The conic-hyd ome e
Shi a P. Pudasaini
Technical Uni e si y o Munich, School o Enginee ing and Design
Ci il and En i onmen al Enginee ing, A ciss asse 21, D-80333, Munich, Ge many
Ka hmandu Ins i u e o Complex Flows
Kageshwo i Manoha a - 3, Bhad abas, Ka hmandu, Nepal
E-mail: shi [email p o ec ed]
Abs ac : He e, I p esen di e en conic-hyd ome e models; he ellip ical-hyd ome e , pa abolic-hyd ome e
and he hype bolic-hyd ome e . The new models p o ide he hyd ome e s and he hyd ome e - eloci ies mea-
su ing he e ol ing dep hs o he ic ional luids and hei exi eloci ies om ou le s in such longi udinally
o ien ed conic-sec ional ese oi s o moun ain lakes. The conic-hyd ome e s a e exp essed as simple di e -
en ial equa ions whose solu ions a e p esen ed. These wo-pa ame e hyd o-mechanical sys ems a e based
on dynamically jus i ied physical p inciples o ene gy dissipa ions h ough he discha ge coe icien and he
complemen a y ic ion. These new models a e undamen ally di e en om he classical To icelli-Be noulli-
law. As he luid su ace a ea e ol es nonlinea ly wi h he hyd aulic head in a complex manne , his becomes
he game-change cha ac e izing he conic-hyd ome e s. I p esen ed se e al inno a i e ese oi p inciples:
The ese oi geome y in insically commands he hyd ome e dynamics, i s iming and he ou le eloci y.
The conic ese oi s a e canonically in a ian unde o a ion and ansla ions wi h he hyd aulic-head a e
o he ese oi p o ile a ea, a g ea new e ela ion. The ellip ical-hyd ome e is weakly con ex, in con as ,
he pa abolic- and he hype bolic-hyd ome e a e mode a ely conca e, and su p isingly hey o e lap. The
conca i y-con exi y and o e lapping o he hyd ome e s a e no el unde s anding o he conic ese oi luid
exi p ocesses. The e exis s a uni e sal hyd o-mechanical p inciple explaining how he po en ial ene gy o he
conic- ese oi sys em is consumed by hyd o-geome ic cons ains in con olling he low. The hyd ome e
commands he exi eloci y o ollow a uni e sal conic-hyd ome e - eloci y ule as he pa abolic-hype bolic
hyd ome e eloci ies o e lap. This is phenomenal. I cons uc he undamen al heo em o ese oi : he a e
o change o he ese oi mass is gi en by he ac ual ese oi luid leng h. The new conic-hyd ome e s can be
applied o hyd aulic, hyd o-mechanical, anspo a ion and en i onmen al enginee ing p oblems. This includes
he con olled and e icien design and eme gency e acua ion o he geome ically di e en ly shaped glacial o
moun ain lakes, hyd opowe ese oi s, luid essels and hei enginee ed discha ges. Fo moun ain lash lood
simula ion, my me hods p o ide cos -e ec i e, e y e icien , p emie solu ions wi h hyd og aphs.
1 In oduc ion
The p ocess o d aining and emp ying o luid essels ( anke s), moun ain lakes and ese oi s is an impo an
hyd o-mechanical p oblem o ci il, hyd aulic, hyd o-mechanical, anspo a ion and en i onmen al enginee s
while loading and un-loading o luid (e.g., oil) anke s and ope a ing hyd opowe plan s, i iga ion sys ems,
lood con ol and decommissioning o endange ed s uc u es. The seminal analy ical models p oposed by To -
icelli (1644) and Be noulli (1738) o he cylind ical o ec angula essels luid exi p ocess ha e been widely
applied and imp o ed o e ime by hyd o-mechanisis s (de Oli ei a e al., 2000; Lin e al., 2008; Ma anni e al.,
2021; Malche ek, 2022). Pudasaini (2025) p esen ed an analy ical model o desc ibe he d aining o a iangula
wedge- ype luid-body. Howe e , he e exis s no dynamical model o desc ibe he d aining o gene al, e i-
cally (longi udinally) o ien ed conic- ese oi s and luid-bodies wi h any ic ional p ope y. These a e o he
spec um o he hyd o-mechanical p oblems wi h ample applica ion po en ial as such ese oi s may possess
na u al, s uc u al and economical ad an ages o e ese oi s wi h ec angula shapes. Howe e , in con as o
he classical To icelli-Be noulli- ype ho izon ally o ien ed ese oi s wi h ixed luid su ace a ea he e ically
o ien ed conic ese oi s pose g ea challenge in cons uc ing he luid mass e olu ion and he luid exi models
due o he nonlinea ly e ol ing dynamic luid su ace a ea as he exi ga e ope a es.
