Location of the continuous spectrum in complex flows of the UCM fluid

J. Non-Newtonian Fluid Mech. 94 (2000) 75–85

Location of the continuous spectrum in complex flows of the UCM fluid Michael Renardy∗ Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA Received 11 January 2000

Abstract In contrast to the Newtonian case, linear stability problems for viscoelastic flows involve continuous as well as discrete spectra, even if the flow domain is bounded. Numerical methods approximate these continuous spectra poorly, and incorrect claims of instability have been published as a result of this on more than one occasion. In this paper, we shall derive some analytical results on the location of the continuous spectrum for linear stability of flows of the upper convected Maxwell fluid. In general, we shall show that in ‘subsonic’ flows, where the fluid speed is always slower than the speed of shear wave propagation, there are only three possible contributions to the continuous spectrum: 1. A part on the line Re λ = −1/W , where W is the relaxation time of the fluid. 2. A part associated with the short wave limit of wall modes which has real parts confined between −1/W and −1/(2W ). 3. A part associated with the integration of stresses in a given velocity field. If the flow is two-dimensional and has no stagnation points, then the latter part also has real part on the line Re λ = −1/W , and hence the continuous spectrum is always stable. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Continuous spectrum; Viscoelastic flows; Maxwell fluid

1. Introduction In contrast to Newtonian flows, the linear eigenvalue problem governing the stability of viscoelastic flows involves continuous spectra in addition to discrete eigenvalues. Although in most situations of interest these continuous spectra are not responsible for flow instabilities, they are hard to resolve numerically. Typically, a line segment in the continuous spectrum is approximated numerically by an oval shaped cluster of eigenvalues, and the width of the oval decreases only slowly as numerical resolution ∗

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is refined (see e.g. the discussion in [1]). The inaccuracy in resolving the continuous spectrum has led to erroneous instability claims. As noted in [3], early studies of stability of plane shear flow incorrectly claimed instability because of poor numerical resolution. A recent example in a complex flow is [9] (a retraction of the instability claim is in press). To assist in the interpretation of numerical stability results, it is clearly advantageous to know a priori where the continuous spectrum ought to be. For parallel shear flows, the linear stability problem for the upper convected Maxwell model is quite well understood [1–4,8,10]. After separation of variables, a system of ordinary differential equations is obtained. For each wavenumber, the continuous spectrum of this system of ODEs consists of a segment on the line Re σ = −1/W , where W is the relaxation time of the fluid. We note that this value is determined by material parameters alone and independent of the flow. This feature does not extend to other constitutive theories, see for instance [6] for the Johnson–Segalman model. We note, however, that the continuous spectrum of the two-dimensional flow is not determined completely by the continuous spectrum of the one-dimensional problems which arise after separating variables. Indeed, in the limit of infinite wavenumber, there may be limit point of Gorodtsov–Leonov wall modes [2], which would be part of the continuous spectrum. We note that all these Gorodtsov–Leonov eigenvalues have real parts between −1/W and −1/(2W ). The objective of this paper is to characterize the location of the continuous spectrum for the UCM fluid for general flows. The main conclusion is that the continuous spectrum is always stable in two-dimensional flows which are subsonic and have no stagnation points. The precise statement of results is in Section 2. The next three sections are concerned with the proofs. The essence of the proof of Theorem 1 is in deriving elliptic estimates. This first of all requires separating the equations of motion into ‘elliptic’ and ‘hyperbolic’ parts. This is based on a transformation of the equations known in numerical simulation as the ‘EEME method’. Ellipticity involves a condition about the partial differential equations as well as the boundary conditions. The violation of the ellipticity condition for the boundary conditions (known as the ‘complementing condition’) is related to the short wave limit of the Gorodtsov–Leonov eigenvalue problem. Once the elliptic estimates are known, Theorem 1 follows by a rather routine application of functional analytic techniques, this part of the argument is carried out in Section 4. The characterization of the spectrum of M requires an analysis of the stress integration along streamlines. The absence of stagnation points plays a crucial role in this analysis. 2. Statement of the problem and main results We consider flows in a domain , which is assumed to be either bounded or periodic. If the region is periodic in one or more direction, we shall look for solutions which are also periodic in these directions. On boundaries, we shall, for the time being, assume zero velocity; later we shall extend our discussion to situations involving rigidly moving boundaries. All boundaries are assumed smooth. We state our equations in dimensionless form. The momentum balance reads   ∂vv R + (vv · ∇)vv = div T − ∇p + f , (1) ∂t the incompressibility condition is div v = 0,

