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ES 212: Fluid Mechanics Tutorial (March 25, 2021) (Submit Problems 2 and 4)

Problem 1

^ ~ 5 ax2 y^i 2 byj^1 cz2 k, Consider the flow field given by V 1

1

where, a = 2 (suitable units), b = 2 (suitable units), and c = 1 (suitable units). Determine (a) the number of dimensions of the flow, (b) if it is a possible incompressible flow, and (c) the acceleration of a fluid particle at point (x, y, z) = (2, 1, 3). Problem 2 Verify if the flow filed is irrotational

(a) (b) (c) (d)

u 5 2x2 1 y2 2 x2y; v 5 x3 1 x(y2 2 2y) u 5 2xy 2 x2 1 y; v 5 2xy 2 y2 1 x 2 u 5 xt 1 2y; v 5 xt2 2 yt u 5 (x 1 2y)xt; v 5 2 (2x 1 y)yt

Problem 3 The velocity field near the core of a cyclone can be approximated as

~ 5 2 q eˆr 1 K eˆθ V 2πr 2πr Is this an irrotational flow field? Obtain the stream function for this flow.

Problem 4 A liquid flows down an inclined plane surface in a steady, fully developed laminar film of thickness h. Simplify the continuity and Navier-Stokes equations to model this flow field. Obtain expressions for the liquid velocity profile, the shear stress distribution, the volume flow rate, and the average velocity. Relate the liquid film thickness to the volume flow rate per unit depth of surface normal to the flow. Calculate the volume flow rate in a film of water h = 1 mm thick, flowing on a surface b = 1 m wide, inclined at θ =150 to the horizontal.

Problem 5 The water flow rate through the siphon is 5 L/s, its temperature is 200C, and the pipe diameter is 25 mm. Compute the maximum allowable height, h, so that the pressure at point A is above the vapor pressure of the water. (Assume the flow is frictionless.)

A

h

D = 25 mm Flow...