The system of equations proposed by Burgers to model turbulent flow in a channel is extended to include viscoelastic affects. The stability and bifurcation properties are examined in the neighborhood of the critical Reynolds number. For highly elastic fluids, the bifurcated state is periodic with a shift in frequency

Lasseigne, D. Glenn

Olmstead, W. E.

English

Old Dominion University

Old Dominion UniversityODU Digital CommonsMathematics & Statistics Faculty Publications Mathematics & Statistics4-1990Stability of a Viscoelastic Burgers FlowD. Glenn LasseigneOld Dominion University, dlasseig@odu.eduW. E. OlmsteadFollow this and additional works at: https://digitalcommons.odu.edu/mathstat_fac_pubsPart of the Applied Mathematics Commons, and the Mathematics CommonsThis Article is brought to you for free and open access by the Mathematics & Statistics at ODU Digital Commons. It has been accepted for inclusion inMathematics & Statistics Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contactdigitalcommons@odu.edu.Repository CitationLasseigne, D. Glenn and Olmstead, W. E., "Stability of a Viscoelastic Burgers Flow" (1990). Mathematics & Statistics FacultyPublications. 33.https://digitalcommons.odu.edu/mathstat_fac_pubs/33Original Publication CitationLasseigne, D. G., & Olmstead, W. E. (1990). Stability of a viscoelastic burgers flow. SIAM Journal on Applied Mathematics, 50(2),352-360. doi:10.1137/0150022SIAM J. APPL. MATH.Vol. 50, No. 2, pp. 352-360, April 1990(C) 1990 Society for Industrial and Applied Mathematics003STABILITY OF A VISCOELASTIC BURGERS FLOW*D. GLENN LASSEIGNE? AND W. E. OLMSTEAD:This paper is dedicated to Edward L. Reiss on the occasion of his 60th birthday.Abstract. The system of equations proposed by Burgers to model turbulent flow in a channel is extendedto include viscoelastic affects. The stability and bifurcation properties are examined in the neighborhoodof the critical Reynolds number. For highly elastic fluids, the bifurcated state is periodic with a shift infrequency.Key words, bifurcation, stability, viscoelastic fluidsAMS(MOS) subject classifications. 35K55, 45KO51. Introduction. In some recent work of Olmstead et al. [8], the model problem(1.1) ut(x, t) f K t, s)uxx(x, s) ds + u(x, t) u3(x, t),d-(1.2) u(0, t) u(Tr, t) 0,was proposed to study the bifurcation and stability properties associated with aviscoelastic fluid. The interpretation of (1.1), (1.2) is that u(x, t) is the velocity perturba-tion for a Bnard type flow in a channel (0=<x=< 7r). The viscoelastic force of anon-Newtonian fluid with memory is modeled by the integral term in (1.1). The constantR-> 0, which corresponds to the Rayleigh number, becomes the bifurcation parameterin the analysis of (1.1), (1.2).The introduction of the model problem (1.1), (1.2) was motivated by the difficultyof identifying viscoelastic phenomena from the full-fledged Navier-Stokes equations(e.g., Joseph [6] and Eltayeb [3]). The analysis of [8] was carried out for a Jeffreyskernel,(1.3) K(t’s)=l-6exp(--)+266A(t-s), A->0, 0-<6<1,where A is the relaxation time, 6 is the ratio of a retardation time to the relaxationtime, and _(t-s) is the Dirac delta function. The limiting case in which 6--> 0, A --> 0is interpreted as a Newtonian fluid.The results of [8] show that, for fixed 6, the relaxation time A controls the behaviorof the bifurcation parameter R. As R is increased, the first bifurcation is to a steadystate for A sufficiently small and to a periodic state for A sufficiently large. This behavioris consistent with the linear