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M. Renardy / J. Non-Newtonian Fluid Mech. 94 (2000) 75–85

and the constitutive relation is   T ∂T T − (∇vv )T T − T (∇vv )T + T = ∇vv + (∇vv )T . + (vv · ∇)T W ∂t

77

(3)

Here R is the Reynolds number and W is the Weissenberg number. f is time-independent, and that there is a smooth time-indepenWe assume that the prescribed body forcef dent solution v = u , T = S . We consider the eigenvalue problem for the linearized system (in which v and T denote the perturbation to the time-independent flow) u · ∇)vv + (vv · ∇)u u) = div T − ∇p, R(λvv + (u div v = 0, T T +(u u · ∇)T T + (vv · ∇)S S − (∇vv )S S −S S (∇vv ) − (∇u u)T T − T (∇u u)T ) + T = ∇vv + (∇vv )T . W (λT

(4)

On the boundary, we have the boundary condition v = 0 . To provide a rigorous definition of eigenvalues and spectra, we define some function spaces and operators. By H k () we denote, as usual, the space of functions which have square integrable derivatives up to order k, with the understanding that periodic boundary conditions are imposed in those directions where  is periodic. Let V denote the solenoidal subspace of (H 0 ())3 , i.e. the space of those v ∈ (H 0 ())3 which satisfy div v = 0 and v · n = 0 on ∂. Moreover, H01 () denotes the space of those functions in H 1 () which vanish on the boundary. We now define an operator w, Q) L(vv , T ) = (w where

(5)

 1 u · ∇)vv − (vv · ∇)u u + div T , w = 5 −(u R u · ∇)T T − (vv · ∇)S S + (∇vv )S S + S (∇vv )T + (∇u u)T T + T (∇u u)T Q = −(u 1 1 − T + (∇vv + (∇vv )T ). W W 

(6)

Here 5 denotes the orthogonal projection from (H 0 ())3 onto V . We shall also need to consider the operator M defined by T ) = −(u u · ∇)T T + (∇u u)T T + T (∇u u)T − M(T

1 T. W

(7)

The eigenvalue problem above has the form L(vv , T ) = λ(vv , T ).

(8)

We also need to consider the inhomogeneous problem L(vv , T ) − λ(vv , T ) = (aa , B ),

(9)

where a ∈ V ∩ (H 2 ())3 , B ∈ (H 2 ())6 . We say that λ is in the resolvent set of L if, for every a ∈ V ∩ (H 2 ())3 and B ∈ (H 2 ())6 , Eq. (9) has a unique solution v ∈ V ∩ (H01 () ∩ H 3 ())3 , T ∈ (H 2 ())6 . We say that λ is an eigenvalue of finite multiplicity if there is a finite-dimensional space

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of solutions of the eigenvalue problem L(vv , T ) = λ(vv , T ) and the inhomogeneous problem (9) is solvable for (aa , B ) in a set of finite codimension. We call an eigenvalue λ isolated if every µ sufficiently close to λ is in the resolvent set. In a finite dimensional situation, every λ is either an isolated eigenvalue or in the resolvent set. In infinite dimensional problems, there are other possibilities leading to ‘continuous’ spectra. In Newtonian flows, such continuous spectra can occur only if the flow domain is unbounded. Our goal is the following result. Theorem 1. Assume that the matrix S+

1 uuT I − Ruu W

(10)

¯ Let m denote the infimum and M the supremum of the real part is strictly positive definite throughout . of the spectrum of M. If Re λ < min(m, −1/W ) or Re λ > max(M, −1/(2W )), then λ is either in the resolvent set of L or an eigenvalue of finite multiplicity. As shown in the final section of this paper, this result remains valid if the flow is two-dimensional and some components of the boundary move rigidly rather than having zero velocity. The preceding result leaves it open to characterize the spectrum of the operator M. This spectrum depends on the nature of the flow. We can claim the following result. Theorem 2. Assume that the flow is two-dimensional and that the velocity field u in the base flow is such that u vanishes at most on a set of measure zero and there is a continuous function φ with φ(0) = 0 and u · ∇)|u u|| ≤ φ(|u u|)|u u|. Then the spectrum of M is contained in the line Re λ = −1/W . |(u We note that the hypothesis of the theorem holds if the velocity vanishes only on the wall, but does not hold if there is a stagnation point in the interior of the flow domain. 3. Some estimates We consider the ‘constitutive’ part of the equation (L − λ)(vv , T ) = (aa , B ), i.e. u · ∇)T T − (vv · ∇)S S + (∇vv )S S + S (∇vv )T + (∇u u)T T + T (∇u u)T −(u 1 1 T + (∇vv + (∇vv )T ) = B . − T − λT W W

