×

The dam-break problem for viscous fluids in the high-capillary-number limit. (English) Zbl 1171.76349

Summary: Experiments were undertaken to investigate dam-break flows where a finite volume of highly viscous fluid (glucose with viscosity \(\mu \approx 350\) Pa s) maintained behind a lock gate was released into a horizontal or inclined flume. The resulting sequence of flow-depth profiles was tracked using a three-dimensional visualization system. In the low-Reynolds-number and high-capillary-number limits, analytical solutions can be obtained from the Navier-Stokes equations using lubrication theory and matched asymptotic expansions. At shallow slopes, similarity solutions can also be worked out. While the variation in the front position scaled with time as predicted by theory for both horizontal and sloping flumes, there was a systematic delay in the front position observed. Moreover, taking a closer look at the experimental flow-depth profiles shows that they were similar, but they noticeably deviated from the theoretical similarity form for horizontal planes. For sloping beds, the flow-depth profile is correctly predicted provided that different scalings are used at shallow and large slopes.

MSC:

76D08 Lubrication theory
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76M55 Dimensional analysis and similarity applied to problems in fluid mechanics
86A05 Hydrology, hydrography, oceanography
Full Text: DOI

References:

[1] DOI: 10.1111/1467-9590.00074 · Zbl 1001.35056 · doi:10.1111/1467-9590.00074
[2] DOI: 10.1017/S002211200600930X · Zbl 1090.76022 · doi:10.1017/S002211200600930X
[3] DOI: 10.1017/S0022112098002390 · Zbl 0941.76509 · doi:10.1017/S0022112098002390
[4] DOI: 10.1017/S0022112007005174 · Zbl 1110.76015 · doi:10.1017/S0022112007005174
[5] DOI: 10.1017/S0022112073002594 · Zbl 0265.76040 · doi:10.1017/S0022112073002594
[6] Smith, Z. Angew. Math. Mech. 20 pp 556– (1969) · Zbl 0175.24003 · doi:10.1007/BF01595050
[7] Simpson, Gravity Currents in the Environment and the Laboratory (1997)
[8] DOI: 10.1017/S0022112082001797 · doi:10.1017/S0022112082001797
[9] DOI: 10.1038/300427a0 · doi:10.1038/300427a0
[10] DOI: 10.1061/(ASCE)0733-9429(1994)120:12(1350) · doi:10.1061/(ASCE)0733-9429(1994)120:12(1350)
[11] DOI: 10.1017/S0022112082001979 · Zbl 0492.76101 · doi:10.1017/S0022112082001979
[12] DOI: 10.1017/S0022112090001616 · Zbl 0686.76027 · doi:10.1017/S0022112090001616
[13] DOI: 10.1002/cpa.3160190405 · Zbl 0145.22701 · doi:10.1002/cpa.3160190405
[14] DOI: 10.1098/rspa.1982.0079 · Zbl 0488.35022 · doi:10.1098/rspa.1982.0079
[15] DOI: 10.1063/1.858233 · doi:10.1063/1.858233
[16] DOI: 10.1093/imamat/30.2.209 · doi:10.1093/imamat/30.2.209
[17] DOI: 10.1017/S0022112082001785 · doi:10.1017/S0022112082001785
[18] DOI: 10.1093/imamat/31.2.121 · doi:10.1093/imamat/31.2.121
[19] DOI: 10.1017/S0022112086000332 · Zbl 0597.76102 · doi:10.1017/S0022112086000332
[20] DOI: 10.1017/S0022112090001240 · Zbl 0686.76024 · doi:10.1017/S0022112090001240
[21] Courant, Supersonic Flow and Shock Waves (1948)
[22] Goodwin, Phys. Fluids 3 pp 515– (1991) · Zbl 0735.76022 · doi:10.1063/1.858113
[23] DOI: 10.1007/s00348-007-0374-3 · doi:10.1007/s00348-007-0374-3
[24] DOI: 10.1017/S0022112077002328 · Zbl 0379.76029 · doi:10.1017/S0022112077002328
[25] Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics (1996) · Zbl 0907.76002 · doi:10.1017/CBO9781107050242
[26] Schenck, Ullmann’s Encyclopedia of Industrial Chemistry pp 457– (2007)
[27] Ockendon, Moving Boundary Problems (1978)
[28] Nsom, J. Hydraul. Res. 38 pp 459– (2000)
[29] DOI: 10.1061/(ASCE)0733-9429(2002)128:5(543) · doi:10.1061/(ASCE)0733-9429(2002)128:5(543)
[30] DOI: 10.1143/JPSJ.37.539 · doi:10.1143/JPSJ.37.539
[31] Mei, J. Maths. Phys. 45 pp 266– (1966) · Zbl 0144.47401 · doi:10.1002/sapm1966451266
[32] DOI: 10.1017/S0022112083000476 · doi:10.1017/S0022112083000476
[33] DOI: 10.1016/j.euromechflu.2005.09.005 · Zbl 1098.76037 · doi:10.1016/j.euromechflu.2005.09.005
[34] DOI: 10.1017/S0022112092002520 · Zbl 0825.76146 · doi:10.1017/S0022112092002520
[35] DOI: 10.1007/BF00129871 · Zbl 0708.76051 · doi:10.1007/BF00129871
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.