Advances in Fluid Mechanics IX 5 Liquid gas inteface unde hydostatic pessue A. Gajewski Bialystok Univesity of Technology, Faculty of Civil Engineeing and Envionmental Engineeing, Depatment of Heat Engineeing, Poland Abstact The suface-tension foces and the pessue foce at a gas liquid inteface wee balanced. This was done at a diffeential contol suface unde hydostatic pessue. As a esult, thee quadatic and two linea equations wee deived epesenting the conditions fo equilibium at the inteface. These equations have eight oots descibing eleven diffeent equilibium states. The most impotant is the Young-Laplace fomula unde hydostatic pessue which is denoted in the spheical coodinate system. Hence, it is a quadatic equation. The fist oot of this equation descibes a pendant dop, dop o bubble, while the second one cicumscibes a sessile dop. Thee ae two solutions fo the unifom pessue, whee one of them is the Young-Laplace fomula. It is concluded that the suface existence depends on the pessue diffeence between both bulk phases. A plane is fomed when the suface-tension foces equilibate themselves. Unde unifom pessue the inteface is a sphee and its othe shapes need a pessue gadient. To fom a dop o a bubble the pessue diffeence must be highe than its bode value. A sessile dop exists if the gauge pessue is negative, while a pendant dop equies a positive value. A compaison with expeimental esults is done fo the bubble. Keywods: Young-Laplace equation in spheical coodinates, static equilibium, dop, bubble. Intoduction To descibe the foce balance on the liquid gas inteface, the Young-Laplace equation is usually applied. The equation must be solved fo each point sepaately. Hence, it is computationally difficult to balance the suface-tension WIT Tansactions on Engineeing Sciences, Vol 74, WIT Pess www.witpess.com, ISSN 74-5 (on-line) doi:.495/afm
5 Advances in Fluid Mechanics IX and hydodynamic foces on a diffeential suface o volume. The Young- Laplace equation was deived fom wok involving the expansion of a soapbubble suface. The same fomula can be also applied as a condition of static equilibium because the suface tension is consideed simultaneously as a suface-tension foce pe unit length and, following Tolman [], eithe as the quotient of the enegy change with the change in suface aea at constant entopy, constant composition and constant volumes fo the two phases (measued at the suface of tension) o as the ate of fee enegy change with the change in suface aea at constant tempeatue, constant composition and constant volumes. The theoy of wetting phenomena on cuved sufaces has been developed to date by the deivation of fomulas o theoems oiginating fom an enegetic definition and thei application in foce balances. Hence, this solution is limited to one scala equation. Because a foce may have thee components, this equation may not be applied as a condition fo equilibium in the case of an unsymmetical shape, no can the shape of the suface be descibed using only the foce balance due to the lack of the coesponding equations along the tangents to the suface diections. The main goal of the cuent wok is to develop a wetting theoy of cuved gas-liquid intefacial sufaces beginning diectly fom the suface tension defined as the unit foce. This was expected to yield two equations of foce balance fo the symmetic shapes fo the doplet o bubble unde hydostatic pessue. Mathematical model The topic of discussion hee is the pediction of the shape of the liquid gas inteface unde hydostatic pessue. To do it, the foces applied on the suface at est ae balanced. As suface foces shape the suface, they should be balanced at the suface. While the fixed coodinate system is used, the hydostatic pessue ceates a changing adius of cuvatue. Theefoe, the suface with a vaying adius along the depth of the liquid must be descibed. The suface pesented in fig. is defined in the spheical paameteisation in which the adial distance changes along the coodinate: f, ( )sincos,( )sinsin,( )cos () The vesos of the suface-coodinate system ae δ cos cos i cos sinj sin k () δ sini cos j () whee is defined in fig.. Moving along the tangential vecto, the suface cuves in the diection of the movement. The cuvatue is descibed by the unit vecto pependicula to the suface. It aises fom the stuctue of the suface, which is a two-dimensional object existing in thee-dimensional space. Taking the above into account, it is necessay to define the nomal unit vecto: δ n sin cos i sin sinj cos k (4) WIT Tansactions on Engineeing Sciences, Vol 74, WIT Pess www.witpess.com, ISSN 74-5 (on-line)
