### Abstract

We employ a homotopy method, rather than conventional stability theory, in order to resolve the degeneracy due to resonance, which exists in fluid motion associated with a channel of infinite extent in ventilated double glazing. The introduction of a symmetry breaking perturbation, in the form of a Poiseuille flow component, alters substantially the resonant bifurcation tree of the original flow. Previously unknown resonant higher order nonlinear solutions, i.e. after the removal of the perturbative Poiseuille flow component, are discovered. A possible extension of the methodology to consider non-Newtonian gradient enhanced incompressible viscous fluids is also briefly discussed.

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## Funding

TA acknowledges the financial support from the Horizon 2020 Marie Sklodowska Curie Programme of the European Union. SG acknowledges financial support from ORDIST of Kansai University. TI acknowledges an International Collaboration fund from Aston University. THB acknowledges a research studentship from EPSRC DtP 2020. This work was also funded by the RISE-2018–824022-ATM2BT of the European H2020-MSCA programme. Finally, ECA acknowledges the support of Friedrich-Alexander University of Erlangen–Nuremberg, grant no. 377472739/GRK 2423/1-2019 and a Mercator Fellow post.

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## GRADIENT FLUIDS

### GRADIENT FLUIDS

At small microscopic and molecular scales, non-local effects are important and a robust way to account for them is to use higher-order spatial gradients to enter in the consitutive equation for the Newtonian stress \(\boldsymbol{\tau}\). One effective way to do this is to employ the internal length gradient (ILG) framework recently reviewed by the last author [21, 22]. In the present case of incompressible Newtonian fluids, this amounts to replacing the standard expression for \(\boldsymbol{\tau}\) by the weakly non-local gradient constant given by the equation below:

where \(l_{\mathbf{D}}\) is an internal length parameter accounting for weakly non-local gradient effects. The introduction of the Laplacian is not arbitrary, but it arises from a Taylor series expansion of a non-local integral expression for the average (macroscopic) stress which, by retaining terms up to the second order in the Taylor series expansion, is replaced by the local (microscropic) stress and its Laplacian multiplied by the internal length parameter \(l_{\mathbf{D}}\) as shown in Eq. (A2).

The determination of the new phenomenological parameter \(l_{\mathbf{D}}\) is left to experiments and simulation, and the same holds for the new type of boundary conditions that the Laplacian term requires. Among the implications of Eq. (A2) are the elimination of singularities and the revision of boundary layer estimates, as well as the interpretation of size effects that classical theory is not able to capture. Equally important is the new possibility that is offered for relocating the Eckhaus stability boundary and to obtain new results by applying the homotopy-SBA method. All this will be a subject of a future series of articles.

For the purposes of the present article, however, we list below the governing equation of motion implied by Eq. (A1), i.e. the modified, gradient enhanced Navier–Stokes (NS) equations of incompressible flow:

where \(\Delta=\nabla^{2}\) and \(\Delta^{2}=\nabla^{4}\) denote the Laplacian and biharmonic operators respectively. It is noted that Eq. (A2) is identical to the equation used to model plane Poiseuille liquid flow at small-length scales as discussed in [20, 21]. A slightly generalized model was also used by the authors to consider turbulence. The governing differential equations for this model reads

where the \(\alpha\) parameter denotes a statistical correlation length and \(\overset{\nabla}{\mathbf{D}}=\dot{\mathbf{D}}+\mathbf{DW}-\mathbf{WD}\) denotes the usual Jaumann rate (\(\mathbf{W}=\frac{1}{2}\left[\nabla{\mathbf{u}}-(\nabla{\mathbf{u}})^{T}\right]\)).

Steady-state solutions of Eq. (A2) may be determined by employing the operator split method (or the use of Ru–Aifantis theorem [20, 21]) utilized to eliminate singularities from dislocation lines and crack tips in the theory of gradient elasticity. This same procedure leads to the cancelation of singularities in typical fluid flow calculations involving immersed objects. It turns out, for example, that the resulting gradient Oseen tensor \(\mathcal{O}_{ij}^{G},\) which generalizes its classical counterpart \(\mathcal{O}_{ij}\)

where \(r_{i}\) denotes the position vector and \(r\) its magnitude, reads

More details on gradient fluids can be found in [21]. In a forthcoming publication, the methodology outlined in Section 2 will be utilised to analyse Eq. (A2) in place of Eq. (1) listed in Section 1.

It is expected that new solutions and stability branches will be found. This expectation is partly suggested by the new shear band solutions obtained within the ILG framework for gradient plasticity by employing the homotopy method [22].

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Akinaga, T., Harvey-Ball, T.M., Itano, T. *et al.* On the Problem of Resonant Incompressible Flow in Ventilated Double Glazing.
*Lobachevskii J Math* **42**, 1753–1767 (2021). https://doi.org/10.1134/S1995080221080035

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DOI: https://doi.org/10.1134/S1995080221080035