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Figure 3: Time-dependent wedge-shaped fluid domain (of infinite extent) formed between two solid surfaces. 2. Streamlines Generated by Rigid Boundaries Rotating Consider a 2D incompressible inviscid irrotational flow u= u(r, 0,t)r+ ue(r, 0, t)0, in the absence of gravity, between two rigid boundaries 0 = tnt rotating with equal and opposite angular velocities with a fluid trapped between them in the time dependent domain -t < 0 < t, 0 <r <∞ (Figure 3). The problem is set up in a polar coordinate system. (a) Since the flow is irrotational, there exists a velocity potential o(r, 0, t). Show that this potential satisfies 1 ə 10² (56) = 0 rər 2 მ02 (6) (b) Noting that the angular speed up at the boundary is given byrd, dt formulate the boundary conditions of impermeability on the solid surfaces in terms of o. (8) (c) Show that a separable solution (i.e. = R(r, t)e(0, t)) to the above equation is = (D₁rk + D₂rk) (C₁ cos ke + C₂ sin ko) where C₁, C2, D1, D2 can be functions of time, but k is a constant. (12) (d) Apply the boundary conditions to show that can be written as = r² cos(20)T (12) 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 ur = 1 and up = , find . By expressing coordinates, draw the streamlines at an instant in time. in Cartesian (12) [50] +