█ 張鈞棣教授 懸浮液滴共振頻譜上失落的拼圖 Part 3
文/圖@台大機械 助理教授 張鈞棣
液滴振盪的相關研究始於Rayleigh和Lamb的小振幅振盪理論,但液滴的大振幅振盪似乎卻跟工程應用更直接相關。譬如,大振幅振盪使人們可將大液滴打散成許多小液滴,藉以在空氣中製造噴霧,或在液體中產生穩定的不互溶懸浮微液滴。反之,若能抑制液滴的大振幅振盪,噴墨印刷的墨滴將不再因不規則地破裂擴散,印刷品質即可提高。下文中的「大振幅」將不限於使液滴破裂的振幅,而是任何觸發液滴非線性振盪的振幅。
除了小振幅實驗,Trinh等人在1982年也發表了以[2, 0]模態為觀察對象的大振幅實驗結果 ADDIN
EN.CITE
<EndNote><Cite><Author>Trinh</Author><Year>1982</Year><RecNum>1</RecNum><DisplayText>[1]</DisplayText><record><rec-number>1</rec-number><foreign-keys><key
app="EN" db-id="drr2t9xdj5drv7evx5o5ztwa999s22seswv2"
timestamp="1550066217">1</key></foreign-keys><ref-type
name="Journal
Article">17</ref-type><contributors><authors><author>Trinh,
E.</author><author>Wang, T.
G.</author></authors></contributors><titles><title>Large-Amplitude
Free and Driven Drop-Shape Oscillations - Experimental-Observations</title><secondary-title>Journal
of Fluid Mechanics</secondary-title><alt-title>J Fluid
Mech</alt-title></titles><periodical><full-title>Journal
of Fluid Mechanics</full-title><abbr-1>J Fluid
Mech</abbr-1></periodical><alt-periodical><full-title>Journal
of Fluid Mechanics</full-title><abbr-1>J Fluid
Mech</abbr-1></alt-periodical><pages>315-338</pages><volume>122</volume><number>Sep</number><dates><year>1982</year></dates><isbn>0022-1120</isbn><accession-num>WOS:A1982PL06400017</accession-num><urls><related-urls><url><Go
to
ISI>://WOS:A1982PL06400017</url></related-urls></urls><electronic-resource-num>Doi
10.1017/S0022112082002237</electronic-resource-num><language>English</language></record></Cite></EndNote>[1]。大振幅實驗揭露了許多懸浮液滴在小振福實驗沒有的行為。首先,小振幅的液滴內部沒有封閉的流線,大振幅的有。其次,振幅夠大的音波除了讓液滴振盪,還會使液滴旋轉,甚至使液滴斷裂成數個小液滴。此外,液滴振幅夠大時,原本瘦長的液滴,振盪週期大半時間呈瘦長狀;原本扁胖的液滴,振盪週期大半時間是扁胖的。很不對稱的是,振幅越大,液滴呈瘦長狀的時間比例會增加。最後,大振幅震盪的液滴還呈現了很多Rayleigh和Lamb的線性理論無法描述的週期性的振動。這些都是液滴非線性振盪的行為。
Tsamopoulos和Brown推導了不帶電 ADDIN EN.CITE
<EndNote><Cite><Author>Tsamopoulos</Author><Year>1983</Year><RecNum>12</RecNum><DisplayText>[2]</DisplayText><record><rec-number>12</rec-number><foreign-keys><key
app="EN" db-id="drr2t9xdj5drv7evx5o5ztwa999s22seswv2"
