Question

(b) Figure Q2b shows two parallel plates with h apart, where an incompressible fluid is injected into the tiny holes as shown. Assume total volume flow rate injected is Q and width of the plates is w. The fluid is only flowing horizontally between the two parallel plates. Gravitational force may be neglected. Obtain an expression for the pressure variation using the control volume method. Provide explanation as you show your steps. (29 marks) V(x) Pi Q Figure Q2b

Answer 1

Solution- Given & Parallel & horizontal plates width=w ya dae contro volume" Let us consider control volume of height dy length da' and unit widt (et er dy) « daxi pxdgxT K (et af dai) x dy xl C. daxD za an * X da we know that for steady uniform flow acceleration in the direction of flow will be zero. p.dy. 1 - (Pt 31 da) dy. +(2+ 27. dy) dd-1-2-da-l sy =0 = - apda.dy taz.dy.dd=0 ay ottdy= 27 dy dal sy ap re za PT sy

dz de da Let us consider the flow be laminar wt, h ly we know that Modu de dy dx. a rede de dy dy len te पवर integrate on both sides with respect to = -1 de dy da g du de dp. da y.to Now, again integrate with respect both sides. TA u= dy for yz 2 te; y TC₂ Now, apply boundary conditions. At walls i.e. boundary i.e-plates gh velocity assumed to be zero.

at y=0 u=0 = > of a + G, Xota and at y = h = "=0 - I - de br. dre +Gchto 2 ap 54 Gt= da den = -1 21 dx. I deky u= I 210 dar 24 da il= 2re. gr - tas [hy - y Let us calcuate discharge formula for average velocity to be determined dy wik T- dQ= U.DA Q = 'u-da here dAz day. I

h O > Q= a-f 1 l - ap) (hy-yg-dy. I * = [ HT (-abh b2 ) - 12 Q-193-] are - aby b. (eds) n 2u 12 M Bnow, Average relocity Vary) (ell y A3 fde da 12r их) Vaug = a (a) where, h= distance between two parallel plates pe=dynamic viscopily of water on fluid. Pq - Pj BA Pe Pi aya da

where, L= Length of plates between point I and point 2 - dp PP2 da L { 8, > P2 12re because I denn Varg- Edel- 12 Mx Varg Flow is from si to P hr P, P₂ 12 px 'Vag hr IP, -P2 = 120 Vargit hr for our case, given that discharge Q and oldit of plates w Varga nXW 12 M. Q x L how 62 -:P, P₂ = 12.Q.L h². W Answer.