In any shear protocol investigated in this thesis, we solve first the flow response to deformation with homogeneity artificially enforced: ˆs(t) — we call this the homo- geneous ‘background’ state. Our aim is then to examine the time-dependent linear stability of this homogeneous background state to the growth of heterogeneous per- turbations. To investigate this, we express the response to deformation as the sum of the time-dependent homogeneous background state plus any (initially) small het- erogeneous perturbations decomposed into Fourier modes15:
s(y, t) = ˆs(t) +X k
δsk(t) cos(kπy/L). (3.4.1)
Here, the homogeneous background is represented by hats ˆs (that we later neglect for clarity of presentation), and k = 1, 2, . . . is the mode number. Our aim is to determine whether the magnitude of the heterogeneous perturbations δsk(t) at any time t have a positive or negative growth rate (indicating instability or stability to heterogeneity, respectively). To do this, we substitute this form of s(y, t) (Eqn 3.4.1) into its constitutive equation: Eqn 3.3.2, and expand and linearise to the first order terms in δsk [neglecting terms O(δs2k) and higher]. Doing so, we find the governing equation for the heterogeneous perturbations:
∂tδsk= P (t) · δsk, (3.4.2)
where we neglect contributions from the diffusive terms16 of Eqn 3.3.2 as the pertur- bations of interest in this thesis have wavelength l. Here, P is the time-dependent ‘stability matrix’:
P (t) = M (t) − G
ηq(t) p, (3.4.3)
15A similar time-dependent linear stability analysis has been done for the Johnson Segalman
model in Ref. [61], the rolie-poly model in Ref. [2] and a shear transformation zone model in Ref. [101]. In this thesis we use use their methods to obtain more general rules for the time- dependent linear stability to shear heterogeneity in complex fluids.
16Which would add a contribution to P of −(`2k2π2/τ
0L)I in Eqn 3.4.2. This means that
diffusion terms provide a stabilising contribution to the system, i.e., they only act to decrease the growth rate of heterogeneous perturbations. If this were not the case (for example, in models with more complicated non-local terms), neglecting the diffusive terms in P might hide linear instability.
with M = ∂sQ|s,ˆ˙ˆγ and q = ∂γ˙Q|ˆs,ˆ˙γ from the homogeneous background state that we recall obeys:
∂ts = Q,ˆ ˆ
Σ = Gˆσ + ηˆ˙γ. (3.4.4)
This separation of P (Eqn 3.4.3) into two terms with partial derivatives ∂s and ∂γ˙ of Q exists because force balance demands uniform total shear stress:
δΣk= G p · δsk(t) + η δ ˙γk(t) = 0. (3.4.5)
We may therefore replace all occurrences of δ ˙γk(t) with δ ˙γk(t) = −Gηp· δsk(t).
The growth of heterogeneous perturbations δsk in this linearised system corre- sponds to the growth of heterogeneity in the full nonlinear system as long as the perturbations remain small. In this case, δ ˙γk of the linear analysis is approximately equal to δ ˙γk ∼ ∆γ˙, where ∆γ˙ is the ‘degree of banding’ (Eqn 3.3.3) measured in the nonlinear simulation. If ∆γ˙ becomes large, nonlinearities neglected in this linear analysis become important and the relation δ ˙γk ∼ ∆γ˙ no longer holds. This linear analysis is therefore capable of capturing just the onset of instability to shear het- erogeneity (rather than the return to stability), which is our aim.
Classical stability theory
A linear stability analysis of a dynamical system is usually performed for a time- independent background state, so that the stability matrix P , which depends on the background state, is time-independent [157]. Solutions to Eqn 3.4.2 then have the form δs(t) = P
kδsk(0) exp(ωkt), where ωk is an eigenvalue of P . Thus a posi- tive, real part17 of an eigenvalue of P leads to exponentially growing heterogeneous
17Note that the imaginary part of any complex eigenvalue results in oscillations in the per-
turbations [whose magnitude may be growing or decaying depending on Re(ωk)] with frequency
perturbations, and the system is linearly unstable. Conversely, if all real parts of all eigenvalues of P are negative, the system is linearly stable as all perturbations decay exponentially18.
Now, how do we use the above classical stability theory with a time-dependent background state, P (t)? One possibility is to ask whether the system is linearly unstable to heterogeneity according to the eigenvalue analysis above at some time instant t∗. To do this, we define ω(t) largest real part of any eigenvalue of P(t) at time t∗, and ask whether ω(t∗) > 0. If so, the system is linearly unstable to the growth of heterogeneous perturbations at that instant in time19, according to classical stability theory.
