Boundary Layer I HW #2 Instructor: M

Boundary Layer I

HW #2 Instructor: M. Behbahani-Nejad.

1. A certain flow given by u = y⁵, v = -x², w = 0 has a temperature field given by T = y + t²x. Calculate the local time-rate of change of T at (1, 2), and also the material derivative of T at (1, 2) at any time t. From this last result, what conclusions can you draw?

2. The trajectories of the fluid particles in a particular flow are specified by

        x = x₀e^t,     y = x₀ + (y₀ – x₀)e^-2t

 (a) Sketch a few trajectories in the x - y plane for different values of x₀ and y₀

(b) Obtain the Lagrangian velocities by differentiating the x and y function above w.r.t time.

(c) By elimination (x₀, y₀) between the trajectory equations above and the velocity components, obtain expressions for u and v in terms of (x, y, t); this constitude the Eulerian velocity components for this flow.

3. A compressible fluid is caused to flow through a tube of constant diameter in such a way that the velocity along the axis is given by

             u = u₁ + u₂/2 + u₂ – u₁/2 tanh x

Where u₁ and u₂ are the velocities when x is minus or plus infinity, respectively. The density does not change with time at any point. The density at x = - ¥ is r = r₁. Obtain an equation for the distribution of density along the tube.

4. Computation algorithms for Navier-Stokes equations are often expressed in column-vector notation. For Cartesian coordinates, the divergence form of the Navier-Stokes equations can be written as:

      ¶U/¶t + ¶E/¶x + ¶F/¶y + ¶G/¶z = 0

Where U, E, F and G are 5-element column vectors. Show that

U =

E =

and give the corresponding forms for F and G.