1
He e, I cons uc some no el physical-ma hema ical models add essing his s anding p oblem by p esen ing
some simple dynamical, and he mos e icien and cos -e ec i e me hods wi h conic-hyd ome e s o ellip ical-
ese oi , pa abolic- ese oi and he hype bolic- ese oi . I p esen some seminal ese oi p inciples in de-
sc ibing he essen ial dynamics o he ese oi luid and i s condui exi p ocess. As in en ed he e, he main
p inciple appea s o be how he ese oi -sys em u ilizes he a ailable po en ial ene gy h ough he hyd o-
mechanical cons ains as guided by he physical and geome ical pa ame e s, he ma e ial ic ions and he
ic ion a he condui . The mos impo an e ela ion is he undamen al heo em o ese oi : he a e o
change o he ese oi mass ( he luid p o ile a ea) is gi en by he ese oi luid leng h. My me hods p o ide
he quickes es ima e o he eno mous o ces ca ied by he exi je s wi h po en ially ca as ophic consequences
downs eam. These also u nish he i s analy ical solu ions o he con olled e acua ion o he luid om
conic- ese oi s h ough he sluice ga es o he dam collapses.
The solu ions demons a e ha he hyd ome e s ages o ellip ical- ese oi dec eases gen ly as he exi ga e
ope a es. Un il he end, he discha ge a e is dec easing slowly, inally comple ing he e acua ion p ocess.
Howe e , he solu ions o he pa abolic- ese oi and he hype bolic- ese oi mani es ha he hyd ome e
s ages o hese ese oi s dec eases gen ly as he exi ga e ope a es. Towa ds he end, he discha ge a e is
dec easing apidly, comple ing he e acua ion p ocess. These wo hyd ome e s i ually o e lap. Wi h his,
I demons a e ha he pa abolic- and hype bolic hyd ome e ollow a single ule. This also applies o hei
hyd ome e - eloci ies. These a e ascina ing non-in ui i e hyd o-mechanical p inciples b ough o wa d he e.
The gene al esul s p esen ed he e indica e he applica ion po en ial o he new conic-hyd ome e s in he el-
e an ci il, hyd aulic and en i onmen al enginee ing, and he ea h sys em sciences in he desi ed, con olled
and e icien design o he ese oi s and he luid essels. Mo eo e , o moun ain sunami and lash lood
simula ions (Me gili e al., 2018; 2020; Pudasaini and Me gili, 2019; Sa a e al., 2025), he new me hods yield
he mos cos -e icien and he bes dynamical solu ions o he ini ial condi ions wi h he conic-hyd og aphs
and he incipien eloci ies o he conic- ese oi s a he dam loca ion (Basel e al., 2021).
2 The conic-hyd ome e s
Conside a conic-shaped luid body ( ese oi ), say in a moun ain alley, wi h he (longi udinal) leng h L, he
luid dep h hand he (la e al) wid h WTa he op ee-su ace along he x-, y- and z-di ec ions, espec i ely,
whe e xand za e he longi udinal and he la e al di ec ions. Also conside an ou le (condui ) wi h dep h
HAand wid h WAa he base ( he on , xz- ace) o he ese oi . Mo eo e , le ρbe he densi y o he luid
in he ese oi and g he g a i a ional accele a ion. Assume ha , as he ou le ope a es, he mass exi s
(ou lows) he ese oi wi h he eloci y uA( he discha ge eloci y) and he luid a he ee su ace mo es
(lowe s down) wi h he eloci y uT. Also assume ha µc ep esen s he o e all (in e nal and he bounda y)
ma e ial ic ion, gene ally a small posi i e numbe . The ic ion µcincludes: (i) he in e nal ic ion due o
he mo ion and de o ma ion o he mobile ( h ee-dimensional) ubing- ype (channel) sub-body in he ese oi ,
(ii) he ic ion o his mobile unneling-body agains he con empo a y immobile su ounding complemen a y
ma e ial, and (ii) when he mobile ma e ial encoun e s he ese oi ( essel) bounda y, inducing he bounda y
ic ion (Pudasaini, 2025).
Ene gy balance
Conside ing he kine ic ene gy, po en ial ene gy and he ic ional ene gy loss, he ene gy balance yields:
1
2ρu2
T+λ ρgh =1
2ρu2
A+λ ρghµc,o , u2
T+ 2λghµp=u2
A,(1)
whe e, µp= 1 −µcis he e ec i e sys em ic ion coe icien , o he complemen a y ic ion coe icien . P ac-
ically, he pa ame e λmos ly akes alue 1 o 1/2 o he condui low o he dam collapse, espec i ely
(Pudasaini, 2025). Howe e , as explained in Pudasaini (2025), my app oach o ic ional ene gy loss appea s
o be dynamically be e jus i ied han o he exis ing app oaches (du Bua , 1786; de Oli ei a e al., 2000) as i
e ol es wi h he ac ual (mobile) ma e ial load.
