1 A Variational Principle for Dissipative Fluid Dynamics Hiroki Fukagawa∗) and Youhei Fujitani∗∗) School of Fundamental Science & Technology, Keio University, Yokohama 223-8522, Japan In the variational principle leading to the Euler equation for a perfect fluid, we can use the method of undetermined multiplier for holonomic constraints representing mass conservation and adiabatic condition. For a dissipative fluid, the latter condition is replaced by the constraint specifying how to dissipate. Noting that this constraint is nonholonomic, we can derive the balance equation of momentum for viscous and viscoelastic fluids by using a single variational principle. We can also derive the associated Hamiltonian formulation by regarding the velocity field as the input in the framework of control theory. §1. Introduction In many areas of physics, variational principles help us choose convenient vari- ables in formulating problems. The least-action principle can lead to the equations of motion not only for microscopic dynamics but also those for nondissipative macro- scopic dynamics. For example, in a perfect fluid, the sum of the kinetic energy and internal energy is conserved, and we can write down the action easily. There are also the conservation laws for mass and entropy; these holonomic constraints can be incorporated into the action by means of the method of undetermined multipli- ers. Thus, the Euler equation can be derived from the stationary condition of the action.1), 2), 3), 4), 5), 6), 7), 8), 9) The Navier-Stokes equation is the equation of motion for the Newtonian viscous fluid, and represents the momentum balance with the dissipative force taken into account in the linear phenomenological law.10), 11) Without this force, the equation is reduced to the Euler equation. The dissipative force can be derived using another variational principle called the maximum dissipation principle, where Rayleigh’s dis- sipation function is minimized.12), 13) Using this principle amounts to assuming the linear phenomenological law. A set of variables with respect to which Rayleigh’s dissipation function is minimized is usually different from a set of variables with respect to which the action is stationary.14), 15) One way of unifying these two variational principles is called Onsager’s varia- tional principle,16) which has been used frequently in formulating dissipative dynam- ics of complex fluids.17), 18), 19), 20), 21) Using this principle, we can obtain the balance equation of momentum with the dissipative force taken into account in the linear phenomenological law. However, applying this principle to the Newtonian fluid in a arXiv:1104.0866v8 [physics.flu-dyn] 29 Apr 2012 straightforward way, we can obtain only a linearized version of Navier-Stokes equa- tion, i.e., the Stokes equation.22) A viscous fluid is not locally adiabatic, and the entropy conservation law cannot ∗ ) E-mail: [email protected] ∗∗) E-mail: [email protected] typeset using PTPTEX.cls hVer.0.9i 2 be assumed. We instead impose a constraint specifying how to dissipate. Noting that this constraint is nonholonomic, unlike in the previous studies, we can derive the balance equation of momentum in the framework of a single variational principle. As mentioned above, this framework for a viscous fluid originates from that for a perfect fluid, which we briefly review in §2. Our main results are shown in §3, where we apply our procedure to viscous and viscoelastic fluids. In §4, for completeness, we derive the associated Hamiltonian formulation in the framework of the control theory. Our study is discussed and summarized in the last section. §2. Perfect fluid We briefly review Ref. 5) with slight modifications. Let us consider the dynamics of a perfect fluid in a fixed container from the initial time tinit to the final time tfin. Let V denote the region occupied by the fluid. The velocity field v satisfies no- penetration condition at the boundary ∂V , i.e., n · v =0 on ∂V , (2.1) where n is the unit normal vector directed outside on ∂V . Let us write ρ for mass per unit volume, and s for entropy per unit mass. The internal-energy density per unit mass, ǫ, is a function of ρ and s because of the local equilibrium. The action is given by the integral of the Lagrangian density with respect to the time