The conic-hydrometer

The conic-hydrometer Shiva P. Pudasaini Technical University of Munich, School of Engineering and Design Civil and Environmental Engineering, Arcisstrasse 21, D-80333, Munich, Germany Kathmandu Institute of Complex Flows Kageshwori Manohara - 3, Bhadrabas, Kathmandu, Nepal E-mail: shiv[email protected] Abstract: Here, I present different conic-hydrometer models; the elliptical-hydrometer, parabolic-hydrometer and the hyperbolic-hydrometer. The new models provide the hydrometers and the hydrometer-velocities measuring the evolving depths of the frictional fluids and their exit velocities from out lets in such longitudinally oriented conic-sectional reservoirs or mountain lakes. The conic-hydrometers are expressed as simple differential equations whose solutions are presented. These two-parameter hydro-mechanical systems are based on dynamically justified physical principles for energy dissipations through the discharge coefficient and the complementary friction. These new models are fundamentally different from the classical Torricelli-Bernoullilaw. As the fluid surface area evolves nonlinearly with the hydraulic head in a complex manner, this becomes the game-changer characterizing the conic-hydrometers. I presented several innovative reservoir principles: The reservoir geometry intrinsically commands the hydrometer dynamics, its timing and the out let velocity. The conic reservoirs are canonically invariant under rotation and translations with the hydraulic-head rate of the reservoir profile area, a great new revelation. The elliptical-hydrometer is weakly convex, in contrast, the parabolicand the hyperbolic-hydrometer are moderately concave, and surprisingly they overlap. The concavity-convexity and overlapping of the hydrometers are novel understanding for the conic reservoir fluid exit processes. There exists a universal hydro-mechanical principle explaining how the potential energy of the conic-reservoir system is consumed by hydro-geometric constrains in controlling the flow. The hydrometer commands the exit velocity to follow a universal conic-hydrometer-velocity rule as the parabolic-hyperbolic hydrometer velocities overlap. This is phenomenal. I construct the fundamental theorem of reservoir: the rate of change of the reservoir mass is given by the actual reservoir fluid length. The new conic-hydrometers can be applied to hydraulic, hydro-mechanical, transportation and environmental engineering problems. This includes the controlled and efficient design and emergency evacuation of the geometrically differently shaped glacial or mountain lakes, hydropower reservoirs, fluid vessels and their engineered discharges. For mountain flash flood simulation, my methods provide cost-effective, very efficient, premier solutions with hydrographs. 1 Introduction The process of draining and emptying of fluid vessels (tankers), mountain lakes and reservoirs is an important hydro-mechanical problem for civil, hydraulic, hydro-mechanical, transportation and environmental engineers while loading and un-loading of fluid (e.g., oil) tankers and operating hydropower plants, irrigation systems, flood control and decommissioning of endangered structures. The seminal analytical models proposed by Torricelli (1644) and Bernoulli (1738) for the cylindrical or rectangular vessels fluid exit process have been widely applied and improved over time by hydro-mechanisists (de Oliveira et al., 2000; Lin et al., 2008; Maranni et al., 2021; Malcherek, 2022). Pudasaini (2025) presented an analytical model to describe the draining of a triangular wedge-type fluid-body. However, there exists no dynamical model to describe the draining of general, vertically (longitudinally) oriented conic-reservoirs and fluid-bodies with any frictional property. These are other spectrum of the hydro-mechanical problems with ample application potential as such reservoirs may possess natural, structural and economical advantages over reservoirs with rectangular shapes. However, in contrast to the classical Torricelli-Bernoulli-type horizontally oriented reservoirs with fixed fluid surface area the vertically oriented conic reservoirs pose great challenge in constructing the fluid mass evolution and the fluid exit models due to the nonlinearly evolving dynamic fluid surface area as the exit gate operates. 