CHAPTER 5 THE DIFFERENTIAL EQUATIONS OF FLOW

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1 CHATER 5 THE DIFFERENTIAL EQUATIONS OF FLOW In Chape 4 we sed he Newon law of consevaion of ene and he definiion of viscosi o deemine he veloci disibion in seadsae nidiecional flow hoh a condi. In his chape we shall eamine he applicaion of he same laws in he eneal case of heedimensional nsead sae flow. We will do so b developin and solvin he diffeenial eqaions of flow. These eqaions ae ve sefl when deailed infomaion on a flow ssem is eqied sch as he veloci empeae and concenaion pofiles. The diffeenial eqaions of flow ae deived b considein a diffeenial volme elemen of flid and descibin mahemaicall a) he consevaion of mass of flid enein and leavin he conol volme; he eslin mass balance is called he eqaion of conini. b) he consevaion of momenm enein and leavin he conol volme; his ene balance is called he eqaion of moion. In descibin he momenm of a flid we shold noe ha in he case of a solid bod is mass is eadil defined and has he dimension M; he same is e fo is momenm which has he dimensions of M L. Howeve in he case of a flid we ae dealin wih a coninm and he onl wa o define mass a an iven locaion is in ems of mass fl i.e. mass anspo ae pe ni coss secional aea hoh which flow occs. This qani is eqal o he podc of densi of he flid imes is veloci () 4

2 and has he dimensions of M L. Fo he same easons he momenm of a flid is epessed in ems of momenm fl ( ) i.e. anspo ae of momenm pe ni coss secional aea (M L ). In heedimensional flow he mass fl has hee componens () and he veloci also hee ( and ); heefoe in ode o epess momenm we need o conside a oal of nine ems. 5. THE EQUATION OF CONTINUITY Le s conside an infiniesimal cbical elemen d.d.d (Fi. 5.) in he heedimensional flow of a flid in a vessel. The mass consevaion of flid pas hoh his elemen is saed as follows: ae of mass accmlaion wihin elemen = = anspo ae of mass in anspo ae of mass o (5..) The anspo ae of mass ino he elemen a locaion and hoh he face d.d of he elemen is eqal o mass fl of flid in diecion sface aea of face d.d i.e. ( ) (d d) (5..) Similal he anspo ae of mass o of he elemen a locaion d and hoh he face d.d is epessed b ( ) (d d) ( ) d (d d) (5..3) d The onl wa ha mass can be accmlaed (o depleed) wihin he conol volme is b a coespondin chane in he flid densi. Mahemaicall he ae of accmlaion is epessed as follows: (d d d) (5..4) B sbsiin fom (5..)(5..4) in he consevaion eqaion (5..) and eliminain edndan ems we obain 43

3 44 = (5..5) In he above deivaion we sed paial diffeenials becase we deal onl wih he componen of veloci. Of cose he same consideaions appl o he and componens. Theefoe he eneal eqaion of conini in heedimensional flow is epessed as follows: = (5..6) o in abbeviaed veco noaion =. (5..7) whee is he veloci veco and. is called he diveence of he qani. B diffeeniain he podc ems on he ih hand side of (5..6) and hen collecin all deivaives of densi on he lef hand side we obain: = (5..8) The lef side of (5..8) is called he sbsanial ime deivaive of densi. In a phsical sense he sbsanial ime deivaive of a qani deaes is ime deivaive (i.e. ae of chane) evalaed alon a pah ha follows he moion of he flid (seamline Chape 6). In eneal ems he sbsanial ime deivaive of a vaiable w is defined b he followin epession: D(w) w = D w w w (5..9) Theefoe (5..8) can be epessed in abbeviaed veco noaion as follows:

4 45 D =. D (5..0) Eqaions (5..8) and (5..0) descibe he ae of chane of densi as obseved b someone who is movin alon wih he flid. Fo sead sae condiions hee is no mass accmlaion and he eqaion of conini becomes = 0 (5..) and fo an incompessible flid (i.e. fo neliible vaiaion in densi of flid) = 0 (5..) 5. THE EQUATION OF MOTION To develop he eqaion of moion we sa fom he Newon law of consevaion of ene: ae of momenm accmlaion = = anspo ae of momenm in anspo ae of momenm o sm of foces acin on elemen (5..) In ode o epess his eqaion mahemaicall we ms conside ha momenm is anspoed in and o of he elemen in wo was: a) convecive anspo of momenm is b means of he kineic ene of he flid mass movin in and o of he si faces of o cbical elemen. Ths he convecive anspo of momenm in he diecion (momenm) consiss of hee ems: ) ( ) ( ( )

5 46 The ne convecive anspo of momenm hoh he faces d.d of he cbical elemen d.d.d (Fi. 5.) is [( ) ( ) ](d d)= (d d d) (5..) d Similal he ne convecive anspo of momenm hoh he faces d.d and d.d of he elemen is epesened b he ems (d d d) (5..3) (d d d) (5..4) Of cose hee ae si moe smmeical ems fo convecive anspo of momenm in he and diecions of flow. b) diffsive anspo of momenm also akes place a he si sfaces of he cbical elemen (Fi. 5.) b means of he viscos shea sesses. Ths he ne diffsive anspo of momenm in he diecion hoh he faces d.d of he elemen is [( ) ( ) ](d d)= (d d d) (5..5) d Similal he ne diffsive anspo of momenm in he diecion hoh he faces d.d and d.d is epessed b he ems (d d d) (5..6) (d d d) (5..7) Thee ae si moe smmeical ems fo diffsive anspo of momenm in he and diecions of flow. Le s now conside he foces acin on he conol volme of Fi.

