Velocity potential

An irrotational velocity field is locally a gradient

u = \nabla ϕ

ϕ

is its velocity potential.

Table of contents
- Potential flow around a circular cylinder Velocity potential
  - Potential flow around a circular cylinder with circulation Potential flow around a circular cylinder
    - Kutta–Joukowski theorem Potential flow around a circular cylinder with circulation
- Radially symmetric incompressible flow in a planar annulus Velocity potential
  - Pressure in radially symmetric annular potential flow Radially symmetric incompressible flow in a planar annulus
    - Small oscillation of a planar gas bubble in an annular liquid Pressure in radially symmetric annular potential flow

Potential flow around a circular cylinder

Uniform irrotational flow of speed

U

past a circular cylinder of radius

a

, with zero circulation, has velocity potential

ϕ = U (r + \frac{a ^{2}}{r}) cos θ .

On the cylinder, the tangential speed is

- 2 U sin θ

, so the Bernoulli equation gives

p (a, θ) = p_{\infty} + \frac{1}{2} ρ U^{2} (1 - 4 sin^{2} θ) .

Potential flow around a circular cylinder with circulation

Adding circulation

κ

gives the possibly multivalued velocity potential

ϕ = U (r + \frac{a ^{2}}{r}) cos θ + \frac{κ}{2 π} θ .

The velocity remains single-valued, and on the cylinder its tangential component is

- 2 U sin θ + κ / (2 πa)

.

Kutta–Joukowski theorem

For steady two-dimensional inviscid flow of density

ρ

, far-field velocity

U

, and circulation

κ

, the force per unit span is perpendicular to the flow. With counterclockwise circulation taken positive and

U = U e_{x}

, it is

- ρ κ U e_{y}

.

Radially symmetric incompressible flow in a planar annulus

For liquid occupying

a (t) < r < b (t)

, radial symmetry and incompressible flow imply

u_{r} (r, t) = \frac{a a ˙}{r}, ϕ (r, t) = a \overset{a}{˙} lo g r + C (t) .

The kinematic boundary condition at the outer interface gives

b^{2} - a^{2} = constant,

which expresses conservation of the liquid's area.

Pressure in radially symmetric annular potential flow

If the pressure at

r = b (t)

is

p_{\infty}

, the Unsteady Bernoulli equation gives

p (a, t) - p_{\infty} = ρ [\frac{d}{d t} (a \overset{a}{˙}) lo g \frac{b}{a} - \frac{( a a ˙ ) ^{2}}{2} (\frac{1}{a ^{2}} - \frac{1}{b ^{2}})] .

Small oscillation of a planar gas bubble in an annular liquid

For

a = a_{0} (1 + ϵ)

, equilibrium outer radius

b_{0}

, and a bubble gas obeying

p_{0} π a^{2} = constant

, the linearization is

ρ a_{0}^{2} lo g \frac{b _{0}}{a _{0}} \overset{ϵ}{¨} = - 2 p_{\infty} ϵ .

The bubble therefore undergoes simple harmonic motion with

ω^{2} = \frac{2 p _{\infty}}{ρ a _{0}^{2} l o g ( b _{0} / a _{0} )} .

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Velocity potential by

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Velocity potential is a scalar function used in fluid dynamics and vector calculus that is associated with the flow of an incompressible, irrotational fluid. The concept arises from the mathematical formulation of fluid motion and is particularly useful in the study of potential flows. In the context of fluid dynamics: - **Definition**: The velocity potential \(\phi\) is defined such that the velocity \(\mathbf{v}\) of the fluid can be expressed as the gradient of this potential function.

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