stability analysis of Sokolov and Tanner [9] for a morerealistic formulation of the B6nard problem.Our goal here is to extend the (weakly) nonlinear bifurcation and stability analysisof [8] to a fluid mechanical model which more closely resembles the Navier-Stokes* Received by the editors June 12, 1989; accepted for publication (in revised form) June 21, 1989. Thisresearch was supported in part by the National Science Foundation under grant DMS-8700962.f Department of Mathematical Sciences, Old Dominion University, Norfolk, Virginia 23508.$ Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston,Illinois 60208.352Downloaded 10/10/17 to 128.82.252.150. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpA VISCOELASTIC BURGERS FLOW 353problem. In place of (1.1) we consider the system(1.4a) U,(t) P U(t) wR*u (’, t) d:,(1.4b) ut(x, t) + 2u(x, t)Ux(X, t) f K(t, S)Ux(X, s) ds + R* U(t)u(x, t).The Newtonian version of this model, where the integral term in (1.4b) is replaced byUxx(X, t), was proposed by Burgers [1] as a model for turbulent flow in a channel(0 _-< x _<- r). In that model, U(t) denotes a mean velocity driven by the pressure gradientP, while R* represents the Reynolds number. The integral term in (1.4a) plays the roleof the Reynolds stress which governs energy conversion between the mean flow andthe velocity perturbation u(x, t). Various aspects of the stability and bifurcationproperties for the Newtonian model have been investigated by Stuart [10], Eckhaus[2], Golia and Abel [4], Horgan and Olmstead [5], and Olmstead and Davis [7].The analysis here of the viscoelastic Burgers system (1.4) will also be for theJeffreys kernel (1.3). It is convenient to define(1.5) S(x, t)= exp u(x, s) ds,(1.6) V(t)-R*[P-U(t)]so that (1.4) can be replaced by the system(1.7a) u, Sxx + SU,,x + Ru uV-2uu,,(1.7b) AS, -S+(1-6)u,(1.7c) Vt=-V+ u2(, i) d,where R PR* will be the bifurcation parameter.The corresponding system for (1.1), treated in [8], involves the same linear termsthat appear in (1.7a) and (1.7b). The two nonlinear terms in (1.7a) are replaced by-u3, and (1.7c) is absent.For the analysis that follows, it will be convenient to define the system operator___,ROt Ox2 Ox20-(1-6) A--Ot0 0(1.8) LR0--+1otThen (1.7) can be expressed in the form(1.9) LR-uV- 2UUxU02/r u2( t) d:We will investigate the solution of (1.9), subject to the boundary conditions(1.10) u(0, t)= u(er, t)= S(0, t)= S(7r, t)=0,Downloaded 10/10/17 to 128.82.252.150. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php354 D. G. LASSEIGNE AND W. E. OLMSTEADand initial conditions(.) u(x, o) h(x), S(x, o) o, v(o) o,where eh(x), 0< e << 1, is the given initial velocity perturbation. The other conditionsin (1.11) correspond to a null history; that is, for all <0 the mean velocity in thechannel is constant (U-= P) with no perturbation (u 0).2. Linear stability analysis. By linearizing the system (1.7), and introducing normalmodes, we can identify the branch points were bifurcation can occur for the nonlinearproblem. We considerI,,e("+i’")tsinnxl(2.1) LR ,, e(’’+’’’’’ sin nx O, n 1, 2,...,n e(, +iw )twhere the or, and o, ate real. It follows that t7, O, n 1, 2,..., while for arbitrary7 and S there is a characteristic equation which separates into real and imaginaryparts as2 2(2.2a) A(o’,-Wn)+O’,(1-,R+n26)+n2 R=0,(2.2b) w,(1 + 2)tr, + ,nZ6 R) =0, n 1, 2. .Solving (2.2) yields two distinct cases. For branch points corresponding to steadybifurcations, we set w, 0, or, 0 and find(2.3) Rs) n 2 n 1 2,...For branch points corresponding to periodic