Fluid Mechanics - Advanced mathematical derivations

Osborne Reynolds' 1883 experiment demonstrates laminar, transitional, and turbulent flow using dye in a water tank and a long glass tube, and defines Reynolds number as inertia over viscous force.

Analyze major energy losses in pipes using the Darcy equation, distinguishing friction losses from minor ones. Relate pressure, velocity, and pipe geometry to head loss via Bernoulli.

Study minor energy losses in pipes, focusing on sudden enlargement and contraction, using Bernoulli and momentum balance to derive expansion and contraction head-loss formulas.

Explore minor energy losses in pipes, including entrance, exit, bends, and fittings, plus gradual contraction or enlargement and obstruction, using velocity, area concepts, and loss formulas.

Explore hydraulic gradient line and total energy line in inclined and varying-diameter pipes, using head concepts to account for major and minor losses, including entry and exit losses.

Compare pipes in series and in parallel, noting equal discharge in series and split discharge in parallel, with head losses and a Jupiter equation to compute an equivalent pipe.

Explore laminar flow between two fixed parallel plates at rest, deriving the parabolic velocity profile under a pressure gradient and analyzing the relationship between shear stress and mean velocity.

Explore viscous flow in a circular pipe, derive parabolic velocity distribution, linear shear stress, and mean and maximum velocities, and relate pressure drop to flow rate using Newton's viscosity concepts.

Explore laminar flow through an inclined circular tube at angle theta, derive a parabolic velocity distribution, and determine the maximum and mean velocities, discharge, and pressure head along the tube.

Introduce turbulent flow concepts, derive friction factor relations, and explain turbulent shear stress using Reynolds stress and mixing length theory.

Derives the turbulent pipe flow velocity distribution using mixing-length theory, with L proportional to y and the kappa constant, leading to Mantle's universal distribution equation and the shear-velocity concept.

Explore hydrodynamically smooth and rough boundaries by analyzing the average height of surface irregularities. Understand how the laminar sublayer, delta dash, and turbulence zones govern velocity distribution in rough pipes.

Examine the logarithmic velocity distribution for turbulent flow in smooth pipes and how boundary conditions define the wall and laminar sublayer, versus rough pipes with protrusions.

Introduce compressible flow where density, pressure, and temperature change, requiring thermodynamic treatment. Derive the ideal gas relation p = ρ R T and discuss isothermal and adiabatic (γ) relations.

Derives the continuity equation for compressible flow and Bernoulli's equation for compressible isothermal and idiomatic processes, linking pressure, density, velocity, and elevation.

Derive the velocity of sound in a fluid from piston compression, continuity, and impulse momentum, showing c = sqrt(dp/dρ) as the wave speed.

Derives the speed of sound in a fluid from bulk modulus and density, giving c = sqrt(K/ρ). Compares isothermal and adiabatic processes and introduces Mach number, v/c.

Explore Mach number and the propagation of pressure waves in compressible fluids, comparing subsonic, sonic, and supersonic regimes and the formation of the Mach cone.

Explore mach number in compressible flow through a pipe with varying cross-section. Derive its relation to inertia and elastic forces via continuity and c equals sqrt gamma p over rho.

Analyze the normal shock in a perfect gas for one-dimensional steady flow, deriving Mach number relations and pressure ratios across the shock from mass and momentum conservation.

Derive stagnation pressure from Bernoulli's equation for adiabatic flow, identifying the stagnation point where velocity drops to zero and kinetic energy converts to pressure energy.

Explore stagnation properties by deriving stagnation pressure, density, and temperature at the stagnation point, and compare subsonic, sonic, and supersonic flows using density and pressure ratios.