Advances in Fluid Mechanics IX 5 z n i k O j y x Figue : The assumed suface coodinates and the vesos.. The foce system No matte how small, a fluid is a substance that defoms continuously unde the application of tangential stess. In the consideed case, the shea stesses equal zeo. Theefoe, only the nomal stesses ae sustained. As a esult, the esultant foce acting on the suface is nomal to the suface. The esultant pessue foce that tends to expand the suface, acts in the cente of the aea. Because the foce system has to satisfy Newton s thid law, the diections and senses of the esultant suface foces peventing expansion must be oiented out fom the suface. Howeve, the components of the esultant suface-tension foces must be applied at the bounday ac of the suface, and these ae tangential to the suface. The foce system is shown in fig.. In this case, the esultant pessue foce is the active foce. Theefoe, the esultant of the tangential suface foces is the eactive foce. A diffeential suface element is defined using two acs with adii and sin (cf. fig. ). Theefoe, this suface is a cuvilinea ectangle (fig. and fig. ). Hence, we have fou infinitesimal suface-tension foces (fig. ). As a esult, we obtain two esultants, one each of each pai of these foces. These esultants equilibate the infinitesimal pessue foce. WIT Tansactions on Engineeing Sciences, Vol 74, WIT Pess www.witpess.com, ISSN 74-5 (on-line)
54 Advances in Fluid Mechanics IX z df sin dl d df O df df p dda dl dl df dl Figue : The foce system fo an infinitesimal element. The pessue foce is expessed by the fomula sindd df p pda p δ n (5) cos The suface tension foces in the component ae pesented df d cos δ (6) df d cos δ (7) Next, the components in the diection ae obtained: df d sin d d δ, (8) df d sin d d δ. (9) The esultant foce in the diection is: d df sin dcos d sincos dδ sin sin ddδ n () and in the diection: d d d d cos sin sinsin n cos cos () The system of the esultant foces and the coss section of the investigated suface ae showed in fig.. Hydostatic pessue is calculated as follows: p g cos. () WIT Tansactions on Engineeing Sciences, Vol 74, WIT Pess www.witpess.com, ISSN 74-5 (on-line)
Advances in Fluid Mechanics IX 55 df df d -d/ d/ d/ df p d Figue : The foce system esolved into meidian and nomal diections.. The foce balance The esultant suface tension foces and pessue foce ae balanced. Respectively, thei components along the meidian and nomal diections to the suface must equal zeo. Fist we obtain the foce balance in tangential diection: d tan d d d () and the balance of the foces in the nomal diection: d d g cos (4) Substituting the elationship g (5) into Eqs. () and (4), we obtain a system of dimensionless equations as follows: d d tan tan d d d d o cos (6) fo which the solution is the poduct of the left sides of the five equations: WIT Tansactions on Engineeing Sciences, Vol 74, WIT Pess www.witpess.com, ISSN 74-5 (on-line)
56 Advances in Fluid Mechanics IX o cos (7) o (8) cos (9) o cos sin () o cos cos () These equations can be expessed in dimensional fom as g cos () () g cos (4) g cossin (5) g coscos (6) Fomulas (7) and () ae the Young-Laplace equation fo the hydostatic pessue witten in spheical coodinates. These fomulas can be solved as a quadatic equation with as the vaiable. Hence, the shape of the suface can be found easily. In contast, thei solution in the Catesian coodinate system, in which the double suface tension is divided by the adius of cuvatue, equies the solution of a second-ode diffeential equation. It can be concluded that the suface can exist if the esultant suface-tension foce equals zeo (cf. Eq. (4)), but it cannot be fomed if the hydostatic pessue is equal to zeo. The last conclusion aises fom the analysis of Eqs. (), (4), (5) and (6) and is expessed implicitly by Eq. ().. The shapes of the intefaces As the shapes of the liquid gas intefaces ae pesented by the solutions of Eqs. (7) (), we can distinguish fou kinds of suface equilibium. The fist is descibed the Young-Laplace fomula, i.e., Eq. (7). The second is the flat suface, expessed by Eq. (9), with a lack of a suface as its paticula case (cf. Eq. (8)). The thid and fouth kinds of equilibia have not been connected with any eal situation so fa. Theefoe, they can be only the solutions of the foce balance without any physical meaning. Although the analysis is made in the suface-coodinate system, the gaphs ae plotted in Catesian coodinates: x sin (7) z cos (8) whee the vaiables ae defined in fig.. Because these solutions have axisymmetic shapes, they ae plotted as coss-sections in the x z plane... The Young-Laplace equation The solution of Eq. (7), i.e., the Young-Laplace fomula with hydostatic pessue in spheical coodinates, has two oots: WIT Tansactions on Engineeing Sciences, Vol 74, WIT Pess www.witpess.com, ISSN 74-5 (on-line)