(11)

We take the divergence of this equation and find u · ∇)(div T ) + (T T : ∂ 2 )u u + (∇u u)div T − (vv · ∇)(div S ) + (S S : ∂ 2 )vv + (∇vv )div S −(u 1 1 − div T − λ div T + 1vv = div B . W W Here the notation S : ∂ 2 stands for X ∂2 Sij . ∂x ∂x i j i,j

(12)

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u · ∇) + (∇u u) − (1/W ) − λ to the momentum equation, Next, we apply the operation −(u 1 1 u · ∇)vv − (vv · ∇)u u + div T − ∇p − λvv = a , (14) −(u R R and find   1 1 1 1 2 S : ∂ v ) + ∇ (u u · ∇)p + p + λp u · ∇)(u u · ∇)vv − 1vv − (S (u WR R R W 1 u) + (∇u u)T )∇p = φ(vv , ∇vv , T , a , ∇aa , λ). − ((∇u (15) R Here the notation φ(vv , ∇vv , T , a , ∇aa , λ) indicates an expression which involves v , T , a and first derivatives of v and a . We take the divergence of Eq. (12) and find 1 u · ∇)(div div T ) − (div div T ) − λ(div div T ) + ψ(T T , ∇T T , v , ∇vv , ∇ 2v , λ)=div div B . (16) −(u W Here the expression ψ depends on derivatives of v up to second and derivatives of T up to first order. Finally, we take the normal component of Eq. (12) on the boundary, taking note of the fact that u and v vanish on the boundary. We find an equation of the form     1 1 T : ∂ 2u ) = n · div B . (17) − − λ n · div T + n · S : ∂ 2v + 1vv + n · (T W W From the momentum equation, we have u · v ) + (vv · u )). div T = ∇p + R(aa + λvv + (u

(18)

We insert this relationship into Eqs. (16) and (17) and obtain an equation of the form 1 T , ∇T T , v , ∇vv , ∇ 2v , a , ∇aa , ∇ 2B , λ), 1p − λ1p = h(T W where on the boundary we have     1 1 2 B , λ). − − λ n · ∇p + n · S : ∂ v + 1vv = k(vv , ∇vv , T , a , ∇B W W u · ∇)1p − −(u

(19)

(20)

We can now solve Eq. (19) for 1p as long as Re λ 6= −1/W . The result is 1p = s,

(21)

where the function s satisfies an estimate of the form B k2 + kvv |2 + kT T k1 ), ksk0 ≤ C(λ)(kaa k2 + kB

(22)

Here | · |k denotes the norm in the Sobolev space H k . We now consider the system of equations 1 1 1 1 S : ∂ 2v ) + ∇q − ((∇u u)+(∇u u)T )∇p = φ(vv , ∇vv , T , a , ∇aa , λ), 1vv − (S WR R R R 1p = s, (23)

u · ∇)(u u · ∇)vv − (u div v = 0,

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with the boundary conditions     1 1 2 B , λ). v = 0, − − λ n · ∇p + n · S : ∂ v + 1vv = k(vv , ∇vv , T , a , ∇B W W

(24)

Our goal is the following result. Lemma 3. Any solution of Eq. (23) with boundary conditions (24) satisfies an estimate of the form kvv k3 + kpk2 + kqk2 ≤ C(λ)(kφk1 + ksk0 + kkk1/2 + kvv k2 + kpk1 + kqk1 ).

(25)

From the form of φ, s and k, it follows immediately that we have an estimate of the form B k2 + kvv k2 + kpk1 + kT T k1 ). kvv k3 + kpk2 + kqk2 ≤ C(λ)(kaa k2 + kB

(26)

The condition that λ is in the resolvent set of M then guarantees an estimate of the form T k2 ≤ C(λ)(kaa k2 + kB B k2 + kvv k2 + kpk1 + kT T k1 ). kvv k3 + kpk2 + kqk2 + kT

(27)