6 5.. The ne pesse foce acin on he faces d.d of he elemen (i.e. he faces nomal o he diecion) is ( )(d d)= (d d d) (5..8) d Thee ae wo moe smmeical ems fo he pesse on he faces d.d and d.d of he elemen. The ohe foce acin on he elemen is avi; his is a bod foce and is eqal o he densi of he flid imes he volme of he elemen (i.e. is mass) imes he aviaional acceleaion. In he diecion he avi foce is epessed as follows: ( )(d d d) (5..9) whee is he componen of he aviaional acceleaion in he diecion. Thee ae wo moe smmeical ems fo he and he componens of avi. In developin he eqaion of conini (see (5..6)) we showed ha he ae of accmlaion of mass in he conol elemen was eqal o he ime diffeenial of densi imes he volme of he elemen. Similal he ae of accmlaion of momenm in he diecion is epessed b 47 (d d d) (5..0) Thee ae wo moe smmeical ems fo he and diecions of momenm. We now have mahemaical epessions fo all he ems of (5..). B sbsiion and eliminaion of edndan ems we obain he followin eqaion fo he eqaion of moion in he diecion of momenm:

7 48 in he diecion of momenm: and in he diecion of momenm: The above eqaions of moion can be epessed moe convenienl in veco noaion as whee he bold pe indicaes veco qaniies. I shold be noed ha if in addiion o avi hee wee ohe "bod" foces acin on he flid (e.. an elecomaneic foce) hei effec wold be added o he aviaional componens. The vecoial shea sess in (5..4) epesens si shea sesses (i.e. acin in he diecion of flow) and hee nomal sesses (i.e. (5..) = (5..) = (5..4).. =

8 49 compessive o ensile sesses nomal o he diecion of flow). Mahemaicall hese sesses ae epessed as follows (3): Shea sesses: Nomal sesses: Fo incompessible flow (i.e. when chanes in densi ae neliible) we can diffeeniae each componen of convecive momenm in he eqaion of moion ((5..)(5..3)) as follows: B eliminain he followin ems (see (5..)) and movin he emainin convecive momenm ems o he lef hand side of (5..)(5..3) we obain he followin simplified eqaion fo he momenm balance = = = = = = µ µ µ _ (5..5). 3 = _. 3 = _ µ µ µ. 3 = (5..6) = = 0

9 50 = (5. The lef hand side of (5..7) will be econied b he eade as he sbsanial ime deivaive of veloci as defined ealie (see (5..8)). Theefoe ( 5..7 fo all hee dimensions is epessed as follows in veco noaion: D =. (5..8) D Table 5. shows he eneal eqaions of moion fo incompessible flow in he hee pincipal coodinae ssems: ecanla clindical and spheical. The anles shown in he las wo ssems ae defined in Fi I can be seen ha he complei of hese eqaions inceases fom ecanla o spheical coodinaes. The eason is obvios: In ecanla coodinaes he cosssecional aea of flow does no chane in all hee dimensions; in clindical coodinaes his aea does no chane in he dimension; and in spheical he cosssecional aea of flow chanes in all hee dimensions.

10 5 TABLE 5.. Geneal eqaions of moion Recanla coodinaes (): Clindical coodinaes ( ): Spheical coodinaes ( ): Fo incompessible Newonian flids a consan viscosi he eneal eqaions of moion can be simplified fhe b eplacin he shea sess fncions b he Newon law of viscosi. Ths ( 5..6 becomes = = = ) ( = ) ( = = ) ( ) ( = = co = co

11 5 = µ (5 This eqaion and he smmeic eqaions fo he and componens of momenm ae called he NavieSokes eqaions of flow. In veco noaion he ae epessed as: D = µ D (5..9) In cases whee he viscos effecs can be consideed o be neliible he NavieSokes eqaions ma be wien as follows: D = D (5..) This eqaion is known as he Ele eqaion. In he case of ve slow moion sch as he flow of lass in a melin fnace he ineia ems in he NavieSokes eqaions ma be neleced o ield: = µ (5..) In he followin chape we shall illsae he manne in which he diffeenial eqaions of flow can be solved o povide deailed infomaion on he micosce of flow ssems. REFERENCES. R.B. Bid W.E. Sewa and E.N. Lihfoo Tanspo henomena Wile. New Yok N.. Cheemioff and R. Gpa edios Handbook of Flids and Moion Bewohs Boson (983).

12 3. H. Schlichin Bonda Lae Theo eamon ess s ediion p. 48 (955). 53