Conse a ion o mass wi h luxes
2
Figu e 1: A: Examples o he e ically-longi udinally o ien ed conic-sec ions gi en by (3), (4), (5), including a
iangula p o ile; and, B: hei espec i e conic- ese oi s oge he wi h he ou le in g ay in he lowe igh .
A B
The lux-based conse a ion o mass implies
uTAT=uAAA,(2)
whe e, ATis he luid su ace a ea, uAis he bulk ou low eloci y (e lux), and AAis he condui a ea.
Combining (2) wi h (1) one ob ains he ou low eloci y once ATis known o modelled. I deal wi h his la e ,
sepa a ely o each conic- ese oi .
The conic- ese oi s
Conside he h ee conic-sec ions, he ellip ical, pa abolic and hype bolic, espec i ely (Downs, 2003; Kendig,
2005):
x2
a2
e
+y2
b2
e
= 1,(3)
y2= 4apx, (4)
x2
a2
h−y2
b2
h
= 1,(5)
whe e he su ices e,p,hin he pa ame e s aand bindica e hei associa ions wi h he ellip ical, pa abolic and
he hype bolic p o iles. Since be=aein (3) u ns i in o a ci cle as a special si ua ion, only he ellip ic p o ile
is conside ed he e.
Di e en conic sec ions, including he iangula p o ile, a e shown in Fig. 1A. No e ha , he le end poin s o
hese p o iles can be shi ed as equi ed in echnical use. O he han ellip ical and ci cula , om he geome ical
poin o iew, all hese ese oi p o iles ( he iangula , ellip ical, pa abolic and hype bolic) a e undamen ally
di e en class o cu es. So, one would an icipa e ha he dynamics o he ese oi emp ying p ocesses wi h
ese oi s based on hese dis inc p o iles will be di e en . The ellip ical- ese oi , pa abolic- ese oi , and he
hype bolic- ese oi oge he a e called he conic- ese oi s, and a e displayed in Fig. 1B. These ese oi s a e
e ically-longi udinally o ien ed downslope.
3
2.1 The ellip ical-hyd ome e
Following Pudasaini (2025), a hyd ome e is de ined as a physical model ha desc ibes he dynamics o he
e ol ing luid le el (hyd og aph) and he exi eloci y once he ese oi ou le ope a es. The iangula -
hyd ome e is p esen ed in Pudasaini (2025). Buil on Pudasaini (2025), he e, I cons uc he o he h ee
hyd ome e s: he ellip ical-hyd ome e , he pa abolic-hyd ome e and he hype bolic-hyd ome e . Howe e ,
he cons uc ion p ocess o hese conic-hyd ome e s is much mo e complex han he iangula hyd ome e as
i in ol es se e al special ma hema ical o mula ions, simpli ica ions, and s uc u al de elopmen s.
2.1.1 The dynamical p o ile a ea
Conside he ellipse in (3). Wi h his, I cons uc a e ically o ien ed, downwa d acing ellip ical ese oi in
he ou h quad an as shown in Fig. 1B, enclosed by he sec ion o he ellipse:
y=−bes1−x2
a2
e
,(6)
below he line y= 0 and bounded by he wo lines x=−aeand x= 0. As he luid exi s he condui , he
ese oi luid le el dec eases om i s ini ial le el h0 o some ac ual alue h. So, he luid le el (ini ially he
ho izon al dashed blue line in Fig. 1B) dec eases as y=−H =−(h0−h), whe e, His an auxilia y unc ion.
A he incep ion (h=h0), he dec ease o he luid le el is H=h0−h0= 0, a he ime o he ull emp ying
o he ese oi (h= 0), he dec ease in he luid le el is H=h0−0 = h0. No e ha , o ellip ical ese oi ,
ollowing Fig. 3, h0=be. Since x=−aep1−H2/b2
e=−Lde ines he p opaga ing le bounda y o he
deg ading ese oi , as he luid le el dec eases, he ac ual p o ile sec ional a ea o he ellip ical ese oi also
dec eases, and (i s magni ude) is gi en by:
Ae=Z0
−L bes1−x2
a2
e−H!dx. (7)
I is c ucial o no e ha as He ol es om 0 o h0,−Lcon ac s i sel om −ae o 0. A e some ma hema ical
in ol emen s and simpli ica ions, I ob ain he e ol ing unc ional sec ional p o ile a ea o he ellip ical ese oi :
Ae=1
2aeh0sin−11
h0qh2
0−(h0−h)2−ae
h0
(h0−h)qh2
0−(h0−h)2.(8)
This shows ha he longi udinal sec ional p o ile a ea o he ellip ical ese oi dec eases om i s maximum
aeh0π/4 (when h=h0) o i s minimum 0 ( o h= 0).
