The circular capillary jump

Circular hydraulic jumps: where does surface tension matter?

“…The last term on the right designates the viscous dissipation by friction on the substrate, while the first one is an additional term compared to previous approaches, which is presumed to be ‘at the origin’ of the new scaling (1.2). This conjecture has been contested (Duchesne, Andersen & Bohr 2019; Bohr & Scheichl 2021) (see also the answer in Bhagat & Linden (2020)), and it is also known that a scaling like (1.2) can also appear without such an assumption, as shown for instance by Button et al. (2010) for liquid bells formed below a ceiling.…”

Surface tension and energy conservation in a moving fluid

“…In order to consider the advection of a point on this segment by a velocity field U(x, y, t ) = (U, 0, W ), we take its x coordinate as time-dependent, thus its z coordinate as h(x(t ), t ). On C s , U = ẋ and W = h t + ẋh x by (12). It is evident that U • ms vanishes on the points of C s with x coordinates located between x 1 and x 2 .…”

“…In the important case of a static control volume, however, its projection B and thus C stay stationary, while its top face A s is moving with the normal component u n of the material flow (when the flow is not stationary), following from comparison of ( 12) with (7). Thus, U = 0 on the lateral sides of V, U C = (U, V, 0) ≡ 0, but C s moves vertically with U = h t ẑ, see (12), and n = ns , dA = dA s ,…”

“…Although this is merely a consequence of the governing equations of motion and conventional kinematic and dynamic boundary conditions, this procedure has been challenged recently in Journal of Fluid Mechanics by Bhagat et al [11] and Bhagat and Linden [12]. These authors claim that the tangential surface tension contributes to a power term in the integral energy budget and should, therefore, be taken into account explicitly.…”

“…The additional energy term (1) for a static control volume in a steady flow was originally postulated by [11] without explanation. In their follow-up paper [12] they argue that it actually stems from the viscous advection terms in (32). Writing the global stress balance (10) in the form [their Eq.…”