Fluids, seen through pressure and flow

Pressure at a point is direction-independent (Pascal); defined as perpendicular force over area, p = F / A

Hydrostatic p = ρgh; depth alone sets it regardless of shape or amount (the paradox), slope ρg

Archimedes F = ρgV; points up from the pressure difference; float or sink by density ratio

Eulerian velocity field + streamlines (tangent to velocity, no crossing) + steady vs unsteady

Continuity A·v = const (mass conservation); narrower means faster inversely, Q = A·v

Bernoulli p + ½ρv² + ρgh = const (one streamline); faster = lower pressure; Venturi, atomizer

Momentum theorem F = ṁ Δv (control volume); jet and bend reaction; fire hose, rocket

No-slip + linear Couette profile + τ = μ du/dy; honey vs water (Newtonian)

Reynolds number Re = ρvD/μ (inertia/viscosity); critical Re~2300 splits laminar from turbulent; dimensionless

Parabolic profile u(r) = u_max(1 − (r/R)²); zero at the wall, max at the center; max = 2 × mean; turbulent is blunt

Darcy–Weisbach h_f = f (L/D)(v²/2g); linear in length L, squares with velocity (turbulent), surges as diameter shrinks; f from ε/D and Re (Moody)

A thin layer δ where no-slip grows the speed from 0 at the wall to free stream U∞; δ ∝ √x (δ/x = 5/√Re_x); laminar to turbulent; wall gradient du/dy sets τ_w

Drag = friction drag + pressure (form) drag; adverse gradient causes separation and a wake; F_D = ½ρv²C_D A; golf dimples delay separation

Units cancel into dimensionless numbers (Buckingham Π); Re, Fr, Ma are force ratios; dynamic similarity lets a model predict the full scale; a single C_D(Re) curve