and space considered. Apart from the constraints mentioned below, the Lagrangian density is given by 1 L(ρ, v,s) ≡ ρ v2 − ǫ(ρ, s) , (2.2) 2 which is the difference between kinetic energy density and internal energy den- sity. The variables are not independent because of the constraints mentioned be- low. Writing T and p for the temperature and the pressure, respectively, we have dǫ = −pdρ−1 + T ds in the thermodynamics, which leads to ∂ǫ ∂ǫ p ≡ ρ2 and T ≡ , (2.3) ∂ρ ∂s s ρ where the subscripts s and ρ indicate variables fixed in the respective partial dif- ferentiations. We write τ for the time in the Lagrangian coordinates although it is equivalent to the time t in the non-relativistic theory. The partial derivatives with respect to τ (∂τ ) and t (∂t) imply the Lagrangian and Eulerian time-derivatives, respectively. In the Lagrangian description, we label a fluid particle with its initial position a = (a1, a2, a3), and write X = (X1, X2, X3) for its position at time τ. The time derivative of X denotes the velocity fields v, i.e., ∂τ X = v . (2.4) The endpoints of the path line are fixed by δX(a,tinit)= δX(a,tfin)= 0 . (2.5) 3 The volume element in the Lagrangian coordinates can be given by the determinant of the Jacobian matrix, ∂(X , X , X ) J(a,τ) ≡ 1 2 3 . (2.6) ∂(a1, a2, a3) We assume that J has no singular points in the space and time considered and that J(a,tinit) is unity. The conservation law of mass is given by ρ(a,τ)J(a,τ) − ρinit(a) = 0 , (2.7) where ρinit(a) is the initial value of mass density. The adiabatic condition implies that the entropy of a fluid particle in the perfect fluid is conserved along a path line, s(a,τ) − sinit(a) = 0 , (2.8) where sinit(a) is the initial value of entropy density per mass. Equations (2.7) and (2.8) give holonomic constraints. Using undetermined multipliers, γ, K and Λ, we can define the action as tfin 3 dτ d a {JL(ρ, s, v)+γ · (∂τ X − v)+K(ρJ − ρinit)+ΛJ(s − sinit)} , (2.9) Ztinit ZV which is a functional of γ, K, Λ, ρ, s, v and X. Taking the variations of Eq. (2.9) with respect to these trajectories, respectively, leads to Eqs. (2.4), (2.7), (2.8), 1 p K = − v2 + ǫ + , (2.10) 2 ρ Λ = ρT , (2.11) γ = ρJv (2.12) and ∂ ∂ 1 ∂J γ = − ρ v2 − ǫ + K . (2.13) ∂τ i ∂a 2 ∂(∂X /∂a ) j i j Roman indices run from 1 to 3, and repeated indices are summed up. The sur- face integral terms, appearing on the way of this calculation, vanish because of the boundary conditions, Eqs. (2.1) and (2.5). Calculating by means of the cofactors yields ∂J ∂a = J j , (2.14) ∂(∂Xi/∂aj) ∂Xi while some algebra yields ∂ ∂a J j = 0 . (2.15) ∂a ∂X j 1 Thus, we can rewrite Eq. (2.13) as ∂ ∂ 1 γ = −J ρ v2 − ǫ + K . (2.16) ∂τ i ∂X 2 i 4 Substituting Eqs. (2.10) and (2.12) into Eq. (2.16), we successfully obtain the Euler equation in the Lagrangian description, ∂v ∂p ρ i = − . (2.17) ∂τ ∂Xi We can replace ∂vi/∂τ by (∂t + v ·∇)vi to obtain the Euler equation in the Eulerian description, which is given by Eq. (2.29) below.3), 4) In the Eulerian description, a fluid particle is labeled by the Lagrangian co- ordinates A ≡ (A1, A2, A3), which depend on the spatial position x. Since the Lagrangian coordinates are conserved along the path line, we have ∂ A = −v ·∇A . (2.18) ∂t i i The endpoints of a path line are fixed by δA(x,tinit)= δA(x,tfin)= 0 , (2.19) instead of Eq. (2.5), as discussed in Ref. 5). The mass conservation law Eq. (2.7) can be rewritten as −1 ρ(x,t)= J (x,t)ρinit(A(x,t)) , (2.20) where the inverse of J is given by − ∂(A , A , A ) J 1(x,t)= 1 2 3 . (2.21) ∂(x1,x2,x3) The action in the Lagrangian description, Eq. (2.9), is rewritten in the Eulerian description as tfin 3 −1 dt d x L(ρ, s, v)+βi(∂tAi + v ·∇Ai)+ K(ρ − ρinitJ )+ Λ(s − sinit) , Ztinit ZV (2 .22) which is a functional of β, K, Λ, ρ, s, v and A. The stationary condition of Eq. (2.22) yields Eqs. (2.8), (2.10), (2.11), (2.18), (2.20), ρv + βi∇Ai = 0 (2.23) and −1 ∂ − ∂ρ ∂ ∂J ∂s β = −∇ · (β v) − KJ 1 init + Kρ − Λ init . (2.24) ∂t i i ∂A ∂x init ∂(∂A /∂x ) ∂A i j i j i Here, ρinit and sinit depend on A, but ρ and s are independent variables. Using Eqs. (2.8), (2.10), (2.11), −1 ∂J − ∂x = J 1 j (2.25) ∂(∂Ai/∂xj) ∂Ai and ∂ − ∂x J 1 j = 0 , (2.26) ∂x ∂A j i 5 we can rewrite Eq. (2.24) as ∂ ∂K ∂s ∂x β = −∇ · (β v)+ ρ − ρT j ∂t i i ∂x ∂x ∂A j j i ρ ∂v2 ∂p ∂x = −∇ · (β v)+ − + j .