🔎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.

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Fluid Momentum and the Continuity Equation-Derivation of the Cauchy and Navier-Stokes Equations | Condensed Notesviadean on Notion