{i}}}\ dV.}

Again doing an Integration by parts gives

∫ ( e i j ′ − e i j ) e i j   d V = − 1 2 μ ∫ p ( u i ′ − u i ) n i   d A + 1 2 μ ∫ p ∂ ( u i ′ − u i ) ∂ x i   d V . {\displaystyle \int (e_{ij}'-e_{ij})e_{ij}\ dV=-{\frac {1}{2\mu }}\int p(u_{i}'-u_{i})n_{i}\ dA+{\frac {1}{2\mu }}\int p{\frac {\partial (u_{i}'-u_{i})}{\partial x_{i}}}\ dV.}

The first integral is zero because velocities are equal and the second integral is zero because the flow in incompressible, i.e., ∇ ⋅ u = ∇ ⋅ u ′ = 0 {\displaystyle \nabla \cdot \mathbf {u} =\nabla \cdot \mathbf {u} '=0} . Therefore we have the identity which says,

∫ ( e i j ′ − e i j ) e i j   d V = 0. {\displaystyle \int (e_{ij}'-e_{ij})e_{ij}\ dV=0.}


The total rate of viscous dissipation energy over the whole volume of the field u ′ {\displaystyle \mathbf {u} '} is given by

D ′ = ∫ Φ ′ d V = 2 μ ∫ e i j ′ e i j ′   d V = 2 μ ∫ [ e i j e i j + e i j ′ e i j ′ − e i j e i j ]   d V {\displaystyle D'=\int \Phi 'dV=2\mu \int e_{ij}'e_{ij}'\ dV=2\mu \int [e_{ij}e_{ij}+e_{ij}'e_{ij}'-e_{ij}e_{ij}]\ dV}

and after a rearrangement using above identity, we get

D ′ = 2 μ ∫ [ e i j e i j + ( e i j ′ − e i j ) ( e i j ′ − e i j ) ]   d V {\displaystyle D'=2\mu \int [e_{ij}e_{ij}+(e_{ij}'-e_{ij})(e_{ij}'-e_{ij})]\ dV}

If D {\displaystyle D} is the total rate of viscous dissipation energy over the whole volume of the field u {\displaystyle \mathbf {u} } , then we have

D ′ = D + 2 μ ∫ ( e i j ′ − e i j ) ( e i j ′ − e i j )   d V {\displaystyle D'=D+2\mu \int (e_{ij}'-e_{ij})(e_{ij}'-e_{ij})\ dV} .

The second integral is non-negative and zero only if e i j = e i j ′ {\displaystyle e_{ij}=e_{ij}'} , thus proving the theorem.

Poiseuille flow theorem

The Poiseuille flow theorem[7] is a consequence of the Helmholtz theorem states that The steady laminar flow of an incompressible viscous fluid down a straight pipe of arbitrary cross-section is characterized by the property that its energy dissipation is least among all laminar (or spatially periodic) flows down the pipe which have the same total flux.

References

  1. ^ Helmholtz, H. (1868). Verh. naturhist.-med. Ver. Wiss. Abh, 1, 223.
  2. ^ von Helmholtz, H. (1868). Zur Theorie der stationären Ströme in reibenden Flüssigkeiten. Verh. Naturh.-Med. Ver. Heidelb, 11, 223.
  3. ^ Lamb, H. (1932). Hydrodynamics. Cambridge university press.
  4. ^ Batchelor, G. K. (2000). An introduction to fluid dynamics. Cambridge university press.
  5. ^ Korteweg, D. J. (1883). XVII. On a general theorem of the stability of the motion of a viscous fluid. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 16(98), 112-118.
  6. ^ Rayleigh, L. (1913). LXV. On the motion of a viscous fluid. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 26(154), 776-786.
  7. ^ Serrin, J. (1959). Mathematical principles of classical fluid mechanics. In Fluid Dynamics I/Strömungsmechanik I (pp. 125-263). Springer, Berlin, Heidelberg.