To prove the lemma, we simply need to verify that Eq. (23) with boundary conditions (24) is an elliptic system in the sense of Agmon–Douglis–Nirenberg (see e.g. [7]). The ellipticity of the partial differential uuT . It equations follows in a straightforward manner from the positive definiteness of S + (1/W ) − Ruu remains to check the complementing condition for the boundary conditions. This requires considering u (note that u = 0 at boundary points). Eqs. (23) and (24) in a halfspace with constant values of S and ∇u u is of the form Without loss of generality, let us consider the half-space x > 0. At a boundary point, ∇u   0 0 0   u =  κ1 0 0  , (28) ∇u κ2 0 0 and the corresponding viscometric stress is   0 κ1 κ2   S =  κ1 2W κ12 2W κ1 κ2  . κ2

2W κ1 κ2

(29)

2W κ22

We now look for a solution of Eq. (23) which has the form v = v (x) exp(iαy + iβz),

p = p(x) exp(iαy + iβz),

q = q(x) exp(iαy + iβz),

(30)

The homogeneous differential system then assumes the form (we set v (x) = (u(x), v(x), w(x)), and, to simplify notation, we set γ 2 = α 2 + β 2 , κ = ακ1 + βκ2 ) 1 00 (u − γ 2 u) − 2iκu0 + 2W κ 2 u + q 0 − iκp = 0, W 1 − (v 00 − γ 2 v) − 2iκv 0 + 2W κ 2 v + iαq − κ1 p0 = 0, W 1 − (w00 − γ 2 w) − 2iκw0 + 2W κ 2 w + iβq − κ2 p0 = 0, W p00 − γ 2 p = 0. u0 + iαv + iβw = 0,

−

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M. Renardy / J. Non-Newtonian Fluid Mech. 94 (2000) 75–85

81

The boundary conditions are u = v = w = 0, and



 1 − − λ p0 + q 0 − iκp = 0. W

(32)

(33)

We need to look for solutions which tend to zero as x → ∞. We find p(x) = p0 exp(−γ x),

(34)

u+ with a constant p0 . Next, we take the divergence of the momentum equation, resulting in 1q = div((∇u u)T )∇p), or (∇u q 00 − γ 2 q = −2κip0 exp(−γ x).

(35)

The solution is q(x) = [p0 iκx + q0 ] exp(−γ x). From the last boundary condition, we find   1 + λ γp0 − γ q0 = 0. W

(36)

(37)

We next substitute the expressions for p and q and solve for u in Eq. (31). We then impose the conditions u(0) = u0 (0) = 0 at the boundary. The outcome of this calculation, carried out by Mathematica, is p −γ + γ 2 + κ 2 W 2 p q0 = p0 . (38) W (−γ + iκW + γ 2 + κ 2 W 2 ) Together with Eq. (37), this yields that either p0 = q0 = 0 (and it is then easy to show that u = v = w = 0 as well) or κ p . (39) λ= −iγ − κW + i γ 2 + κ 2 W 2 A simple calculation shows that λ as given by Eq. (39) lies in the strip −

1 1 ≤ Re λ ≤ − , W 2W

(40)

which was excluded in the assumptions of the theorem. 4. Proof of the main result Suppose λ satisfies the hypotheses of Theorem 1. Let N be the nullspace of L − λ in X = V ∩ (H 2 ())3 × (H 1 ())6 ,

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and let N ⊥ be its orthogonal complement. Let B be the unit ball in N . From Eq. (27), it follows that B is bounded in X = V ∩ (H 3 ())3 × (H 2 ())6 .

(42)

By the compactness of embeddings between Sobolev spaces, this implies that N is finite-dimensional. Moreover, the range of L−λ is closed. To see this, let (aa n , B n ) be in the range of L−λ, and let (vv n , T n ) be such that (L − λ)(vv n , T n ) = (aa n , B n ). Now suppose that (aa n , B n ) → (aa , B ) in V ∩ (H 2 ())3 × H 2 ()6 . T n k1 Without loss of generality, we may assume (vv n , T n ) ∈ N ⊥ . It follows from Eq. (27) that kvv n k2 + kT is bounded. Because otherwise a subsequence of (vv n , T n ) T n k1 kvv n k2 + kT

(43)

T n k1 , Eq. (27), converges in X to an element of N , a contradiction. Using the boundedness of kvv n k2 + kT and the compactness of embeddings between Sobolev spaces, we can now conclude that a subsequence of (vv n , T n ) converges in X. The limit is the solution of (L − λ)(vv , T ) = (aa , B ). Hence the operator L − λ is semi-Fredholm. We claim that its index is zero. If λ is in the resolvent set of M, it suffices to show that the index for the elliptic problem consisting of Eqs. (23) and (24) is zero. This index is constant in each half-plane Re λ < min(m, −1/W ) and Re λ > max(M, 1/(2W )). Moreover, it is easy to see that the index is zero if |Re λ| is sufficiently large. Consequently L − λ is a Fredholm operator. Now consider the equation (L − λ)(vv , T ) = (aa , B ).