1 Here, I construct some novel physical-mathematical models addressing this standing problem by presenting some simple dynamical, and the most efficient and cost-effective methods with conic-hydrometers for ellipticalreservoir, parabolic-reservoir and the hyperbolic-reservoir. I present some seminal reservoir principles in describing the essential dynamics of the reservoir fluid and its conduit exit process. As invented here, the main principle appears to be how the reservoir-system utilizes the available potential energy through the hydromechanical constrains as guided by the physical and geometrical parameters, the material frictions and the friction at the conduit. The most important revelation is the fundamental theorem of reservoir: the rate of change of the reservoir mass (the fluid profile area) is given by the reservoir fluid length. My methods provide the quickest estimate of the enormous forces carried by the exit jets with potentially catastrophic consequences downstream. These also furnish the first analytical solutions for the controlled evacuation of the fluid from conic-reservoirs through the sluice gates or the dam collapses. The solutions demonstrate that the hydrometer stages for elliptical-reservoir decreases gently as the exit gate operates. Until the end, the discharge rate is decreasing slowly, finally completing the evacuation process. However, the solutions for the parabolic-reservoir and the hyperbolic-reservoir manifest that the hydrometer stages for these reservoirs decreases gently as the exit gate operates. Towards the end, the discharge rate is decreasing rapidly, completing the evacuation process. These two hydrometers virtually overlap. With this, I demonstrate that the parabolicand hyperbolic hydrometer follow a single rule. This also applies to their hydrometer-velocities. These are fascinating non-intuitive hydro-mechanical principles brought forward here. The general results presented here indicate the application potential of the new conic-hydrometers in the relevant civil, hydraulic and environmental engineering, and the earth system sciences in the desired, controlled and efficient design of the reservoirs and the fluid vessels. Moreover, for mountain tsunami and flash flood simulations (Mergili et al., 2018; 2020; Pudasaini and Mergili, 2019; Sattar et al., 2025), the new methods yield the most cost-efficient and the best dynamical solutions for the initial conditions with the conic-hydrographs and the incipient velocities for the conic-reservoirs at the dam location (Baselt et al., 2021). 2 The conic-hydrometers Consider a conic-shaped fluid body (reservoir), say in a mountain valley, with the (longitudinal) length L, the fluid depth hand the (lateral) width WTat the top free-surface along the x-, yand z-directions, respectively, where xand zare the longitudinal and the lateral directions. Also consider an out let (conduit) with depth HAand width WAat the base (the front, xz-face) of the reservoir. Moreover, let ρbe the density of the fluid in the reservoir and gthe gravitational acceleration. Assume that, as the out let operates, the mass exits (outflows) the reservoir with the velocity uA(the discharge velocity) and the fluid at the free surface moves (lowers down) with the velocity uT. Also assume that µcrepresents the overall (internal and the boundary) material friction, generally a small positive number. The friction µcincludes: (i) the internal friction due to the motion and deformation of the mobile (three-dimensional) tubing-type (channel) sub-body in the reservoir, (ii) the friction of this mobile tunneling-body against the contemporary immobile surrounding complementary material, and (ii) when the mobile material encounters the reservoir (vessel) boundary, inducing the boundary friction (Pudasaini, 2025). Energy balance Considering the kinetic energy, potential energy and the frictional energy loss, the energy balance yields: 1 2ρu2 T+λ ρgh =1 2ρu2 A+λ ρghµc,or, u2 T+ 2λghµp=u2 A,(1) where, µp= 1 −µcis the effective system friction coefficient, or the complementary friction coefficient. Practically, the parameter