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Hydrodynamic Stability of Parallel Flow and Rotating Flow of Inviscid Fluid Dynamics

Manju Bala



Stream unsteadiness and fierce change can be all around clarified utilizing another proposed
hypothesis - Energy slope hypothesis (Dou, 2005). In this hypothesis, the strength of a stream
relies upon the general size of vitality slope in streamwise heading and that transverse way, if
there is no work input. In this note, it is indicated dependent on the vitality slope hypothesis
that inviscid non-uniform stream is flimsy if the vitality transverse way isn't consistent. This
new discovering breaks the old style direct hypothesis from Rayleigh that inviscid stream is
precarious if the speed profile has an emphasis point in equal streams and inviscid stream is
steady if the speed profile has no enunciation point in equal stream. At that point, security of
turning gooey and inviscid streams is examined, and two instances of pivoting streams
(pivoting inflexible body movement and free vortex movement) are appeared, separately.
Keywords: Energy Gradient; Energy loss; Inviscid Instability; Non-uniform flow, Rotating
flow; Viscous Instability.

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A. J. Rosenthal, R. S. L. (1983).

Instabilities in a stratified fluid having one

critical level: Part II. Explanation of

gravity wave instability as overrflected

waves. J. Atmos. Sci., 40, 521–529.

Chandrasekhar, S. (1953). The instability

of a layer of fluid heated below and

subject to Coriolis forces. Proceedings of

the Royal Society of London. Series A.

Mathematical and Physical Sciences,

(1130), 306–327.


Craik, A. D. D. (2004). THE ORIGINS


Review of Fluid Mechanics, 36(1), 1–28.


Darbyshire, A. G., & Mullin, T. (1995).

Transition to turbulence in constant-mass-

flux pipe flow. Journal of Fluid

Mechanics, 289, 83–114.


Dou, H.-S. (2005). Energy Gradient

Theory of Hydrodynamic Instability.

Drazin, P. G. (2002). Introduction to

Hydrodynamic Stability. In Introduction to

Hydrodynamic Stability. Cambridge

University Press.


Eckhardt, B., Grossmann, S., & Lohse, D.

(2007). Torque scaling in Taylor-Couette

flow. Advances in Turbulence XI -

Proceedings of the 11th EUROMECH

European Turbulence Conference,



Grossmann, S. (2000). The onset of shear

flow turbulence. Reviews of Modern

Physics, 72(2), 603–618.


Hof, B., Van Doorne, C. W. H.,

Westerweel, J., Nieuwstadt, F. T. M.,

Faisst, H., Eckhardt, B., Wedin, H.,

Kersweli, R. R., & Waleffe, F. (2004).

Experimental observation of nonlinear

traveling waves in turbulent pipe flow.

Science, 305(5690), 1594–1598. https://

Lessen, M., Singh, P. J., & Paillet, F.

(1974). The stability of a trailing line

vortex. Part 1. Inviscid theory. Journal of

Fluid Mechanics, 63(4), 753–763. https://

Roy, A., & Govindarajan, R. (2010). An

introduction to hydrodynamic stability. In

Rheology of Complex Fluids (pp.

–147). Springer New York. https://

Schmitz, G. (1979). Lighthill, J., Waves in

Fluids. Cambridge-London-New York-

Melbourne, Cambridge University Press

XV, 504 S., £ 17.50 A. ZAMM -

Zeitschrift Für Angewandte Mathematik

Und Mechanik, 59(11), 671–671. https://

Trefethen, L. N., Trefethen, A. E., Reddy,

S. C., & Driscoll, T. A. (1993).

Hydrodynamic Stability without

Eigenvalues. Science, 261(5121).

H-S Dou, Energy gradient theory of

hydrodynamic instability, The Third

International Conference on Nonlinear

Science, Singapore, 30 June -- 2 July,

, revised version also in: International

Journal of Non-Linear Mechanics,

accepted and in press (2005).

Rayleigh, L. On the stability or instability

of certain fluid motions. Proc. Lond.

Maths. Soc., 1880, 11: 57-70

Tollmien,W., 1935, Ein allgemeines

Kriterium der Instabilitat laminarer

Gescgwindigkeitsverteilungen., Nachr.

Wiss fachgruppe, Gottingen, math. Phys.,

, 79- 114. Translated as, General

Journal of Aerospace Engineering & Technology

Volume 10, Issue 2

ISSN: 2231-038X (Online), ISSN: 2348-7887 (Print)

JoAET (2020) 1–14 © STM Journals 2020. All Rights Reserved Page 13

instability criterion of laminar velocity

disturbances, NACA TM-792, 1936.

Fjφrtoft R. Application of integral

theorems in deriving criteria of stability

for laminar flows and for the baroclinic

circular vortex. Geofys. Publ., 1950, 17: 1-

P.J.Schmid, and D.S.Henningson.,

Stability and transition in shear flows,

New York, Springer-Verlag, 2000.

P. G. Drazin and W. H. Reid,

Hydrodynamic stability, Cambridge

University Press, 2nd Ed., Cambridge,

England, 2004, 69-123.

