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2017 Exam.pdf - All questions may be attempted but only...

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All questions may be attempted but only marks obtained on the bestfoursolutions willcount.The use of an electronic calculator isnotpermitted in this examination.The fluid is incompressible, inviscid and has constant densityρ. Gravitational accel-eration is denoted bygthroughout.1.(a) Define the terms streamline, particle path and streakline.(b) A fluid moves two-dimensionally so that its velocityuis given byu= e2ti+ e3tj,whereiandjare the unit vectors for the Cartesian coordinates (x, y) andtistime.Obtain equations in terms ofxandyalone for the following:(i) the streamline through (1,1) at timet= 0,(ii) the particle path for a particle released from (1,1) at timet= 0,(iii) the streakline, at timet= 0, through (1,1) formed by particles releasedfrom (1,1) at timest60.(c) Sketchcarefullythe three loci describing the fluid motion identified in (b) onthe same diagram in the (x, y) plane. Where two or more curves coincide, markthe tangents to the curves. Indicate clearly the start and end points, or extent,of each curve.(d) Show that in steady flow the quantityH=p+12ρ|u|2+ρGis constant along streamlines.You may use the momentum equationρDuDt=-∇p+ρF,wherepis the pressure. HereFis a conservative external force per unit masswith potentialGdefined throughF=-∇G.You may also use the result(u.)u=12∇|u|2+ (∇ ×u)×u.

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