2.1.2 The exi eloci y
Wi h he ape u e ic ion µA(Be noulli, 1738) and dynamically e ol ing (longi udinal) leng h o he ellip ical
ese oi L= 0−(−L) = L=aep1−H2/b2
e, he con ac ing luid su ace a ea, and he condui a ea a e gi en
by:
AT=LWT, AA=HAWAµA.(9)
I is impo an o no e ha , in con as o he classical p oblems, whe e he luid su ace a ea emains cons an
because o he conside a ion o he ho izon ally o ien ed cylind ical o a ec angula essel (To icelli, 1644;
Be noulli, 1738; de Oli ei a e al., 2000; Lin e al., 2008; Ma anni e al., 2021), in he p esen se ing, he
luid su ace a ea ATe ol es (declines) wi h he hyd aulic head ha he ou le . This plays a c ucial ole in
he desc ip ion o he ellip ical-hyd ome e . µAis he empi ical ou low coe icien , o con ac ion a io o he
je behind he o i ice (Be noulli, 1738; Liu e al., 2008). In p inciple, µAmay be modelled in di e en ways
(Pudasaini, 2025; Malche ek, 2022).
4
Now, combining (1), (2) and (9), a e simpli ica ion, I ob ain an exp ession o he exi (discha ge) eloci y
uAe o he ellip ical- ese oi as:
uAe=qh2
0−(h0−h)2
u
u
βh
hh2
0−(h0−h)2i−α2
,(10)
whe e, α=WA
WTe
h0
ae
HAµAand β= 2λgµp. This p o ides a ela ionship be ween he discha ge eloci y and he
hyd aulic head ha he dam, including he collec i e physical pa ame e s αand β. This is an ex ension o he
classical To icelli-Be noulli-law (To icelli, 1644; Be noulli, 1738; Malche ek, 2022). Howe e , he appea ance
o qh2
0−(h0−h)2in he nume a o and hh2
0−(h0−h)2iin he denomina o wi h α2on he igh hand side
o (10) becomes he game-change as bo h eme ge due o he dynamically changing luid su ace a ea wi h luid
dep h a he dam. These would o he wise be uni y wi h he cylind ical o he ec angula essels o ese oi s
in To icelli-Be noulli- ype-laws. The To icelli p inciple co esponds o he mos simple si ua ion: WA≪WT,
which simply says ha he ou low eloci y uAis he ee all eloci y. This is equi alen o se ing hese e ms
o uni y in (10) and se ing λ= 1, µp= 1 in he nume a o , which e ec i ely means no ic ional conside a ion.
2.1.3 Conse a ion o mass wi h olume change and ou lux
Nex , I de i e an e olu ion equa ion o he hyd aulic head a he dam. This equi es he ac ual in o ma ion
on he ese oi luid olume Ve, which, wi h (8), is gi en by:
Ve=AeWT.(11)
This shows ha , he luid su ace a ea dec eases as he mass d ains ou h ough he ou le (o i ice). This
is a undamen ally di e en si ua ion as compa ed o he classical To icelli-Be noulli- ype models in which
he su ace a ea emains cons an (To icelli, 1644; Be noulli, 1738; de Oli ei a e al., 2000; Lin e al., 2008;
Ma anni e al., 2021; Malche ek, 2022). This is a game-change .
The educ ion o he luid heigh hin he ese oi ( essel) is desc ibed by he mass balance. As he a e o
change o he ese oi luid olume and he olume lux h ough he exi balance, we ha e:
dVe
d =−AAuAe.(12)
This essen ially means ha he a e by which he mass in he ese oi dec eases is exac ly balanced by he
mass ha lows ou h ough he o i ice.
To implemen (12), we need in o ma ion on how Aee ol es wi h h. Fo his, di e en ia e (8) wi h espec o
h. Then, he change o he p o ile a ea wi h espec o he e ol ing luid le el is gi en by
dAe
dh =ae
h0qh2
0−(h2
0−h)2,(13)
which is he ese oi leng h L. The ela ion (13) w i en in e ms o Lyields:
dAe
dh =L,(14)
which, I call he undamen al p inciple o ese oi . I s a es: he a e a which he ese oi mass ( luid p o ile
a ea) dec eases is exac ly he ese oi luid leng h.
I deno e he igh hand side o (13) by P:
P=ae
h0qh2
0−(h0−h)2.(15)
5
Since h0=be, and (h0−h) = H, his can be e-w i en as
P2
a2
e
+H2
b2
e
= 1,(16)
which is an ellipse. I call (16) he dynamic P-ellipse (o , Pe). This p o ides geome ically impo an in o ma-
ion. The a e o change o he ellip ical-p o ile luid a ea wi h espec o he educing luid dep h is ep esen ed
by ano he ellipse. So, as seen in (16), he ascina ing ac is ha , wi h espec o he ellip ical-op ics, he
ellip ical ese oi is in a ian unde a o a ion and a ansla ion (shi ) wi h i s canonical coun e pa dAe/dh.
