3 fa velocity field is given by u ay compute the

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Unformatted text preview: given by u = ay, compute the circulation around a circle of radius r = i about the origin. Check the result by using Stokes' theorem. 4. Consider a plane Couette flow of a viscous fluid confined between two flat plates at a distance b apar (see Figure 9.4c). At steady state the velocity distrbution is u = Uy/b v = w = 0, 11 Problem 3.3 12 Show how we can define" a streamfnction so that the equation of contiuity is satisfied automatically. Problem 3.4 3. !fa velocity field is given by u = ay, compute the circle circulation around a of radius r = i about the origin. Check the result by using Stokes' theorem. 4. Consider a plane Couette flow of a viscous fluid confined between two flat plates at a distance b apar (see Figure 9.4c). At steady state the velocity distrbution is u = Uy/b v = w = 0, Kinef1atic.~ where the upper plate at y = b is moving parallel to itself at speed U, and the lower plate is held stationary. Find the rate of linear strain, the rate of shear strain, and vorticity. Show that the streamfunction is given by Uy2 1/ = - +const. 2b 5. Show that the vortcity for a plane flow on the xy-plane is given by Wz - - ax2 + ay2 . _ (a21/ a21/) Using this expression, find the vorticity for the flow in Exercise 4. 6. The velocity components in an unsteady plane flow are given by 13 Problem 3.4 14 plate is held stationary. Find the rate of linear strain, the rate of shear strain, and vorticity. Show that the streamfunction is given by Problem 3.5 Uy2 1/ = - +const. 2b 5. Show that the vortcity for a plane flow on the xy-plane is given by Wz - - ax2 + ay2 . _ (a21/ a21/) Using this expression, find the vorticity for the flow in Exercise 4. 6. The velocity components in an unsteady plane flow are given by u=x 1+ t 2y and v = -. 2+t Describe the path lines and the streamlines. Note that path lines are found by following the motion of each paricle, that is, by solving the differential equations dx/dt = u(x, t) and dy/dt = vex, t), subject to x = Xo at t = O. 15 7. Determne an expression for 1/ for a Rankine vortex (Fi...
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