advanced34 min studyLesson 175 in Physics

Potential flow and inertia in the surrounding fluid

Reduce irrotational incompressible flow to a boundary problem and calculate the added inertia of an ideal accelerating sphere.

Before you begin

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The question for today

What you’ll understand

Derive Laplace’s equation for an incompressible velocity potential.
Distinguish no penetration from no slip at a moving body.
Compute ideal sphere added mass and distinguish it from drag or displaced mass.

A body accelerating through water changes the motion of water around it as well as its own motion. Even an ideal fluid with no viscous drag can therefore resist acceleration. Potential flow makes that additional inertia calculable by solving a spatial boundary problem for an irrotational, incompressible velocity field. The model is powerful precisely when its boundary conditions and missing physics are stated explicitly.

Replace a vector field with a potential on an allowed domain

If velocity has zero curl in a simply connected smooth region, write it as the gradient of a scalar velocity potential. Incompressibility then requires the potential to satisfy Laplace’s equation. If the domain is multiply connected, local irrotationality need not imply one globally single-valued potential without further circulation conditions. Superposition works for potentials satisfying this linear spatial equation, but boundary conditions must still be met for the actual geometry.

A solid impermeable boundary requires equal fluid and body normal velocity. It does not require equality of tangential velocity in the inviscid model. Imposing viscous no slip on a generic potential-flow solution overconstrains that model. At infinity one might prescribe rest or a uniform stream. Stating the body frame and laboratory frame prevents accidentally combining a rest-at-infinity condition from one frame with a moving-wall condition from another.

\mathbf u=\nabla\phi,\qquad \nabla^2\phi=0,\qquad (\nabla\phi-\mathbf U_{\mathrm b})\cdot\mathbf n=0\ \text{on the body}

The potential equation applies in the fluid region. Ub is the local boundary velocity, including rotation if present; the last relation imposes no penetration rather than no slip.↔ Scroll wide equations horizontally.

Accelerating the body stores kinetic energy outside it

For a sphere translating in an unbounded incompressible ideal fluid initially at rest, the induced velocity field is proportional to the instantaneous sphere velocity. Integrating one half density times speed squared over the fluid gives a kinetic energy proportional to the sphere speed squared. Its coefficient defines added mass. The sphere value is one half the displaced fluid mass, a geometry-specific result rather than a rule for every object.

Added mass is not extra material attached permanently to the body. It is a way of representing the inertia of the coordinated surrounding flow. In more general geometries it becomes a matrix coupling translational and rotational motions, and nearby boundaries change its entries. The ideal unbounded-fluid result also excludes wave radiation, compressibility, viscosity, and history-dependent viscous forces. These omissions matter when interpreting measurements in a real tank.

T_{\mathrm f}=\frac12m_{\mathrm a}U^2,\qquad m_{\mathrm a}=\frac12\rho V=\frac23\pi\rho a^3\quad\text{(sphere)}

Here a is the sphere radius and V its displaced volume. For this translational ideal-fluid problem the acceleration-dependent fluid reaction is −ma dU/dt; steady added-mass force vanishes.↔ Scroll wide equations horizontally.

Zero ideal steady drag does not imply zero fluid force

A steady irrotational ideal flow past a suitable body can predict zero drag, the classical d’Alembert paradox. This does not mean the pressure is uniform around the body, and it does not rule out a nonzero acceleration force in an unsteady problem. Real separated wakes and viscous stresses produce drag absent from the ideal model. An ideal prediction that fails is useful when it identifies which omitted mechanism must enter next.

For a sphere subject to a horizontal applied force in the stated ideal model, combine body mass and added mass before calculating acceleration. Gravity and buoyancy are separate forces if vertical motion is studied. Do not subtract displaced mass from inertial mass merely because buoyancy reduces a static weight. In an experiment, change density, body geometry, or distance to a wall independently to distinguish inertial fluid loading from a constant friction force or a velocity-dependent drag.

Work it through

An ideal sphere with a specified displaced volume

A sphere has body mass $3\,\mathrm{kg}$ and displaced volume $0.004\,\mathrm{m^3}$ . It moves horizontally in an unbounded incompressible ideal fluid of density $1000\,\mathrm{kg\,m^{-3}}$ . A horizontal force of $10\,\mathrm N$ acts, with other horizontal forces neglected. Find acceleration and the fluid kinetic energy at $2\,\mathrm{m\,s^{-1}}$ .

The displaced fluid mass is $\rho V=1000(0.004)=4\,\mathrm{kg}$ . The sphere’s added mass is one half of that value, $m_{\mathrm a}=2\,\mathrm{kg}$ , using the sphere-specific unbounded potential-flow result.
The applied force accelerates both body inertia and the induced fluid motion: $(3+2)a=10$ . Therefore $a=2\,\mathrm{m\,s^{-2}}$ . The body alone would accelerate faster, showing that no viscous drag is needed for this inertial effect.
At speed two, the induced fluid kinetic energy is $T_{\mathrm f}=m_{\mathrm a}U^2/2=2(4)/2=4\,\mathrm J$ . The body kinetic energy is separately six joules, so total kinetic energy is ten joules.

The result

The ideal acceleration is 2 m/s² and the induced fluid kinetic energy at the stated speed is 4 J. Buoyancy is irrelevant to this horizontal force balance and has not been reinterpreted as inertial mass.

The living notebook

Explore the model

Interactive lab

Interactive lab · function

Read the slope of a quadratic energy

Restrict coefficient a to positive values and position x to the nonnegative interval 0 to 2. Use the normalized curve y = ax² as an energy-versus-speed analogy, with a representing half an effective mass. Start at a = 0.5 and predict that changing a to 1 at fixed x doubles both energy and tangent slope. Change x to compare energy growth with its slope.

y = ax²TangentSigned integral

y at chosen x: 0.608
Tangent slope · 2ax: 1.35
Signed area · ax³/3: 0.182

Coefficient a0.75Position x0.9

t = 0 s

This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe quadratic identity illustrates kinetic-energy scaling only. The sliders do not solve Laplace’s equation, change a body shape, or compute an added-mass coefficient.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

The ideal sphere coefficient is one half, so its fluid reaction is proportional to acceleration through that added mass. Other geometries and boundaries have different coefficients; the result is neither a universal displaced-mass rule nor steady viscous drag.

Your turn

Try it yourself

The same ideal sphere is placed in fluid of density $500\,\mathrm{kg\,m^{-3}}$ , retaining body mass 3 kg, volume 0.004 cubic metres, and horizontal applied force 10 N. Find added mass and acceleration. Explain whether a steady speed produces an added-mass force in this model.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Added mass scales with fluid density at fixed geometry. The acceleration term depends on the derivative of speed, not its value.

Compare with the solution

The added mass is $\rho V/2=500(0.004)/2=1\,\mathrm{kg}$ . The acceleration is $10/(3+1)=2.5\,\mathrm{m\,s^{-2}}$ . At strictly constant translational speed in this unbounded ideal setting, the added-mass acceleration force is zero, although induced fluid kinetic energy remains nonzero. Real viscous drag or wave-making would require additional physics.

Keep the idea

Take it with you

An irrotational incompressible field reduces to a Laplace boundary problem with domain qualifications.
No penetration is the ideal boundary condition; viscous no slip belongs to another model.
Added mass represents surrounding-flow inertia and depends on geometry, boundaries, and fluid density.

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