On moment relations on surfaces of discontinuity in dissipative media

Баренблатт, Г. И.; Chernyi, G. G.

doi:10.1016/0021-8928(63)90063-7

Cited by 17 publications

(6 citation statements)

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“…Instead, the velocity on the solid‐facing side of the liquid–solid interface U is known. A link between the two is obtained by relating the velocity difference across the interface to the difference in tangential stress acting on the interface, that is, the torque on the interface, which gives

\frac{1}{2} {bold-italicn}_{2} MathClass-bin\cdot ((), {bold-italicP}^{MathClass-bin+} MathClass-bin+ {bold-italicP}^{MathClass-bin-}) MathClass-bin\cdot ((), bold-italicI MathClass-bin- {bold-italicn}_{2} {bold-italicn}_{2}) MathClass-rel= \overset{β}{true} ((), {bold-italicu}_{MathClass-rel| MathClass-rel|}^{MathClass-bin+} MathClass-bin- {bold-italicU}_{MathClass-rel| MathClass-rel|}) MathClass-punc.

This condition is considered, for example, in and later derived thermodynamically in . Upon elimination of the stress on the solid‐facing side of the interface, n 2 · P − from Equation using Equation , we arrive at the so‐called generalized Navier condition [, p.198], which has the form

{bold-italicn}_{2} MathClass-bin\cdot {bold-italicP}^{MathClass-bin+} MathClass-bin\cdot ((), bold-italicI MathClass-bin- {bold-italicn}_{2} {bold-italicn}_{2}) MathClass-bin+ \frac{1}{2 italicC a} {MathClass-rel\nabla}^{s} σ_{2} MathClass-rel= \overset{β}{true} ((), {bold-italicu}_{MathClass-rel| MathClass-rel|}^{MathClass-bin+} MathClass-bin- {bold-italicU}_{MathClass-rel| MathClass-rel|}) MathClass-punc,

and can see that when the surface tension is a constant, this generalized condition reduces to the standard Navier condition .…”

Section: Finite Element Methods Implementation Of Interfacial Physicsmentioning

confidence: 99%

“…This condition is considered, for example, in [70] and later derived thermodynamically in [2]. Upon elimination of the stress on the solid-facing side of the interface n 2 • P − from ( 48) using (47), we arrive at the so-called generalized Navier condition [2, p.198], which has the form…”

Section: Fem Implementation Of Interfacial Physicsmentioning

confidence: 99%

“…Therefore, given values of (χ, ζ), we can determine (x, y) from ( 70) and hence a position in the (r, z)-plane from (72). One can still use (67) to define the distance along the x-axis (ζ = 0) of a given spine from the contact line, except one must now use (70) and (72) to determine the appropriate value of χ k from R k . For our particular geometry we take R max such that the y-axis passes through the apex of the free surface at z = z a and choose the angle Ω such that the distance along y to the apex is the same as the distance along y to the solid (Figure 11).…”

Section: Bipolar Spinesmentioning

confidence: 99%

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Finite element framework for describing dynamic wetting phenomena

Sprittles

Shikhmurzaev

2011

Numerical Methods in Fluids

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SUMMARYThe finite element simulation of dynamic wetting phenomena, requiring the computation of flow in a domain confined by intersecting a liquid-fluid free surface and a liquid-solid interface, with the three-phase contact line moving across the solid, is considered. For this class of flows, different finite element method (FEM) implementations have been used in the literature, and in some cases, these produced apparently contradictory results. In the present paper, a robust framework for the FEM simulation of dynamic wetting flows is developed, which, by consistently adhering to the FEM methodology, leaves no room for ad hoc 'optional' variations in the numerical handling of these flows. The developed approach makes it possible to conduct a convergence study, assess the spatial resolution required to achieve a preset accuracy and provide the corresponding benchmark calculations. This analysis allows one to identify numerical artefacts, which had previously been interpreted as physical effects, and demonstrates that suppressing numerical errors using a 'strong' implementation of a boundary condition creates bigger and less detectable errors elsewhere in the computational domain. We provide practical recommendations on the spatial resolution required by a numerical scheme for a given set of non-dimensional similarity parameters and give a user-friendly step-by-step guide specifying the entire implementation, which allows the reader to easily reproduce all presented results including the benchmark calculations. It is also shown how the developed framework accommodates generalizations of the mathematical model accounting for additional physical effects, such as gradients in surface tensions.

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\frac{1}{2} {bold-italicn}_{2} MathClass-bin\cdot ((), {bold-italicP}^{MathClass-bin+} MathClass-bin+ {bold-italicP}^{MathClass-bin-}) MathClass-bin\cdot ((), bold-italicI MathClass-bin- {bold-italicn}_{2} {bold-italicn}_{2}) MathClass-rel= \overset{β}{true} ((), {bold-italicu}_{MathClass-rel| MathClass-rel|}^{MathClass-bin+} MathClass-bin- {bold-italicU}_{MathClass-rel| MathClass-rel|}) MathClass-punc.

{bold-italicn}_{2} MathClass-bin\cdot {bold-italicP}^{MathClass-bin+} MathClass-bin\cdot ((), bold-italicI MathClass-bin- {bold-italicn}_{2} {bold-italicn}_{2}) MathClass-bin+ \frac{1}{2 italicC a} {MathClass-rel\nabla}^{s} σ_{2} MathClass-rel= \overset{β}{true} ((), {bold-italicu}_{MathClass-rel| MathClass-rel|}^{MathClass-bin+} MathClass-bin- {bold-italicU}_{MathClass-rel| MathClass-rel|}) MathClass-punc,

and can see that when the surface tension is a constant, this generalized condition reduces to the standard Navier condition .…”

Section: Finite Element Methods Implementation Of Interfacial Physicsmentioning

confidence: 99%

Section: Fem Implementation Of Interfacial Physicsmentioning

confidence: 99%

Section: Bipolar Spinesmentioning

confidence: 99%

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Finite element framework for describing dynamic wetting phenomena

Sprittles

Shikhmurzaev

2011

Numerical Methods in Fluids

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“…It is important to emphasize that in the above equations the mass exchange between the interfaces and the bulk is taken into account not only in the surface mass balance equations (34) and (35), as in earlier works [31,32], but also in the conditions for the normal component of the bulk velocity (32) and (33). In earlier works [31,32], the latter was neglected, which essentially corresponds to Q 1 = 0.…”

Section: The Modelmentioning

confidence: 98%

“…no slip) can be explained as follows. As is known, the velocity difference across any thin layer, in our case across the liquid-solid interface, is proportional to the torque formed by external forces acting on it [35]. In our case, the torque is formed by the drag force acting on the solid-facing side of the interface and the shear stress acting on its liquid-facing side.…”

Section: The Modelmentioning

confidence: 98%