**3.2 Shear viscosity as a consequence of the angular momentum relaxation for the hydrodynamical description of continuum mechanics**

As shown in the previous section, it is possible to derive the system of hydrodynamical equations on the GVP basis for viscous, compressible fluid in the form of Navier-Stokes equations. However for the account of terms responsible for viscosity it is required to introduce some tensor internal parameter *ik* ξ in agreement with Mandelshtam-Leontovich approach (Mandelshtam & Leontovich, 1937). Relaxation of this internal parameter provides appearance of viscous terms in the Navier-Stokes equation. It is worth mentioning that the developed approach allowed to generalize the Navier-Stokes equation with constant viscosity coefficient to more general case accounting for viscosity relaxation in analogy to the Maxwell's model (Landau & Lifshitz, 1972). However the physical interpretation of the tensor internal parameter, which should be enough universal due to general character of the Navier-Stokes equation, requires more clear understanding. On the intuition level it is clear that corresponding internal parameter should be related with neighbor order in atoms and molecules placement and their relaxation. In the present section such physical interpretation is represented.

As was mentioned in Introduction the system of hydrodynamical equations in the form of Navier-Stokes is usually derived on the basis of conservation laws of mass *M* , momentum *P* and energy *E* . The correctness of equations of the traditional hydrodynamics is confirmed by the large number of experiments where it is adequate. However the conservation law of angular momentum *M* is absent among the mentioned balance laws laying in the basis of traditional hydrodynamics. In this connection it is interesting to understand the role of conservation law of angular momentum *M* in hydrodynamical description. It is worth mentioning that equation for angular momentum appeared in hydrodynamics early (Sorokin, 1943; Shliomis, 1966) and arises and develops in the momentum elasticity theory. The Cosserat continuum is an example of such description (Kunin, 1975; Novatsky, 1975; Erofeev, 1998). However some internal microstructure of medium is required for application of such approach.

In the hydrodynamical description as a partial case of continuum mechanics the definition of material point is introduced as sifficiently large ensemble of structural elements of medium (atoms and molecules) that on one hand one has to describe properties of this ensemble in statistical way and on the other one has to consider the size of material point as small in comparison with specific scales of the problem. A material point itself as closed ensemble of particles possesses the following integrals of motion: mass, momentum, energy and angular momentum.

The basic independent variables, in terms of which the hydrodynamical description should be constructed, are the values which can be determined for separate material point in

Generalized Variational Principle for Dissipative Hydrodynamics: Shear Viscosity

ϕ

Its solution can be represented in the form:

ρ

For the case of large times / 1 *t*

condition has to be satisfied

computing it by parts

viscosity.

representation ( *t* →

The zeros of the denominator

Substitution (47a) in (44c) leads to the following result

σ γ

 λμ

contribution and equation reduces to the form

ρ

ρ

ω)  λμ

The corresponding estimation for the large time limit / *t* >>

ρ

 λμ

ϕ

1,2

ω

equation (45c) is also local in space and it can be resolved for the function

 λμ ϕ

δ

γ

 μ −∞

from Angular Momentum Relaxation in the Hydrodynamical Description of Continuum Mechanics 47

 δ

 γ

*t t t dt e u* σ

γ

( 2 ) ( ) [ [ ]] [ [ ]]

*u uu dt e u*

<sup>2</sup> *u u uu* ( 2 ) ( ) [ [ ]] [ ] δ

By the reason that the medium at large times should behave like a fluid then the following

2 0

δ

σ

Taking into account condition (49) let's make more accurate estimation of the integral,

( 2)( ) [ [ ]]

−∞

 γ δ

<sup>2</sup> [ ] *u*

( ) <sup>2</sup>

σ

<sup>1</sup> <sup>4</sup>

*i I I*

 = −± − γ γ

*u udt e u*

δ

σ

<sup>2</sup> *u uu* ( 2 ) ( ) [ [ ]]

which coincides with the structure of Navier-Stokes equation in the presence of shear

Let's consider the case with non zero moment of inertia 0 *I* ≠ . For this case the second

δ

<sup>−</sup> = ∇ −+ +

*I i*

ω ωγ σ

2

μ

 − + ∇∇ + − ∇∇ = ∇ 

σ

 μ − − ′

σ

γ

 ϕ [ ] *u*

( ) [ ]

