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.

Demonstrates using a manometer and barometer to compute gas and atmospheric pressures, derive hydrostatic relations, and explain how high altitude lowers air density, affecting electronic cooling.

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.

Analyze hydrostatic forces on curved surfaces by projecting onto horizontal and vertical planes, use a free-body diagram to resolve into x and y components and locate the resultant.

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 flow visualization to reveal flow patterns in experiments and simulations. Define streamlines, stream tubes, streak lines, pathlines, and timelines, and relate them to velocity components u and v.

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.

Explore the Reynolds transport theorem to connect control mass and control volume formulations, deriving conservation of mass, momentum, and energy using an intensive property B and its flux through boundaries.

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.

Derive the control volume form of the momentum equation from Newton's second law, applying Reynolds transport theorem, accounting for gravity and pressure, and beta correction for nonuniform velocity.

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.

Examine laminar and turbulent pipe flow, relate pressure drop to head loss via the energy equation, and derive laminar fully developed flow with a parabolic profile and development length.

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 minor losses from fittings, using loss coefficients k_l and head loss h_l = k_l v^2/(2g). Calculate total head loss as the sum of major and minor losses.

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.

Learn to analyze parallel piping systems by solving for q1 and q2 from qa, using equal head loss and pressure drop with nonlinear equations solved by fixed-point or Newton methods.

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.