![By makinga mass balance overa volume element(Ar)(rAO)(Az)derive the equationof continuityin
cylindrical coordinates.
What isthe physical meaningof the termAx Ay(pv,v,)l,inEq.3.2-2? What isthe physical
meaningof [V.pwl?
rate of momentum rate ofmomentum additionbyconvectio per unit volume
[ .ρv v]isa vectorcalledthe "divergence of (thedyadicproduct) pw . "
What happenswhenf issetequal to unityinEq. 3.5-4?
EquationB inTable 3.5-1 isnot restrictedtofluidswithconstantdensity,eventhoughpisto
the leftof the substantial derivative.Explain.
In the tangential annularflowprobleminExample3.5-3,wouldyouexpectthe velocityprofilesrelative
to the innercylindertobe the same in the followingtwosituations:(i)the inner
cylinderisfixedandthe outercylinderrotateswithanangularvelocity fl;(ii)the outercylinderisfixed
and the innercylinderrotateswithanangularvelocity -a?Bothflowsare presumedtobe laminarand
stable.
Suppose that,inExample 3.6-4,there were twoimmiscibleliquidsinthe rotatingbeaker.
What wouldbe the shape of the interface betweenthe twoliquidregions?
Wouldthe systemdiscussedinExample3.6-5be useful asaviscometer?
In Eq. 3.6-55, explainbymeansof acarefullydrawnsketchthe choice of limitsinthe integrationandthe
meaningof eachfactor inthe firstintegrand.
What factors wouldneedtobe takenintoaccountin designingamixingtankforuse onthe
moonby usingdata froma similartankon earth?](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Questionsfor discussion
- 1. QUESTIONSFOR DISCUSSION What isthe physical meaningof the termAx Ay(pv,)l, inEq.3.1-2?What isthe physical meaningof (V.v)?of (V- pv)? Here (V .pv) is calledthe "divergence of pv,"sometimeswrittenas"divpv."The vector pv isthe mass flux,anditsdivergence hasasimple meaning:itisthe netrate of mass efflux perunit volume.The derivationinProblem3D.1usesavolume elementof arbitrary shape;itis notnecessaryto use a rectangularvolume elementaswe have done here. A veryimportantspecial formof the equationof continuityisthatfora fluidof constantdensity,for whichEq. 3.1-4 assumesthe particularlysimpleform (incompressible fluid) (V .v) = 0 (3.1-5) Of course,nofluidistrulyincompressible,butfrequentlyinengineeringandbiological applications,the assumptionof constantdensityresultsinconsiderable simplification and verylittle error.'r2
- 2. By makinga mass balance overa volume element(Ar)(rAO)(Az)derive the equationof continuityin cylindrical coordinates. What isthe physical meaningof the termAx Ay(pv,v,)l,inEq.3.2-2? What isthe physical meaningof [V.pwl? rate of momentum rate ofmomentum additionbyconvectio per unit volume [ .ρv v]isa vectorcalledthe "divergence of (thedyadicproduct) pw . " What happenswhenf issetequal to unityinEq. 3.5-4? EquationB inTable 3.5-1 isnot restrictedtofluidswithconstantdensity,eventhoughpisto the leftof the substantial derivative.Explain. In the tangential annularflowprobleminExample3.5-3,wouldyouexpectthe velocityprofilesrelative to the innercylindertobe the same in the followingtwosituations:(i)the inner cylinderisfixedandthe outercylinderrotateswithanangularvelocity fl;(ii)the outercylinderisfixed and the innercylinderrotateswithanangularvelocity -a?Bothflowsare presumedtobe laminarand stable. Suppose that,inExample 3.6-4,there were twoimmiscibleliquidsinthe rotatingbeaker. What wouldbe the shape of the interface betweenthe twoliquidregions? Wouldthe systemdiscussedinExample3.6-5be useful asaviscometer? In Eq. 3.6-55, explainbymeansof acarefullydrawnsketchthe choice of limitsinthe integrationandthe meaningof eachfactor inthe firstintegrand. What factors wouldneedtobe takenintoaccountin designingamixingtankforuse onthe moonby usingdata froma similartankon earth?