INTERMEDIATE FLUID MECHANICS

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Bruce Spencer
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1 INTERMEDITE FLUID MEHNIS enot shman-rosn Thaer School of Engneerng Dartmoth ollege See: Kn et al. Secton 3.4 pages 76-8 Lectre : Stran Vortct rclaton an Stress The ector eloct fel has 3 components each epenng on 3 spatal coornates. There s therefore a total of 9 possble spatal erates characterng the araton of the flo across space: r It can be shon that ths 33 table forms a 33 tensor. In other ors t possesses the reqre nmber of narants th respect to rotaton of the coornate aes. One mportant narant s the trace hch s the sm of the agonal elements: hch s none other than the ergence of the flo fel. r 3 3 th mple sm oer repeate nces

2 Of nterest also are: the smmetrc part of the tensor: the ant-smmetrc part of the tensor: R S In D th an these sb-tensors are: S R 3 stnct components onl component 6 components n 3D 3 components n 3D Phscal nterpretaton represents stretchng or sqeeng n the -recton represents stretchng or sqeeng n the -recton Smlarl + f f + < tme tme

3 The terms an relate to eformaton: t t Pont moes from to t t Pont moes from to Pont moes from to t t t t t t t t tan t t t Ths e see that: epresses trnng aa from the -as t t tan t t t epresses trnng aa from the -as. 3

4 4 t t t t No e see hat the combnatons of erates represent: For pre rotaton t t t an Eample of sol bo-rotaton: Therefore the epresson th the fference measres the amont of trnng n the flo. t t t n n the opposte case hen there s no net rotaton: Therefore the epresson th the sm measres the amont of stran case b the flo. In three mensons there are three stran-rate components: In a stranng flo fel an ntall sphercal fl parcel storts nto an ellpso. The three stan rates correspon to the rates of elongaton or shortenng of each maor as.

5 5 onser no all three possble ant-smmetrc components of the erate tensor: These can be collecte no longer n a tensor bt n a ector: = crl of the eloct ector s calle the ortct ector. It measres the amont of trnng n the flo. In D the ector ortct has a sngle component stckng ot of the plane of flo: General propert of the ortct ector: ecase the ortct ector s the crl of a ector ts ergence anshes. Inee = Note: No phscal prncple has been noke here onl mathematcs. Ths means that the ortct ector no matter hat the flo ma be alas carres thn tself a constrant. Ths s fferent from a ergence-free flo fel hch s the reslt of mass conseraton for an ncompressble fl.

6 Vortct an rclaton For an close loop n 3D e efne s rclaton s pplcaton of Stokes Theorem: The ntegral of a ector along a close cre = fl of the crl of that ector throgh the srface spanne b the close loop. Ths s nˆ nˆ nˆ nother of pttng ths s to sa that the crclaton s eqal to the ortct fl. pplcaton to sol-bo rotaton Flo fel s: long crclar loop of ras r the amthal eloct s V r It follos that the crclaton aron that loop s s V r V r r The ortct ot of the plane s: n the fl of ortct s r r Same as crclaton nee. 6

7 Proof n the flat plane: For a lttle sqare D s s s D s s ortct area D n no for man sqares fllng the area nse the loop No o sm oer the man small rectangles. ontrbtons of all nteror ses cancel one another ot. Onl the contrbtons along the otse segments remans. Ths s Vorte Tbe Take a close loop --- an from each pont along t ra a lne segment algne th the local ortct ector. Termnate each lne arbtrarl. The en ponts form a ne loop an the set of ortct lnes from startng loop to enng loop forms a orte tbe. learl ortct s tangent to the lateral srface bt not to the base an top. No appl Stokes Theorem to the follong close contor hch ncles gong from a pont on the base loop p along a ortct lne - aron the top loop n the conterclockse recton back on the same lne n reerse - an fnall aron the base loop n the clockse recton ---. Ths path oes form a close loop. The crclaton along ths loop s nl becase the ortct s eerhere tangent to the srface spannng ths loop: s ' ' ' ' ' s s s ' ' ' ' s ' ' ' ' ' s s s s Helmholt Theorem 7

8 Stress Tensor fl s sbecte to stresses especall hen t flos. Stress s force per area lke pressre bt s more general. It can hae an recton on an srface on hch t bears. Ths on an srface of a fl parcel there s a ector stress. If e no conser a small cbe of fl e hae a most general stress confgraton as epcte n the se fgre th a three-component ector stress on three fferent srface orentatons. eng a ector of ectors stress forms a tensor: The agonal components an are normal stresses; all others are shear stresses. The stress tensor mst be smmetrc: = = = Inee f t ere not the fferences ol create anomalosl large torqes. Take for eample the torqe aron the -as hch s torqe The mass of the fl parcel s so that ts moment of nerta th respect to the -as s on the orer of. Snce the rotatonal acceleraton mst reman phscall fnte t s necessar that the torqe be on the same orer as the moment of nerta. Ths reqres that the stress fference be on the orer of an ths mch less than each term alone. The to terms mst ths be eqal th an possble fference attrbtable to spatal graents small change oer a small stance. 8

9 Pressre as part of the stress tensor We sa earler than pressre s one of the forces per area actng on a fl parcel. Ths pressre s a part of the stress. Snce pressre - s a normal stress - has the same ale rrespecte of recton an - s measre postel n compresson t follos that the pressre part of the stress tensor s of the form: pressre part of stress tensor p p p 9

Kinematics of Fluid Motion

Kinematics of Fluid Motion Knematcs of Flu Moton R. Shankar Subramanan Department of Chemcal an Bomolecular Engneerng Clarkson Unversty Knematcs s the stuy of moton wthout ealng wth the forces that affect moton. The scusson here