THE UNIVERS I T Y OF M I C H I GAN COLLEGE OF ENGINEERING Department of Engineering Mechanics Technical Report THE STABILITY OF PARALLEL FLOWS OF FLUIDS WITH MEMORIES Dean To Mook W. P. Graebel ORA Project 06505 under contract with: DEPARTMENT OF THE NAVY OFFICE OF NAVAL RESEARCH CONTRACT NO. Nonr-1224(49), NR NO. 062-342 CHICAGO, ILLINOIS administered through. OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR September 1967 Distribution of this document is unlimited.

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iv ABSTRACT v I. INTRODUCTION 1 II. GOVERNING EQUATIONS 8 IlIo ASYMPTOTIC SOLUTIONS 20 A. The Determination of the Characteristic Equation for a Na-vier-Stokes Li quid 20 B. The Determination of the Characteristic Equation for a Viscoelastic Liquid of Type A' or B' 29 Co Solution of the Characteristics Equation 31 Do The Stability of a Viscoelastic Liquid of Type C' 36 BIBLIOGRAPHY 42 APPENDIX 44 iii

LIST OF ILLUSTRATIONS Table Page lo Velocity and Stresses for Primary Flow of Materials A', B' and C' 10 Figure 1. Neutral stability curves for various values of \. 37 2. Critical Reynolds number versus the parameter Rlo 38 3. Neutral stability curves and curves of constant c for various values of \. 39 iv

The Stabilitty of Parallel Flows of Flu-id$ with Memories D0 T. Mook Virginia Polytechnic Institute and W, P., Graebel The University of Michigan Ab stract The equatione governing the sta&bittv of plane parallel flows are developed for three models of fluids with memories, Asymptotic solutions valiid for large Reynolds numbers are obtained and the effect of the memory are shown to be destablizing, Ttie approach to the problem allows evaluation of how fast a memory must fade to allow evaluation of the stresses in power series in the time interval, An alternate approach to inverting convected derivatives is also presented..V

L0 Introduction A recent paper by Chan Man Fong and Walters i 1965) considered the stability of parallel flows of two visco-elastic f~1uds with very short memories,. The present work extends their analysis to such fluids with long but still fading memory and also extends the analysis to a newer model which has been proposed by Goddard and Miller (1966). A discussion of the various convected derivatives is also presented in a manner which allows more ready physical interpretation as well as a quicker way of obtaintirng forms for convected integrals, Quasi- inear models of visco-elastic fluids are usually written in the forrm - A t Lt +) =d+i A r where is related to the total stress t by d is the rate of deformation tensors, and L is a tim-e derivative operator satisfying the prtncipal of material indifference, The constants X and T are related to the stress relaxation times and a m rate of deformation relaxation tmaes. respectively, Several forms for these operators have been proposed in the past (See Oldroyd 1968) for a review); we present here briefly three of these definitions in a somewhat different maanpe which facilitates their physical interpretation. If 0 is a convected material reference frame, and? s a set of covariant base vectors defined by (r a position srector) so that they are tangent to the frame bsee for examwple Sokolnikofi (1951)9 1~

Chapter 3), then a second order tensor can be written as or y being the contravariant base vectors defined by The absence of a dot or cross between two vectors indicates the indefinite, or dyadic, product. Oldroyd (1950) proposed two separate definitions for Lt; denoting time differentiation with material coordinates held fixed by D/Dt, they are -t^ TDt - - for the model he called type A, and L 5 e Dr ^ y y aa7J for the model he caled type B atio t indices and base vectors here refer to a space fixed reference frame. It is readily found by the normal tensor transformation laws that