λmostly takes value 1 or 1/2 for the conduit flow or the dam collapse, respectively (Pudasaini, 2025). However, as explained in Pudasaini (2025), my approach of frictional energy loss appears to be dynamically better justified than other existing approaches (du Buat, 1786; de Oliveira et al., 2000) as it evolves with the actual (mobile) material load. Conservation of mass with fluxes 2 Figure 1: A: Examples of the vertically-longitudinally oriented conic-sections given by (3), (4), (5), including a triangular profile; and, B: their respective conic-reservoirs together with the out let in gray in the lower right. A B The flux-based conservation of mass implies uTAT=uAAA,(2) where, ATis the fluid surface area, uAis the bulk outflow velocity (efflux), and AAis the conduit area. Combining (2) with (1) one obtains the out flow velocity once ATis known or modelled. I deal with this later, separately for each conic-reservoir. The conic-reservoirs Consider the three conic-sections, the elliptical, parabolic and hyperbolic, respectively (Downs, 2003; Kendig, 2005): x2 a2 e +y2 b2 e = 1,(3) y2= 4apx, (4) x2 a2 h−y2 b2 h = 1,(5) where the suffices e,p,hin the parameters aand bindicate their associations with the elliptical, parabolic and the hyperbolic profiles. Since be=aein (3) turns it into a circle as a special situation, only the elliptic profile is considered here. Different conic sections, including the triangular profile, are shown in Fig. 1A. Note that, the left end points of these profiles can be shifted as required in technical use. Other than elliptical and circular, from the geometrical point of view, all these reservoir profiles (the triangular, elliptical, parabolic and hyperbolic) are fundamentally different class of curves. So, one would anticipate that the dynamics of the reservoir emptying processes with reservoirs based on these distinct profiles will be different. The elliptical-reservoir, parabolic-reservoir, and the hyperbolic-reservoir together are called the conic-reservoirs, and are displayed in Fig. 1B. These reservoirs are vertically-longitudinally oriented downslope. 3 2.1 The elliptical-hydrometer Following Pudasaini (2025), a hydrometer is defined as a physical model that describes the dynamics of the evolving fluid level (hydrograph) and the exit velocity once the reservoir out let operates. The triangularhydrometer is presented in Pudasaini (2025). Built on Pudasaini (2025), here, I construct the other three hydrometers: the elliptical-hydrometer, the parabolic-hydrometer and the hyperbolic-hydrometer. However, the construction process of these conic-hydrometers is much more complex than the triangular hydrometer as it involves several special mathematical formulations, simplifications, and structural developments. 2.1.1 The dynamical profile area Consider the ellipse in (3). With this, I construct a vertically oriented, downward facing elliptical reservoir in the fourth quadrant as shown in Fig. 1B, enclosed by the section of the ellipse: y=−bes1−x2 a2 e ,(6) below the line y= 0 and bounded by the two lines x=−aeand x= 0. As the fluid exits the conduit, the reservoir fluid level decreases from its initial level h0to some actual value h. So, the fluid level (initially the horizontal dashed blue line in Fig. 1B) decreases as y=−H =−(h0−h), where, His an auxiliary function. At the inception (h=h0), the decrease of the fluid level is H=h0−h0= 0, at the time of the full emptying of the reservoir (h= 0), the decrease in the fluid level is H=h0−0 = h0. Note that, for elliptical reservoir, following Fig. 3, h0=be. Since x=−aep1−H2/b2 e=−Ldefines the propagating left boundary of the degrading reservoir, as the fluid level decreases, the actual profile sectional area of the elliptical reservoir also decreases, and (its magnitude) is given by: Ae=Z0 −L bes1−x2 a2 e−H!dx. (7) It is crucial to note that as Hevolves from 0 to h0,−Lcontracts itself from −aeto 0. After some mathematical involvements and simplifications, I obtain the evolving functional sectional profile area of the elliptical reservoir: Ae=1 2aeh0sin−11 h0qh2 0−(h0−h)2−ae h0 (h0−h)qh2 0−(h0−h)2.(8) This shows that the longitudinal sectional profile area of the elliptical reservoir decreases from its maximum aeh0π/4 (when h=h0) to its minimum 0 (for h= 0). 