Non-normal growth
There is one problem with the classical stability theory approach described above: the assumption that ω describes the decay of perturbations is only valid if the eigen- vectors of P are orthogonal [148]. We outline the reason for this and its implications below, for further details we refer the reader to Schmid et al. [147–149]. To begin, we note that the solution to the time-independent system has the form:
δsk(t) = exp(t P ) · δsk(0), (3.4.6)
where exp(t P ) is the matrix exponential. Classical stability theory uses the eigen decomposition of P = H · Ω · H−1, where H is the matrix whose columns are the eigenvectors of P , and Ω the matrix whose diagonals are the corresponding eigenval- ues. We consider the norm k exp(tP )k2, which describes the growth of perturbation energy from linear stability analysis, whose lower bound is given by the eigenvalue
18A simple example of instability involves a ball at the top of a hill: a small perturbation causes
it to move away from its original position. Similarly, if the ball were in a dip, after a small perturbation is applied the ball simply rolls back to its original position, i.e, a stable system.
19In our ball-on-a-hill example the time-dependent background state could be represented by a
time-dependent landscape for the ball. This linear stability analysis then asks whether a configu- ration at time t∗ of ball location and current landscape is stable.
analysis from classical stability theory described above [148]:
exp(2ωt) ≤ k exp(tP )k2. (3.4.7)
This means that if ω > 0 the system is linearly unstable, regardless of the orthogo- nality of the eigenvectors of P . The upper bound of the norm k exp(tP )k2is obtained by expanding P to find:
k exp tP k2 = k exp H.Λ.H−1t k2 ≤ kHk2kH−1k2exp (2ωt) . (3.4.8)
For orthogonal eigenvectors: kHk2kH−1k2 = 1 and the upper and lower bounds agree, so that the eigenvalue analysis described above returns the growth rate of perturbations in the linearised system. This is the case for perturbations of the time-independent system [147–149].
For the time-dependent system the eigenvectors may not be orthogonal, resulting in kHk2kH−1k2 > 1. This means that the eigenvalue analysis described above only provides a lower bound for stability: positive eigenvalues ω(t∗) > 0 result in linear instability, but negative eigenvalues ω(t∗) < 0 do not guarantee stability. I.e., a positive real part of an eigenvalue of P is sufficient but not necessary for the growth of heterogeneous perturbations. This can lead to ‘transient’ or ‘non-normal’ growth of perturbations at times for which all real parts of the eigenvalues of P are negative. Indeed, we have seen such non-normal growth in the models described in Section 3.2. However, in general we find that any non-normal growth of perturbations is small relative to the growth associated with an unstable eigenvalue ω(t∗) > 0, and never results in ‘significant’ shear banding (that is, shear banding large enough to be detected in experiment).
The above shows that caution must be used whenever considering the largest real part of an eigenvalue from stability analysis alone. We therefore integrate Eqn 3.4.2 to directly determine the solutions δsk(t) whenever the eigenvalue analysis is used in this thesis. This obviates the need for considering a time-dependent eigenvalue and the danger associated with it described above. We note that neither criteria for the step stress or strain ramp protocols depend on this eigenvalue — they directly
use the condition for the growth of heterogeneous perturbations ∂tδsk> 0.
A remark on the eigenvalues of M
For the derivation of the shear startup criterion in Chapter 5 it will be useful to understand the properties of the eigenvalues of M . Thus we now pause briefly to consider these. We begin by considering the linear stability of the steady state homogeneous system described by s(t → ∞) to homogeneous perturbations under the constant shear rate protocol. This corresponds to the k = 0th mode perturbation of the above analysis, with the constraint of δ ˙γk=0 = 0 (due to the fixed shear rate condition). That is, the full response is written as a sum of the background state plus (initially) small homogeneous perturbations δΣk=0, δsk=0:
Σ = Σ + δΣˆ k=0, (3.4.9)
s = ˆs + δsk=0, (3.4.10)
δΣk=0 = G δσk=0 (due to force balance). (3.4.11)
Following the protocol of the above analysis, the perturbations then obey:
∂tδsk=0 = Mk=0· δsk=0, (3.4.12)
with M = ∂sQ|s,ˆ˙ˆγas before. In this thesis we shall consider models with constitutive curves that have a unique value of the total shear stress for a given shear rate Σ( ˙γ). Such systems must be linearly stable to homogeneous perturbations (in the constant shear rate protocol) if they are to be physically meaningful, as linear instability can only result in divergence of the total shear stress Σ → ±∞. Therefore, for physically meaningful systems, all real parts of the eigenvalues of M must be negative. This leads to the results: trM < 0 and20 (−1)n|M | > 0, which we will use in Chapter 5.