bifurcations, we set w, # 0, on 0 and find1 2,(2.4) R(f’=l--+ n28, to. =-[(1-6)n 1]Awith the stipulation that1(2.5) , >,,(1-6)n 2"These results suggest that the critical value of R where the null solution losesstability is given by(2.6) R= R]p) 1 1 A>A1,where A1 1/(1-6). The physical implication of (2.6) is that A represents a measureof how much elasticity the fluid must have for the loss of stability to have an oscillatorynature.Since the linearized versions of (1.4) and (1.1) are identical, it is to be expectedthat the bifurcation sites given by (2.3) and (2.4) are the same as those found in [8].However, we can anticipate some differences in the evolutionary nature ofthe nonlinearproblem.3. Bifurcation analysis for k<k. In view of (2.6), we anticipate the nonlinearproblem (1.7) to exhibit its first bifurcation, as R increases from zero, at R Rs= 1when A < A1. Moreover, we expect that an initial perturbation of the null solution willDownloaded 10/10/17 to 128.82.252.150. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpA VISCOELASTIC BURGERS FLOW 355evolve slowly to a steady birucated state for R > Rs. To analyze that bifurcation, weset(3.1) R R]S) + e2= 1 + e 2,where e is the same small parameter that scales the initial data in (1.11). We alsointroduce the slow time(3.2) 7. e 2t,and seek an asymptotic solution of (1.9)-(1.11) through the expansionU-- 6J+luj(x, t, 7"),j=0(3.3) S= Y eJ+lSj(x, t, 7"),j=0V= 2 eJ+’ vj( t, 7").j=0By introducing (3.1)-(3.3), our problem (1.9)-(1.11) takes the form-uV-2uux ]U 117 U(3.4)L, 2/’n-fi=u2d, -e:z ASv, .,//’uj=S=0 for j_->0 atx=0,uo=h, So =Vo=0, uj=Sj=V=O forj->l at t=7"=0.(3.)(3.6)At O(e) it follows from (3.4) thatIul(3.7) L1 So =0.VoUpon solving (3.7) subject to (3.5) and (3.6), while retaining only the terms that surviveas --> , we find/1 C0(7") sin x](3.8) So Co(7")sin x + O(e-yt)Vofor some 3’ > 0. Here(3.9) Co(7")where(3.10a)Ao(7") ABo(7.)Ao(7") _2 Uo(X, O, 7") sin x dx,Ao(0) _2 h(x) sin x dx ho(3.10b) Bo(7") _2 So(x, O, 7") sin x dx, Bo(0) 0.Downloaded 10/10/17 to 128.82.252.150. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php356 D. G. LASSEIGNE AND W. E. OLMSTEADAt O(e2) it follows from (3.4) thatul -A Co(z) sin 2x(3.11) L, S, 0 +O(e-W).VlUpon solving (3.11) subject to (3.5) and (3.6), while retaining only the terms thatsurvive as t-- o, we find(3.12) S1 C(r) sin x-1/2AiCg(r) sin 2x +At O(e 3) it follows from (.3.4) that//2(3.13) L S + O(e-),where(3.14) G(t, r, x)= {-h,Cg(z)+ Co(r) h3 C3o(Z)[1 ](cos 2x +cos2 x)]4A C(z) Co(’) cos x} sin x,Q(t, z, x)= -hCg(z) sin x.(3.15)For the solution of (3.13), subject to (3.5) and (3.6), to remain bounded in t, werequire the solvability condition,(3.16) lim 1/o’/o,_ [G(s, z,x)-Q(s, z,x)]sinxdxds=O.This yields the nonlinear differential equation(3.17) (A, 1)C;(7) 1,Co(7 43 0,subject to the initial condition Co(O)= ho/(,-,) from (3.9). It follows that(3.18) Co(r) x/ ho{E3(A1 A) 4hA] e-2X’/(x’-’x)+4A2h2o}-’/2.1This, together with (3.8), determines the slow evolution toward the steady bifurcatedstate. It follows that as(3.19) S 1 esinx+O(e2).V 04. Bifurcation analysis for h > k. As suggested by (2.6), when the fluid elasticityis sufficiently large, i.e., A > A1, the nonlinear problem (1.7) will exhibit its firstbifurcation at R RP)< 1. Moreover, we expect that an initial perturbation of thenull solution will evolve slowly to a periodic bifurcated state for R > Rp. To analyzethat bifurcation, we set(4.1) R=R]P)+e2=I -- +eAs in 3, we introduce the slow time z= ezt and seek an asymptotic solution of(1.9)-(1.11) in the form of the expansions (3.3).Downloaded 10/10/17 to 128.82.252.150. