What is fluid mechanics?
A fluid is anything that flows — liquids and gases both — and the mechanics of fluids is the physics of matter treated as a continuum rather than as individual particles. Two conservation laws carry almost everything: conservation of mass (the continuity equation, Q = A·v with constant density) and conservation of energy along a streamline (Bernoulli's equation, P + ½ρv² + ρgh = constant). On top of those, real fluids dissipate energy through viscosity, and the ratio of inertia to viscosity — the Reynolds number Re = ρvL/μ — decides whether the flow stays orderly (laminar) or breaks into eddies (turbulent). The six calculators on this panel cover the working equations: Archimedes' buoyant force, hydrostatic pressure, Bernoulli along a streamline, flow rate, Stokes drag for spheres in creeping flow, and Reynolds number for regime classification.
Two conservation laws do the heavy lifting: mass conservation gives continuity (Q = A·v, A1v1 = A2v2), energy conservation along a streamline gives Bernoulli (P + ½ρv² + ρgh = constant). Everything else — buoyancy, hydrostatic pressure, drag — is a special case or a consequence.
Archimedes' buoyancy — displaced weight
Any body submerged in a fluid feels an upward force equal to the weight of the fluid it displaces: Fb = ρ·V·g, where ρ is the fluid density, V the displaced volume, and g the gravitational acceleration. The default sample (ρ = 1000 kg/m³ fresh water, V = 0.01 m³, g = 9.81 m/s²) yields Fb ≈ 98.1 N — the weight of 10 kg of water. That is exactly why a 10 kg block of wood (density 700 kg/m³) sits half-submerged: half its volume displaces half its weight. An object floats when its average density is less than the fluid's; it sinks when greater; it hangs in suspension at exactly equal. For gases the same relation works with ρ around 1.2 kg/m³ at sea level, which is why a helium balloon lifts only about 1 gram per liter — the density contrast is tiny.
Hydrostatic pressure — weight of the column
In a static fluid, pressure grows linearly with depth: P = P0 + ρg·h, where P0 is the pressure at the surface (atmospheric, 101,325 Pa at sea level). The default sample (20 m of fresh water) gives a gauge pressure of ρg·h = 196,200 Pa and a total pressure of 297,525 Pa (about 2.94 atm, 2.98 bar). Pressure depends only on depth and fluid density, not on the shape of the container — the "hydrostatic paradox" — because every element of fluid at depth h has the same weight column above it. Pascal's principle: a change in pressure applied to an enclosed fluid is transmitted undiminished to every part, which is why hydraulic lifts multiply force: a small force on a small piston creates the same pressure that pushes up a large piston with proportionally large force.
Bernoulli — energy along a streamline
For steady, incompressible, non-viscous flow, energy per unit volume is conserved along any streamline: P + ½ρv² + ρgh = constant. The default sample starts at atmospheric pressure, v1 = 2 m/s, h1 = 10 m, and ends at v2 = 4 m/s, h2 = 5 m — the panel returns P2 ≈ 144,375 Pa with a constant of about 201,425 Pa. The drop in elevation adds potential pressure; the rise in speed removes static pressure. This is the working principle behind the venturi (a constriction in a pipe speeds the flow and drops the pressure), the carburetor (a throat in the air channel lowers pressure and draws fuel in), the airplane wing (the upper surface accelerates the air, lowering its pressure and lifting the wing), and the pitot tube (pressure difference measures velocity). Bernoulli fails wherever flow is not steady, not incompressible, or not inviscid — notably at high Reynolds numbers where turbulence makes pressure noisy and the formula averages over time. An airfoil's lift is mostly Newton's third law — deflecting airflow downward — with Bernoulli's pressure difference as the bookkeeping, not the cause.
Flow rate — how much per second
The volumetric flow rate is Q = A·v, with A the cross-sectional area and v the mean velocity. The default sample (A = 0.05 m², v = 3 m/s) gives Q = 0.15 m³/s = 150 L/s, or 9 m³ per minute. For incompressible flow through a varying pipe, mass conservation forces A1·v1 = A2·v2: halving the cross-section doubles the velocity. This is why a garden hose with a thumb over the end shoots farther, and why narrowing a river at rapids makes whitewater. The continuity rule is local at any instant; the speed does not "remember" earlier sections. For compressible flow (gas pipelines, jet engines) the right invariant is mass flow rate ρA·v, which is constant when no fluid is added or removed.
Stokes' drag — the laminar-flow end of fluid resistance
For a sphere of radius r moving at low speed through a fluid of viscosity μ, the drag force is Fd = 6πμr·v. The default sample (μ = 0.001 Pa·s water, r = 1 cm, v = 0.5 m/s) gives Fd ≈ 9.42×10−5 N (about 94 μN). The relation is linear in v because at low Reynolds number the wake is symmetric and viscosity dominates; the work the sphere does against drag goes into viscous heating of the fluid. Stokes drag only works when Re < 1 — for a 1 cm sphere in water this means velocities below about 10−4 m/s. At higher velocities the drag coefficient becomes roughly constant and the force scales as v² instead of v. Stokes drag is what controls settling of dust in air, swimming of bacteria, and falling of fine sediment through water.
Reynolds number — order versus chaos
The dimensionless ratio Re = ρvL/μ compares inertial forces to viscous forces. The default sample (water, v = 1 m/s, L = 0.1 m) gives Re ≈ 100,000, well into the turbulent regime. Empirically: Re < ~2,000 means laminar flow (smooth, predictable, parallel streamlines); Re > ~4,000 means turbulent (chaotic, mixing, eddies); the narrow band between is transitional. Blood flow in the aorta (Re ~ 4,000) is at the boundary; in capillaries (Re ~ 0.001) it is firmly laminar and Stokes drag applies. Air over an airplane wing is intentionally turbulent to delay flow separation. The Reynolds number is the reason pipes have a "critical velocity" above which pumping costs jump, and the reason smoke from a candle rises as a smooth laminar plume for a few centimeters before breaking into chaotic curls.
Common misconceptions
- Archimedes' force comes from the bottom pushing up. It comes from the integral of pressure over the entire wetted surface. The pressure is higher at the bottom, but the side contributions are nonzero too, and the net is exactly the weight of displaced fluid — which is why the geometry of the container does not change the answer.
- Airplane wings lift because the upper surface is longer and air "has to catch up". The lift comes from a pressure differential generated by circulation and the shape of the wing; equal-transit-time is a folk derivation that happens to give the right numerical answer for the wrong reason. For low-speed wings the dominant mechanism is the pressure distribution, not any "speed up over the top".
- Stokes' law works at any speed. Only when Re < 1. A 1 cm steel ball falling in water reaches Re = 1 at a few tenths of a millimeter per second; above that the wake becomes asymmetric and drag climbs toward v² behavior. Stokes underpredicts drag for anything resembling a marble.
- Pressure always pushes outward from the fluid. Pressure is isotropic in a static fluid: it pushes equally in every direction. The "push from below" on a submerged surface is just the bottomward component of the same pressure that pushes sideways and upward on every square centimeter.
Related tools: Mechanics for force, work, and energy, Thermodynamics for compressible flow and gas dynamics, and Unit Converter for switching between Pa, bar, atm, and psi.
