An accurate and comprehensive model of thin fluid flows with inertia on curved substrates

Roberts, A. J.; Li, Zhenquan

doi:10.1017/s0022112006008640

Cited by 40 publications

(54 citation statements)

References 58 publications

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“…When the topography is arbitrary, this approach has been generalized using the centre manifold framework [9,10]: in that case the computations are carried out in the neighbourhood of the Nusselt parabolic flow. Using the centre manifold reduction, Roberts and co-workers obtained formally lubrication models [9] and shallow water type models [10] under the hypothesis that the curvature of the underlying bottom surface is small.…”

Section: Introductionmentioning

confidence: 99%

Shallow water viscous flows for arbitrary topopgraphy

Boutounet¹,

Chupin²,

Noble³

et al. 2008

Communications in Mathematical Sciences

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Abstract. In this paper, we obtain new models for gravity driven shallow water laminar flows in several space dimensions over a general topography. These models are derived from the incompressible Navier Stokes equations with no-slip condition at the bottom and include capillary effects. No particular assumption is made on the size of the viscosity and on the variations of the slope. The equations are written for an arbitrary parametrization of the bottom and an explicit formulation is given in the orthogonal courvilinear coordinates setting and for a particular parametrization so called "steepest descent" curvilinear coordinates.

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Section: Introductionmentioning

confidence: 99%

Shallow water viscous flows for arbitrary topopgraphy

Boutounet¹,

Chupin²,

Noble³

et al. 2008

Communications in Mathematical Sciences

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“…Following similar modelling for Newtonian thin films [2,10], this innovation of modifying the free surface condition to (13) places the modelling of a physically important class of nonNewtonian fluids upon the powerful and sound basis of centre manifold theory, e.g. [15][16][17].…”

Section: Resultsmentioning

confidence: 99%

“…Then the systematic analysis developed in this Letter supports the nondimensional model of the flow ( where Re is the nondimensional Reynolds number, c s is the coefficient of proportionality in the nonlinear stress-strain relation, and where g 1 and g 2 are the nondimensional components of gravity along and normal to the flat substrate, respectively. This model generalises the model of Newtonian fluids [2]. Fluid is conserved through (1).…”

Section: Introductionmentioning

confidence: 88%

“…Fluid is conserved through (1). The momentum equation (2) incorporates effects of inertia,ū t , self-advection, uū x andū 2 η x , bed drag, (ū/η) s , and gravitational forcing, (g 1 − g 2 η x ); the dependence of the coefficients upon s models the subtle effects of the power law rheology. For example, for flow down an inclined flat plate with lateral gravity g 1 , the nonlinear bed friction may balance gravitational forcing whence the above model predicts the equilibrium flow to have mean velocity…”

Section: Introductionmentioning

confidence: 99%

“…But the lubrication approximation, that creates models expressed only in terms of the fluid thickness η(x, t), does not model inertial dynamics and so cannot resolve any wave-like dynamics. To model faster flows, potentially with wave effects, we must resolve the dynamics of both the fluid thickness and a measure of horizontal momentum [2,10], we used η andū in (1)-(2). For example, Harris et al [11] modelled particle driven gravity currents using shallow water equations that resolve the dynamics of both the fluid thickness and the mean lateral velocity.…”

Section: Introductionmentioning

confidence: 99%

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The inertial dynamics of thin film flow of non-Newtonian fluids

Roberts

2008

Physics Letters A

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Consider the flow of a thin layer of non-Newtonian fluid over a solid surface. I model the case where the viscosity depends nonlinearly on the shear-rate; power law fluids are an important example, but the analysis here is for general nonlinear dependence. The modelling allows for large changes in film thickness provided the changes occur over a relatively large enough lateral length scale. Modifying the surface boundary condition for tangential stress forms an accessible foundation for the analysis where flow with constant shear is a neutral critical mode, in addition to a mode representing conservation of fluid. Perturbatively removing the modification then constructs a model for the coupled dynamics of the fluid depth and the lateral momentum. For example, the results model the dynamics of gravity currents of non-Newtonian fluids when the flow is not creeping.

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Modeling Methodologies for Moderate Reynolds Number Flows

Kalliadasis

Ruyer-Quil

Scheid

et al. 2012

Applied Mathematical Sciences

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Nearly all low-dimensional models for isothermal films at moderate Reynolds numbers found in the literature rely on a fundamental closure assumption for the streamwise velocity field: a simple self-similar velocity profile with the variables (x, t) and y/ h separated. This is the basis for the classical Kapitza-Shkadov model. The velocity profile in this model is a self-similar semi-parabolic velocity profile in which the variables are separated as above and which trivially satisfies the x component of the momentum equation at zero Reynolds number (in which case the interface is flat). In this chapter we discuss a systematic methodology to relax the self-similar assumption while maintaining separation of variables as in the long-wave theory: it is based on a combination of an expansion for the velocity field in terms of polynomial test functions, the gradient expansion and an elaborate averaging technique that utilizes the method of weighted residuals. The averaging can be justified by the indepth coherence of the flow ensured by the action of viscosity. The result is two "optimal" models in the sense that the models are always the same independently of the particular averaging methodology employed (provided of course that a sufficiently large number of test functions is used in each case). The two models are: A twoequation system consistent at O(ε), referred to as the first-order model, and a fourequation system consistent at O(ε 2 ), referred to as the full second-order model. An ad-hoc compromise between the two in both complexity and accuracy is provided by the simplified second-order model, whereas a regularization procedure enables us to reduce the dimension of the four-equation system and to obtain a two-equation model consistent at O(ε 2 ) which we refer to as the regularized model. The weighted residuals formulation developed here is compared to the center-manifold analysis by Roberts. The momentum equation in Robert's model contains all the terms of the momentum equation of the first-order model but with different coefficients. However, it also contains additional terms including high-order nonlinearities which then necessarily restrict the applicability of the model in the drag-gravity regime. On the other hand, the average models we obtain from the weighted residuals formulation, are capable of describing the drag-inertia regime, even though the formulation presumes that inertia effects are weak corrections to the balance of viscous drag and gravity, which strictly speaking holds only in the drag-gravity regime. The reason S. Kalliadasis et al., Falling Liquid Films, Applied Mathematical Sciences 176,

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An accurate and comprehensive model of thin fluid flows with inertia on curved substrates

Cited by 40 publications

References 58 publications

Shallow water viscous flows for arbitrary topopgraphy

Shallow water viscous flows for arbitrary topopgraphy

The inertial dynamics of thin film flow of non-Newtonian fluids

Modeling Methodologies for Moderate Reynolds Number Flows

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