Fluid Momentum and the Continuity Equation-Derivation of the Cauchy and Navier-Stokes Equations

The complex topic of fluid dynamics is fundamentally rooted in the continuity equation, which, when applied to momentum density ( $\rho v$ ), leads to the derivation of the governing differential equations for fluid flow. The resulting Cauchy momentum equation expresses that the material acceleration of a fluid element (represented by the material derivative $\frac{D v}{D t}$ ) is balanced by external body forces (like gravity) and internal forces stemming from the stress tensor ( $\sigma_{i j}$ ). The nature of this stress determines the specific flow model: for an inviscid fluid (zero shear stress), the stress simplifies to just pressure, yielding the Euler equations; however, for real, viscous fluids, the stress must be modeled as linearly dependent on the strain rate tensor, incorporating viscosity ( $\mu$ ) and leading to the comprehensive Navier-Stokes equations. Mathematically, solving these equations is often simplified using the superposition principle for linear systems, which allows complex problems with inhomogeneous boundary or source conditions to be split into easier-to-solve homogeneous and non-homogeneous sub-problems, such as separating the transient flow from the steady-state solution.

Key takeaways

The Cauchy Momentum Equation
- Derivation Foundation: The Cauchy momentum equations are derived by applying the continuity equation to momentum density ( $P=\rho v$ ) and accounting for the momentum current density ( $j_{i j}$ ).
- Total Momentum Current $\left(j_{i j}\right)$ : This current is the sum of convective transport $\left(\rho v_i v_j\right)$ and the contribution from surface forces, which is defined by the stress tensor $\left(\sigma_{i j}\right)$.
$$ j_{i j}=\rho v_i v_j-\sigma_{i j} $$
- Final Form: The Cauchy equation relates the material acceleration of the fluid to external body forces ( $g$ ) and internal stresses $\left(\sigma_{i j}\right)$ :
$$ \frac{D v_i}{D t}=g_i+\frac{1}{\rho} \partial_j \sigma_{i j} $$
The Material Derivative
- The Material Derivative ( $D / D t$ ) represents the rate of change of a property (like velocity $v_i$ ) for a specific point moving with the fluid. It links changes in time ( $\partial / \partial t$ ) and changes in space due to advection/convection $(v \cdot \nabla)$.
$$ \frac{D f}{D t}=\frac{\partial f}{\partial t}+(v \cdot \nabla) f $$
Euler and Navier-Stokes Equations
- Euler Equations (Inviscid Flow): These apply to inviscid fluids (where $\sigma_{i j}$ is purely isotropic and equals $-p \delta_{i j}$ ). They describe the motion under the influence of pressure and body forces only.
$$ \frac{\partial v}{\partial t}+(v \cdot \nabla) v=-\frac{1}{\rho} \nabla p+g $$
- Navier-Stokes Equations (Viscous Flow): These apply to viscous fluids and require modeling the stress tensor $\sigma_{i j}$ as linearly dependent on the strain rate tensor $\left(\varepsilon_{i j}\right)$, using the dynamic viscosity ( $\mu$ ) and bulk viscosity ( $\lambda$ ). This adds terms representing viscous dissipation to the Cauchy equation.
$$ \rho \frac{D v}{D t}=\rho g-\nabla \tilde{p}+\mu \nabla^2 v+\frac{\mu}{3} \nabla(\nabla \cdot v) $$
- Incompressible Flow: The condition for incompressible flow is that the volume of a material element does not change with time, which simplifies to the solenoidal condition on the velocity field: $\nabla \cdot v = 0$.
Superposition Principle in PDEs
- The Superposition Principle applies to linear differential equations and allows solutions to be added together. If $u_1$ solves a problem with inhomogeneity $f_1$ and $u_2$ with $f_2$, then $u_1+u_2$ solves the problem with $f_1+f_2$.
- Practical Use: This principle is used to simplify complex problems by:
  - Homogenizing Boundary Conditions (BCs): By setting $u=v+u_0$, where $u_0$ satisfies the non-homogeneous BCs. The resulting problem for $v$ has homogeneous BCs but a modified PDE inhomogeneity.
  - Separating Steady-State Solutions: By setting $u=v+u_0$, where $u_0$ is the stationary/steady-state solution. The new problem for $v$ becomes homogeneous in both the PDE and BCs, leaving only the Initial Condition (IC) as non-homogeneous.

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