2.1.2 The exit velocity With the aperture friction µA(Bernoulli, 1738) and dynamically evolving (longitudinal) length of the elliptical reservoir L= 0−(−L) = L=aep1−H2/b2 e, the contracting fluid surface area, and the conduit area are given by: AT=LWT, AA=HAWAµA.(9) It is important to note that, in contrast to the classical problems, where the fluid surface area remains constant because of the consideration of the horizontally oriented cylindrical or a rectangular vessel (Torricelli, 1644; Bernoulli, 1738; de Oliveira et al., 2000; Lin et al., 2008; Maranni et al., 2021), in the present setting, the fluid surface area ATevolves (declines) with the hydraulic head hat the out let. This plays a crucial role in the description of the elliptical-hydrometer. µAis the empirical outflow coefficient, or contraction ratio of the jet behind the orifice (Bernoulli, 1738; Liu et al., 2008). In principle, µAmay be modelled in different ways (Pudasaini, 2025; Malcherek, 2022). 4 Now, combining (1), (2) and (9), after simplification, I obtain an expression for the exit (discharge) velocity uAefor the elliptical-reservoir as: uAe=qh2 0−(h0−h)2v u u t βh hh2 0−(h0−h)2i−α2 ,(10) where, α=WA WTe h0 ae HAµAand β= 2λgµp. This provides a relationship between the discharge velocity and the hydraulic head hat the dam, including the collective physical parameters αand β. This is an extension to the classical Torricelli-Bernoulli-law (Torricelli, 1644; Bernoulli, 1738; Malcherek, 2022). However, the appearance of qh2 0−(h0−h)2in the numerator and hh2 0−(h0−h)2iin the denominator with α2on the right hand side of (10) becomes the game-changer as both emerge due to the dynamically changing fluid surface area with fluid depth at the dam. These would otherwise be unity with the cylindrical or the rectangular vessels or reservoirs in Torricelli-Bernoulli-type-laws. The Torricelli principle corresponds to the most simple situation: WA≪WT, which simply says that the outflow velocity uAis the free fall velocity. This is equivalent to setting these terms to unity in (10) and setting λ= 1, µp= 1 in the numerator, which effectively means no frictional consideration. 2.1.3 Conservation of mass with volume change and out flux Next, I derive an evolution equation for the hydraulic head at the dam. This requires the actual information on the reservoir fluid volume Ve, which, with (8), is given by: Ve=AeWT.(11) This shows that, the fluid surface area decreases as the mass drains out through the out let (orifice). This is a fundamentally different situation as compared to the classical Torricelli-Bernoulli-type models in which the surface area remains constant (Torricelli, 1644; Bernoulli, 1738; de Oliveira et al., 2000; Lin et al., 2008; Maranni et al., 2021; Malcherek, 2022). This is a game-changer. The reduction of the fluid height hin the reservoir (vessel) is described by the mass balance. As the rate of change of the reservoir fluid volume and the volume flux through the exit balance, we have: dVe dt =−AAuAe.(12) This essentially means that the rate by which the mass in the reservoir decreases is exactly balanced by the mass that flows out through the orifice. To implement (12), we need information on how Aeevolves with h. For this, differentiate (8) with respect to h. Then, the change of the profile area with respect to the evolving fluid level is given by dAe dh =ae h0qh2 0−(h2 0−h)2,(13) which is the reservoir length L. The relation (13) written in terms of Lyields: dAe dh =L,(14) which, I call the fundamental principle of reservoir. It states: the rate at which the reservoir mass (fluid profile area) decreases is exactly the reservoir fluid length. I denote the right hand side of (13) by P: P=ae h0qh2 0−(h0−h)2.(15) 5 Since h0=be, and (h0−h) = H, this can be re-written as P2 a2 e +H2 b2 e = 1,(16) which is an ellipse. I call (16) the dynamic P-ellipse (or, Pe). This provides geometrically important information. The rate of change of the elliptical-profile fluid area with respect to the reducing fluid depth is represented by another ellipse. So, as seen in (16), the fascinating fact is that, with respect to the elliptical-optics, the elliptical reservoir is invariant under a rotation and a translation (shift) with its canonical counterpart dAe/dh. This is a great new revelation. From (11) and (13), with the chain rule of differentiation, it follows: dVe dt =ae h0qh2 0−(h2 0−h)2WT dh dt .