A Potential Field Description for Gravity-Driven Film Flow over Piece-Wise Planar Topography

Marner

2020

Water

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The use of potential fields in fluid dynamics is retraced, ranging from classical potential theory to recent developments in this evergreen research field. The focus is centred on two major approaches and their advancements: (i) the Clebsch transformation and (ii) the classical complex variable method utilising Airy’s stress function, which can be generalised to a first integral methodology based on the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Basic questions relating to the existence and gauge freedoms of the potential fields and the satisfaction of the boundary conditions required for closure are addressed; with respect to (i), the properties of self-adjointness and Galilean invariance are of particular interest. The application and use of both approaches is explored through the solution of four purposely selected problems; three of which are tractable analytically, the fourth requiring a numerical solution. In all cases, the results obtained are found to be in excellent agreement with corresponding solutions available in the open literature.

Section: Integration Of the Dynamic Boundary Conditionmentioning

confidence: 99%

Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances

Marner

2020

Water

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“…Next, by neglecting ∂v y /∂ x compared to ∂v x /∂ y and omitting inertial terms, the set of PDEs is reduced to one that can be solved by successive integration, as shown in detail in [24], leading to the following general solution:…”

Section: Lubrication Approximationmentioning

confidence: 99%

Multilayer Modelling of Lubricated Contacts: A New Approach Based on a Potential Field Description

Mellmann

Springer Tracts in Mechanical Engineering

et al. 2020

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A first integral approach, derived in an analogous fashion to Maxwell’s use of potential fields, is employed to investigate the flow characteristics, with a view to minimising friction, of shear-driven fluid motion between rigid surfaces in parallel alignment as a model for a lubricated joint, whether naturally occurring or engineered replacement. For a viscous bilayer arrangement comprised of immiscible liquids, it is shown how the flow and the shear stress along the separating interface is influenced by the mean thickness of the layers and the ratio of their respective viscosities. Considered in addition, is how the method can be extended for application to the more challenging problem of when one, or both, of the layers is a viscoelastic material.

“…a 11 , a 12 , become integrable by: (i) neglecting inertia; (ii) applying an asymptotic analysis, underpinned by the long-wave approximation, as demonstrated in [4] for planar film flows. a 11 , a 12 , become integrable by: (i) neglecting inertia; (ii) applying an asymptotic analysis, underpinned by the long-wave approximation, as demonstrated in [4] for planar film flows.…”

Section: Implementation For Axisymmetric Flowsmentioning

confidence: 99%

“…The two relevant field equations for ψ and Φ, derived as Euler-Lagrange equations of (4) w.r.t. a 11 , a 12 , become integrable by: (i) neglecting inertia; (ii) applying an asymptotic analysis, underpinned by the long-wave approximation, as demonstrated in [4] for planar film flows. Following the same steps as in [4], a single equation for the local film thickness results.…”

Section: Implementation For Axisymmetric Flowsmentioning

confidence: 99%

“…a 11 , a 12 , become integrable by: (i) neglecting inertia; (ii) applying an asymptotic analysis, underpinned by the long-wave approximation, as demonstrated in [4] for planar film flows. Following the same steps as in [4], a single equation for the local film thickness results. Neglecting capillary effects and taking r 0 as an appropriate length scale, leads to the following equation:…”

Section: Implementation For Axisymmetric Flowsmentioning

confidence: 99%

See 1 more Smart Citation

Thin liquid film formation on hemispherical and conical substrate

Marner

Proc Appl Math and Mech

2019

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The deposition and coating of thin films onto curved rigid substrate, involving displacement of air by a liquid, has numerous applications within the technology sectors but faces two major challenges: (i) control of the local film thickness; (ii) ensuring that the coating remains stable. The work reported here investigates the full coverage of three-dimensional curved geometries, of hemispherical and conical shape, by a continuously fed, gravity-driven, thin liquid layer. The modelling approach adopted utilises a first integral formulation [1, 2] of the Navier-Stokes equations leading to a variational formulation in the case of steady flow and an advantageous re-formulation of the dynamic boundary condition at the free surface [3]. Asymptotic analysis, underpinned by the long-wave approximation, enables analytic solutions for the local film thickness to be obtained. Formulation Problem definitionConsider the case of a continuous, steady, gravity-driven, incompressible film flow over a hemispherical or conical surface, as illustrated in Fig. 1, formed via a Newtonian liquid fed from above, with particular interest directed at the resulting free surface profile. (a) z r r 0 ϑV (b) z r y x α V Fig. 1 Gravity driven film flow over: (a) a hemisphere of radius r0; (b) a cone with inclination angle α.Both are continuously fed, with volumetric flow rateV , from above. Spherical (r, ϑ) and cylindrical (r, z) coordinate systems are employed for the hemisphere and cone, respectively; the equations of motion, once formulated, are transformed to natural coordinates as follows x = −z sin α+r cos α, y = z cos α + r cos α. First Integral variational approach for steady 3D flowIn [3] a first integral of the Navier-Stokes equation is derived in an analogous fashion to Maxwell's use of potential fields when developing his classical electro-magnetic theory. For steady flow, a traceless symmetric tensor potentialā ij and a scalar potential Φ are introduced as auxiliary unknowns allowing derivation of the equations of motion from the variational principle:in terms ofā ij , Φ and the streamfunction vector Ψ k ; the velocity components given via u i = ε ijk ∂ j Ψ k , with ε ijk Levi-Civita symbol, fulfil the continuity equation ∂ i u i = 0 automatically. Taking the divergence of the tensor equation resulting via variation with respect toā ij , recovers the Navier-Stokes equations exactly [3]. Likewise, the dynamic boundary condition related to stress equilibrium at a free surface can be recasted in the form of a first integral of the tensor potential:where σ s is the surface tension and U the potential energy density of an external conservative force. The mathematical formulation is completed by the kinematic boundary condition at a free surfaces and the no-slip/no-penetration condition at a solid boundary. Implementation for axisymmetric flowsIn terms of both coordinate systems, q 1 = r, q 2 = ϑ, q 3 = ϕ for the hemisphere and q 1 = z, q 2 = r, q 3 = ϑ, for the cone, the flow is clearly independent of the azimuthal angle ϕ, allowing fo...