GIA r ~JSVt* (d: D c J T, -) dT daT';' ~' _ (,,u + Evi T~ and that -o th ( c < cKs T; D t (- d Ad1 where g is the vorticity tensor, Thus Oldroyd's definition of the convected rate is the material rate of those tensor components which an observer would measure with respect to a coordinate system both rotating and deforming with tAh, materials fcr type A the components are measured wteh respect to the contravariant base vectors, white for type B the covariant base vecrtors are used, the base vectors in both cases being both stretched and rotated with the matera,!a Th-e pre se nt Zform of writing thfe conveacted der ivative allows ready inversion, for letting then s ince

and since the O are constant In, t.lme for a material particle, or and amailrly fiW the type B derivative, These integrala were preo.tod first by Oldroyd (1950); they were used by Walters (1962) In nmodels de.ignted as Al and B' by writing +pK i 2 8(a) d ^ (x',tJ dl' and -'j 1-IkI~itI L2 (',,d l (3) respcltively where (t) is a material relaxation function, * When 4(t) eonllst* of a co mbination of exponential), and Dirac delta functionsD *( it l. freqi.enty more convenient to w)rk with N9 the ditr ibution of r elaatilo tims t t defnedby i(t d r AIc(p ) eXpuht /p ) Jp/p )

equations (2) and (3) are exactly equivalent to equation (1); otherwise they are generalizations cf equation (1). -(For example, when N = M equation (1) is obtained by taking N(p) P S (p) +(l- Ao " A p-1. Jaumann (1911) proposed a different definition of L t which we shall designate as L tC Introducing new base vectors d( by where sBatisf ies the equation and reduces to the identity matrix as an initial condition, then9 denoting the inverse tensor with a minus unity superscript, Irr, T'P T,:-T. and as before it ie readily found from the transformat.ion laws that and that

Since by liccs' theorem and tho above rales g and lowering of tIndice commutes with the operatlOn of Jaauann differentia. tion. The above resalts can be put in a simnpler appearing form oy introduciag a further tensor defined by then - C)(4) and where R to equal initially to the identity matrixo Thus 1it the tensor rotating the material base vectors y into the material base vectors, a&nd R Is the tensor rotating the material base vectors rlt into the fLxed base vectors i A has been shown by Goddard and Mller (1965),. corresponds to an orthoional transformation. and hence its inverse and transpeose are equivalento Thus the Jaunann derivative s the material tlnme rate of those tensor components which

an observer would measure with respect.to the base vectors r rotating locally with the same rate as the vorticity, io e, moving with the principal axes of do The length of these base vectors changes also, but not directly with the material Inversion of the derivative again follows readily from the definition- if now then Yop y =dt fWfi A and. - i t =(. 5, R, (et' Ot' ~~ R RL~~~nRS~\(oW()Rllt~~~~~dE,)O 7~~~~~~~~~T'

the initial conditions on R.. being imposed at time t', A material of type C' could now be defined by -2.'(k- t'h a'd" (xj)gx (5) We note that equation (5) is the constitutive equation presented by Goddard and Miller (1966), their integration being presented by other arguments.. No simple relation has so far beeen found relating the various Oldioyd and Jaumann integrals. IL. Governing equations The solution for steady flow between stationary parallel plates is next presented for materials of type A', B', and C'. The stability problem for parallel flows is then formulated for each by superimposing a wavy infinitesimal disturbance on the primary flow and then determining under what conditions this disturbance will grow. Cartesian coordinates are used, the z-axis and the y-axis being chosen parallel and perpendicular to the plates, respectively. For the steady flow the velocity components are assumed in the form with W 0= at y +ko By inspection the motion is primed coordinates denoting the position of a particle at time t'. The only non-hero component of the rate of.deformation is 8

dz3 = ~ DWI where D represents differentiation with respect to y. The non-zero displacement gradients and rotation components are aI1; I - C) - as - t' )Ha R11= 1 ) Rz?33 Co$[IZt3Dt Ij a<d R 3 =; R,,- - so (t Ad (tt-i.)V I Substitution of these into equations (2), (3) and (5) and the equations of motion yields the non-zero partial stress components and velocity as shown in Table I, where:,, jiO T^ cr) r- Jo (6) and s - _o T" N(') [,(rOP d, (7) Hence, fluids A', B', and C' all predict different normal stresses. Normal stress differences are consistent with the sudden expansion or contraction of the stream when some non-Navier-Stokes liquids suddenly emerge from a tube into the atmosphere (the Merrington effect, which would occur in A' and C') as well as with the differences in the shape of the free surface for such different liquids undergoing

p 0x 0 o 216 252 DW 0-j D p 0 2 1 2 BP 2 2 ( ( BDW Y L W$ z B (DW zz 2j( py BDWI DW 0 2 i DW~0 0 ~ ~ ~ ~ ~ ~ ~ ~~~ ~~~~~~~~~P~ ~ ~ ~ ~~O ap 1 BPB DW B 2W (__ 2J(DW) I C 0~~~~~ h 0 2 _a 2 BD 1 2 2TABLE I Velocity and Stresses for primary flow of miater'i&l A' BI. and C',