> δ

> > γ

−∞ − + ∇∇ + ∇∇ =− ′ ∇ ∇ (48a)

2 2

2 ( )

2

μ

*t t t*

σ

− − ′

− + ∇∇ =− ′ ∇ ∇ (48c)

γ σ

− + ∇∇ = ∇∇ (48d)

(50)

γ

(48b)

=− − ∇ (46)

= − ′ <sup>∇</sup> (47a)

2 ( )

*t t t*

σ

− − ′

>> the upper limit of integration gives the principal

 δ

σ

− = (49)

reduces to the equation

ϕ

(51)

in Fourier

γ

γ

accordance with its integrals of motion: mean mass displacement vector *u* (velocity of this displacement / *v ut* =∂ ∂ is determined by integrals of motion / *vPM* <sup>=</sup> ), rotation angle ϕ (angular velocity of rotation Ω = ϕ is determined by integrals of motion / Ω = *<sup>M</sup> <sup>I</sup>* , where *I* - inertia moment) and heat displacement *uT* , determining variation of temperature and related with integral of energy *E* .

In accordance with the set of independent field variables we can represent the kinetic *K* and the free *F* energies as corresponding quadratic forms

$$\mathbf{\hat{ZK}} = \rho \dot{\bar{u}}^2 + \mathbf{I} \dot{\bar{\phi}}^2 \tag{41}$$

$$2\mathcal{F} = (\mathcal{X} + 2\mu)(\nabla \bar{u})^2 + \mu[\nabla \bar{u}]^2 + 2\delta \bar{\phi}[\nabla \bar{u}] + \sigma(\bar{\phi})^2 + \varepsilon(\nabla \bar{\phi})^2 + \xi[\nabla \bar{\phi}]^2 \tag{42}$$

Taking into account that the dissipation dealt only with field of micro rotations, and omitting for shortness dissipation of mean displacement field, described by heat conductivity, we can write the dissipation function in the following form

$$
\mathcal{D}D = \mathcal{\mathcal{Y}} \dot{\mathfrak{P}}^2 \tag{43}
$$

Equations of motion derived from GVP without temperature terms have the forms:

$$\frac{d}{dt}\frac{\partial \mathcal{K}}{\partial \dot{\vec{u}}} - \nabla \frac{\partial F}{\partial \nabla \vec{u}} - [\nabla \frac{\partial F}{\partial [\nabla \vec{u}]}] = -\frac{\partial D}{\partial \dot{\vec{u}}}\tag{44a}$$

$$\frac{d}{dt}\frac{\partial \mathcal{K}}{\partial \dot{\vec{\phi}}} + \frac{\partial \mathcal{K}}{\partial \vec{\phi}} - \nabla \frac{\partial F}{\partial \nabla \vec{\phi}} - [\nabla \frac{\partial F}{\partial [\nabla \vec{\phi}]}] = -\frac{\partial D}{\partial \dot{\vec{\phi}}}\tag{45a}$$

Without dissipation 0 β = the motion equations obtained with use of quadratic forms (41)- (43) correspond to the ones for Cosserat continuum (Kunin, 1975; Novatsky, 1975; Erofeev, 1998). Indeed for this case the equations (44) have forms:

$$
\rho \frac{d}{dt} \dot{\bar{u}} - (\dot{\mathcal{A}} + 2\mu) \nabla(\nabla \bar{u}) + \mu [\nabla[\nabla \bar{u}]] - \delta[\nabla \bar{\phi}] = 0 \tag{44b}
$$

$$\mathrm{i}\hbar\frac{d}{dt}\dot{\phi} - \varepsilon\nabla(\nabla\bar{\phi}) + \mathcal{J}[\nabla[\nabla\bar{\phi}]] + \sigma\bar{\phi} + \mathcal{J}[\nabla\bar{u}] = 0\tag{45b}$$

The explicit form of these equations confirms that they are indeed the Cosserat continuum. If one sets formally 0 δ = , then equations (44b) and (45b) are split and the equation (44b) reduces to ordinal equation of the elasticity theory and the equation (45b) represents the wave equation for angular momentum.