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpA VISCOELASTIC BURGERS FLOW 357For this case, our problem (1.9)-(1.11) takes the form (3.4) with L replaced byL1/(1/,)-(/,,. The boundary conditions (3.5) and initial condition (3.6) again apply.At O(e) it follows that(4.2) Ll+(/,)_(/,,) So O.VoUpon solving (4.2) subject to (3.5) and (3.6), while retaining only the terms that surviveas t-+ oo, we find(4.3)1 1 [aU0 COS 0) +sin 0) sin 0) 0 o(’r)/(.o 0)1So /lhO.)ll sinwt cos tolt---sintotAwl I[B%r sinxVo 0 0+ O(e-r’).Considerable simplification of (4.3), as well as other results to follow is obtainedby introducing the polar representation(4.4a)(4.4b)Ao(z) R(-) sin (’)[1___ sin (z)-cos xI*(r)].Bo(z) 031R (z) Ato,Then (4.3) becomesUo(4.5) So sin[wVoAt O(e2) it follows that(4.6) LI+(1/A)_(1/A1) Svlsin [o91t + xI*(7")]it + xI*(7-)]- o, cos [olt + aIY(’r)]0R(z) sin x + O(e-r’).-sin 2x01R2(-) sin [(t, z)]+ O(e-V’),where(4.7) (I)(t, "/’) tO + (’/"Upon solving (4.6) subject to (3.5) and (3.6), while retaining only the terms that surviveas t-+ oo, we find(4.8)’/1 1 F --{a + b cos 2(t, 7")+ c sin 2(t, "r)} sin 2xS --{a + b cos 2(t, 7") + g sin 2(t, "r)} sin 2xV1 1 (1 + 4w)-1 cos 2(t, z) 2w(1 + 4w)- sin 2(t, z)+ O(e-’),Downloaded 10/10/17 to 128.82.252.150. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php358 D.G. LASSEIGNE AND W. E. OLMSTEADwhere(4.9)(4.10)1 -a(1 + 4A 2w2) 6Wl(1 + 4A 2to2)b= c=3 + Ato’ O2+ 36w c2 + 36to21-6 /= b 2AtOlC g= 2Awlb+c3+ AtOl AI(1 + 4A2tOl)’ AI(1 + 4A:to])a =(l+4A2to) 3A1At O(e3) it follows thatLl+(/x)-(1/a,) $2 Q + O(e-Vt),v owhereG {-R’ sin @ + R(sin @-’ cos @)R3[1/2- 2(a + b cos 2@+ c sin 2@)(cos 2x +cos2 x)] sin @(4.11) R+-- (1 + 4OZl)-l(cos 2 + 2Wl sin 2)]} sin xQ {-R’(sin Atol cos ) R’(cos + Aw sin )} sin x.For the solution of (4.10), subject to (3.5) and (3.6), to remain bounded in t, werequire the solvability conditions (see [8])G(s, z, x) cos (s, z)+[Q(s, z, x)-G(s, z, x)](s, )} sin x dx ds 0,sin(4.12)G(s, , x) sin (s, r)+[Q(s, , x)-G(s, , x)]1cos (s, z)} sin x ds ds 0.These solvability conditions yield the nonlinear differential equations(4.a) 2g’() ()-(),(4.13b) ,(r)=_( 1 )R’() fiL, g +-’where(4.14)+ a b c (A 2)--+ >0,u,2 4 4htOl 4A (1 + 4Wl)4A 1 +4to2The initial conditions for (4.13) follow from (4.4) and (3.10) as(4.15) hoR(O)(’01%//1’ XI)’(O)tan (Awl).Downloaded 10/10/17 to 128.82.252.150. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpA VISCOELASTIC BURGERS FLOW 359Solving (4.13) subject to (4.15) yields(4.16a) R(z)= R(O){tzR2(O)+[1-txR2(O)] e-}-1/2(4.16b) (z) (0) + (/3 1) R(’r)Ao, o R(0) +-- "This determines the slow evolution towards the bifurcated state which is periodic int. It follows that as z ,(4.17)sin (a3t +)Esin (o3t +) wl cos (o3t +) sin x + O(e ),v0where(4.18) 0. (.01"- 2 (/31)--e *=*(0)- log (R(0)v).We note that the bifurcated state is periodic with a frequency 03 that contains a shiftaway from the frequency w given by the linear analysis.5. Discussion. We have examined the (weakly) nonlinear stability properties ofthe null state for a viscoelastic version of the Burgers system. This analysis representsan extension of the work in [8] to a model which more closely resembles the Navier-Stokes problem.We find that for a non-Newtonian fluid of relatively small elasticity (A < A) thefirst instability is encountered at R R= 1, where there is a bifurcation to a steadystate. This is in accord with the results for the Burgers model with a Newtonian fluid(A - 0, - 0), as