(44)

For any λ0 satisfying the hypotheses of Theorem 1, there is a decomposition (V ∩ (H 3 () ∩ H01 ())3 ) × H 2 ()6 = U1 × U2 ,

(V ∩ H 2 ()3 ) × H 2 ()6 = V1 × V2 ,

(45)

such that U1 is the nullspace of L − λ0 , U1 and V1 are finite dimensional and L − λ0 is a bijection from U2 to V2 . Let now (vv , T ) = u1 + u2 , (aa , B ) = v1 + v2 , where ui ∈ Ui , vi ∈ Vi . Moreover, let P and Q denote the projections onto V1 and V2 , respectively. Eq. (44) becomes −P (λ − λ0 )(u1 + u2 ) = v1 ,

(L − λ0 )u2 − Q(λ − λ0 )(u1 + u2 ) = v2 .

(46)

If λ is close to λ0 , we can eliminate u2 from the second equation, and we obtain −P (λ − λ0 )u1 − P (λ − λ0 )(L − λ0 − Q(λ − λ0 ))−1 (v2 + Q(λ − λ0 )u1 ) = v1 .

(47)

This is now an equation in a finite-dimensional space. Eigenvalues are determined by the roots of a determinant, which is an analytic function of λ. The final claim of the theorem now follows from the fact that roots of an analytic function are isolated unless the function is identically zero. 5. The spectrum of M To prove Theorem 2, we first prove that the resolvent of M in the function space H 0 ()6 exists for Re λ 6= −1/W . To this end, consider the equation u · ∇)T T − (∇u u)T T − T (∇u u )T + (u

1 T = B. T + λT W

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M. Renardy / J. Non-Newtonian Fluid Mech. 94 (2000) 75–85

Following [5], we define   u v , w= − 2 , u + v 2 u2 + v 2

83

(49)

where u = (u, v), and we rewrite T in terms of the basis given by u and w uw T + wu T ) + νww ww T . uuT + µ(uw T = κuu

(50)

We then obtain the transformed equation u · Bu 1 κ + λκ = , u|4 W |u 1 u · ∇)µ + ν div w + µ + λµ = u · Bw + w · Bu Bu, (u W 1 u|4w · Bw u · ∇)ν + ν + λν = |u (u Bw. W u · ∇)κ + 2µ div w + (u

(51)

u|2 /( + |u u|2 ). We u|2 , and the first equation by |u In this system, we multiply the last equation by 1 + /|u set κ˜ =

u |2 κ|u , u |2  + |u

ν˜ =

u|2 ) ν( + |u . u |2 |u

(52)

The result is the system u · ∇)|u u|2 u |2 u · Bu (u |u 1 = κ˜ + λκ˜ −  κ˜ 2 + 2µ div w , 2 2 2 u| ( + |u u| ) u| u| ( + |u u|2 ) W |u  + |u |u u |2 |u 1 u · ∇)µ + ν˜ (u div w + µ + λµ = u · Bw + w · Bu Bu, 2 u  + |u | W u · ∇)|u u|2 (u 1 u|2 )w w · Bw u|2 ( + |u u · ∇)˜ν + ν˜ + λ˜ν +  ν˜ 2 Bw. = |u (u u| ( + |u u |2 ) W |u u · ∇)κ˜ + (u

u|2 div w is bounded, and, moreover, that We note now that |u (u u|2 u|) 2φ(|u u · ∇)|u . ≤ |u 2 2 u| ( + |u u| ) u |2  + |u We now choose  such that u|) 1 2φ(|u < λ + u |2  + |u W

(53)

(54)

(55)

¯ We can then solve Eq. (53) and obtain a unique solution which satisfies a bound of the throughout . form B k0 . kκk ˜ 0 + kµk0 + k˜ν k0 ≤ C()kB

(56)

By transforming back to the original variables, we obtain T k0 ≤ C()kB B k0 . kT

(57)

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u · ∇) to Eq. (48). The result is To derive estimates for derivatives of T , we first apply the operation (u u · ∇)(u u · ∇)T T − (∇u u)(u u · ∇)T T − [(u u · ∇)T T ](∇u u)T + (u u · ∇)(∇u u)]T T + T [(u u · ∇)(∇u u)T ] + (u u · ∇)B B. = [(u

