Chapter 40: Flow of Dry Water – Hydrostatics & Circulation

Flow of Dry Water – Hydrostatics & Circulation lays the foundation for fluid dynamics by first examining hydrostatics, which describes fluids at rest where the primary force is pressure acting perpendicular to any surface. The fundamental law governing static pressure variation states that the sum of pressure and the product of density, gravitational acceleration, and height is constant. The transition to hydrodynamics simplifies the complexity of real fluids by introducing the theoretical model of "dry water," which is defined as an ideal fluid that is both incompressible (meaning its density is constant throughout) and completely inviscid (meaning it has zero internal friction or viscosity). The key mathematical requirement for incompressibility is the continuity equation, which dictates that the divergence of the velocity vector is equal to zero. The rigorous derivation of the equations of motion for a fluid element, based on Newton's laws, incorporates the effects of pressure gradients and external forces on the fluid's acceleration. A significant simplification arises when analyzing steady flow, where the velocity field at any fixed location does not change with time. Under these conditions, the principles of energy conservation lead directly to Bernoulli's theorem, which states that the quantity formed by pressure divided by density, plus half the velocity squared, plus the potential energy function remains a constant value along any specific streamline. Furthermore, if the flow is also irrotational (meaning the fluid element has zero curl of the velocity, or zero vorticity), this constant applies everywhere in the fluid. The chapter then introduces the concepts of vorticity, defined as the curl of the velocity, and circulation, defined as the line integral of velocity around a closed loop, linking the two using Stokes' theorem. This ideal model allows for the discussion of Helmholtz's theorem, which asserts that vortex lines move along with the fluid, and for dry water, the circulation surrounding a vortex ring is conserved. The analysis concludes by highlighting the critical failure of the inviscid approximation: while the model predicts that a fluid starting irrotationally must remain irrotational, real fluids, due to viscosity, easily generate vorticity, as seen in phenomena like vortex rings or draining water.