Couette flow (the Weissenberg effect, which would also occur in A' and CG)o Only C' shows a variable effective viscosity. In the development of the Orr-Sommerfeld equation (the stability equation for Navier-Stokes liquidWs) consideratfion is llmwtd4 to a disturbance- that corresponds to a velocity field which is both temporally and spatially (in t) periodic, Subject to this limitation9 Squire (1933) has shown that it is sufficient to consider a disturbance that corresponds to a twodimensional velocity field0 In the viscoelastic case, only such disturbances will be considered also, a.~lthQugh no proof of the sufficiency of this exists for these fluids, {In fact, Listrov (1966) has shown that at least for a Stokesian fluid three dimensional disturbances are less stable than two dimensional disturbances,) Accordingly, the disturbance velocity components are taken in the form m a o) 11* V(7 E ) (7v0 L - a where E = exp iA(z Ct); A is the (real) wave number, and c the (complex) wave speed, We first derive the stability equation for materials A' and BK' The total displacemrent is assumed to be the sum of the primary flow displacement and the dsplacernent resulting from the disturbance9 name ly, I _ I'4;(j|( t 11

Taking and similar expressions for. and G,.ince IDt t Dt the oQlutions are readily obtained as',- (-D +L.C/^(~- c), ( -] - (- + 4),A(- C%),II A i X (Wv-Xi~~~~~~~~d~~t'A(W~~~~n j

Expanding all quantities about values at time t, it follows that to the lowest order E (1-F) -- t.A(W-C) and al' (-tiwJDER (t) uW.here F f exp iA(C.W) (t-t)., If consideration is restricted to a short time interval, then aquations (8) and (9) can be approximated by expanding F in a power series in (t-t') and retaining only the first order terms. Hence, a.... aad 2' w- (W Pt'). The'a. the expressions used by Walters (t962)} The more general forms given by equations (8) and (9) will, howev ert be used here, We note also that equations (8}) and (9) are well-behaved at the critical point where W and C have the samre value, Applying the limiting process W4C results in 13

and ^& C wiC -IvE DW (r-t)2 in agreement with the limit as t approaches t' to the order of the linear terms in tot'. The nonzeiro r-ate of deformnatin components at time t are dlY — a,, - LE Dw and d'-t -(A- t,X To the same approximation used for the displacements, the non-zero rate of deformation components at time t' are'd' =-dl, =Su E F and d - = I [DW' (:A^,-') E r(F -. V X 4E(1- F) Making use of the continuity equation, the non-zero derivatives of the, displacements can thus be written as

___ DiA e (i4.).\ E(- T ),.j4 _- 3 c IWO I" Combining these results with the conrstltuiye equations for A' (equatio.' (.Z) leads to the disturbance stresses Z = z <DIaJ - ZtADwk F). p>;~ j c,,A 3r 21 WDvD ~tXAK ih~u- c

and p -*A(K + K + 2 (DWYL A(- C)( K'+ K' 3 v %s K +ZlDWf'W (zvt * Kx) The fuinctions Kn (y), are generalizations of Waltera' constants and are defined by. 0 (10) Upon considering the equations of motion, one obtains, after subtracting the equations satisfied by the velocity components of the primary flow and eliminating the prsosure by crosasdifierentiation, F(W- CXD'-A') -D ]V. F)(A') Pr~O In order to wilte equatlon (11) in terms of non.dlmensional variables, the following dlmenaionles. functions of y are introduced: 16

