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Unformatted text preview: Fluid Dynamics 3 - Solutions to Sheet 2 1. If the point x travels at constant speed V , i.e. x = V t , the transformation x ′ = x − V t implies that x ′ = 0. This means the transformation corre- sponds to the fluid as seen by an observer moving with velocity V . As a result, the velocity has to be transformed to the comoving coordinate system, u ′ = u − V , while the pressure is simply convected along: u ′ ( x ′ , t ) = u ( x , t ) − V , p ′ ( x ′ , t ) = p ( x , t ) . Now u ′ ( x ′ , t ) = u ( x ′ + V t, t ) − V , and thus ∂ u ′ ∂t = ∂ u ∂t + ( V · ∇ ) u . In addition, we have ( u ′ ·∇ ) = ( u ·∇ ) − ( V ·∇ ) and ∇ p ′ = ∇ p . Thus ∂ u ′ ∂t + ( u ′ · ∇ ) u ′ − ∇ p/ρ = ∂ u ∂t + ( u · ∇ ) u − ∇ p/ρ. Thus the Euler equation remains invariant as seen by an observer moving at constant speed: it is Galilean invariant. 2. (i) u = ( αx, − αy, 0). So streamlines are d x αx = d y − αy Solution | y | = C | x | for C constant: (ii) Dye injected along the circle x 2 + y 2 = c 2 at t = 0. Consider a general point on the circle ( x , y ) at t = 0. Particle paths are given by d x d t = αx, d y d t = − αy so that x = x e αt , y = y e − αt y x Figure 1: Streamlines for α > 0. But x , y satisfy x 2 + y 2 = c 2 , so substituting in x 2 e − 2 αt + y 2 e 2 αt = c 2 This is the equation of an ellipse (( x/a ) 2 +( y/b ) 2 = 1) with semi-major axis a = c e αt and semi-minor axis b = c e − αt . So as time increases the dye forms an increasingly stretched (in the direction of x ) el- lipse y x t = 0 t = 1 t = 2 Figure 2: Evolution of circle of dye with time in the flow. (iii) The area of an ellipse is πab , which in this case is πc 2 e αt e − αt = πc 2 . Thus the area does not change in time....
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This note was uploaded on 01/20/2012 for the course MATH 33200 taught by Professor Eggers during the Fall '11 term at University of Bristol.
- Fall '11