ISEGORIA / MATH ENCYCLOPEDIA
Fluid dynamics: flow, pressure, and vorticity
Streamlines, Bernoulli pressure, and rotating flow.
Before you begin: Vectors and calculus
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Streamlines
A uniform stream U is combined with a source and a sink, the building blocks of potential flow. Drag the two singularities and change their strengths. The streamlines are integrated from the exact velocity field, the shading shows the speed, and the gold points are where the velocity vanishes. With equal and opposite strengths in a stream, the orange dividing streamline closes into a Rankine oval: a body shape the flow goes around.
Worked example. A streamline is tangent to the velocity field at every point, so no fluid crosses it; the dividing streamline can be replaced by a solid wall.
Watch out. A streamline is not necessarily the path of one marked fluid parcel in unsteady flow.
What does the shading encode?
The local speed |u|. By Bernoulli, pressure is lowest where the flow is fastest, at the sides of the oval.
2. Bernoulli in a venturi
Water flows through a horizontal pipe that narrows to a throat and widens again. Continuity fixes the speed, v = v₁A₁/A, so tracers released at equal time intervals spread apart where the flow is fast. Bernoulli then fixes the pressure: the columns in the tubes drop at the throat by exactly the rise in ½ρv². Change the throat area and the inlet speed.
Worked example. With v₁ = 3 m/s and A₂/A₁ = 0.5, continuity gives v₂ = 6 m/s, and the pressure at the throat is lower by ½ρ(v₂² − v₁²) = 13.5 kPa.
Watch out. The ideal relation assumes steady, incompressible, inviscid flow along a streamline, and the flow is treated as one-dimensional across each section.
Which term rises in a constriction?
The kinetic term ½ρv² as speed increases.
3. Vorticity and circulation
Choose a flow and drag the loop C. The circulation Γ is computed as the line integral of velocity around the loop, and it always equals the flux of vorticity through it (Stokes’ theorem). In solid-body rotation the vorticity is 2Ω everywhere and the paddle wheels turn with the flow. In the free vortex the fluid circles faster near the centre, yet the vorticity is zero away from the axis: the wheels orbit without turning, and a loop that misses the axis has Γ = 0. The Rankine vortex joins the two, a rotating core inside a free vortex.
Worked example. Solid-body rotation has uniform vorticity twice the angular speed, so a centred loop of radius R has Γ = 2Ω·πR².
Watch out. Circular streamlines alone do not determine vorticity: the solid-body and free vortices have the same circular streamlines, yet one has uniform vorticity and the other none away from the axis. The free vortex is singular on its axis, where all its vorticity is concentrated.
What does circulation measure?
The line integral of velocity around the closed loop.