Advances in Fluid Mechanics IX 57 o o 8cos (9) cos o o 8cos () cos Because these solutions have discontinuities at, the limits of the functions at this point must be calculated. To do so, we epesent the squae oots in Eqs. (9) and () as Taylo seies. If, the limits of the oots and ae calculated as follows: lim () o lim () If, these limits ae pesented below: lim, () lim (4) o Hence, in the case of a positive hydaulic head,, the oot is a continuous function. If the hydaulic head has a negative value, the solution fo has no discontinuity. We can summaise that the hydaulic head is the asymptote fo some solutions. Note that and can geneally be complex numbes. To etun a eal numbe fo the squae oot ove the inteval cannot be less than. Ove the domain cosine is less than zeo, and so the disciminant has a positive value. The solutions of these equations fo this positive bode values ae pesented in fig. 4. The gaphs fo and fo ove the inteval show the shapes of the intefaces when the pessue inside is highe than thei suoundings. The plots fo ove the inteval pesent the suface limited in volume with intenal pessue lowe than the suoundings. In the case of highe gauge pessue, the dop has no tangent point with the uppe laye, but its shape is almost spheical because the vetical diamete is slightly longe than the hoizontal one. These diametes will be equal if 55., and so the dop will then be a sphee. The sample solution fo is pesented in fig. 4b). The intevals in which the eal oots of eqs. (9) and () can be found ae noted in this figue. The esults fo model the case of a pendant dop that is still connected to the solid phase. We can conclude that the dop will fom if the dimensionless gaugepessue value eaches. Although gaps in the plots ae obseved fo in WIT Tansactions on Engineeing Sciences, Vol 74, WIT Pess www.witpess.com, ISSN 74-5 (on-line)
58 Advances in Fluid Mechanics IX the domain, this does not indicate a lack of the suface at the gap peiod in this case; athe, it only means the lack of a state of the equilibium fo the suface with the assumed foce system. The suface could thus acceleate, o othe foces could be intoduced to the analysis. A diffeent situation is obseved in the inteval. Because cosine is less than zeo hee, the disciminant in Eq. () is a positive value. Thus, thee is no gap fo within this inteval. 5 z 4 - hydaulic head - - - x 5 z 4 - hydaulic head - - - x a) b) Figue 4: The plots of eqs. (9) and () fo: a), b). The plots fo ae the mio image of the plots fo thei modulus values. Thee occus the shift between solutions fo Eqs. (9) and (). The second oot descibes the sessile dop well. Theefoe, the plots fo ae the mio image of the plots in fig. 4b), whee the hydaulic head line is the axis of symmety... The flat suface and the lack of suface The flat suface is the solution of Eq. (9): o 4 (5) cos The height of the suface o the depth of the liquid is equal to hydaulic head,. As the ight side of Eq. (9) is zeo, thee is no esultant suface-tension foce. Howeve, this does not mean that these foces ae not pesent but athe that the suface-tension foces achieve equilibium on the plane and only thei esultant does not exist. The most exteme state is the lack of a suface if the hydaulic head equals zeo, which is expessed by Eq. (). This is an impotant finding of this wok. The suface can exist if the suface-tension foces balance each othe, but the suface cannot be shaped if thee is no pessue diffeence between two sides of the suface (i.e., between the two bulk phases). This conclusion is confimed by WIT Tansactions on Engineeing Sciences, Vol 74, WIT Pess www.witpess.com, ISSN 74-5 (on-line)
Advances in Fluid Mechanics IX 59 the limit of the function calculated by Eq. (). A liquid gas suface will not each the level of the hydaulic head if the suface-tension foces do not achieve equilibium. These situations ae pesented in figs. 4, 5. 