timestamp="1550066848">12</key></foreign-keys><ref-type
name="Journal
Article">17</ref-type><contributors><authors><author>Tsamopoulos,
J. A.</author><author>Brown, R.
A.</author></authors></contributors><titles><title>Non-Linear
Oscillations of Inviscid Drops and
Bubbles</title><secondary-title>Journal of Fluid Mechanics</secondary-title><alt-title>J
Fluid
Mech</alt-title></titles><periodical><full-title>Journal
of Fluid Mechanics</full-title><abbr-1>J Fluid
Mech</abbr-1></periodical><alt-periodical><full-title>Journal
of Fluid Mechanics</full-title><abbr-1>J Fluid
Mech</abbr-1></alt-periodical><pages>519-537</pages><volume>127</volume><number>Feb</number><dates><year>1983</year></dates><isbn>0022-1120</isbn><accession-num>WOS:A1983QF03700029</accession-num><urls><related-urls><url><Go
to ISI>://WOS:A1983QF03700029</url></related-urls></urls><electronic-resource-num>Doi
10.1017/S0022112083002864</electronic-resource-num><language>English</language></record></Cite></EndNote>[2]與帶電 ADDIN EN.CITE
<EndNote><Cite><Author>Tsamopoulos</Author><Year>1984</Year><RecNum>13</RecNum><DisplayText>[3]</DisplayText><record><rec-number>13</rec-number><foreign-keys><key
app="EN" db-id="drr2t9xdj5drv7evx5o5ztwa999s22seswv2"
timestamp="1550066848">13</key></foreign-keys><ref-type
name="Journal Article">17</ref-type><contributors><authors><author>Tsamopoulos,
J. A.</author><author>Brown, R.
A.</author></authors></contributors><titles><title>Resonant
Oscillations of Inviscid Charged Drops</title><secondary-title>Journal
of Fluid Mechanics</secondary-title><alt-title>J Fluid Mech</alt-title></titles><periodical><full-title>Journal
of Fluid Mechanics</full-title><abbr-1>J Fluid
Mech</abbr-1></periodical><alt-periodical><full-title>Journal
of Fluid Mechanics</full-title><abbr-1>J Fluid
Mech</abbr-1></alt-periodical><pages>373-395</pages><volume>147</volume><number>Oct</number><dates><year>1984</year></dates><isbn>0022-1120</isbn><accession-num>WOS:A1984TT66100018</accession-num><urls><related-urls><url><Go
to
ISI>://WOS:A1984TT66100018</url></related-urls></urls><electronic-resource-num>Doi
10.1017/S0022112084002135</electronic-resource-num><language>English</language></record></Cite></EndNote>[3]液滴的非線性振盪模型。針對不帶電、無黏性的球狀懸浮液滴,他們透過微擾理論導出軸對稱模態的形狀與頻率。結果顯示液滴振盪時,液滴呈瘦長狀的時間比扁胖狀的時間要長,且振幅越大,瘦長狀的時間佔振盪週期的比例就越大。此外,他們也探究懸浮液滴各模態之間的互動關係。若液滴不帶電,[2, 0]模態可能觸發[4, 0];若液滴帶電,[4, 0]可觸發[6, 0];若帶電量恰當,[3, 0]會觸發[5, 0],而[4, 0]會觸發[8, 0]。由這些預測結果,兩位作者斷定模態間的能量傳遞具有專一性:只有特定模態能互傳能量。Trinh等人幾年後的實驗明顯地支持模態間能量傳遞的專一性。然而,對於哪些模態會互相觸發,實驗結果似乎與Tsamopoulos和Brown的預測不盡相同。
Trinh等人1998年的新實驗結果顯示次數高的模態(高次模態)若振幅夠大,次數低的模態(低次模態)可能也會被觸發,且兩者的振盪頻率都會是各自的自然振盪頻率 ADDIN EN.CITE
<EndNote><Cite><Author>Trinh</Author><Year>1998</Year><RecNum>11</RecNum><DisplayText>[4]</DisplayText><record><rec-number>11</rec-number><foreign-keys><key
app="EN" db-id="drr2t9xdj5drv7evx5o5ztwa999s22seswv2"