Schlichting H, and Gersten K. Boundary

Layer Theory. Berlin: Springer, 8th Ed.,

Lin C C. The Theory of Hydrodynamic

Stability. Cambridge: Cambridge Press,

H-S Dou, Viscous instability of

inflectional velocity profile, Recent

Advances in Fluid Mechanics, Proc. of the

th Inter. Conf. on Fluid Mech., July

~23, 2004, Dalian, China; Tsinghua

University Press & Springer-Verlag, 2004,


H.-S. Dou, B.C.Khoo, and K.S.Yeo, Flow

transition in plane Couette flow, Technical

Report of National University of

Singapore, 2003.


S. Grossmann, The onset of shear flow

turbulence. Reviews of Modern Physics,

(2000), 603-618.

L.N.Trefethen, A.E. Trefethen, S.C.

Reddy, T.A. Driscoll, Hydrodynamic

stability without eigenvalues, Science, 261

(1993), 578-584.

A.G.Darbyshire and T.Mullin, Transition

to turbulence in constant-mass-flux pipe

flow, J. Fluid Mech, 289 (1995), 83-114.

M. Nishioka, S Iida, and Y.Ichikawa, An

experimental investigation of the stability

of plane Poiseuille flow, J. Fluid Mech.,

(1975), 731-751.

H. Wedin, and R.R. Kerswell, Exact

coherent structures in pipe flow: travelling

wave solutions, J. Fluid Mech. 508 (2004),


B. Hof, C.W. H. van Doorne,

J.Westerweel, F.T. M. Nieuwstadt,

H.Faisst, B.Eckhardt, H.Wedin, R.R.

Kerswell, F.Waleffe, Experimental

observation of nonlinear traveling waves

in turbulent pipe flow, Science, 305

(2004), Issue 5690, 10 September 2004,

- 1598.

H.-S. Dou, B. C. Khoo, and K. S. Yeo,

Energy loss distribution in Taylor-Couette

flow between concentric rotating

cylinders, Technical Report of National

University of Singapore, 2004.

Bale R, Govindarajan R (2010) Transient

Growth and why we should care about it,

Resonance, 15:441–457

Batchelor GK (1967) Introduction to fluid

dynamics. Cambridge University Press,


Chandrasekhar S (1981) Hydrodynamic

and hydromagnetic stability. Dover

Publications, New York

Chomaz JM (2005) Global instabilities in

spatially developing flows: Non-normality

and non- linearity. Annu Rev Fluid Mech


Craik ADD (2004) The origins of water

wave theory. Annu Rev Fluid Mech


An Introduction to Hydrodynamic

Stability 147

Craik ADD (2005) George Gabriel Stokes

on water wave theory. Annu Rev Fluid

Mech 37: 23–42

de Gennes PG, Brochard F, Quere D

(2004) Capillarity and wetting

phenomena: Drops, bubbles, pearls,

waves. Springer, New York

Drazin PG (2002) Introduction to

hydrodynamic stability. Cambridge

University Press, Cambridge

Eckhardt B, Grossman S, Lohse D (2007)

Torque scaling in turbulent

Taylor–Couette flow between

independently rotating cylinders. J Fluid

Mech 581:221–250

Hydrodynamic Stability of Parallel Flow and Rotating Flow Manju Bala

JoAET (2020) 1–14 © STM Journals 2020. All Rights Reserved Page 14

Eckhardt B, Schneider TM, Hof B et al

(2007) Turbulence transition in pipe flow.

Annu Rev Fluid Mech 39:447–468

Farrell BF, Ioannou PJ (1993) Optimal

excitation of three-dimensional

perturbations in viscous constant shear

flow. Phys Fluids A 5:1390–1400

Friedman B (1990) Principles and

techniques of applied mathematics. Dover,

New York

Groisman A, Steinberg V (2000) Elastic

turbulence in a polymer solution flow.

Nature 405:53–55

Ho CM, Huerre P (1984) Perturbed free

shear layers. Annu Rev Fluid Mech


Hof B, van Doorne CWH, Westerweel J et

al (2004) Experimental observation of

nonlinear traveling waves in turbulent pipe

flow. Science 305:1594–1598

Huerre P, Monkewitz PA (1990) Local

and global instabilities in spatially

developing flows.

Annu Rev Fluid Mech 22:473–537

Huerre P, Rossi M (1998) Hydrodynamic

instabilities in open flows. In: Godreche C,

Manneville P (eds) Hydrodynamics and

nonlinear instabilities 81–294, Cambridge

University Press, Cambridge

Huerre P (2000) Open shear flow

instabilities. In: Batchelor GK, Moffatt

HK, Worster MG (eds) Perspectives in

fluid dynamics: A collective introduction

to current research, Cambridge University

Press, London

Landahl MT (1980) A note on an algebraic

instability of inviscid parallel shear flows.

J Fluid

Mech 98:243–251

Lighthill MJ (1978) Waves in fluids.

Cambridge University Press, London

Lindzen RS (1988) Instability of plane

parallel shear flow (toward a mechanistic

picture of how it works). PAGEOPH


Morozov AN, van Saarloos W (2007) an

introductory essay on subcritical

instabilities and the transition to

turbulence in visco-elastic parallel shear

flows. Phys Rep 447:112–143

Shaqfeh ESG (1996) purely elastic

instabilities in viscometric flows. Annu

Rev Fluid Mech


Waleffe F(1995) Transition in shear flows.

Nonlinear normality versus non-normal


Phys Fluids 7:3060–3066

Waleffe F (1997) on a self-sustaining

process in shear flows. Phys Fluids




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