Answer Happy

WU MA3D1, Please answer with clear and full solutions for all
parts of the question for a thumbs up.
MA 3D1 4. Consider a 2D incompressible inviscid irrotational flow u = U_ (1,0,1)e, +uar, 0,t)eo in the absence of gravity, between two rigid boundaries 6 = +Nt rotating with equal and opposite angular velocities with a fluid trapped between them in the time- dependent domain - <<St, 0 <r< (Figure 2). The problem is set up in a polar coordinate system so that the Given Information for cylindrical coordinates may prove useful. Z Figure 2: Time-dependent wedge-shaped fluid domain (of infinite extent) formed be- tween two solid surfaces. a) Since the flow is irrotational, there exists a velocity potential o(r, 0,t). Show that this potential satisfies дф 1 Ꭷ rar = 0. ar +720 [3] [6] b) Noting that the angular speed up at the boundary is given by reformulate the boundary conditions of impermeability on the solid surfaces in terms of p. c) Show that a separable solution (i.e. o = R(r. t)e(e,t)) to the above equation is 0 = (D.+*+ Dar-)(Ccos ko + C sin ko) where C1, C2, D., D, can be functions of time, but k is a constant d) Apply the boundary conditions to show that can be written as o=r* cos(20)T, where you need to find the function T(t) in terms of t and e) Noting that in polar coordinates the streamfunction is defined by vy and us = re.find y. By expressing v in Cartesian coordinates, draw the streamlines at an instant in time. [6] [6] 7 CONTINUED