Putting these and the previous expressions for p p arid z into equation (11) results in l-u-<,i: DuI -0 - V2 OU4"uT - (U-cJ03taU - D42h uDV)'"- O'U(tJ -c~p -(z DU-U\JDvir wiere a' is the non -dirmeensional wave number Ah. c is the naQna dimnensional wave speed C/Wo U is now the non-dimensiona4 primary flow velocity, W/W and P) represents differentiation with respect to the non-dimensional y, For a Navier-Stokes liquid all the p are zero except P, and equation (12) would then reduce to the Orr-Sommerfeld equation. Walters (1962) has suggested that for.son e viscoelast[c liquidsa called "!lightly viscoelastic,'' it i8 reasonalbe to expeet the {(r.) to 1''T

vanish rapidly as Tr increases. This is the justification he gave for using the approi-ambt forms for equations (8) and (9), This is equivalent to replacing the upper limit in the expressions for the br a finite- limit (say T) which may even be quite small, Assumag *At- )\CC T. everywhere in the flow field and neglectg all B for n > 2, equation (12) reduces to L ['(,u- D- ) -DL1 ~~Ut where and R, is the equation givn by Wal which is the equation given by Walters, Two remarks are appropriate. Equation (12) was derived without specifying the form of W(y) and hence is not restricted to the primary flow here consideredo Also, if equation (3) (the constitutive equation for material B-') had been used in place of:equation (2) (the conastitutive equation for material A'), the resulting equation for v would have been exactly equation (12) -a perhaps surprising result, considering the primary flows involving the two constitutive equations give quite different normal stress results, 18

The stability equation for the C' material is developed in much the same way. For the perturbation velocities, the solution to equation (4) is found to be, to the order of the linearization, where (F From equation {5), the disturbance stresses are p~ l^ ~,[A'(W-Ctl4 -t'WL 4 0 H "rLW.'!CL A C ( t IA

'+'t.DW L~.,I) A - L whexre(jAr Because of the complexity of these stress terms we de not write out the stability equation here, but consider the appropriate approximation in the next section. III. Asymptotic Solutions An approximate solution to the stability equation (12} for the primary flow U = 1 - y is next obtained. The determination of the characteristics equation for a Navier-Stokes liquid is first briefly discussed in Part A, the procedure used being that presented by Graebel (1966) with only slight modifications, The counterpart characteristics equation for viscoelastic liquids Al and B' is next presented in Part B, and the procedure used to actually solve this characteristic equation and to determine the points on the neutral stability curve is presented in Part Co Viscoelastic liquid C' is discussed in Part D, A, The Determination of the Characteristic Equation for a NavierStokes Liqu id It is anticipated that both c and 1/aR will be small for the case of interest, which suggests a solution in terms of matched asymptotic expansions. The flow region is first divided into an inner and an outer region~ The inner region-is a strip that includes the rigid bottom boundary at y = 1 and the "crltical point" at y = Yc' 20'

where U(yc) - c-. The outer region extends from the inner region to the centerline between the plates at y 0 which, as a result of the symmetry of the geometry and equations, serves as the other boundary, It is assumed that the two regions overlap, The procedure is to obtain a solution valid in the inner region and a solution valid in the outer region and then to merge the two solutions. The inner solution is made to satisfy the boundary conditions at y;-1 and the outer solution the conditions at y = O0 The mer'ing then produces a characteristic equation which gives a as a function of R and c the plot of a versus R for c, = 0 being the neutral stability curve. In the inner region v is obtained by introducing the change in variable (coordinate stretching) and by putting where ~ is a function of aR3 expected to be small, but unknown at this point. Substitution of th e abov e into equation (1 1) with all of the R n for n L 1 set equal to zero suggests that the proper choice for A. is (aR) / and equation (12) becomes, X at-o d~g-, 21

1C(7 and (1),are the solutions of and Hence, (0: tO) X? - II and where 2r~. 2 a, E~~~~~~~~~~~~d~.

