Kinematics of Fluid Motion

Transcription

1 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 s of lmte scope an for more etals, the reaer s encourage to consult any of the references lste at the en. Our focus here s on flu moton. We shall use rectangular Cartesan coornates,,, jk,. ( y, ) along wth the assocate bass set of mutually orthogonal unt vectors ( ) The poston vector s labele. Imagne a tny lne element, labele PQ n the sketch, at some nstant of tme. After a small amount of tme t, the two ens have move to new locatons because of flu moton, an the new lne element s labele PQ. Q Q P P We can see that f the velocty were to be the same at both ens of the element, t woul change nether ts length, nor ts orentaton. Therefore, n a unform velocty fel, there s smple translaton of flu elements wth no eformaton or rotaton. To cause ether, v must be non-unform. To unerstan the nature of the changes n flu the velocty ( ) elements brought about by the flow, we must, therefore, nvestgate the velocty graent, v, whch s a secon orer tensor. From calculus, we know that the fferental change y v = + y + v can be wrtten as an smlar results can be wrtten for the changes v y an fferental change n the vector velocty, v, s gven by v. It follows that the

2 v = v + j v + k v y y y = + y + + j + y + + k + y + y y y y v y v y = + j + k + + j + k y + + j + k y y y = + y + = v y Thus, the relatve velocty of a pont a stance from any gven locaton s gven by the ot prouct of the tensor v an the fferental lne element. Ths tensor can be wrtten as follows. v y y y y = y y Any tensor can be wrtten as the sum of a symmetrc an an antsymmetrc tensor. Let us o ths wth the velocty graent tensor, wrtng t as v = E + Ω where the (symmetrc) rate of stran or rate of eformaton tensor E s gven by E= v+ v T ( ) an the (antsymmetrc) vortcty tensor Ω s gven by Ω = T ( v- v ) The acton of each of these contrbutons to the velocty graent wll be eplore n etal net. Frst, we conser the vortcty tensor. Vortcty Tensor Ω The vortcty tensor Ω s a skew-symmetrc tensor. We can wrte ts components n terms of the components of the velocty graent as follows.

3 y 0 y y y Ω = 0 y y y 0 y A skew-symmetrc tensor Aj can be forme from a vector a k by wrtng Aj = εjkak. The vector assocate wth the vortcty tensor n ths manner s ω, where ω = v s known as the vortcty vector. Usng the relatonshp between Ω an ω, we obtan Ω j j = εjk jωk or n Gbbs notaton, Ω = ω= ω Ths means that the relatve moton that s contrbute by the vortcty tensor at a pont an nfntesmal stance away from a reference pont n a flu s that cause by a rg rotaton wth an angular velocty equal to ω. Because a flu oes not usually rotate as a rg boy n the manner that a sol oes, we shoul nterpret the above statement as mplyng that the average angular velocty of a flu element locate at a pont s one-half the vortcty vector at that pont. To prove ths clam, conser a surface forme by an nfntesmal crcle of raus a locate at a pont. Let the unt normal vector to the surface (perpencular to the plane of the paper) be n, an the unt tangent vector to the crcle at any pont be t. n a t Apply Stokes s theorem to the velocty fel n ths crcle. S ( ) v ns = v t s C 3

4 Here, S s an area element on the surface of the crcle S an s s a lne element along the crcle C. Because v t s the component of the velocty along the perphery of the crcle, we can wrte the average lnear velocty along the crcle as s π a v t an therefore the average angular velocty as π a C C v t s. From Stokes s theorem, we see that ths s equal to the average value of v n over the surface of the crcle. Thus, n the lmt as the raus of the crcle approaches ero, we fn that the average angular velocty aroun the crcle approaches the value of one-half the component of the vortcty vector n a recton perpencular to the surface of the crcle. We also can show (see Batchelor, page 8) that the angular momentum of a sphercal element of flu s equal to one-half the vortcty tmes the moment of nerta of the flu, just as t s for a rg boy. Vortcty Vector The vortcty ω = v, s an mportant entty n flu mechancs. It s transporte from one place to another n a flu by convectve an molecular means, just as energy an speces are, an an approprate partal fferental equaton that governs ts transport can be wrtten. In aton, vortcty also s ntensfe by the stretchng of vorte lnes, a mechansm that s not present n the transport of energy an speces. One reason for workng wth the equatons of vortcty transport s that pressure s absent as a epenent varable n those equatons. It can be shown that f a flu mass begns wth ero vortcty, an the flu s nvsc (meanng the vscosty s ero), the vortcty wll reman ero n that flu mass. A flow n whch the vortcty s ero s known as an rrotatonal flow. Vortcty s generate at flu-sol nterfaces an at flu-flu nterfaces. Vortcty cannot be generate nternally wthn an ncompressble flu. Ths s the reason why, n a hgh Reynols number flow (mplyng weak vscous effects) past a rg boy, most of the flow can be escrbe by usng the equatons that apply to rrotatonal flow, wth the vortcty beng confne to a bounary layer near the surface of the boy. Vorte Lnes an Tubes Just as a streamlne s a curve to whch the velocty vector s tangent everywhere, we can efne a vorte lne as a curve to whch the vortcty s tangent everywhere. If the components of the vortcty = v ω, ω, ω, then we can wrte the equatons of the space curves that are vorte lnes as y = =. ω ω ω y ω are ( y ) 4

