Question

Water can be considered as a non-viscous incompressible fluid of density p. A laboratory set-up is such that water flows through a pipe, exhibiting a laminar and steady-state flow. At the top end of the pipe, the flow tube has a cross-sectional area A and point 1 (located on the central streamline) is exposed to the ambient environment. The pipe drops through a A vertical distance h7 while its area decreases to when it reaches point 2 (also on the central streamline). A small vertical tube extends from the lower portion, as shown in Fig.2, and is open to atmospheric pressure. Point 1 A = 4 Point 3 v=V3 hi h2 ruler y Point 2 A2 = 1.4 Bucket х a) Calculate the ratio between the two speeds vị and v2 of the water, at point 1 and point 2 respectively. Explain which conservation principle the continuity equation reflects. b) Given the height h2 of water measured in the narrow tube at the low end of the pipe above point 2, determine the static pressure of the fluid right at the bottom of the vertical standpipe. c) Assuming that the pressure at point 2 can be approximated by the pressure at the bottom of the standpipe, determine the speed v2 of the fluid in the narrower section of the tube. Express your answer in terms of hi, h2 and any relevant physical constants. Explain which conservation law is reflected by Bernoulli's equation. Beyond point 2, the tube continues to taper down until point 3, with a slight vertical drop. However the cross-sectional area and height are unknown, so the speed v3 at which the water flows out of the tube will be determined differently. Using this set-up and a bucket that collects the water at the lower end of the pipe, we can measure experimentally the horizontal distance x at which the water reaches the top of the bucket, for a given height y between the lower end of the pipe and the top of the bucket. d) Determine the velocity v3 of the water as it exits the horizontal pipe in terms of x, y and 8.

Answer 1

a) A, v = A₂ V2 V 1 2 mом. Av = ( 4 ) Vz Continuity equations reflects law of conservation of P. = atmospheric pressure] b) Pr = Po tlghz water density c) P + Pg H, + Levy? = P₂ + Pg Hz + į pre ² Po t eg (H, -H₂) Po + ska) +że(v Igh, Ighz + 3 lv = 2 V₂ 2 = e alt, he d) Time of flight = / 27 x V₂/24 g > V₃ = х 2y Bernoulli's equation represents law of conservation of energy.