Fluid Mechanics - Basic mathematical derivations

Understand streamline, stream tube, stream filament, path line, and streak line in two-dimensional flow, where velocity is tangent to the streamline. Dye traces reveal particle paths.

Differentiate steady and unsteady flows by whether flow patterns and velocity directions stay constant over time, and distinguish uniform from non-uniform and one-, two-, three-dimensional flows via streamlines.

explain rotational flow by defining angular velocity components omega x, omega y, and omega z and noting that true rotation requires ideal fluids with no tangential and steady stresses.

Define the velocity potential function as a scalar field whose negative spatial derivatives yield fluid velocity, derive its Laplace equation for steady incompressible flow, and note implications for rotational flow.

Derive forced vortex flow in a rotating fluid and P2 - P1 = 1/2 rho omega^2 (r2^2 - r1^2) - rho g (z2 - z1); the surface is a parabola.

Derive the venturi meter’s operation from Bernoulli and continuity, linking pressure difference to velocities and throat area, and express theoretical versus actual discharge using the discharge coefficient.

Explore the venturi meter with a u-tube manometer, deriving head differentials for heavy and light liquids, and analyze inclined venturi conditions with height relations and pressure terms.

Learn how a hot wire hardware anemometer measures velocity by heating a wire and sensing resistance changes from heat loss in air or gas, constant current and constant temperature modes.

Analyze how head measurement errors affect discharge through rectangular and triangular notches, and compare their sensitivity. Explore viscometers - coaxial cylinder, falling sphere, and capillary tube - and their Stokes-based viscosity relations.