5 The surface that s forme by all the vorte lnes passng through a close reucble curve s known as a vorte tube. If we construct an open surface S boune by ths close curve C, we can efne the strength of the vorte tube as theorem, we can see that ths s the crculaton C S ω. By usng Stokes s S v t s where C s any close curve aroun the vorte tube, t s a unt tangent vector to the curve at any pont, an s s a lne element. Rate of Stran or Rate of Deformaton Tensor E From the above scusson of the vortcty tensor, you can see that the role of that tensor s to escrbe the nstantaneous angular velocty of a flu element, but that t contrbutes nothng to eformaton of elements. Now, we move on to scuss the sgnfcance of the rate of stran tensor, whch contans all the nformaton about the eformaton. The rate of stran tensor s a symmetrc tensor. We can wrte ts components n terms of the components of the velocty graent as follows. y + + y y y y Ε = + + y y y y + + y The agonal elements of E Conser a lne element wth a length s. t t ( s ) = ( ) Usng the fact that t = v, the above result can be rewrtten as s ( s) = v= ( v ) = E + Ω t The secon term n the far-rght-se s ero because Ω s an antsymmetrc tensor. To see ths, we wrte 5

6 Ω = Ω = Ω = Ω so that Ω = 0. j j j j j j j j In the above result, after wrtng the result n ne notaton, we frst echange the nces an j to obtan an ntermeate result, an then use the antsymmetry property to wrte Ω = Ω. j j Therefore, we fn that s ( s) = E from whch, by vng through by t ( ) s = E s t s s s we can wrte The vector / s s a unt vector pontng n the recton of the nfntesmal vector. Therefore, we can thnk of the rght se of the above result as the ouble projecton of the tensor E n that recton. The term projecton s use n a loose sense here. The physcal meanng s clear. The rate of stran of a lne element pontng n any recton at a gven pont (whch s the tme rate of change of length, ve by the length) s the ot prouct of a unt vector n that recton wth the ot prouct of the rate of stran tensor wth the same unt vector. Let us choose the recton to be the recton. In ths case, the rate of stran of a lne element n that recton s smply E, whch s equal to. In a lke manner, the rate of stran of a lne element n the y recton s y, an that n the recton s. Ths s the physcal y nterpretaton of the agonal elements of the rate of stran tensor. The sum of the agonal elements of E, known as the trace of E s v. Ths s known as the rate of lataton of a flu element at the gven locaton. To see why, conser a materal boy occupyng a volume V enclose by the surface S. Let us nqure how V changes wth tme. We can wrte the rate of change of the volume of a materal boy wth tme as the ntegral of S v over the surface. V t = S v = v V by the vergence theorem. S V From the above, we can see that V lm = lm V = V t V v v. So, the trace of V 0 V 0 E s the rate of ncrease n the volume of an nfntesmal element, ve by ts volume, an s calle the rate of lataton. When the flow s ncompressble, the rate of lataton s ero. V 6

7 The off-agonal elements of E Now, conser two lne elements an at a gven pont an let the angle between them be θ. Q P θ Q Let us nvestgate the tme rate of change of the ot prouct of the vectors an. t s s cosθ = = v + v t ( ) ( ) = j + j j j In wrtng the result n the secon term n the secon lne, we have use the fact that the nfntesmal change v s the change n the velocty over an nfntesmal stance n the recton of the vector. Interchangng the nces an j n that secon term permts us to combne the two terms. j ( s s cosθ ) = + = E t j Dvng both ses by s s yels j j j j ( s s cosθ ) = Ej = E s s t s s s s So, we see that f we take the ot prouct of E wth unt vectors n two fferent rectons n successon (the orer s mmateral because E s symmetrc), the result s the left se of the above equaton. Let us work out the fferentaton n the left se. θ ( s s cosθ) = cosθ ( s) + ( s ) snθ s s t s t s t t The term n square brackets n the rght se s the sum of the nvual rates of stran of the two lne elements. We can see that the above result reuces to the earler result we obtane when the two vectors an are the same. Let us conser the case when the two vectors are orthogonal to each other. In ths case, we obtan 7

8 E = s s θ t So, the sequental ot proucts of E wth unt vectors n two orthogonal rectons yels one-half the rate of ecrease of the angle between unt vectors n those rectons. If we choose these two orthogonal rectons to conce wth any two coornate rectons, then the ot proucts yel the off-agonal elements of E. For eample, f we use E = E. Smlar physcal nterpretatons can be an y rectons, the element s ( ) gven to the other off-agonal elements of the rate of stran tensor. Thus, the offagonal elements escrbe shear eformaton of the flu. There are three mutually orthogonal rectons assocate wth the symmetrc tensor E that are known as ts egenvector or prncpal rectons. We can use a bass set bult from these prncpal rectons to escrbe the components of the tensor. If we o, the tensor wll be agonal. The off-agonal elements wll be ero, so that the rate of change of the angles between the prncpal rectons s ero; of course the entre set of prncpal aes can rotate, an n fact t oes, wth the angular velocty ω= v. Instantaneous Deformaton of a Flu Element Base on all of the above materal on knematcs, we can conclue that n a flow, an nfntesmal sphercal element of flu unergoes translaton, rotaton, an eformaton n general. It eforms nto an ellpso whose aes are algne wth the prncpal aes of the rate of stran tensor. Ths ellpso also rotates wth an nstantaneous angular velocty that s equal to the one-half of the vortcty of the flu at the gven pont. Some goo sources for further stuy are lste below. References. R. Ars, Vectors, Tensors, an the Basc Equatons of Flu Mechancs, Dover Publcatons, 989, Chapter 4 (orgnal by Prentce-Hall, 96).. G.K. Batchelor, An Introucton to Flu Dynamcs, Cambrge Unversty Press, 967, Chapter C. Truesell, Knematcs of Vortcty, Inana Unversty Press, J. Serrn, Mathematcal Prncples of Classcal Flu Dynamcs, Hanbuch er Physk VIII/, E. S. Flugge, P.G. Saffman, Vorte Dynamcs, Cambrge Unversty Press, 99. 8