This is a g ea new e ela ion.
F om (11) and (13), wi h he chain ule o di e en ia ion, i ollows:
dVe
d =ae
h0qh2
0−(h2
0−h)2WT
dh
d .(17)
F om (17), i is e iden ha he nonlinea change (diminish) in olume is p opo ional o he change in he
luid dep h wi h he p opo ionali y Pbeing a nonlinea unc ion o he luid dep h. This is in con as o he
classical To icelli-Be noulli cylind ical o ec angula ese oi p oblem o which he p opo ionali y ac o
is a cons an , namely he op sec ional a ea o he essel, equi alen ly i s diame e . These special ea u es
cha ac e ize he hyd o-mechanics o he ellip ical ese oi . Addi ionally, he a e dVe/d also depends on he
ese oi wid h and he ba hyme y.
The ellip ical-hyd ome e
Wi h he e ol ing exi eloci y (10) and he luid olume (17), wi h (12), I cons uc
dh
d =−α
u
u
βh
hh2
0−(h0−h)2i−α2
,(18)
whe e, α=WA
WTe
h0
ae
HAµA,β= 2λgµp. I call (18) he ellip ical-hyd ome e as i desc ibes he dynamically
e ol ing luid dep h o he ellip ical- ese oi . Consequen ly, I call (10) he ellip ical hyd ome e - eloci y. Com-
pa ed o he classical To icelli-Be noulli-law (To icelli, 1644; Be noulli, 1738), and he iangula -hyd ome e
(Pudasaini, 2025), (18) e eals impo an hyd o-mechanical in o ma ion. This cons i u es a unique mech-
anism on how o u ilize he e ec i e po en ial ene gy o he ese oi sys em (in he o m Ee
po =√β h)
in con olling he low wi h he ese oi geome y, and he luid and ou le ic ions, in he o m Ee
con =
α 1/hh2
0−(h0−h)2−α2i. This p ese e s a uni e sal hyd o-mechanical p inciple as he same also ap-
plies o he pa abolic-hyd ome e and he hype bolic-hyd ome e de eloped in Sec ion 2.2 and Sec ion 2.3,
espec i ely.
I is impo an o no e ha al hough he hyd ome e (18) is de eloped wi h he help o he exi eloci y uAe
in (10), he hyd ome e is comple ely decoupled om he exi eloci y. Howe e , he exi eloci y depends
ully and explici ly on he hyd ome e . Mo eo e , he pa ame e αin he nume a o on he igh hand side o
(18) eme ges collec i ely om he physical-geome ical pa ame e s associa ed wi h he ese oi olume and
he ou le eloci y. Howe e , his pa ame e appea s o be exac ly he same as ha αin he denomina o
on he igh hand side o (18) ha o igina es om he mass ou lux, o he exi eloci y. This simpli ies he
model s uc u e. This also applies o he pa abolic-hyd ome e and he hype bolic-hyd ome e cons uc ed in
Sec ion 2.2 and Sec ion 2.3.
6
2.2 The pa abolic-hyd ome e
2.2.1 The dynamical p o ile a ea
Conside he pa abola in (4). Wi h his, one can cons uc a e ically o ien ed, downwa d acing pa abolic
ese oi as shown in Fig. 1B, enclosed by he sec ion o he pa abola:
y=−2√apx, (19)
below he line y= 0 and bounded by he wo lines x= 0 and x=L=h2
0/4ap. The luid le el (ini ially he
ho izon al dashed blue line in Fig. 1B) dec eases as y=−H =−(h0−h). O he p ocedu es a e as o he
ellip ical-hyd ome e in Sec ion 2.1. Since Ll=H2/4apde ines he mobile le bounda y o he luid body,
as he luid le el dec eases, he ac ual p o ile sec ional a ea o he pa abolic ese oi also dec eases, and (i s
magni ude) is gi en by:
Ap=ZL
Ll2√apx−Hdx. (20)
I is c ucial o no e ha as He ol es om 0 o h0,H2/4apcon ac s i sel om 0 o L=h2
0/4ap. A e some
ma hema ical ope a ions and simpli ica ions, I ob ain he ac ual e ol ing sec ional p o ile a ea o he pa abolic
ese oi :
Ap=1
12ap2h3
0+ (h0−h)3−3h2
0(h0−h).(21)
This shows ha he a ea o he pa abolic ese oi dec eases om h3
0/6ap(when h=h0) o 0 ( o h= 0).