H(M) and H(2) are Hankel functions of order one-third, Since h2 increases exponentially with large positive at a much faster rate than does the outer solution, 0t) cannot be merged with the outer solution and is therefore discarded. For 2( ) we have mo = - eIDwI Graebel (1966) gives the solutions for V } and as 2 2 ex2-p - Lo [ ( iS3 and 3. /0(~ + a3 3 where ~ - oL7830 (Duc~/' 0, 39oqa, * (oZ 9 / 6O (vDU,/'- 0, 6784 9). For large

In the outer region v is obtained by introducing the expansion v(/X24 - Eor () (g)+61(r)IJ(e0 Substituting this into equation (12) gives ( U-C.t( (-) U )~ -" (: O, and v) v() for all n such that 0(n(()) > Q(. ) The Tollmien solutions (1936) are hcO (3 - F1(,) =) LTC. and where A - (n w i DUc -c4 Aand t o-2 iUL, B 1:O, B2 l - U Itc. 2?4

In the inner region G raebel used where the C. are arbitrary congstants, For the inner and outer solutions to merge for large positive it the proper choice is 1/,t for to, lni. ior 1 i and I for i 2 In the outer region v is given by vc>dT:,'- r J + C I) whexe A l. a constant that cannot be determlned until higher order terms are considered. ~0} decreases exponentially with large positive and, hence, no terms are needed to merge with it. The boundary conditions are amd D9Du 3 C3 o # 0.' ~: 0 If v is an even function of y or if V is an odd function of y0 The Orr'Sommerfeld equation allows separation into even and odd parts when U is an even function of y; 6iLnce v i[ t;~e oUter region satisfes a second order equation

which is even in y, satisfaction of only one boundary condition at y = 0 is sufficient, Using symmetrical disturbances since they are less stable than asymmetrical disturbances, the boundary conditions give and to the lowest non-trivial order in, 9 where! I (1 + Y )/R a zl/ and z. y ~ For the existence of a non-trivial solution, CI1' C0,. 3' 040 D_. -... which is the characteristic equation obtained byGraebel. The plot of a vs R for c, = 0 obtained from this, however, does not contain the characteristic loop of a neutral stability curve, but gives only the lower branch of the curve. This shortcoming is explained by considering the magnitudes of the (1) terms in the Inner expansion, which in turn requLres some knowledge of q1 With Lines (1945) results (1 2o.5., j 5 1/ /20) as a guide it appears that 2(0) does not adequately describe the solution near y w.1,- for Im({ 1) )'e 0Q 3 as compared with the Im'07( ) 0 The real parts of anhd 26

'2 1 are however small compared to and respectively, and )is small compared to?() Thus by including the 1() terms the imaginary part of ~2 is altered significantly, this additional term apparently being responsible for generating the loop in the stability curve, It seems then that a better choice for the solution in the inner:egion is +(l(r"C2 ) IC?3 (16) For merging equation (16) with the outer solution, again the proper cholce is 1/, for G. lnFA for x i' and I for t2 as before. Then v(y) in the outer region is given by Substituting the appropriate boundary conditions leads to,,Io) 1(lr r -A i~CzE(s1 jUc o,.)(,(~,X

+ R2(OUc\2 t + T P) and ~i ~ R, (whee (sCT ) 2 where D U a D U evalzated at y = y * These are valid in the neigIhborhood of the critical point for any distribution function that vanishes as T beconmes large, specifically for N(T) neglgible when T >T where iA(W-C)T i< 1 A change to a stretched variable and inner and outer expansions are' Introduced as in part A. When all these are substituted into equation (1j), the stability equation in the inner region is now modified to

and thus to the characteristics equation The (I- vr The plot o$f ac versus R resulting from this does have a loop and the equation ls essentially that used by LAn, although he elected to express the auter soiution as an expansion in powers of a. With present computers the expansion in terms of the coordinate rather than a seems to be much simpler and more accurateo B, The Deterrmination of the Characteristic Equation for a Viscoelastic Liquid of Type A' or B' Fox a viscoelastic liquid whick, does not depart too drastically from. a Navier-Stokes.iqu.id the solution of equation (1{l) can be carried oQt in a ma nner analogcues to the solution presented in part A, Spec.if{i cal~y1 it is anticipated that in equation (12) both ci andi IP/a | will be rsmll0 Hence, the flow region can be divided Into an inner and an outer region as previously dorie, In the inner region the P can be expressed as series in z by expanding the denominators and then integrating term by term. The results are 29

