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Hydrodynamics and Elasticity: Class 2

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Hydrodynamics and Elasticity: Class 2

1. Consider a 2D flow which has the Cartesian coordinates:

u =

 x

1 + t, y

 ,

where t > −1.

Determine the equation describing:

(a) the streamline passing through the point (1,1),

(b) the trajectory of a particle which started at t = 0 from the point (1, 1),

(c) the position at t = 1 of a streak of dye released from the point (1, 1) in the interval t ∈ (0, 1).

Sketch the obtained curves when possible.

2. Consider the following steady 2D flows (a) u = (kx, −ky) (extensional flow), (b) u = (ky, 0) (shear flow),

(c) u = (ky, −kx) (what is this flow?), where k is a real constant.

(i) Check, whether these flows are solenoidal (e.g. divergence-free). For each flow, find the associated stream function and use it to sketch the streamlines. Find the vorticity.

(ii) Imagine that at t = 0 particles positioned on a circle of radius a around the origin, given by x2+ y2= a2, were marked with a colored dye. Find the equation for this material curve at any time t > 0.

How does the surface area enclosed by this curve change in time?

Cytaty

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