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.

Explore laminar and turbulent flows, contrasting smooth layers with chaotic vortex structures. Learn Reynolds numbers and the transition range 2000–4000, plus critical velocity and real-world examples.

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.

Explain the continuity equation from conservation of mass, showing constant mass flow across cross sections with discharge Q = A V; in incompressible flow, density is constant, reducing dimensional forms.

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.

Explore the stream function in two-dimensional, incompressible flows, linking it to velocity components, continuity, and the Laplace equation, and relate it to the velocity potential function.

Derive Bernoulli's equation on a streamline using a cylindrical fluid element, then compare compressible and incompressible flow and explain pressure, kinetic, and potential heads.

Derives the momentum equation from conservation of momentum and applies it to a pipe bend, resolving fluid forces into x and y components to obtain resultant force and its direction.

Derive the momentum equation from conservation of momentum and apply it to jet propulsion, showing the jet reaction equals twice the hydrostatic force for an ideal fluid.

Derive the momentum equation for an orifice tank on frictionless wheels and apply it to jet propulsion, showing that maximum efficiency is 50% when the jet speed equals tank speed.

Derive the vortex flow from radial and vertical pressure variations, with dP/dR = rho v^2 / R and dP/dZ = -rho g, giving pressure as a function of radius and height.

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.

Explore surface and body forces, including gravity, pressure, viscous, and turbulence, and discuss the Navier-Stokes equation of motion for laminar flow.

Explore how the orifice meter, using an orifice plate, measures flow by deriving discharge with the coefficient of velocity, coefficient of contraction, and coefficient of discharge via Bernoulli head differences.

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.

Explore how a pitot tube measures flow velocity by comparing stagnation and static pressures, using Bernoulli's equation and a velocity coefficient to account for losses.

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.

Discover how flow through notches and weirs is classified into rectangular, triangular, trapezoidal, and stepped types, and derive discharge using elemental strips, area, and sqrt(2 g h) with cd.

Explore flow through notches and weirs, deriving discharge for trapezoidal and step notches by combining rectangular and triangular portions, and applying coefficients of discharge cd1 and cd2.

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.

Explore flow through nozzles by applying continuity, head loss, and kinetic head concepts to derive inlet and outlet powers, efficiency, and the nozzle diameter for maximum power transfer.

Explore similarity laws for distorted models, distinguishing model from prototype, and analyze geometric, kinematic, and dynamic similarity plus key force ratios like Reynolds and Froude numbers.

Explore similarity laws in fluid mechanics, including Reynolds, Mach, and Weber models, and distinguish undistorted from distorted models with their advantages, limitations, and practical dam-scale testing.