Concept

Singular perturbation — where it appears

A problem in which setting the small parameter to zero changes the character of the equations rather than merely perturbing the answer. The reduced problem then fails in a region that shrinks with the parameter, and a matched expansion with its own scaling is needed there — a boundary layer, a triple deck, or a far field.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

One more condition, and the price of it. The model problem ε u″ + u′ = 0 with a condition at each end, at three values of ε. The outer solution is the flat line at one — that is the whole of the answer when ε is zero, and it is a first-order equation that can meet one condition, so it meets the one at the far end and misses the one at the wall by the whole range. Restoring ε restores the second condition and pays for it with a layer of thickness ε, inside which the gradient is of order 1/ε. The product of those two — which is what a stress is — does not depend on ε at all: it is 1.0000 at every value tried, to nine decimal places. Drag does not vanish as viscosity does. It converges.

How many things a flow must be told

The equations of motion do not have one answer. They have as many as the conditions on the edge allow, and the number of those is decided by the order of the equation — which is why viscosity does not make the same problem harder, it makes a different problem.

kinematics · Boundary conditions

The only candidate the far field allows, and the wall it slips past. The general Stokes solution has four constants; the condition at infinity kills two of them and fixes a third, leaving one to satisfy two conditions at the wall. Setting the stream function to zero there uses it up, and the tangential velocity that remains is exactly twice the free stream — for every radius, every speed, and every fluid.

The flow with no solution

Creeping flow past a sphere has a solution and everybody knows it. Creeping flow past a cylinder has none — not a difficult one, not one needing a clever method. The equations, the no-slip condition and the uniform stream at infinity are inconsistent, and the residual is exactly twice the free stream.

viscous · Stokes' paradox

Named alongside it

The objects these essays reach for when they reach for this one.

Boundary condition Boundary layer Creeping flow d'Alembert's paradox Drag Inviscid Laplace's equation Logarithm Mass conservation Matched asymptotics Model limit Navier–Stokes equations

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