关于Chorin《流体力学数学导论》等熵流动欧拉方程推导的疑问

Great question—let's walk through this derivation step by step, and break down why each part holds, especially since Chorin's text takes a pretty standard (but rigorous) approach to connecting energy conservation to the Euler equations for isentropic flow.

First, let's anchor ourselves to the starting point you cited—this is the integral form of the energy conservation equation for a material volume $W_t$ (a region of fluid moving with the flow):

$$\frac{d}{dt}\int_{W_t} \left(\frac{1}{2} \rho ||\vec{u}||^2 + \rho \epsilon \right) dV = -\int_{\partial W_t} p \vec{u}\cdot \vec{n} dA + \int_{W_t}\rho \vec{u}\cdot \vec{b}dV $$

1. Apply the Reynolds Transport Theorem to the Left-Hand Side (LHS) #

Since $W_t$ is a material volume (it moves with the fluid particles), we use the Reynolds Transport Theorem to convert the time derivative of a moving integral into a volume integral of local derivatives:
$$\frac{d}{dt}\int_{W_t} f dV = \int_{W_t} \frac{\partial f}{\partial t} + \nabla \cdot (f \vec{u}) dV$$
Here, $f = \frac{1}{2}\rho ||\vec{u}||^2 + \rho \epsilon$ (kinetic energy per unit volume plus internal energy per unit volume). Expanding this gives:
$$\text{LHS} = \int_{W_t} \frac{\partial}{\partial t}\left(\frac{1}{2}\rho u^2 + \rho \epsilon\right) + \nabla \cdot \left[\left(\frac{1}{2}\rho u^2 + \rho \epsilon\right)\vec{u}\right] dV$$

2. Convert the Surface Integral to a Volume Integral (Gauss Divergence Theorem) #

For the pressure term on the right-hand side (RHS), we use the Gauss Divergence Theorem to turn the surface integral over $\partial W_t$ into a volume integral over $W_t$:
$$\int_{\partial W_t} p \vec{u} \cdot \vec{n} dA = \int_{W_t} \nabla \cdot (p \vec{u}) dV$$
Substituting this back into the original equation gives an integral equation where all terms are volume integrals:
$$\int_{W_t} \left[ \frac{\partial}{\partial t}\left(\frac{1}{2}\rho u^2 + \rho \epsilon\right) + \nabla \cdot \left(\left(\frac{1}{2}\rho u^2 + \rho \epsilon\right)\vec{u}\right) + \nabla \cdot (p \vec{u}) - \rho \vec{u} \cdot \vec{b} \right] dV = 0$$

3. Localize the Equation (Set the Integrand to Zero) #

Since this integral holds for any arbitrary material volume $W_t$, the integrand must be zero everywhere (this is the "localization principle," valid for smooth solutions with no singularities):
$$\frac{\partial}{\partial t}\left(\frac{1}{2}\rho u^2 + \rho \epsilon\right) + \nabla \cdot \left(\left(\frac{1}{2}\rho u^2 + \rho \epsilon\right)\vec{u}\right) + \nabla \cdot (p \vec{u}) = \rho \vec{u} \cdot \vec{b}$$

4. Expand and Simplify Using the Continuity Equation #

Now expand all terms, and group terms involving the continuity equation ($\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0$, mass conservation):

• The time derivative and divergence terms will combine to give terms like $\left(\frac{1}{2}u^2 + \epsilon\right)\left(\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u})\right)$, which vanish entirely because of the continuity equation.
• What's left are terms involving the velocity and thermodynamic quantities:
  $$\rho \vec{u} \cdot \frac{\partial \vec{u}}{\partial t} + \rho \frac{D\epsilon}{Dt} + \rho \vec{u} \cdot \nabla\left(\frac{1}{2}u^2\right) + p \nabla \cdot \vec{u} + \vec{u} \cdot \nabla p = \rho \vec{u} \cdot \vec{b}$$
  Here, $\frac{D\epsilon}{Dt} = \frac{\partial \epsilon}{\partial t} + \vec{u} \cdot \nabla \epsilon$ is the material derivative of internal energy (tracking changes for a moving fluid particle).

5. Use Isentropic Thermodynamic Relations #

For isentropic flow (reversible, adiabatic, no dissipation), the thermodynamic relation simplifies to:
$$T dS = d\epsilon + p d\left(\frac{1}{\rho}\right) = 0$$
Taking the material derivative of this gives $\frac{D\epsilon}{Dt} = \frac{p}{\rho} \nabla \cdot \vec{u}$. Substituting this into our equation cancels the $p \nabla \cdot \vec{u}$ term:
$$\rho \vec{u} \cdot \frac{\partial \vec{u}}{\partial t} + \rho \vec{u} \cdot \nabla\left(\frac{1}{2}u^2\right) + \vec{u} \cdot \nabla p = \rho \vec{u} \cdot \vec{b}$$

6. Apply Vector Identities and Rearrange #

Divide through by $\rho$ (valid since fluid density is non-zero), and use the vector identity:
$$\nabla\left(\frac{1}{2}u^2\right) = (\vec{u} \cdot \nabla)\vec{u} + \vec{u} \times (\nabla \times \vec{u})$$
Substituting this in, we get:
$$\vec{u} \cdot \left( \frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\vec{u} + \vec{u} \times (\nabla \times \vec{u}) + \frac{1}{\rho}\nabla p - \vec{b} \right) = 0$$

Now, Chorin's final result is:
$$\frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\vec{u} = -\nabla \omega + \vec{b}$$
This implies that $\nabla \omega = \frac{1}{\rho}\nabla p - \vec{u} \times (\nabla \times \vec{u})$. In isentropic flow, $\frac{1}{\rho}\nabla p = \nabla h$ where $h = \epsilon + \frac{p}{\rho}$ is the specific enthalpy. If the flow is irrotational ($\nabla \times \vec{u} = 0$), then $\nabla \omega = \nabla h$, so $\omega$ is just the specific enthalpy (up to a constant). Check Chorin's symbol glossary to confirm this definition!

Let's break down the key assumptions that make each step rigorous:

• Reynolds Transport Theorem: Requires the velocity field to be smooth enough that the material volume's boundary moves continuously—standard for classical fluid mechanics.
• Gauss Divergence Theorem: Requires pressure $p$ and velocity $\vec{u}$ to be continuously differentiable on $W_t$ and its boundary—again, a standard assumption for inviscid flow.
• Localization Principle: Only holds for smooth solutions with no discontinuities (e.g., shocks). If shocks are present, we'd need to use weak solutions instead, but Chorin's text focuses on smooth flows here.
• Isentropic Assumption: This is critical—we're assuming no heat transfer, no viscous dissipation, and reversible flow. If any of these fail, we'd need to add terms for heat flux or viscous stress to the energy equation.
• Continuity Equation: Mass conservation is a fundamental law of fluid mechanics, so using it to simplify terms is always valid (as long as we're dealing with a single-phase fluid with no mass sources/sinks).

内容的提问来源于stack exchange，提问作者Aiman Al-Eryani