2.2.2 The exi eloci y
Wi h he ac ual leng h o he pa abolic ese oi L=L−Ll, he luid su ace a ea and he condui a ea a e
known:
AT=LWT, AA=HAWAµA.(22)
This means, ATis dec easing nonlinea ly wi h h. Now, combining (1), (2) and (22), a e simpli ica ion, I
ob ain an exp ession o he exi eloci y uAp o he pa abolic- ese oi as:
uAp=hh2
0−(h0−h)2i
u
u
βh
hh2
0−(h0−h)2i2
−α2
,(23)
whe e, α= 4 WA
WTp
HAapµA,β= 2λgµp. This p o ides a ela ionship be ween he discha ge eloci y and he
hyd aulic head ha he dam, including he physical pa ame e s αand β. The appea ance o hh2
0−(h0−h)2i
in he nume a o and hh2
0−(h0−h)2i2
in he denomina o wi h α2on he igh hand side o (23) becomes
a game-change as bo h eme ge due o he dynamically changing (sh inking) luid su ace a ea wi h educing
luid dep h a he dam.
2.2.3 Conse a ion o mass wi h olume change and ou lux
F om (21), he ese oi luid olume is gi en by:
Vp=ApWT.(24)
The luid su ace a ea dec eases as he mass d ains ou h ough he ou le . This is a game-change .
7
Figu e 2: A: The P-ellip ical and, B: he P-pa abolic conics, ollowing he o mal ule o o a ion and ansla-
ions, gi en espec i ely by (16) and (29). The abscissae and o dina es a e (x, y) o he ese oi -conics, and
(h, P) o he P-ellipse and he P-pa abola, espec i ely.
A B
As he a e o change o he ese oi luid olume and he olume lux h ough he exi balance, we ha e:
dVp
d =−AAuAp.(25)
To implemen (25), we need in o ma ion on how Ape ol es wi h h. Fo his, di e en ia e (21) wi h espec o
h. Then, he change (decline) o he p o ile a ea wi h espec o he e ol ing luid le el is gi en by
dAp
dh =1
4aph2
0−(h0−h)2,(26)
which is he ese oi luid leng h L. The ela ion (26) w i en in e ms o Lyields:
dAp
dh =L,(27)
which is he undamen al p inciple o ese oi (pa abolic).
I deno e he igh hand side o (26) by P:
P=1
4aph2
0−(h0−h)2,(28)
o ,
H2=−4ap(P −L),(29)
which is a pa abola. I call (29) he dynamic P-pa abola (o , Pp). This p o ides geome ically impo an in o -
ma ion. The a e o change o he pa abolic-p o ile luid a ea wi h espec o he educing luid dep h is ep e-
sen ed by ano he pa abola. So, as seen in (29), he appealing ac is ha , wi h espec o he pa abolic-op ics,
he pa abolic ese oi is in a ian unde a o a ion and ansla ions (shi s) wi h i s canonical coun e pa
dAp/dh. This is a g ea new unde s anding.
The P-ellip ical and he P-pa abolic conics a e p esen ed in Fig. 2 which a e in a ian o hei espec i e
ese oi -conics. They ollow he p inciple o o a ion and ansla ions. The P-hype bola can be displayed
simila ly wi h i s ep esen a ion de i ed la e .
I is impo an o no e ha he P-ellipse equi ed one o a ion and one ansla ion. Howe e , he P-pa abola
8
equi ed one o a ion and wo ansla ions. Such beha io s can be explained by he na u al, s uc u al cha ac-
e is ics o he espec i e conic- ese oi s. Because, he ellip ical-conic is a closed cu e, wi h one o a ion and
jus one ansla ion, i ans o ms he pa en ese oi -ellipse in o a closed P-ellipse. Howe e , he pa abolic-
conic is an open cu e; so, i needed one o a ion and wo ansla ions in o de o de elop i in o a closed egion
enclosed by a pa abola desc ibing he hyd ome e a e o ese oi luid p o ile a ea, and he axis desc ibing
he hyd ome e . This explains he in insic mechanisms o he P-ellipse and he P-pa abola. Mo eo e , Fig. 2
displays comple ely di e en hyd ome e a es o ese oi luid p o ile a ea o he ellip ical- ese oi and he
pa abolic- ese oi , essen ially di e en ly con olling hei hyd ome e dynamics.
F om (24) and (26), wi h he chain ule o di e en ia ion, i ollows:
dVp
d =1
4aph2
0−(h0−h)2WT
dh
d .(30)
F om (30), i is e iden ha he change in olume is p opo ional o he change in he luid dep h wi h he
p opo ionali y Pbeing a nonlinea unc ion o he luid dep h.
The pa abolic-hyd ome e
Wi h he e ol ing exi eloci y (23) and he luid olume (30), wi h (25) I cons uc
dh
d =−α
u
u
βh
hh2
0−(h0−h)2i2
−α2
,(31)
whe e, α= 4 WA
WTp
HAapµA,β= 2λgµp. I call (31) he pa abolic-hyd ome e . Consequen ly, I call (23) he
pa abolic hyd ome e - eloci y. Compa ed o he classical To icelli-Be noulli-law (To icelli, 1644; Be noulli,
1738), and he iangula -hyd ome e (Pudasaini, 2025), (31) e eals impo an hyd o-mechanical in o ma ion
ha will be clea e la e .
