Question

Problem 2: Using a control volume in Cartesian coordinates, derive the 2-D momentum equation in the y direction. (20 points)

Answer 1

Conservation of Momentun :- a) It is conserved with b) stress is farce per force balance a force wit area 8 since and area are vector both a tensor - stress is Nomenelature: a) In general storess Tij is to aseumed to ith plane & jth disceetion b) TijdAi u in "tve" j direetien far tve and in "-ve" j direction for -ve dAi act on Tzy dudy -> > Tyy dxdz Ex Txydy dz In the control volume shoon with sides dx,dy dz notice that positive disreetiah for z =0, plane t towards left & for Z= dz plane is towands seight This is coulistent with noment leture ado pted: for Tz dxdy as showh. we will desive the balance equation for a small control valume. in cartesian coorrdinates. Mometum Now of

> The primci ple of Momentum is:- accumlation Momentum = Net grate of influx of Mometum due to vebcity t Net srate of influn due to fortes Rate oF Considesn Y disection The for ces due to stresses are written taces. The tay lar series exponion with leading term are taken for y= dy & Z =dz face exit but are omitted foces. Terms on Teydudy)dz E Tzydudy + Ty dndz +2 (tydelaldy Tyy Tzy disrection : force (ty) in ( Ty ty Ty dy -Ty)dudz +(Tay t Zeydz-Tryldndy +(oyt Iseydx- Txy) dy dz Эй Simplify above cxpression gives : - 6 - Ty dudy dz + zyda dy dz + * Tuydudy de Similarly foxe can be derived in 8Z direetion.