Assume that the pressure far from the bubble is p0

Hydrodynamics and Elasticity: Class 4

1. Determine the shape of the surface of an incompressible fluid subject to a gravitational field contained in a cylindrical vessel which rotates about its (vertical) axis with a constant angular velocity Ω. Thus the velocity field is

u(x, y) = (−Ωy, Ωx, 0).

2. Consider the irrotational fluid motion outside a spherical bubble of (a perfect) gas in an incompressible liquid. Neglect the effect of surface tension and mass diffusion. Find an equation for the radius R(t) of the bubble. Assume that the pressure far from the bubble is p₀.

3. The path of a fluid particle, that was at (X, Y, Z) at time t = 0, is described by x(t) =



X cos Ω

2α(e^2αt− 1)



− Y sin Ω

2α(e^2αt− 1)



e^−αt,

y(t) =



Y cos Ω

2α(e^2αt− 1)



+ X sin Ω

2α(e^2αt− 1)



e^−αt, z(t) = Ze^2αt.

(a) Show that the flow velocity reads

u = −αx − Ωye^2αt, −αy + Ωxe^2αt, 2αz
, (1)

where Ω and α are constants. Verify that ∇ · u = 0.

(b) Show that the flow obeys the vorticity equation for an incompressible and ideal fluid

Dξ

Dt = (ξ · ∇)u.

(c) Consider a material curve, described by the equation X²+ Y² = a², Z = 0,

at time t = 0. Obtain the equation that describes the shape of this material curve at any time t > 0.

Hint. Note that the velocity field in Eq. (1) has the form u = u1+ u₂, where u1 = (−αx, −αy, 2αz) , u2 = −Ωye^2αt, Ωxe^2αt, 0
,

∇ × u₁ = 0 and u₂ is a rigid body motion about the z axis.

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4. For an inviscid fluid the Euler’s equation reads

∂u

∂t + ξ × u + ∇ 1 2u²



= −∇h0− ∇χ.

We also have conservation of mass, dρ

dt + ρ∇ · u = 0.

Show that d dt

 ξ ρ



= ξ ρ · ∇

 u.

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