For a given hypothetical flow the velocity at time t = 0 to t = 10 s was u = 0, v = 4m/s. Then from time t = 10 s to t = 15 s, the velocity was u = - 2 m/s, v = 2 m/s. A dye streak was started at a point in the flow field at time t = 0, and the path of a particle in the fluid was also traced. Draw to scale the streakline, pathline of the particle and streamlines at t = 15 s.

1. A piston-cylinder is producing a viscous oil flow, which has the flow rate of Q, radius R=1.5 mm, and stroke L-5 mm. The fluid density is p = 800 kg/m³ and the viscosity is μ = 0.5 Pa-s. The pressure to drive the flow is p=6.25 Pa. a. How many non-dimensional groupings can be obtained? Derive them. b. If the flow transitions at Re = PV(2R) =2300. Is the generated flow laminar or turbulent given Q=20 mL/s? LL c. Assume the pressure gradient is uniform on the cross-section and can be calculated using OP=-. The velocity profile in the cylinder is u(r) = ax - [1-(²]. Sketch the velocity profile. Calculate the shear stress at the R² ap 2UÔI cylinder wall and label the direction. d. Calculate the flow rate Q with the velocity profile in (c).. e. If the shear stress is uniform over the cross-section, i.e. T = C, what's the pressure gradient distribution and sketch it? L L. R

Assumptions The flow is steady. The flow is incompressible. The flow is two-dimensional in the x-y plane V = (u, v) = (U, + bx) ỉ - byj %3D We are to calculate the material acceleration for a given velocity field. (None = b( U, +bx) a. y b? y = b( U, +by) ax b2 x (Uo +bx) = b y a, II

The velocity for a steady, incompressible flow in the xy plane is given by u=Ax², v= Ayx, where A= 1.2 m1s, x and y in meter. Find the the curl of the velocity vector at the point (0.04 m, 0.01 m): O 0.0000 O 2.3999 O 0.0120 O 0.0005 O 0.0019 O 1.1999 O 0.0000 O 1.2000