When dissipation exists the system of equations (44)-(45) contains additional terms responsible for this dissipation

$$
\rho \ddot{\vec{\mu}} - (\dot{\lambda} + 2\mu) \nabla(\nabla \vec{\mu}) + \mu [\nabla[\nabla \vec{\mu}]] - \delta [\nabla \vec{\phi}] = 0 \tag{44c}
$$

$$I\ddot{\vec{\phi}} - \varepsilon \nabla(\nabla \vec{\phi}) + \zeta[\nabla[\nabla \vec{\phi}]] + \sigma \vec{\phi} + \delta[\nabla \vec{u}] = -\gamma \dot{\vec{\phi}}\tag{45c}$$

For the case 0 ε = , 0 ς = and 0 *I* = the second equation (45c) reduces to the pure relaxation form:

$$
\dot{\vec{\varphi}} = -\frac{\sigma}{\gamma} \vec{\varphi} - \frac{\delta}{\gamma} [\nabla \vec{u}] \tag{46}
$$

Its solution can be represented in the form:

46 Hydrodynamics – Advanced Topics

is determined by integrals of motion / Ω = *<sup>M</sup> <sup>I</sup>* , where

ε ϕ

∂ ∂ ∂∇ ∂ ∇ (44a)

 ϕ

∂ ∂ ∂ ∂∇ ∂ ∇ (45a)

(42)

, determining variation of temperature and

(41)

 ςϕ

(43)

displacement / *v ut* =∂ ∂ is determined by integrals of motion / *vPM* <sup>=</sup> ), rotation angle

In accordance with the set of independent field variables we can represent the kinetic *K* and

2 2 2*K uI* = + ρ

2 2 <sup>222</sup> 2 ( 2 )( ) [ ] 2 [ ] ( ) ( ) [ ] *F uu u* = + ∇ + ∇ + ∇+ +∇ +∇

δϕ

Taking into account that the dissipation dealt only with field of micro rotations, and omitting for shortness dissipation of mean displacement field, described by heat

> <sup>2</sup> 2*D* = γϕ

*dK F F D dt u u u u* ∂∂ ∂ ∂ −∇ − ∇ =−

*dK K F F D*

∂∂ ∂ ∂ ∂ + −∇ − ∇ =−

(43) correspond to the ones for Cosserat continuum (Kunin, 1975; Novatsky, 1975; Erofeev,

( 2 ) ( ) [ [ ]] [ ] 0 *<sup>d</sup>*

( ) [ [ ]] [ ] 0 *<sup>d</sup> I u*

The explicit form of these equations confirms that they are indeed the Cosserat continuum.

reduces to ordinal equation of the elasticity theory and the equation (45b) represents the

When dissipation exists the system of equations (44)-(45) contains additional terms

 μ

> σϕ δ

− ∇∇ + ∇∇ + + ∇ =−

 ϕ

 μ

 ϕ σϕ δ− ∇∇ + ∇∇ + + ∇ = (45b)

Equations of motion derived from GVP without temperature terms have the forms:

ϕϕ

*u uu*

[ ] [ ]

[ ] [ ]

 ϕ

= the motion equations obtained with use of quadratic forms (41)-

 δϕ− + ∇∇ + ∇∇ − ∇ = (44b)

= , then equations (44b) and (45b) are split and the equation (44b)

 δϕ*u uu* − + ∇∇ + ∇∇ − ∇ = ( 2 ) ( ) [ [ ]] [ ] 0 (44c)

= and 0 *I* = the second equation (45c) reduces to the pure relaxation

(45c)

γϕ

 ϕ

> σ ϕ

(velocity of this

ϕ

accordance with its integrals of motion: mean mass displacement vector *u*

ϕ

> μ

conductivity, we can write the dissipation function in the following form

(angular velocity of rotation Ω =

related with integral of energy *E* .