timestamp="1550066848">11</key></foreign-keys><ref-type
name="Journal
Article">17</ref-type><contributors><authors><author>Trinh,
E. H.</author><author>Thiessen, D.
B.</author><author>Holt, R. G.</author></authors></contributors><auth-address>CALTECH,
Jet Prop Lab, Pasadena, CA 91109 USA
Washington State Univ, Dept Phys,
Pullman, WA 99164 USA
Boston Univ, Dept Aerosp & Mech Engn,
Boston, MA 02215 USA</auth-address><titles><title>Driven and
freely decaying nonlinear shape oscillations of drops and bubbles immersed in a
liquid: Experimental results</title><secondary-title>Journal of
Fluid Mechanics</secondary-title><alt-title>J Fluid
Mech</alt-title></titles><periodical><full-title>Journal
of Fluid Mechanics</full-title><abbr-1>J Fluid
Mech</abbr-1></periodical><alt-periodical><full-title>Journal
of Fluid Mechanics</full-title><abbr-1>J Fluid
Mech</abbr-1></alt-periodical><pages>253-272</pages><volume>364</volume><keywords><keyword>immiscible
liquid</keyword><keyword>viscous
droplets</keyword><keyword>surface-tension</keyword><keyword>inviscid
drops</keyword><keyword>water</keyword><keyword>field</keyword></keywords><dates><year>1998</year><pub-dates><date>Jun
10</date></pub-dates></dates><isbn>0022-1120</isbn><accession-num>WOS:000074611000011</accession-num><urls><related-urls><url><Go
to
ISI>://WOS:000074611000011</url></related-urls></urls><electronic-resource-num>Doi
10.1017/S0022112098001153</electronic-resource-num><language>English</language></record></Cite></EndNote>[4]。譬如在實驗中,[3, 0]觸發了非軸對稱的[2, 2],而[6, 0]觸發了[3, 0],且它們都以各自的特徵頻率振動。就此,被外力觸發的高次模態會是諧波,而被高次模態觸發的低次模態則是次諧波。即使高、低次模態的頻率不是準確的整數倍,高次模態還是可以觸發低次模態。相對的,當低次模態被觸發時,高次模態雖然也可能被觸發,但兩者都會是諧波,且振幅相對小很多。實驗中,當[2, 0]觸發[3, 0]和[4, 0]時,所有模態都以外力的頻率振盪。就此,被外力觸發的低次模態是諧波,而被低次模態觸發的高次模態也是諧波;超諧波似乎從沒在這些實驗中被觀察到。對此,Trinh等人認為低次模態容易被觸發,因為他們因黏滯性造成的能量耗損較低。
對照Trinh等人的實驗觀察與Tsamopoulos和Brown的理論,實驗凸顯了理論的不足。首先,理論預測了與實驗不一致的模態配對關係。其次,理論模擬液滴的自然振盪,並未包含振盪液滴的外力,所以無法指出各模態究竟是諧波、次諧波或超諧波。對任何需要精確觸發或抑制液滴模態的應用而言,這些資訊將是非常重要的控制參數。
過去半個世紀,相關研究陸續揭露了懸浮液滴的各種共振行為,但許多面相仍舊撲朔迷離。就小振幅振盪而言,Rayleigh和Lamb預測的特徵頻率得到了實驗的初步驗證。實驗中,液滴變形改變模態特徵頻率的現象也有了理論的解釋。這些理論更預測了各模態在變形液滴上的特徵頻率。至今,最低次的幾個zonal和sectoral模態已經被成功地觸發。然而,tesseral模態幾乎都沒被觀察到。相關模型通常不包含振盪液滴的外力,所以實驗者欲以單頻振盪觸發特定模態時,外力的頻率和振幅只在實驗中以嘗試錯誤的方式來決定。也因為理論的不足,各模態究竟如何能以諧波、次諧波甚至超諧波的形態出現,至今仍是個謎。最後,大振幅振盪顯然缺乏完整的理論模型,所以既有的理論難以解釋實驗中觀察到的諸多現象。自從Rayleigh於1879年發表懸浮液滴的第一套理論,人類拼湊懸浮液滴的共振頻譜已屆一百四十年。時至今日,頻譜上仍存在著諸多失落的拼圖,有待相關學者努力探尋,繼續在這個基礎科問題上發掘更多的驚喜。
ADDIN EN.REFLIST 1.Trinh, E. and T.G. Wang, Large-Amplitude Free and Driven Drop-Shape Oscillations - Experimental-Observations. Journal of Fluid Mechanics, 1982. 122(Sep): p. 315-338.
2.Tsamopoulos, J.A. and R.A. Brown, Non-Linear Oscillations of Inviscid Drops and Bubbles. Journal of Fluid Mechanics, 1983. 127(Feb): p. 519-537.
3.Tsamopoulos, J.A. and R.A. Brown, Resonant Oscillations of Inviscid Charged Drops. Journal of Fluid Mechanics, 1984. 147(Oct): p. 373-395.
4.Trinh, E.H., D.B. Thiessen, and R.G. Holt, Driven and freely decaying nonlinear shape oscillations of drops and bubbles immersed in a liquid: Experimental results. Journal of Fluid Mechanics, 1998. 364: p. 253-272.