(17) From (17), it is evident that the nonlinear change (diminish) in volume is proportional to the change in the fluid depth with the proportionality Pbeing a nonlinear function of the fluid depth. This is in contrast to the classical Torricelli-Bernoulli cylindrical or rectangular reservoir problem for which the proportionality factor is a constant, namely the top sectional area of the vessel, equivalently its diameter. These special features characterize the hydro-mechanics of the elliptical reservoir. Additionally, the rate dVe/dt also depends on the reservoir width and the bathymetry. The elliptical-hydrometer With the evolving exit velocity (10) and the fluid volume (17), with (12), I construct dh dt =−αv u u t βh hh2 0−(h0−h)2i−α2 ,(18) where, α=WA WTe h0 ae HAµA,β= 2λgµp. I call (18) the elliptical-hydrometer as it describes the dynamically evolving fluid depth of the elliptical-reservoir. Consequently, I call (10) the elliptical hydrometer-velocity. Compared to the classical Torricelli-Bernoulli-law (Torricelli, 1644; Bernoulli, 1738), and the triangular-hydrometer (Pudasaini, 2025), (18) reveals important hydro-mechanical information. This constitutes a unique mechanism on how to utilize the effective potential energy of the reservoir system (in the form Ee pot =√β h) in controlling the flow with the reservoir geometry, and the fluid and out let frictions, in the form Ee con = αr1/hh2 0−(h0−h)2−α2i. This preservers a universal hydro-mechanical principle as the same also applies to the parabolic-hydrometer and the hyperbolic-hydrometer developed in Section 2.2 and Section 2.3, respectively. It is important to note that although the hydrometer (18) is developed with the help of the exit velocity uAe in (10), the hydrometer is completely decoupled from the exit velocity. However, the exit velocity depends fully and explicitly on the hydrometer. Moreover, the parameter αin the numerator on the right hand side of (18) emerges collectively from the physical-geometrical parameters associated with the reservoir volume and the out let velocity. However, this parameter appears to be exactly the same as that αin the denominator on the right hand side of (18) that originates from the mass out flux, or the exit velocity. This simplifies the model structure. This also applies to the parabolic-hydrometer and the hyperbolic-hydrometer constructed in Section 2.2 and Section 2.3. 6 2.2 The parabolic-hydrometer 2.2.1 The dynamical profile area Consider the parabola in (4). With this, one can construct a vertically oriented, downward facing parabolic reservoir as shown in Fig. 1B, enclosed by the section of the parabola: y=−2√apx, (19) below the line y= 0 and bounded by the two lines x= 0 and x=L=h2 0/4ap. The fluid level (initially the horizontal dashed blue line in Fig. 1B) decreases as y=−H =−(h0−h). Other procedures are as for the elliptical-hydrometer in Section 2.1. Since Ll=H2/4apdefines the mobile left boundary of the fluid body, as the fluid level decreases, the actual profile sectional area of the parabolic reservoir also decreases, and (its magnitude) is given by: Ap=ZL Ll2√apx−Hdx. (20) It is crucial to note that as Hevolves from 0 to h0,H2/4apcontracts itself from 0 to L=h2 0/4ap. After some mathematical operations and simplifications, I obtain the actual evolving sectional profile area of the parabolic reservoir: Ap=1 12ap2h3 0+ (h0−h)3−3h2 0(h0−h).(21) This shows that the area of the parabolic reservoir decreases from h3 0/6ap(when h=h0) to 0 (for h= 0). 2.2.2 The exit velocity With the actual length of the parabolic reservoir L=L−Ll, the fluid surface area and the conduit area are known: AT=LWT, AA=HAWAµA.