Without dissipation 0

If one sets formally 0

For the case 0

form:

ε

 = , 0 ς

responsible for this dissipation

*I* - inertia moment) and heat displacement *uT*

λ

the free *F* energies as corresponding quadratic forms

μ

*dt* ϕ

1998). Indeed for this case the equations (44) have forms:

 λ μ

*dt* ρ

> *dt*ϕ ε ϕ ς

ρ

 λ μ

 ( ) [ [ ]] [ ] *I u* ϕ

ϕς

ε

β

δ

wave equation for angular momentum.

$$\vec{\varphi} = -\frac{\delta}{\mathcal{Y}} \int\_{-\infty}^{t} dt' e^{-\frac{\mathcal{F}}{\mathcal{Y}}(t-t')} [\nabla \vec{u}] \tag{47a}$$

Substitution (47a) in (44c) leads to the following result

$$
\rho \ddot{\vec{u}} - (\lambda + 2\mu) \nabla(\nabla \vec{u}) + \mu[\nabla[\nabla \vec{u}]] = -\frac{\delta^2}{\mathcal{T}} \int\_{-\infty}^t dt' e^{-\frac{\sigma}{\mathcal{T}}(t - t')} [\nabla[\nabla \vec{u}]] \tag{48a}
$$

For the case of large times / 1 *t*σ γ >> the upper limit of integration gives the principal contribution and equation reduces to the form

$$
\rho \ddot{\vec{u}} - (\dot{\lambda} + 2\mu) \nabla(\nabla \vec{u}) + \left(\mu - \frac{\delta^2}{\sigma}\right) [\nabla[\nabla \vec{u}]] = \gamma \frac{\delta^2}{\sigma^2} [\nabla \dot{\vec{u}}] \tag{48b}
$$

By the reason that the medium at large times should behave like a fluid then the following condition has to be satisfied

$$
\mu - \frac{\delta^2}{\sigma} = 0 \tag{49}
$$

Taking into account condition (49) let's make more accurate estimation of the integral, computing it by parts

$$
\rho \ddot{\vec{u}} - (\lambda + 2\mu) \nabla(\nabla \vec{u}) = -\frac{\delta^2}{\sigma} \int\_{\to} dt' e^{-\frac{\sigma}{\mathcal{T}}(t - t')} [\nabla[\nabla \dot{\vec{u}}]] \tag{48c}
$$

The corresponding estimation for the large time limit / *t* >> γ σreduces to the equation

$$
\rho \ddot{\vec{u}} - (\mathcal{A} + 2\mu) \nabla(\nabla \bar{u}) = \mathcal{\gamma} \frac{\mu^2}{\mathcal{\delta}^2} [\nabla[\nabla \dot{\bar{u}}]] \tag{48d}
$$

which coincides with the structure of Navier-Stokes equation in the presence of shear viscosity.

Let's consider the case with non zero moment of inertia 0 *I* ≠ . For this case the second equation (45c) is also local in space and it can be resolved for the function ϕ in Fourier representation ( *t* → ω)

$$\vec{\phi} = \frac{-\delta}{-I\alpha^2 + i\alpha\gamma + \sigma} [\nabla \vec{u}] \tag{50}$$

The zeros of the denominator

$$
tau\_{1,2} = \frac{1}{2I} \left( -\gamma \pm \sqrt{\gamma^2 - 4\sigma I} \right) \tag{51}$$

Generalized Variational Principle for Dissipative Hydrodynamics: Shear Viscosity

with the Navier – Stokes equation system.

chapter.

points on the kinetic level.

**5. Acknowledgment** 

978-3-540-88466-8, Berlin.

*Fluctuations*, Wiley, New York.

Berlin, Springer-Verlag.

**6. References** 

from Angular Momentum Relaxation in the Hydrodynamical Description of Continuum Mechanics 49

forms of time or space derivatives of the mean mass and the heat displacement fields. The generalized system of hydrodynamical equations is then evaluated on the basis of the GVP. At low frequencies this system corresponds to the traditional Navier – Stokes equation system. It allowed us to determine all coefficients of quadratic forms by direct comparison

The second part of the chapter is devoted to consistent introduction of viscous terms into the equation of fluid motion on the basis of the GVP. A tensor internal parameter is used for description of relaxation processes in vicinity of quasi-equilibrium state by analogy with the Mandelshtam – Leontovich approach. The derived equation of motion describes the viscosity relaxation phenomenon and generalizes the well known Navier – Stokes equation. At low frequencies the equation of fluid motion reduces exactly to the form of Navier – Stokes equation. Nevertheless there is still a question about physical interpretation of the used internal parameter. The answer on this question is presented in the last section of the