1Uc18 and, thus where E *,,aR, The outer equation is unchanged. Since the term coptaziig. R1 multiplLee the fourth derivative,. and 2 remain as in part A. TAking * to be simall but still of larler order than u 9 ~3 ts approxrnmated by'. wher e Iow 3a

and Then I fs the same as in Part A, and I ='6 S A;-+i ). 1 5 1 dr dj Thus for the solution L the inner regi.orn and g2 as given in Part A are used, and ~o+ I s used for Because of the additional, linearizat.on introdutced by equation (19), the present results are limited to small e and serve mainly to indicate a trend. For larger values of r *.-he perturbation scheme used to solve equation 18) may not be adequate. In this cased one. would have to resort to an exact solution of equatiosn (ll8I as presernt;ed Ir, the Appendix0 Since the outer solution remailns uncrrhanged and,..s not directly involved'in th.e merging, the chara:testic eguation ueta'n the same form as equation (1 7), thon dn t e dce being s row given by equation (19). C, Solution of; h.e? Characteristics Equat'oFor cal clation pruposes, it.s onevenIernt to >:trLoduce the change in vari able where e~~~~~~~~~a

Then the left hand side of equation (17) becomes ( 00 (o~ - The functions h and h are discussed, and tables of h and h and 51 11 (2 a their derivatives are given In Annals (1945), Putting x I it follows that () +Cf t1 -3 t(21 -)

and'-.., r- -. ( ".I~ I _ ~+ o ----— I "A 3.O)''Y" 3/ where +200) 0 A 0/ 3) vv/ are also tabulatedd For the integration indicated on the left hand side of equation (20) the above series were integrated term by term for $ 4 5, For ~ > 5, the integration was continued numerically usirg Strnpsons rulo and the asymptotic expansion of hI until there were no noticeable changes, tn the value of the integral in tihe seventh significant figure. These asymptotic expansions are 33 5als

and whlewe s fop 21 &, hlt5)s)-S-e _s S,( where S i(LtJ)*/{;9 and d 4 arf S (it, The sraultV of the integration are Qbd. kl h(cs) o0.5O 43 6 s < 1'*:3 er~ Hatfdt r( A +5)(Z oaS tL"" t zl a (-1)M avut S2 (bhc~f z OV! 001K~ z ts^+ ttoo _ ( t ~~~~~2 -,-8A (bvnaa) (2400 53 dsiidt Rh,( 5) 0 Q 39 c5 92 rm cf O b~ )6~+d1g 3 l(6vY+"9u+S)(2)t~oo} 4. (6^*X + )( 6,u+ 3) 12oo~Pt^3J 4 3

c',, w.(,O?) - o, g73588 i(-r31 BA,.o &6o,,*) (z200)) +0,' 7 3I" 8(8tn 3 (l,()+(. Z(to o +3)()* -A (,i_* 5* (33,Zzo _ O.9735,rz I- () 1+1 6t $- --- ------— AP VMtibsPN_0.

For given values of X and c the. solution of equation (Z0) was obtained by plotting the real part against the imaginary part for each side, The intersections of these two families of curves gave a and i, R was computed from The results are shown in Figure 1. The graph for X a 0O which corresponds to a Navier-Stokea liquid, is seen to be in close agreement with Lin's results, and the preceodlag stataments regawrd tig the anticipated se* of the various paramaters are seemLingly cos slteotvt with the final results0 The results are qualitatively in agree. ment with those of Chan Man Fong and Walters, shown also in Figure 10 (The X in their paper is defined as five times the value of the present ne, ) The quaintitative disagreement of the two results is not und*rstood; it is noted, however, that Chan Man Foang a nd Walters, results lor X s 0, departing considerably from the results of Lian do agree with the results of Stuart (1954)0 On this basis, it is believed that ths present results are the more accurate ones, D.'The Stability of a Viscoe*astic 14 uld of T e' C' Introducing stretchd eecoordinates again as in part B, avear the critical point the stresses, beceme -2 DLJc w Ae. -., i- ~02__~1d... l

