Fluid Mechanics Fundamentals

Learn fluid statics and pressure as a scalar in stationary fluids, differentiate absolute and gauge pressure, and apply p = p0 + rho g h with a hydraulic press.

Determine hydrostatic forces on submerged planes using centroid pressure and area. For a rectangular plate, the resultant acts two-thirds down from the surface; for a triangular section, one-third depth.

Generalize the forces on submerged bodies to arbitrary shapes using gauge pressure, centroid, and the center of pressure, with the parallel axis theorem for the moment.

Explore buoyancy and stability of floating bodies using Archimedes principle, center of buoyancy, center of gravity, and the metacenter, through a rotating block example that illustrates stable and unstable configurations.

Explore vorticity as the curl of velocity, distinguishing irrotational, potential flows from rotational flows, and examine Rankine and Burgers vortices with boundary layer implications for lift.

Apply Reynolds transport theorem to mass conservation, deriving the Eulerian form for a fixed control volume and relating unsteady and convective terms.

Apply the energy equation in rate form to pump water from a pond to a 15 m higher tank, using shaft work and 85% efficiency to determine volume flow rate.

Explore momentum equation applications with moving control volumes, using relative jet velocities and angle components to compute forces on a cart and turning plate, 18.75 N and 10,000 N.

Learn how to nondimensionalize the unsteady diffusion and Navier-Stokes equations, defining dimensionless temperature, time, and space, and revealing the thermal diffusivity alpha and Reynolds number as key parameters.

Present similarity—geometric, kinematic, and dynamic—alongside Buckingham pi theorem to form dimensionless drag groups for a sphere, linking drag to density, velocity, diameter, and viscosity via Reynolds number.

Integrate the fully developed pipe flow equation with a constant axial pressure gradient to obtain u(r). Apply boundary conditions to solve constants and derive Q, v̄, Vmax = 2 v̄.

Derive the Darcy friction factor for fully developed laminar pipe flow by equating the pressure drop to a pressure loss form, yielding f = 64 mu / rho v d.

Examine turbulent pipe flow by detailing the viscous sublayer, overlap layer, and outer layer, and apply the law of the wall and log law with u plus and y plus.

Present the power law velocity profile for pipe flow, typically with n around seven, used to initialize CFD velocity fields, though it yields infinite wall shear stress.

Explore piping fundamentals by solving pressure changes and head losses in abrupt expansions and fittings using energy and Bernoulli equations, estimating friction factors with Moody diagrams.

Define a stream function for two-dimensional incompressible flow; contours of constant psi coincide with streamlines. For irrotational flow, psi satisfies Laplace's equation, with flow between lines equal to psi difference.

Construct a stream function plot for u = c x, v = - c y, verify continuity, derive psi = c x y, and visualize streamlines with a central stagnation point.

Derive the differential conservation of linear momentum using an infinitesimal control volume, balancing rate of change and flux with surface and body forces to obtain the momentum equations and Navier-Stokes.

Turn momentum equations into 2D incompressible Navier–Stokes equations for a Newtonian fluid with constant viscosity, deriving and simplifying viscous terms, yielding three equations in u, v, p, and continuity.

The modified pressure p tilde = p - rho g y buries gravity terms in the pressure, simplifying the y momentum equation for incompressible flow.

Explore Prandtl's boundary layer model: a thin viscous layer near walls with an inviscid outer flow, using a parabolic marching equation and Reynolds-number dependent thickness to capture transition.