It is shown that the internal parameter responsible for shear viscosity can be interpreted as a consequence of relaxation of angular momentum of material points constituting a mechanical continuum. Due to angular momentum balance law the rotational degree of freedom as independent variable appears additionally to the mean mass displacement field. For the dissipationless case this approach leads to the well-known Cosserat continuum. When dissipation prevails over momentum of inertion this approach describes local relaxation of angular momentum and corresponds to the sense of the internal parameter. It is important that such principal parameter of Cosserat continuum as the inertia moment of intrinsic microstructure can completely vanish from the description for dissipative continuum. The independent equation of motion for angular momentum in this case reduces to local relaxation and after its substitution into the momentum balance equation leads to the viscous terms in Navier – Stokes equation. Thus, it is shown that the nature of viscosity phenomenon can be interpreted as relaxation of angular momentum of material

The work was supported by ISTC grant 3691 and by RFBR grant №09-02-00927-а.

Biot M. (1970). *Variational principles in heat transfer*. Oxford, University Press.

Kunin I.A. (1975). *Theory of elastic media with micro structure* , Nauka, Moscow.

*Society of America*, Vol.29, pp.204-209, ISSN 0001-4966.

Berdichevsky V.L. (2009). *Variational principles of continuum mechanics*, Springer-Verlag, ISBN

Deresiewicz H. (1957). Plane wave in a thermoplastic solids. *The Journal of the Acoustcal* 

Erofeev V.I. (1998). *Wave processes in solids with microstructure,* Moscow State University, Moscow. Glensdorf P., Prigogine I., (1971). *Thermodynamic Theory of Structure, Stability, and* 

Gyarmati I. (1970). *Non-equilibrium thermodynamics. Field theory and variational principles*.

Landau L.D., Lifshitz E.M. (1986). *Theoretical physics. Vol.6. Hydrodynamics*, Nauka, Moscow.

determine two modes of angular momentum relaxation. Under condition <sup>2</sup> *I* < γ /(4 ) σ both zeros are real and have the following asymptotics for small momentum of inertia 0 *I* → :

$$
\dot{a}a\partial\_1 = -\frac{\sigma}{\mathcal{N}} \qquad \qquad \dot{a}a\_2 = -\frac{\mathcal{N}}{I} \tag{52}
$$

The first zero does not depend on momentum of inertia *I* and the second root goes to infinity when 0 *I* → . Under condition <sup>2</sup> *I* = γ /(4 ) σ the zeros coincide and have the value <sup>1</sup> *i* 2 σ ω γ ≈ − , and under the condition <sup>2</sup> *<sup>I</sup>* <sup>&</sup>gt; γ /(4 ) σ the zeros are complex conjugated with negative real part, which decreases with increase of *I* . The last case corresponds to the resonant relaxation of angular momentum.

In the time representation the solution of the equation (50) can be written in the form

$$\Phi = -\int\_{-\infty}^{t} dt' e^{-\frac{\mathcal{V}}{2I}(t-t')} [\nabla \vec{n}] \left\{ \frac{2\mathcal{S}}{\sqrt{\ldots}} \text{sh} \left( \frac{\sqrt{\ldots}}{2I} (t-t') \right) \right\} \tag{47b}$$

here the notation <sup>2</sup> ... 4 = − γ σ *I* is used. For the case of resonant relaxation <sup>2</sup> *I* > γ /(4 ) σ the corresponding expression has the form

$$\Phi = -\int\_{-\infty}^{t} dt' e^{-\frac{\mathcal{V}}{2I}(t-t')} [\nabla \vec{u}] \left| \frac{2\mathcal{S}}{\sqrt{\|\cdot\|}} \sin\left(\frac{\sqrt{\|\cdot\|}}{2I}(t-t')\right) \right| \tag{47c}$$

Substitution of the explicit expressions (47b) or (47c) in the equation (44c) gives the generalisation of the Navier – Stokes equation for a solid medium with local relaxation of angular momentum. As was mentioned above under special condition (49) and in the limiting case (52) this equation reduces exactly to the form of Navier – Stokes equation.

Thus, it is shown that relaxation of angular momentum of material points consisting a continuum can be considered as physical reason for appearance of terms with shear viscosity in Navier-Stokes equation. Without dissipation additional degree of freedom dealt with angular momentum leads to the well known Cosserat continuum.