5 z 4 5 z 4 hydaulic head hydaulic head 5 6 7 8 - - - - x - - - - x a) b) Figue 5: a) The gaphs of Eqs. (6) and (7) fo b) The gaphs of Eqs. (8) and (9) fo.,.. The thid kind of equilibium state This solution has not been connected with any expeimental obsevations so fa, and its popeties have not been descibed in detail. The thid equilibium state, defined by Eq. (), has two oots: o o 8sin cos 5 (6) cos o o 8sin cos 6 (7) cos The gaphs of Eq. (6), plotted in fig. 5a) have shapes simila to a hon tous. The plots of Eq. (7) ae tangent to the line of the hydaulic head because sine fo in Eq. () is equal to zeo. Fo that eason, this equation is decomposed into two: (8) and (9) (i.e., the lack of a suface and a flat suface, espectively) in this point. The solution fo the plane is plotted in fig. 5a) at which makes this solution diffeent fom the cases descibed in Sections.. and..4. The depth of the lowe laye (i.e., the diffeence between the hydaulic head and the vetical coodinate of the lowe suface) is tangent to the uppe plate (the solution fo and it inceases up to its maximum value (fo =.955) then appoaches the hydaulic-head line. The height of the uppe laye (the solution fo ises fom (fo = ) to its maximum (fo =.86) and then appoaches the hydaulic-head line. WIT Tansactions on Engineeing Sciences, Vol 74, WIT Pess www.witpess.com, ISSN 74-5 (on-line)
6 Advances in Fluid Mechanics IX..4 The fouth kind of equilibium state The fouth equilibium state, given by Eq. (), has two oots: o o 8cos 7 (8) cos o o 8cos 8 (9) cos The seventh oot of the geneal foce balance given by Eq. (8) has the shape of a thin dop suspended within anothe one, cf. fig. 5b). The uppe laye (the solution fo is lagest fo =, and then it deceases and appoaches the hydaulic-head line. The lowe laye (the solution fo is deepest fo, and it monotonically appoaches the hydaulic-head line...5 Suface unde unifom pessue Eq. (4) unde unifom pessue takes the fom d, d p (4) while eq. () does not change. Thee ae two solutions of eqs. () and (4): p, (4) p cos. (4) Fomula (4) is well known Young-Laplace equation and descibes bubble unde unifom pessue that shape is spheical. Although, it is common known esult, fo the fist time this shape was deived fom the complete foce balance diectly. Eq. (4) seems to be only mathematical solution that has not been obseved expeimentally, which coss-section is two equal bubbles whee second bubble is attached unde the fist one. Summay and discussion It was poven that it is possible to obtain a complete system of equations fo the static equilibium of a suface unde hydostatic pessue fo all fluids that do not sustain shea stesses when at est. To pedict the shape of this suface, a new diffeential suface with a vaying adial distance along the meidian diection was defined. The pessue foce and suface-tension foces wee balanced on this suface. In the method pesented heein, it was possible to balance the suface foces and the suface-tension foces on the contol suface. This had not peviously been possible, and it epesents the majo finding of this wok. As a esult, a system of two equations was obtained: one each in the adial and tangential diections. Thei solutions wee decomposed into thee quadatic and two linea equations with eight oots. Due to the existence of discontinuities, they can descibe eleven diffeent situations. The fist solution is the Young- WIT Tansactions on Engineeing Sciences, Vol 74, WIT Pess www.witpess.com, ISSN 74-5 (on-line)
Advances in Fluid Mechanics IX 6 Laplace equation with the inteface unde hydostatic pessue. To obtain the esults in the taditional way, a second-ode diffeential equation must be solved. In the achieved method, a simple quadatic equation is instead solved, which makes the calculations quite simple. The linea solutions ae the flat suface and the lack of a suface. Although they ae both tivial, thei pesence poves the geneality of the developed model. Futhemoe, two new solutions wee obtained that seem to be unconnected with any physical situation.. Conclusions We dew the following conclusions fo a suface unde pessue and sufacetension foces only: A suface unde unifom pessue is a sphee. A plane is fomed if the suface-tension foces equilibate themselves. Othe shapes occu if a pessue gadient exists and the suface-tension foces equilibate pessue foce. A suface cannot exist without a pessue diffeence acoss both of its sides. The pessue foces ae the active foces, wheeas the suface-tension foces ae the eactive foces, which only act to cuve the suface. To fom a closed inteface (e.g., a dop o a bubble), the inteio pessue cannot be less than its bode value; i.e., the modulus of the hydaulic head must be a facto of o moe times highe than the capillay constant. Acknowledgement This scientific eseach was financed fom the esouces of the National Cente fo Science. Refeences [] R.C. Tolman, Consideation of the Gibbs Theoy of Suface Tension, The Jounal of Chemical Physics, Vol. 6, No. 8, 948, 758-774. [] Thooddsen S.T., Etoh T.G., Takehaa K., High-Speed Imaging of Dops and Bubbles, Annu. Rev. Fluid Mech. 8, 4:57-85. WIT Tansactions on Engineeing Sciences, Vol 74, WIT Pess www.witpess.com, ISSN 74-5 (on-line)