# Linear stability of a generalised Newtonian fluid down an incline: unifying the roll waves

Fabien Depoilly, François Rousset, Séverine Millet, Hamda Ben Hadid, Simon Dagois-Bohy

## ► To cite this version:

Fabien Depoilly, François Rousset, Séverine Millet, Hamda Ben Hadid, Simon Dagois-Bohy. Linear stability of a generalised Newtonian fluid down an incline: unifying the roll waves. Fluids and Complexity III, Dec 2023, Nice, France. hal-05266696

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Submitted on 18 Sep 2025

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# Linear stability of a generalised Newtonian fluid down an incline: unifying the roll waves

CETHIL
UMR 5008

<figure><img src="image_3.png" /></figure>

Fabien Depoilly, François Rousset, Séverine Millet,
Hamda Ben Hadid, Simon Dagois-Bohy

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Lyon 1

INSA

INSTITUT NATIONAL
DES SCIENCES
APPLIQUES
LYON

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ÉCOLE
CENTRALE LYON

## 1 - Roll waves on fluids

* Roll waves are met in a wide variety of contexts:

- Civil engineering

- Industrial processes

- Geophysical flows

* Authors have studied the stability of films down an incline for various rheologies:

Newtonian:

Shear thinning:

Viscoplastic:

- Benjamin [1]

- Ng & Mei [4]

- Balmforth [7]

- Smith [2]

- Ruyer-Quil [5]

- Noma [8]

- Liu & Gollub [3]

- Allouche [6]

- Calusi [9]

...but a common framework is missing.

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**Question:** How to unify the theories of roll waves stability for different rheologies?

## 2 - Rheological model

* Generalised Newtonian fluids:

$$ \tau(\dot{\gamma}) = \eta(\dot{\gamma})\dot{\gamma} $$
Viscosity $\xrightarrow{\searrow}$

* And also:

$$ \dot{\gamma}(\tau) = \Phi(\tau)\tau $$
Fluidity $\xrightarrow{\searrow}$

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## 4 - Linear stability model

Generalised Orr-Sommerfeld equation:

$$ \left\{ \begin{array}{l} i\alpha\text{Re}((\bar{u}-c)(\psi''-\alpha^2\psi)-\bar{u}''\psi) = \\ \qquad (\bar{\gamma}\psi'')'' + \alpha^2((\bar{\gamma}\psi)'' - 4(\bar{\gamma}\psi')' + \bar{\gamma}\psi'') + \alpha^4\bar{\gamma}\psi, \\ \text{and Boundary Conditions} \end{array} \right. $$

with $\bar{\gamma}(y) = \frac{-1}{\bar{u}''(y)}$ and $\bar{\eta}(y) = \frac{1}{\Phi(1-y)}$

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Analytical results of long waves asymptotic expansion:

$$ O(1): c_0 = 1 $$

$$ O(\alpha): \text{Re}_c^\theta = \frac{\int_0^1 \int_0^y \bar{u}''(y_1) (1-y_1) dy_1 dy_1}{\int_0^1 \int_0^y \bar{u}''(y_1) [(1-y_1)(\bar{u}(y_1)+c_0) + 2 \int_1^{y_1} \bar{u}(y_2) dy_2] dy_1 dy_1} $$

## 6 - Conclusion

* Unified Orr-Sommerfeld equation by using a generalised Newtonian rheological law

* Analytic stability thresholds with asymptotic expansion

* Good agreement between model and experimental data

* Easy study for new rheological models (not shown here)

## 7 - Perspectives

* Dispersion relations for viscoplastic fluids

- Numerical resolution of the O.S problem

- Regularised rheological laws

* Generalised model for non linear stability

* Experiments

- Stability thresholds

- Experimental dispersion relations

- Non linear evolution

<figure><img src="image_11.png" /></figure>

## 3 - Dimensionless numbers built from typical scales

* Natural scales:

- Flow depth: $h_0$

- Bottom shear stress: $\tau_b = \rho g h_0 \sin(\theta)$

* Scales built from natural scales:

- Bottom shear rate: $\dot{\gamma}_b = \Phi(\tau_b)\tau_b$

- Velocity: $u_0 = \dot{\gamma}_b h_0$

- Pressure: $p_0 = \rho g h_0 \cos(\theta)$

* Dimensionless numbers:

$$ \text{Re} = \frac{\rho u_0 h_0}{\eta(\dot{\gamma}_b)} \quad \text{and} \quad \text{Rheology induced dimensionless numbers} $$

* Dimensionless base flow:

$$ \bar{u}(y) = \int_{1-y}^{1} \Phi(\bar{\tau}) \bar{\tau} \, d\bar{\tau} \quad \text{with } \bar{\tau} = 1 - y $$

## 5 - Validation from experimental results

* Shear-thinning [6] : Power-law : $\tau(\dot{\gamma}) = \dot{\gamma}^n$

Carreau : $\tau(\dot{\gamma}) = \left( \frac{1 + (\dot{\gamma}/\dot{\gamma}^*)^2}{1 + (1/\dot{\gamma}^*)^2} \right)^{\frac{n-1}{2}} \dot{\gamma}$

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* Viscoplastic [8] : HB : $\tau(\dot{\gamma}) = \text{Bi} + (1 - \text{Bi})\dot{\gamma}^n$

Papanastasiou :

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## 8 - Bibliography

[1] Benjamin, JFM, 1956

[2] Smith, JFM, 1990

[3] Liu & Gollub, JFM, 1993

[4] Ng & Mei, JFM, 1994

[5] Ruyer-Quil & al., JFM, 2012

[6] Allouche & al., JFM, 2017

[7] Balmforth & al., JFM, 2004

[8] Noma at al., JFM, 2021

[9] Calusi & al., POF, 2022

[10] Darbois-Texier & al., JFM, 2023