Fluids-HW2

# Fluids-HW2 - 4 EE’Ill IZIT'ES FR TEI 986216625591...

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Unformatted text preview: 4 EE’Ill IZIT'ES FR TEI 986216625591 E'JZIl/IZIE SEP 1 - t. ttevmit the ﬂow between two parallel plates problem (HWI, prob 8) now that you‘ve learned the differential form of conservation of mass. A perfect ﬂuid is contained between two horizontal, parallel plates of length Land infinite extent into the page. The top plate moves toward the ﬁxed bottom plate at speed U from an initial height at time t—-—0 of h=ha Derive an expression for the horizontal velocity of the ﬂuid in the gap as a function of r, the distance from the center of plates. The assumption that the velocity proﬁle should be uniform at all cross sections is still valid. ‘ 2. Rewrite the following in both vector and tensor notation. Prove that these equations are correct using index notation: a. div(¢v)=¢divv+v*grad¢ b. div(uxv)=v-curIu-u-curlv c. curl(nxv)=v~gradu-u-gradv+udivv—vdivn 3. Re-Writc, using index notation, the general integral relations for: a. conservation of mass b. conservation of linear momentum c. conservation of angular momentum 4. Show that Eianijm : 25m in 3 dimensions. 5. Considering the differential continuity equation in cylindrical and spherical coordinates, Write out the general form of a purely radial incompressible flow in each coordinate system. 6. The incompressible ﬂow around a circular cylinder of radius n; is given in cylindrical coordinates as 7r/2, Snwith B=nl2 as rs SrERn, Sm withr=ro as—rt/Z S 65 m2, Swwith Elmer/2 as I.) 5 r 5 R0. These surfaces form a control volume. Compute the following quantities and explain their physical signiﬁcance: (a) LP pvde , (b) L pnividS , (c) in nxpdS, (:1) [SH pawns, (c) {Sm was, (1?) {SW apes. ' D 986B6685591 P.82/82 ‘ _ H _ ‘ .. _l___.....- u...- rvluwll-J' alaupuuculﬁ Elven Dy 1 ’7' 3 3 7" l r 3 3 r ur=UcosB 1+—- i -— i] and uangina -14... _DJ+_[.1] 2 r 2 r 4 r 4 r Compute all the components of the viscous stress tensor in r, 6, ¢coordinatcs. 3. In the problem above ﬁnd the maximum or and compare it with the dynamic pressure head rape: by forming their ratio. What inference can you make about low Reynolds number ﬂows ﬂ'om this ratio? 9. Consider the following incompressible plane unsteady ﬂow 2 u,=0 and u9=E[l—exp[—;—w-I‘ r where C is a constant, the kinematic viscosity vis constant, and gravity is neglected. Plot some velocity proﬁles for this ﬂow. Demonstrate whether this is an exact solution of the continuity and conservation of linear momentum equations or not. Hw1,?aue 3' I a ‘. u s. ammoniaismmmed ‘ a y , , a , hemthohonmal, “stakes Dfla'nsih I. and a extent into the page; mny'pnemj toward‘the ‘ ﬁxed batten: plate at speed U. ‘ DetfiVe ahfexpressionfor the ‘ Yeloeityjofthe-ﬁhidinthe slot as assessors. the-distance Wimmcﬁlﬂm- ** TDTHL PHEE.E2 ** ...
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## This note was uploaded on 10/01/2011 for the course ME 509 taught by Professor Wereley during the Spring '11 term at Purdue.

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Fluids-HW2 - 4 EE’Ill IZIT'ES FR TEI 986216625591...

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