seen in [7]. On the other hand, for a fluid of sufficiently large elasticity(A > A), the first instability is encountered at R Rp) 1 where there is a bifurcationto a periodic state. These results are qualitatively consistent with the linear stabilityanalysis of [9] for a Navier-Stokes problem and with the (weakly) nonlinear analysisof [8] for the model equation (1.1).Our results do reveal an interesting feature not suggested by previous investigations.For a fluid of sufficiently large elasticity (A > A), the evolution to the periodic bifurcatedstate is accompanied by a frequency shift. As seen in (4.17), (4.18), this shift Aw isgiven by(5.1) Ato =fl 2where/3 and/x are given by (4.14).The ratio /3//x can be expressed in terms of only A and 6, which are the twoviscoelastic parameters of the Jettreys kernel (1.3). Figure 1 provides a graphicalillustration of the dependence of fl/ix on these parameters. The fact that fl/tz-as A A should be ignored since the analysis leading to (5.1) is not valid in this limit.It is significant that for sufficiently small, the frequency shift is (essentially) alwayspositive; whereas for 6 sufficiently large, the shift is always negative.Since these results were obtained from the Burgers model, the quantitative valueof the frequency shift cannot be regarded as significant. Nevertheless, the fact thatthere is a frequency shift may provide some insight into future investigations of thefull-fledged Navier-Stokes model.Downloaded 10/10/17 to 128.82.252.150. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php360 1. G. LASSEIGNE AND W. E. OLMSTEAD0.5Bill0.0--0.5IIIIIIIIIb .25b--- 750III!111110.0 5.0 10.0 15.0 20.0FIG.Acknowledgement. On this occasion the authors are reminded how much Ed Reisshas enlightened us by his scientific expertise and encouraged us by his friendship.REFERENCES[1] J. M. BURGERS, Mathematical example illustrating relations occurring in the theory of turbulent fluidmotions, Trans. Roy. Neth. Acad. Sci. Amsterdam, 17 (1939), pp. 1-53.[2] W. ECHKAUS, Studies in nonlinear stability theory, Springer Tracts Nat. Philos. 6, Springer-Verlag,Berlin, 1965.[3] I. A. ELTAYEB, Nonlinear thermal convection in an elasticoviscous layer heatedfrom below, Proc. Roy,Soc. London A, 356 (1977), pp. 161-176.[4] C. GOLIA AND J. M. ABEL, Path integral synthesis of Lyapunov functionals for partial differentialequations, Internat. J. Non-Linear Mech., 10 (1975), pp. 333-344.[5] C. O. HOGAN AND W. E. OLMSTEAD, Stability and uniqueness for a turbulence model of Burgers,Quart. Appl. Math., 36 (1978), pp. 121-128.[6] D. D. JOSEPH, Slow motion and viscoelastic motion; stability and bifurcation of the rest state ofa simplefluids, Arch. Rational Mech. Anal., 56 (1974), pp. 99-157.[7] W. E. OLMSTEAD AND S. H. DAVlS, Stability and bifurcation in a modulated Burgers system, Quart.Appl. Math., 39 (1982), pp. 467-477.[8] W. E. OLMSTEAD, S. H. DAVIS, S. ROSENBLAT, AND W. L. KATH, Bifurcation with memory, SIAMJ. Appl. Math., 46 (1986), pp. 171-188.[9] M. SOKOLOV AND R. I. TANNER, Convective stability of a general viscoelastic fluid heatedfrom below,Phys. Fluids, 15 (1972), pp. 534-539.10] J. T. STUART, On the effects of the Reynolds stress on hydrodynamic stability, Z. Angew. Math. Mech.,10 (1956), pp. 32-38.Downloaded 10/10/17 to 128.82.252.150. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

Stability of a Viscoelastic Burgers Flow

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Stability of a Viscoelastic Burgers Flow

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