{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

worksheet1 - 4 The Euler equation is ∂ u ∂t u ∇ u...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Fluid Dynamics 3 2011/12 Sheet 1 Homework to be handed in Friday 21st October: 1,3,4,9. 1. Let a closed loop of particles C ( t ) be defined by x = a (cos s + αt sin s, sin s, 0) , 0 s < 2 π, where each value of s corresponds to a different fluid particle, and a, α > 0. (i) Sketch how C ( t ) changes as function of time, starting from t = 0. Which points remain stationary ? (ii) Find the Lagrangian velocity field as function of the particle marker s . (ii) Show that the Eulerian velocity field is u = ( αy, 0 , 0) . 2. Find the general equation and sketch (a) the streamlines (at t = 0 , 1 2 , 1); and (b) the particle paths (for particles released at the point (1 , 1 , 0) at t = 0 and t = 1), for the following time dependent, two-dimensional flow fields, given in a Cartesian coordinate system u = ( u, v, 0) (i) u = a cos( πt ), v = a sin( πt ) ( a constant) (ii) u = x - V t , v = y ( V constant) (iii) u = ktx , v = - kty ( k constant) (iv) u = xt , v = - y 3. A two-dimensional fluid velocity field u = ( u, v, 0) has components given by u = u 0 sin ω ( t - y/v 0 ) , v = v 0 . Find and sketch (i) the pathlines for a particle released into the flow from ( x, y ) = (0 , 0) at t = t 0 . Consider different values of
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 4. The Euler equation is ∂ u ∂t + ( u · ∇ ) u =-1 ρ ∇ p, where u = ( u,v,w ) is the velocity and p the pressure. Write out the second component of the Euler equation in components. 5. Simplify (i) ± ijk ± ijm ; (ii) ± ijk ± ijk . 6. Demonstrate that for φ ( x ), a scalar function of position ∇ × ∇ φ = 0 7. Prove the vector identity, for vector field u ( x ) and v ( x ), ∇ × ( u × v ) = u ( ∇ · v )-v ( ∇ · u ) + ( v · ∇ ) u-( u · ∇ ) v 8. Show that for a vector field u ( x ) u · ∇ ( | u | 2 / 2) = u · ( u · ∇ ) u 9. For each of the following functions, φ ( x ), find ∇ φ and ∇ · ∇ φ ≡ ∇ 2 φ ≡ 4 φ using suffix notation and the summation convention. (i) φ = f ( r ); f is an arbitrary scalar function (ii) φ = a · x /r 3 , where a is a constant vector and r 2 = x · x ....
View Full Document

{[ snackBarMessage ]}