(22) This means, ATis decreasing nonlinearly with h. Now, combining (1), (2) and (22), after simplification, I obtain an expression for the exit velocity uApfor the parabolic-reservoir as: uAp=hh2 0−(h0−h)2iv u u t βh hh2 0−(h0−h)2i2 −α2 ,(23) where, α= 4 WA WTp HAapµA,β= 2λgµp. This provides a relationship between the discharge velocity and the hydraulic head hat the dam, including the physical parameters αand β. The appearance of hh2 0−(h0−h)2i in the numerator and hh2 0−(h0−h)2i2 in the denominator with α2on the right hand side of (23) becomes a game-changer as both emerge due to the dynamically changing (shrinking) fluid surface area with reducing fluid depth at the dam. 2.2.3 Conservation of mass with volume change and out flux From (21), the reservoir fluid volume is given by: Vp=ApWT.(24) The fluid surface area decreases as the mass drains out through the out let. This is a game-changer. 7 Figure 2: A: The P-elliptical and, B: the P-parabolic conics, following the formal rule of rotation and translations, given respectively by (16) and (29). The abscissae and ordinates are (x, y) for the reservoir-conics, and (h, P) for the P-ellipse and the P-parabola, respectively. A B As the rate of change of the reservoir fluid volume and the volume flux through the exit balance, we have: dVp dt =−AAuAp.(25) To implement (25), we need information on how Apevolves with h. For this, differentiate (21) with respect to h. Then, the change (decline) of the profile area with respect to the evolving fluid level is given by dAp dh =1 4aph2 0−(h0−h)2,(26) which is the reservoir fluid length L. The relation (26) written in terms of Lyields: dAp dh =L,(27) which is the fundamental principle of reservoir (parabolic). I denote the right hand side of (26) by P: P=1 4aph2 0−(h0−h)2,(28) or, H2=−4ap(P −L),(29) which is a parabola. I call (29) the dynamic P-parabola (or, Pp). This provides geometrically important information. The rate of change of the parabolic-profile fluid area with respect to the reducing fluid depth is represented by another parabola. So, as seen in (29), the appealing fact is that, with respect to the parabolic-optics, the parabolic reservoir is invariant under a rotation and translations (shifts) with its canonical counterpart dAp/dh. This is a great new understanding. The P-elliptical and the P-parabolic conics are presented in Fig. 2 which are invariant to their respective reservoir-conics. They follow the principle of rotation and translations. The P-hyperbola can be displayed similarly with its representation derived later. It is important to note that the P-ellipse required one rotation and one translation. However, the P-parabola 8 required one rotation and two translations. Such behaviors can be explained by the natural, structural characteristics of the respective conic-reservoirs. Because, the elliptical-conic is a closed curve, with one rotation and just one translation, it transforms the parent reservoir-ellipse into a closed P-ellipse. However, the parabolicconic is an open curve; so, it needed one rotation and two translations in order to develop it into a closed region enclosed by a parabola describing the hydrometer rate of reservoir fluid profile area, and the axis describing the hydrometer. This explains the intrinsic mechanisms of the P-ellipse and the P-parabola. Moreover, Fig. 2 displays completely different hydrometer rates of reservoir fluid profile area for the elliptical-reservoir and the parabolic-reservoir, essentially differently controlling their hydrometer dynamics. From (24) and (26), with the chain rule of differentiation, it follows: dVp dt =1 4aph2 0−(h0−h)2WT dh dt .(30) From (30), it is evident that the change in volume is proportional to the change in the fluid depth with the proportionality Pbeing a nonlinear function of the fluid depth. The parabolic-hydrometer With the evolving exit velocity (23) and the fluid volume (30), with (25) I construct dh dt =−αv u u t βh hh2 0−(h0−h)2i2 −α2 ,(31) where, α= 4 WA WTp HAapµA,β= 2λgµp. I call (31) the parabolic-hydrometer. Consequently, I call (23) the parabolic hydrometer-velocity. Compared to the classical Torricelli-Bernoulli-law (Torricelli, 1644; Bernoulli, 1738), and the triangular-hydrometer (Pudasaini, 2025), (31) reveals important hydro-mechanical information that will be clearer later. 