2.3 The hype bolic-hyd ome e
2.3.1 The dynamical p o ile a ea
Wi h he hype bola (5), one can cons uc a e ically o ien ed, downwa d acing hype bolic ese oi as shown
in Fig. 1B, enclosed by he sec ion o he hype bola:
y=−bhsx2
a2
h−1,(32)
below he line y= 0 and bounded by he wo lines x=ahand x=L=ah
bhqh2
0+b2
h. As he luid exi s he
condui , he ese oi luid le el dec eases om i s ini ial le el h0 o some ac ual alue h. So, he luid le el
dec eases as y=−H =−(h0−h). Since x=Ll=ah
bhq(h0−h)2+b2
hde e mines he mo ing le bounda y o
he deg ading ese oi , as he luid le el dec eases, he ac ual p o ile sec ional a ea o he hype bolic ese oi
also dec eases, and (i s magni ude) is gi en by:
Ah=ZL
Ll bhsx2
a2
h−1−H!dx. (33)
As He ol es om 0 o h0,Llcon ac s i sel om ah o L. A e some ma hema ical ope a ions, I ob ain he
ac ual e ol ing sec ional p o ile a ea o he hype bolic ese oi :
Ah=−1
2ahbh anh−11
h0qh2
0+b2
h+1
2
ah
bh
h0qh2
0+b2
h−ah
bh
(h0−h)qh2
0+b2
h
+1
2ahbh anh−11
h0−hq(h0−h)2+b2
h−1
2
ah
bh
(h0−h)q(h0−h)2+b2
h+ah
bh
(h0−h)q(h0−h)2+b2
h,
(34)
9
These esul s lead o he ollowing ou Rese oi P inciples and Theo em.
P inciple 1. The Rese oi Ro a e-T ansla e P inciple: The a e o change o he ese oi luid olume wi h
espec o he hyd aulic-head o a conic- ese oi is ob ained by a o a ion and some ansla ions.
Ro a e he ese oi -conic by an amoun π
2. Then, ollowing he p ocedu e in Sec ion 2, ansla e i as sugges ed
by (16), (29) and (43), espec i ely, o he ellip ical- ese oi , pa abolic- ese oi and he hype bolic- ese oi .
P inciple 2. The Rese oi Eccen ici y P inciple: The conic-hyd ome e is con ex o he conic ese oi
wi h eccen ici y less han uni y. O he wise, he conic-hyd ome e is conca e.
The p ocedu e ollows om Sec ion 2. Fo he ellip ical- ese oi , he hyd ome e is con ex. Fo he pa abolic-
ese oi and hype bolic- ese oi he hyd ome e s a e conca e.
P inciple 3. The Rese oi Quali y-Ra e o Volume Change P inciple: The hyd ome e s o he pa abolic-
ese oi and he hype bolic- ese oi a e i ually he same.
The p ocedu e ollows om Sec ion 2. This is due o he simila quali y- a e o luid olume change in ime
o he pa abolic- ese oi and he hype bolic- ese oi .
Theo em 1. The Fundamen al Theo em o Rese oi : The a e o change o he ese oi mass ( luid p o ile
a ea) is he ac ual ese oi leng h.
P oo : The p ocedu e ollows om Sec ion 2 wi h ela ions (14), (27) and (41). This p o es he heo em.
3.7 Classi ica ion o ese oi
Wi h espec o he esul s p esen ed in Fig. 3, Fig. 4, Fig. 5 and Fig. 6, I classi y he (hyd ome e s) ese -
oi s in o ou : The cylind ical-, iangula -, ellip ical- and pa abolic-hype bolic ese oi s, espec i ely, wi h
cons an ese oi leng h, linea ese oi leng h, ellip ical ese oi leng h and pa abolic-hype bolic ese oi
leng h. Essen ially hese (dynamic) ese oi leng hs cha ac e ize and ully con ol he co esponding hyd ome-
e s and he hyd ome e - eloci ies o he cylind ical-, iangula -, ellip ical- and pa abolic-hype bolic ese oi s;
a new in en ion in hyd o-mechanics.
3.8 Ex ensions o o he conic- ese oi o ien a ions and shapes
The conic-hyd ome e s p esen ed he e can di ec ly be ex ended o o he conic- ese oi o ien a ions and shapes.
