🧄Fluid Mechanics Integrals for Mass and Motion

The solutions demonstrate how fundamental physical quantities in a continuous fluid are determined by integrating their respective densities over a given volume V. In all cases, the mass density $\rho(x)$ is crucial as it scales the quantity per unit mass to the quantity per unit volume, $d V$ . The total kinetic energy is a scalar, found by integrating the kinetic energy density $\frac{1}{2} \rho|v|^2$. In contrast, both the total momentum and total angular momentum are vector quantities. Total momentum is the integral of the linear momentum density $\rho v$. Total angular momentum, which must be defined relative to a specific reference point $x_0$, is the integral of the angular momentum density $\rho\left(x-x_0\right) \times v$.

Fluid Mechanics Integrals for Mass and Motion | Proof and Derivationviadean on Notion