2.3 The hyperbolic-hydrometer 2.3.1 The dynamical profile area With the hyperbola (5), one can construct a vertically oriented, downward facing hyperbolic reservoir as shown in Fig. 1B, enclosed by the section of the hyperbola: y=−bhsx2 a2 h−1,(32) below the line y= 0 and bounded by the two lines x=ahand x=L=ah bhqh2 0+b2 h. As the fluid exits the conduit, the reservoir fluid level decreases from its initial level h0to some actual value h. So, the fluid level decreases as y=−H =−(h0−h). Since x=Ll=ah bhq(h0−h)2+b2 hdetermines the moving left boundary of the degrading reservoir, as the fluid level decreases, the actual profile sectional area of the hyperbolic reservoir also decreases, and (its magnitude) is given by: Ah=ZL Ll bhsx2 a2 h−1−H!dx. (33) As Hevolves from 0 to h0,Llcontracts itself from ahto L. After some mathematical operations, I obtain the actual evolving sectional profile area of the hyperbolic reservoir: Ah=−1 2ahbhtanh−11 h0qh2 0+b2 h+1 2 ah bh h0qh2 0+b2 h−ah bh (h0−h)qh2 0+b2 h +1 2ahbhtanh−11 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 results lead to the following four Reservoir Principles and Theorem. Principle 1. The Reservoir Rotate-Translate Principle: The rate of change of the reservoir fluid volume with respect to the hydraulic-head of a conic-reservoir is obtained by a rotation and some translations. Rotate the reservoir-conic by an amount π 2. Then, following the procedure in Section 2, translate it as suggested by (16), (29) and (43), respectively, for the elliptical-reservoir, parabolic-reservoir and the hyperbolic-reservoir. Principle 2. The Reservoir Eccentricity Principle: The conic-hydrometer is convex for the conic reservoir with eccentricity less than unity. Otherwise, the conic-hydrometer is concave. The procedure follows from Section 2. For the elliptical-reservoir, the hydrometer is convex. For the parabolicreservoir and hyperbolic-reservoir the hydrometers are concave. Principle 3. The Reservoir Quality-Rate of Volume Change Principle: The hydrometers for the parabolicreservoir and the hyperbolic-reservoir are virtually the same. The procedure follows from Section 2. This is due to the similar quality-rate of fluid volume change in time for the parabolic-reservoir and the hyperbolic-reservoir. Theorem 1. The Fundamental Theorem of Reservoir: The rate of change of the reservoir mass (fluid profile area) is the actual reservoir length. Proof: The procedure follows from Section 2 with relations (14), (27) and (41). This proves the theorem. 3.7 Classification of reservoir With respect to the results presented in Fig. 3, Fig. 4, Fig. 5 and Fig. 6, I classify the (hydrometers) reservoirs into four: The cylindrical-, triangular-, ellipticaland parabolic-hyperbolic reservoirs, respectively, with constant reservoir length, linear reservoir length, elliptical reservoir length and parabolic-hyperbolic reservoir length. Essentially these (dynamic) reservoir lengths characterize and fully control the corresponding hydrometers and the hydrometer-velocities of the cylindrical-, triangular-, ellipticaland parabolic-hyperbolic reservoirs; a new invention in hydro-mechanics. 3.8 Extensions to other conic-reservoir orientations and shapes The conic-hydrometers presented here can directly be extended to other conic-reservoir orientations and shapes. I give one example here. Consider the elliptical reservoir in Fig. 1. By also including the section in the fourth quadrant, one can construct the semi-elliptical-reservoir below the line y= 0 that defines the initial fluid surface. For this, one can consider the out let in the front (base) in the xy-face as the reservoir extends laterally in the z-direction. This is an open reservoir with the free surface of the fluid. However, one can also consider the closed semi-elliptical-reservoir above the line y= 0 that defines the base of the reservoir and the initial fluid body is contained by the upper semi-elliptical vessel body. Moreover, one can further consider the entire closed elliptical reservoir with its lateral extent in the xz-plane. The closed reservoirs have engineering applications, however, natural reservoirs and mountain lakes are open. Yet, for all these reservoirs, the hydrometers can easily be constructed by simply following the methods presented at Section 2 by just unfolding the evolving sectional profile area and the fluid surface area in the reservoir. The same methods can be extended in constructing hydrometers for differently oriented and shaped parabolicand hyperbolichydrometers. However, this requires the properly selected orientation of the initial conics. 