2.2 I I I I I I 2.1 I 2.0 / / N / N 1.9- / // N / 1.8 - / 1.7 C i I N I~~~~~~~~~~~~~ I.? N I' N 1.6 1.5 X:0.02 1.4 1.3 - ~ ~' = 0.015 1.3 \ / / 2 a\ 1.2 - \ o o I \ X=,.0 X=o 1.0 \ - X 0.004 0.9 ~~~ \`t~~~`~~`\ ~NEUTRAL STABILITY CURVES 0.8 - CHAN MAN FONG 8 WALTERS (1965) 00000 LIN (1945) 0.7 PRESENT RESULTS 0.60.5 0.4 - - 0.3 1 1 I 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 R 1/3 Figure 1. Neutral stability curves for various values of X. 57

5500,, i i, t t, 5000 4500 Rcr 4000 3500 X 3000 2500! i l i 2500 1 I I 1 1I I I i 0 I 2 3 4 R,x 103 Figure 2. Critical Reynolds number versus the parameter R1. 38

1.8 I I I I I C0.30 Cu0.29 1.7 C 0.28 C=0.32 C=00.2 6 1.6 - C=0.25 1.5 - /.5 X =0.02 1.4 X..'o.o,5 1.3X-o.o, 1.2 - X =0.005 2 a I.I - 1.00.9 0.8 0.7 0.6 0.5 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 R /3 Figure. Neutral stability curves and curves of constant c for various values of.

-1 J c ( L, 1 JD ~W J - 8 (p Wc3 Dx4 J+C + iA l(2 ziJ -J -1- (ta c (J 3C2 ~-+J3I2 ic 2 W 3 CWe c 4 where the subecrlpt appended to the Jr Is dicates that teey are evaluated at y; the Jm's and K's shave been -assumed to be all of n n the same order of magnitude. The parameter'j ie defined by r ~ ( iP jt}''h, where D s h Ho DX, (Pit.2= ap nMaking the distances and velocities dimensionless as before, and defining t=~ ( l /pf )>L J 4' (4DU\j C Ic 3i subptitution of these stresses into equatorn (1L) along with use of the equation for the primary flow results in.~-'C -- all d3 V40

Proceeding further as in case.]E, with r1 Q /A=R l, )2 /4 Is. we have ) L''Y;DW )U.;: C(22) a the the govering equttion in the intner regiono It is seen then that if 1 &nd;Z' are bath small compared to unity, but larger that., aglL.(O) and $(2 remain unchanged, but a first approximation to _... ~* now where o, 91 are as given ta part Bo Away from the critical region, the inviscid equation will again hold, but the primary velocity profile is of course d~~ferent from the p&rabotoc one. (For the model typified by equation (t) with M - N! 1 and L a LtC for instance, DW satisfies a ecubic equation where the t t. coe$ficlents are linear functions of y.) Comparing a flutd of type A' (or B,) with type C', if N and ~ p/az are the same, for both cases, the flfid of type C' will have a steeper velocity profile than the parabolic one, To carry out the details of the Solution,, it is necessary to specify N. Whhn this [i known, W can read[ily be found by numerical methods, " Vt ter us tit-tmss tire neut ral s7abihity curve, 4.1

