Pressure distribution in a fluid:

Transcription

1 18/01/2016 LECTURE 5 Preure ditribution in a fluid: There are many intance where the fluid i in tationary condition. That i the movement of liquid (or ga) i not involved. Yet, we have to olve ome engineering problem involving tationary liquid (imilar to what you have tudied in tatic of the coure Engineering Mechanic). The condition of fluid, for uch cae i termed a hydrotatic. Recall, a fluid at ret cannot reit hear. Therefore, the normal tre on any plane through a fluid element at ret i called fluid preure, p. (Recall you have tudied normal tre & hear tre in Solid Mechanic coure). Thi normal tre, when fluid i at ret, i taken poitive for compreion (conventionally taken through & acro variou coure). To decribe about thi preure, let u decribe through equilibrium of forced in a mall wedge of fluid at ret. ( again recall principle of tatic) Δy Δ Δ θ b For a body in tatic condition you hould atify F = 0 Fy = 0 F = 0 The above fluid element i alo in tatic equilibrium. A thi element i iolated from it urrounding, we need to incorporate correponding force (Free body diagram).

2 By convention the compreive force may be acting on the wall of the fluid element, i.e., force act into the plane in repective direction. Let p be the force per unit area acting on the plane b. Let p be the force per unit area acting on the plane b. Let py be the force per unit area acting on the plane. Let p be the force per unit area acting on the plane b. For F = 0; For F = 0; For Fy = 0; From (1), From (2), p b p b in 0 (1) 1 pb pb co gb 0. (2) py py 0.. (3) 2 2 p b p b in 0 p b p b p p 1 pb pb gb p p g 2 For limit ( 0; 0) p p So, p = p = p = py = p = preure At a tatic point, preure i a calar property without any orientation. P(,y,,t)

3 Preure force on fluid element: Due to preure, force act on the repective plane of interet. Let u conider a rectangular elemental prim of volume y Net force on an element due to preure variation (Source: Fluid Mechanic by F.M. White) Force due to preure i called preure force. At any mathematical point preure i decribed a p(,y,,t) a it i a calar quantity. Therefore, let u ugget that in direction there are two plane normal. On the left plane, let p be the preure. On the right plane, preure will be p So, net force due to preure acting in the direction will be: Similarly, net force in y direction: Fp py ( p ) y = - y Fpy y y

4 Similarly, net force in direction due to preure Fp y Force i a vector & the net preure force i given by, F F iˆ F ˆj F kˆ p p py p [ p ˆ i - p ˆ j p k] ˆ y y A the volume y i arbitrary & chooen by u, the net preure force per unit volume f p [ ˆ i - ˆ j k] ˆ y = = gradient of preure i.e., it i the gradient of preure that caue force in the fluid. Gage Preure & Vacuum Preure: In the phyic clae, you might have already een how preure i epreed. The unit are N/m 2 or Pa or kpa etc. Might have heard about 1. Abolute preure, 2. Gage preure, 3. Vacuum preure Hydrotatic Preure Condition: If a fluid i in ret, it will not have any hear force. Similarly, in a tatic liquid, there will not be any vicou force. For the ame rectangular prim element of the fluid in ret (or tatic): We hould apply principle of tatic F = 0

5 A the fluid i in tatic, the force acting on it will be preure force gravity force ρg g 0iˆ 0ˆj gkˆ i.e. F = 0 mean y 0 Or, 0 Similarly, Fy = 0 implie 0 y Similarly, F = 0 mean p y g y 0 Or, g It can be alo epreed a, F = 0, y gy 0 g i.e., g or p g In hydrotatic condition, it i now clear that, 0 and 0 y

6 That i, there won t be variation of preure in horiontal direction for the ame fluid. So, we can hence write dp g d a dp d in tatic condition. dp g d B A B dp gd A 2 B A p p g ( ) B A 2 1 p p g ( ) A B If B i water urface, then PB = atmopheric preure = 0 (gage preure) So, pa = ρg(2 1) = ρgh, where h = height of water urface level from urface.