What are good books on entropy transport in fluid mechanics?

Question

Most fluid mechanics textbooks deal with mass, momentum and energy transport in fluid flows. Are there any books that deal with entropy transport in fluid flows?

Answer 1

Differential equations

From the principle differential equations in form:
$D_t \rho = -\rho \nabla \cdot \mathbf{u} \qquad \qquad \qquad \qquad \qquad$ (mass)
$\rho D_t \mathbf{u} = \rho \mathbf{g} + \nabla \cdot \mathbb{T} \qquad \qquad \qquad \qquad \ $ (mometum)
$\rho D_t e^{t} = \rho \mathbf{g} \cdot \mathbf{u} + \nabla \cdot ( \mathbb{T} \cdot \mathbf{u}) -\nabla \cdot \mathbf{q}\quad \ \ $(total energy)

it's possible to write some balance equations for derived physical quantities, like kinetic energy, internal energy and the entropy:

• kinetic energy, doing $\mathbf{u} \cdot$ the momentum equation, to get a balance for the kinetic energy $k = \frac{|\mathbf{u}|^2}{2}$:
  $\rho D_t k = \rho \mathbf{g} \cdot \mathbf{u} + \nabla \cdot \mathbb{T} \cdot \mathbf{u}$;

• internal energy, $e = e^{t} - k$, subtracting the kinetice energy equation from the total energy:
  $\rho D_t e = \nabla \mathbf{u} : \mathbb{T} - \nabla \cdot \mathbf{q}$

• entropy, putting together the internal energy and the reversible contribution of the internal energy, $\nabla \mathbf{u} \cdot \mathbb{T} = \nabla \mathbf{u} : \left( -p \mathbb{I} + 2\mu \mathbb{D} + \lambda (\nabla \cdot \mathbf{u} ) \mathbb{I} \right) = - p \nabla \cdot \mathbf{u} + 2 \mu |\mathbb{D}|^2 + \lambda (\nabla \cdot \mathbf{u})^2$,

  so that

  $\rho D_t e + p \nabla \cdot \mathbf{u} = 2 \mu |\mathbb{D}|^2 + \lambda (\nabla \cdot \mathbf{u})^2 - \nabla \cdot \mathbf{q}$
  $\rho T D_t s = 2 \mu |\mathbb{D}|^2 + \lambda (\nabla \cdot \mathbf{u})^2 - \nabla \cdot \mathbf{q}$

  to get:

  $\rho D_t s = \dfrac{2 \mu |\mathbb{D}|^2 + \lambda (\nabla \cdot \mathbf{u})^2}{T} - \dfrac{\nabla \cdot \mathbf{q}}{T} = \\ \qquad \ = \dfrac{2 \mu |\mathbb{D}|^2 + \lambda (\nabla \cdot \mathbf{u})^2}{T} - \dfrac{\mathbf{q} \cdot \nabla T}{T^2} - \nabla \cdot \left(\dfrac{\mathbf{q}}{T}\right)$

  and if we add the Fourier assumption on the heat conduction flux $\mathbf{q} = -k \nabla T$, we can recast the last term as

  $\rho D_t s = \dfrac{2 \mu |\mathbb{D}|^2 + \lambda (\nabla \cdot \mathbf{u})^2}{T} + k\dfrac{|\nabla T|^2}{T^2} + \nabla \cdot \left( \dfrac{ k \nabla T}{T} \right)$

  where we can recognize two contributions at the right-hand side, namely:

  • a volume entropy source,

    $\dfrac{2 \mu |\mathbb{D}|^2 + \lambda (\nabla \cdot \mathbf{u})^2}{T} - \dfrac{\mathbf{q} \cdot \nabla T}{T^2} = \dfrac{2 \mu |\mathbb{D}|^2 + \lambda (\nabla \cdot \mathbf{u})^2}{T} + k\dfrac{|\nabla T|^2}{T^2}$

    always positive, because of the norms, of the positive values of absolute temperature and the constraints on the viscosity coefficients and the conductivity.

  • a flux contribution, that should remind you the contribution in Clausius formulation of the second principle of thermodyamics, $dS \ge \frac{\delta Q}{T}$,

    $ - \nabla \cdot \left(\dfrac{\mathbf{q}}{T}\right) = \nabla \cdot \left( \dfrac{ k \nabla T}{T} \right)$

Integral equations

Integrating over a material volume, we get the integral balance for a closed system,

$\dfrac{d}{dt} \displaystyle \int_{V_t} \rho s = \int_{V_t} \dfrac{2 \mu |\mathbb{D}|^2 + \lambda (\nabla \cdot \mathbf{u})^2}{T} + k\dfrac{|\nabla T|^2}{T^2} - \oint_{\partial V_t } \dfrac{\mathbf{q}}{T} \cdot \mathbf{\hat{n}} $

As you can see, there is a non-negative volume source due to non-rigid motion in viscous fluids, and due to heat transfer between different regions of the domain: viscosity and heat transfer are causes of irreversibility; the other term is the entropy flux through the boundary of the domain, and reminds the expression of Clausius statement of the second principle of thermodynamics.

Exploiting Reynolds' transport theorem we get the balance for an arbitrary domain,

$\dfrac{d}{dt} \displaystyle \int_{v_t} \rho s + \oint_{\partial v_t} \rho s (\mathbf{u} - \mathbf{u}_s) \cdot \mathbf{\hat{n}} = \int_{v_t} \dfrac{2 \mu |\mathbb{D}|^2 + \lambda (\nabla \cdot \mathbf{u})^2}{T} + k\dfrac{|\nabla T|^2}{T^2} - \oint_{\partial v_t } \dfrac{\mathbf{q}}{T} \cdot \mathbf{\hat{n}} $.

where some flux of entropy is associated with the flux of mass through the boundaries of the domain.

Answer 2

For an in-depth resource on entropy and fluid dynamics, you may want to look at Evans' Entropy and PDEs (NB: PDF link to author's website), specifically Chapter 5 (starting on page 106). The text itself is written in a more mathematically formal manner than one might find in a physics textbook, though it is done in way that should be understood by anyone with vector calculus under their tool belt.