4 Summary Following some complex procedures, I constructed three conic-hydrometers: the elliptical-hydrometer, the parabolic-hydrometer and the hyperbolic-hydrometer. These conic reservoirs are canonically invariant under 16 rotation and translations with the hydraulic-head rate of the reservoir profile area. This is a great new revelation. The nonlinear change in reservoir fluid volume is proportional to the change in the fluid depth. These special features characterize the hydro-mechanics of the conic reservoirs. The conic-hydrometers display several distinct behaviours with their implications in controlled and efficient design of the reservoirs and fluid vessels. Elliptical-hydrometer is weakly nonlinear whereas the parabolicand hyperbolic-hydrometers are weakly to strongly nonlinear. The hydrometers are convex for the reservoirs with the eccentricity <1 and the hydrometers are concave for the reservoirs with eccentricity ≥1. The conic-hydrometer rate quality controls the dynamics of the elliptical-, parabolicand hyperbolicreservoirs; the parabolicand hyperbolicrate quality curves overlap. The concavity-convexity, and merging of the parabolicand hyperbolichydrometers (hydrographs) for the fundamentally different reservoirs are novel understanding for the conic reservoir fluid exit processes. I presented some phenomenal reservoir principles: (I) There exists a universal hydro-mechanical principle on how the available potential energy is consumed by hydro-geometric constrains including the physical parameters of the conic-reservoirs in architecting the hydrometers. (II) The rate of change of the reservoir fluid volume with respect to the hydraulic-head of a conic-reservoir is obtained by a rotation and some translations of the conic reservoir-profile. So, the conic-hydrometers are invariant with respect to their parent conic-reservoirs. (III) For the elliptical-reservoir, the hydrometer is convex, and for the parabolic-reservoir and the hyperbolicreservoir the hydrometers are concave. (IV) As the hydrometer commands the exit velocity to follow the universal parabolic-hyperbolic hydrometer-velocity rule, the hydrometers for the parabolic-reservoir and the hyperbolic-reservoir are virtually the same. (V) The fundamental theorem of reservoir, stating that, the rate by which the reservoir mass (fluid profile area) reduces is given by the actual length of the reservoir. This is the most astonishing revelation characterizing the reservoir dynamics. This also proves the physical-mathematical consistency of the reservoir principles prevented here. My models provide important hydro-mechanical information with their implications in the desired, controlled and efficient design of the reservoirs and the fluid vessels to the hydraulic, hydro-mechanical, transportation and environmental engineers in selecting the reservoir geometry that intrinsically commands the dynamics, the hydrometer, its timing and the out let velocity. Depending on the operation and evacuation strategy, I provided some basic plausible design guidelines for practical use. Acknowledgment: The financial support is provided by the German Research Foundation (DFG) through the research project: Landslide mobility with erosion: Proof-of-concept and application - Part I: Modeling, Simulation and Validation; Project number 522097187. References [1] Baselt, I., de Oliveira, G.Q., Fischer, J.-T., Pudasaini, S.P. (2021): Evolution of stony debris flows in laboratory experiments. Geomorphology 372, 107431. [2] Bernoulli, D. (1738): Hydrodynamica sive de viribus motibus fluidorum commentarii. Johann Reinhold Dulsecker, Strassburg, Austria. [3] de Oliveira, P.M.C., Delfino, A., Costa, E.V., Leite, C.A.F. (2000): Pin-hole water flow from cylindrical bottles. Phys. Educ. 35. [4] Downs, J.W. (2003): Practical Conic Sections: The geometric properties of ellipses, parabolas and hyperbolas. Dover, USA. [5] du Buat, P.L.G. (1786): Principes d’Hydrauliques, Tome I et II. Imprimerie de Monsieur, Paris. [6] Kendig, K. (2005): Conics. 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