I gi e one example he e. Conside he ellip ical ese oi in Fig. 1. By also including he sec ion in he ou h
quad an , one can cons uc he semi-ellip ical- ese oi below he line y= 0 ha de ines he ini ial luid
su ace. Fo his, one can conside he ou le in he on (base) in he xy- ace as he ese oi ex ends
la e ally in he z-di ec ion. This is an open ese oi wi h he ee su ace o he luid. Howe e , one can
also conside he closed semi-ellip ical- ese oi abo e he line y= 0 ha de ines he base o he ese oi
and he ini ial luid body is con ained by he uppe semi-ellip ical essel body. Mo eo e , one can u he
conside he en i e closed ellip ical ese oi wi h i s la e al ex en in he xz-plane. The closed ese oi s ha e
enginee ing applica ions, howe e , na u al ese oi s and moun ain lakes a e open. Ye , o all hese ese oi s,
he hyd ome e s can easily be cons uc ed by simply ollowing he me hods p esen ed a Sec ion 2 by jus
un olding he e ol ing sec ional p o ile a ea and he luid su ace a ea in he ese oi . The same me hods
can be ex ended in cons uc ing hyd ome e s o di e en ly o ien ed and shaped pa abolic- and hype bolic-
hyd ome e s. Howe e , his equi es he p ope ly selec ed o ien a ion o he ini ial conics.
4 Summa y
Following some complex p ocedu es, I cons uc ed h ee conic-hyd ome e s: he ellip ical-hyd ome e , he
pa abolic-hyd ome e and he hype bolic-hyd ome e . These conic ese oi s a e canonically in a ian unde
16
o a ion and ansla ions wi h he hyd aulic-head a e o he ese oi p o ile a ea. This is a g ea new e e-
la ion. The nonlinea change in ese oi luid olume is p opo ional o he change in he luid dep h. These
special ea u es cha ac e ize he hyd o-mechanics o he conic ese oi s.
The conic-hyd ome e s display se e al dis inc beha iou s wi h hei implica ions in con olled and e icien
design o he ese oi s and luid essels. Ellip ical-hyd ome e is weakly nonlinea whe eas he pa abolic-
and hype bolic-hyd ome e s a e weakly o s ongly nonlinea . The hyd ome e s a e con ex o he ese oi s
wi h he eccen ici y <1 and he hyd ome e s a e conca e o he ese oi s wi h eccen ici y ≥1. The
conic-hyd ome e a e quali y con ols he dynamics o he ellip ical-, pa abolic- and hype bolic- ese oi s; he
pa abolic- and hype bolic- a e quali y cu es o e lap. The conca i y-con exi y, and me ging o he pa abolic-
and hype bolic- hyd ome e s (hyd og aphs) o he undamen ally di e en ese oi s a e no el unde s anding
o he conic ese oi luid exi p ocesses.
I p esen ed some phenomenal ese oi p inciples: (I) The e exis s a uni e sal hyd o-mechanical p inciple on
how he a ailable po en ial ene gy is consumed by hyd o-geome ic cons ains including he physical pa ame-
e s o he conic- ese oi s in a chi ec ing he hyd ome e s. (II) The a e o change o he ese oi luid olume
wi h espec o he hyd aulic-head o a conic- ese oi is ob ained by a o a ion and some ansla ions o he
conic ese oi -p o ile. So, he conic-hyd ome e s a e in a ian wi h espec o hei pa en conic- ese oi s.
(III) Fo he ellip ical- ese oi , he hyd ome e is con ex, and o he pa abolic- ese oi and he hype bolic-
ese oi he hyd ome e s a e conca e. (IV) As he hyd ome e commands he exi eloci y o ollow he
uni e sal pa abolic-hype bolic hyd ome e - eloci y ule, he hyd ome e s o he pa abolic- ese oi and he
hype bolic- ese oi a e i ually he same. (V) The undamen al heo em o ese oi , s a ing ha , he a e
by which he ese oi mass ( luid p o ile a ea) educes is gi en by he ac ual leng h o he ese oi . This is he
mos as onishing e ela ion cha ac e izing he ese oi dynamics. This also p o es he physical-ma hema ical
consis ency o he ese oi p inciples p e en ed he e.
My models p o ide impo an hyd o-mechanical in o ma ion wi h hei implica ions in he desi ed, con olled
and e icien design o he ese oi s and he luid essels o he hyd aulic, hyd o-mechanical, anspo a ion
and en i onmen al enginee s in selec ing he ese oi geome y ha in insically commands he dynamics,
he hyd ome e , i s iming and he ou le eloci y. Depending on he ope a ion and e acua ion s a egy, I
p o ided some basic plausible design guidelines o p ac ical use.
Acknowledgmen : The inancial suppo is p o ided by he Ge man Resea ch Founda ion (DFG) h ough
he esea ch p ojec : Landslide mobili y wi h e osion: P oo -o -concep and applica ion - Pa I: Modeling,
Simula ion and Valida ion; P ojec numbe 522097187.
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