BIBLJIOGRAPHY 1. The Annals of the Computation Laboratory of Harvard UniverPasty, Volume II, Harvard University Press, Cambridge, Mass, 1945 2. Chan Man Fong, C. F0, and Walters, Ko, "The.olution of flow problems. in the case of materials with minemory, Part II, " Journal de Mechanique 4, 1965, 439=453~ 3. Goddard, J. D,, and Miller9 C0, "An inverse ior the Jaumann derivative and some applications to the theology of viscoelatict fluids, " Rheologica Acta 5, 1966, 177=183. 4, Graebel, W. P., "On the determination of the characteristic equations for the stability of parallel flow, " Journal of Fluid Mechanics 24, 1966, 497 508, 5. Jaumann, G,,, "Geschlossenes System physikalischer und chem[scher Differenzialgesetze," "Sitzbero Akad. Wiss. Wien (IIa) 120, 19119 385=5300 6.4 Ljin, C. C, "On the stability of two dimensional parallel flows," Quarterly of Applied Mathematics 3, 1945, 277-301. 7, Listrov, A, T., "Parallel flow stability of non-Newtonian media," Scviet Physics, Dokl. 10, 1966, 912-914. 8 Gldroyd, J. G., "On the formulation of rheological equations of state," Proceedings of the Royal Society (A) 200, 1950, 523-541. 9Q Oldroyd, J. G., "Non-Newtonian effects in steady motion of some Pdealized elaattico visco. lis t[quids'" Proce edings of the Royal Society (A) 245, 1958, 278-297, XJ0 Sokolnikoff, Io S., Tensor Analysis, John Wiley and Sons, Inc.. New York, 19510 L,4 Squire, H. Bo, "I On the stability for three-dimensional disturbances of viscous f1lud flow between parallel walls," Proceedings of the Royal Society (A) 142, 1933, 621t6258

12. Stuart, J. T.., Proceedings of the Royal Society (A) 221, 1954, 189 205. 13. Tollmien, W, "General instability criterion of laminar velocity distributions," Technical Memorandum 792, NACA, 1936. 14. Walters, K,i "The solution.of flow problems in the case of materials with memory, Part I, " Journal de Mechanique 2, 1962, 479=486. l3

APPENDIX An exact solution of equation (18) is possible and has, in fact, been given by Chan Man Fong and Walters (19'65). A modified and more complete version of their results is presented here to show its use in the present method. Equation (18) is of the form Writing Z2X.1)' j1 XX ) and the equation for f is the confluent hypergeometric form with solutions eP U0e59 Z3 44

and As z approaches ioo, U(z) approaches z and V(z) approaches z e This suggested that U must be the solution corresponding to 23 and V to 24o To verify that this is indeed the case, replace v in equation (23) by 1/2(=1 + 5 f)o Then in the limit as X approaches zero with arg z = and 2 f 2/3,,,,,. (') (-1. e (J(U ) ooe0 +X(s~-s #7d'.-' ]s, or, upon expressing the integral in terms of Hankel functions,,:" r(/) (-lj' P/2 U(-)'ago 4iT. ( dr where lT;/3: -T erpy 7-c-3 [ds

This is the desired result apart from a multiplicative constant. Thus the general form of (0) for arbitrary X is given by integrating l = oe K lK1 X 46 3

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UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA- R&D (Security claafittcation of title. body of abstract and Indexing annotation muet be entered when the overall report ia cleasilfed) 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION The University of Michigan Unclassified Department of Engineering Mechanics 2b GROUP Ann Arbor, Michigan 3. REPORT TITLE The Stability of Parallel Flows of Fluids with Memories 4. DESCRIPTIVE NOTES (Type of report and lncfuesve datea) Technical Report 5. AUTHOR(S) (Laet name, fitt name, Initial) Mook, Dean T. Graebel, W. P., Project Director 6. REPORT DATE 7,. TOTAL NO. OF PAGES 7b. NO. OF REFS September 196 51 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) Nonr-1224( 49)) 06505-3-T b. PROJECT NO. NR 062-342 C. b. OTHER R~ PORT NO(S) (AnY other numbers that may be aalgned thil report) d. 10. A VA IL ABILITY/LIMITATION NOTICES Distribution of this document is unlimited. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Department of the Navy Office of Naval Research Washington, D.C. 13. ABSTRACT The equations governing the stability of plane parallel flows are developed for three models of fluids with memories. Asymptotic solutions valid for large Reynolds numbers are obtained and the effect of the memory are shown to be destabilizing. The approach to the problem allows evaluation of how fast a memory must fade to allow evaluation of the stresses in power series in the time interval. An alternate approach to inverting convected derivatives is also presented. oD D*JAN 1473 6CmSSIFI Security Classification

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