1 u · ∇)T T + λ(u u · ∇)T T (u W (58)

w · ∇) to Eq. (48), and we observe the vector identities Next, we apply the operation (w w · ∇)(u u · ∇)T T = (u u · ∇)(w w · ∇)T T + ([(w w · ∇)u u − (u u · ∇)w w ] · ∇)T T, (w w · ∇)u u − (u u · ∇)w w = −(div w )u u. (w

(59)

As a result, we obtain 1 w · ∇)T T + λ(w w · ∇)T T (w W w · ∇)(∇u u)]T T + T [(w w · ∇)(∇u u)T ] + (w w · ∇)B B + (div w )(u u · ∇)T T. = [(w

u · ∇)(w w · ∇)T T − (∇u u)(w w · ∇)T T − [(w w · ∇)T T ](∇u u )T + (u

(60)

u|)/|u u| and Eq. (60) by |u u|/( + |u u|). As a result, we obtain equations Next, we multiply Eq. (58) by ( + |u for u|)(u u · ∇)T T ( + |u (61) u| |u and u|(w w · ∇)T T |u , u|  + |u

(62)

which are of the same type as the equation for T above. We can, therefore, obtain L2 bounds for these quantities, and, as a consequence, an H 1 bound for T . We can then repeat the argument for second derivatives of T . This completes the proof of the theorem. 6. Moving boundaries Many flows of physical significance are driven at least in part by motion of the wall rather than by pressure gradients. Our analysis above assumes that u vanishes on the wall; this was used in an essential way in arriving at the boundary condition (20). If there are moving boundary conditions, we need to modify our arguments. The following result addresses one particular case, where motion of the boundary actually makes the problem easier. Theorem 4. Assume that the flow is two-dimensional and some components of the boundary ∂ are straight lines or circles moving rigidly. Then the conclusions of Theorem 1 remain valid. If there is a rigidly moving component of the boundary, then the velocity on that part of the boundary is nonzero throughout (note that this is not the case in three dimensions). On such a rigidly moving boundary, we simply replace Eq. (20) with the condition u · ∇)p + (u

1 p + λp = q. W

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This boundary condition, in conjunction with the equation 1p = s, leads to an estimate of the form kpk2 ≤ C(ksk0 + kpk1 + kqk1 ).

(64)

We can then use this estimate in the equations for the velocity and still arrive at the estimate (27), actually in a simpler fashion than before. This simplification occurs because Gorodtsov–Leonov eigenvalues on moving walls cannot have limit points since their imaginary parts tend to infinity with increasing wave number. Acknowledgements This research was supported by the National Science Foundation under Grant DMS-9870220. References [1] M.D. Graham, Effect of axial flow on viscoelastic Taylor–Couette instability, J. Fluid Mech. 360 (1998) 341. [2] V.A. Gorodtsov, A.I. Leonov, On a linear instability of a plane parallel Couette flow of a viscoelastic fluid, J. Appl. Math. Mech. (PMM) 31 (1967) 310. [3] T.C. Ho, M.M. Denn, Stability of plane Poiseuille flow of a highly elastic liquid, J. Non-Newtonian Fluid Mech. 3 (1977) 179. [4] M. Renardy, Y. Renardy, Linear stability of plane Couette flow of an upper convected Maxwell fluid, J. Non-Newtonian Fluid Mech. 22 (1986) 23. [5] M. Renardy, How to integrate the upper convected Maxwell (UCM) stresses near a singularity (and maybe elsewhere, too), J. Non-Newtonian Fluid Mech. 52 (1994) 91. [6] Y. Renardy, Spurt and instability in a two-layer Johnson–Segalman liquid, Theor. Comput. Fluid Dynamics 7 (1995) 463. [7] M. Renardy, R.C. Rogers, An Introduction to Partial Differential Equations, Springer, Berlin, 1993. [8] R. Sureshkumar, A. N.Beris, Linear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm, J. Non-Newtonian Fluid Mech. 56 (1995) 151. [9] R. Sureshkumar, M.D. Smith, R.C. Armstrong, R.A. Brown, Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations, J. Non-Newtonian Fluid Mech. 82 (1999) 57. [10] H.J. Wilson, M. Renardy, Y. Renardy, Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids, J. Non-Newtonian Fluid Mech. 80 (1999) 251.