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Lagrangian and Eulerian Representations of Fluid Flow: Part I,
Kinematics and the Equations of Motion
James F. Price
MS 29, Clark Laboratory
Woods Hole Oceanographic Institution
Woods Hole, MA, 02543

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Lagrangian and Eulerian Representations of Fluid Flow: Part I,Kinematics and the Equations of Motion
James F. PriceMS 29, Clark LaboratoryWoods Hole OceanographicInstitutionWoods Hole, MA, 02543
http://www.whoi.edu/science/PO/people/jprice
jprice@whoi.eduVersion 7.4September 13, 2005
Summary:
This essay introduces the two methods that are commonly used to describe ﬂuid ﬂow, byobserving the trajectories of parcels that are carried along with the ﬂow or by observingthe ﬂuid velocity atﬁxed positions. These yield what are commonly termed Lagrangian and Eulerian descriptions. Lagrangianmethods are often the most efﬁcient way to sample a ﬂuid domain and the physical conservation laws areinherently Lagrangian since they apply to speciﬁc material parcels rather than pointsin space. It happens,though, that the Lagrangian equations of motion applied to a continuum are quite difﬁcult, and thus almostall of the theory (forward calculation)in ﬂuid dynamics is developed withinthe Eulerian system. Euleriansolutionsmay be used to calculate Lagrangian properties, e.g., parcel trajectories, which is often a valuablestep in the description of an Eulerian solution. Transformation to and from Lagrangian and Eulerian systems— the central theme of this essay — is thus the foundationof most theory in ﬂuid dynamics and is a routinepart of many investigations.The transformation of the Lagrangian conservationlaws into the Eulerian equations of motion requiresthree key results. (1) The ﬁrst is dubbed the Fundamental Principle of Kinematics; the velocity at a givenpositionand time (the Eulerian velocity) is identicallythe velocity of the parcel (the Lagrangian velocity)that occupies that positionat that time. (2) The material time derivative relates the time rate of changeobserved followinga moving parcel to the time rate of change and advective rate of change observed at aﬁxed position;
D
. /=
Dt
D
@. /=@
t
C
V
r
. /
. (3) And ﬁnally, the time derivative of an integral over amoving ﬂuid volume can be transformed intoﬁeld coordinatesby means of the Reynolds Transport Theorem.1
Once an Eulerian velocity ﬁeld has been found, it is more or less straightforwardto computeLagrangian properties, e.g., parcel trajectories, which are often of practical interest. The FPK assures that theinstantaneousEulerian and Lagrangian velocities are identically equal. However, when averaging orintegrating takes place, then the Eulerian mean velocity and the Lagrangian mean velocity are not equal,except in the degenerate case of spatially uniform ﬂows. If the dominant ﬂow phenomenon is wavelike, thentheir difference may be understood as Stokes drift, a correlation between displacement and velocitydifferences.In an Eulerian system the local (at a point) effect of transport by the ﬂuid ﬂow is represented by theadvective rate of change,
V
r
. /
, the product of an unknown velocity and the ﬁrst partial derivative of anunknown ﬁeld variable. This nonlinearityleads to much of the interestingand most of the challengingphenomenon of ﬂuid ﬂows. We can put some useful boundsupon what advection alone can do. For variablesthat can be written in conservation form, e.g., mass and momentum, advection alone can not be a net sourceor sink when integrated over a closed or inﬁnite domain. Advection represents the transport of ﬂuidproperties at a deﬁnite rate and direction, that of the ﬂuid velocity, so that parcel trajectories are thecharacteristics of the advection equation. Advection by a nonuniform velocity may cause importantdeformation of a ﬂuid parcel, and it may also cause rotation, an analog of angular momentum, and thatfollows a particularly simple and useful conservation law.
Cover page graphic:
SOFAR ﬂoat trajectories (green worms) and horizontal velocity measured by a currentmeter (black vector) during the Local Dynamics Experiment conducted in the Sargasso Sea. Click on theﬁgure to start an animation. The ﬂoat trajectories are ﬁve-day segments, and the current vector is scaledsimilarly. The northeast to southwestoscillationseen here appears to be a barotropic Rossby wave; see Price,J. F. and H. T. Rossby, ’Observationsof a barotropic planetary wave in the western North Atlantic’,
J. Marine Res.
,
40
, 543-558, 1982. An analysis of the potentialvorticity balance of this motion is in Section6.4.4. These data and much more are available online from http://ortelius.whoi.edu/ Some animations of theextensive ﬂoat data archive from the North Atlantic are athttp://www.phys.ocean.dal.ca/ lukeman/projects/argo/.
Contents
1 Kinematics of Fluid Flow. 4
1.1 Physical properties of material; how are ﬂuids different from solids? . . . . . . . . . . . . . . 41.1.1 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.2 Shear deformation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Fluid ﬂow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Two ways to observe ﬂuid ﬂow and the Fundamental Principle of Kinematics . . . . . . . . . 101.4 About this essay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 The Lagrangian (or Material) Coordinate System. 14
2
32.1 Material coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Forces on a parcel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 Stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Stress components in an ideal ﬂuid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Stress components in a viscousﬂuid . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 The Lagrangian equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 The Eulerian (or Field) Coordinate System. 25
3.1 The material derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Reynolds Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 The Eulerian equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.2 Momentum conservation;Newton’sSecond Law. . . . . . . . . . . . . . . . . . . . . 313.3.3 Energy conservation;the First Law of Thermodynamics. . . . . . . . . . . . . . . . . 323.3.4 State Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 Remarks on the Eulerian balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Depictions of Fluid Flow. 35
4.1 Trajectories, or pathlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Streaklines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Eulerian to Lagrangian Transformation by Approximate Methods. 38
5.1 Tracking parcels around a steady vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Tracking parcels in gravity waves; Stokes drift . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6 Aspects of Advection. 46
6.1 Modes of an advection equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 Fluxes in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2.1 Global conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2.2 Control volume budgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.3 The method of characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.4 Advection of a ﬁnite parcel; the Cauchy-Stokes Theorem . . . . . . . . . . . . . . . . . . . . 586.4.1 The rotation rate tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.4.2 The strain rate tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.4.3 The Cauchy-Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.4.4 Rotation and divergence of a geophysicalﬂow; the potential vorticity . . . . . . . . . 67
7 Concluding Remarks. 71
1 KINEMATICS OFFLUID FLOW.
4
1 Kinematics of Fluid Flow.
The broad aim of this essay is to introduce and exercise a few of the concepts and mathematical tools thatmake up the foundation of ﬂuid mechanics. Fluid dynamics is a vast subject, encompassing widely diversematerials and phenomenon. This essay emphasizes aspects of ﬂuid dynamics that are relevant to thegeophysicalﬂows of what one might term ordinary ﬂuids, air and water, that make up the Earth’s ﬂuidenvironment. The physicsthat govern geophysicalﬂow is codiﬁed by the conservation laws of classicalmechanics; conservation of mass, (linear) momentum, angular momentum and energy. TheLagrangian/Euleriantheme of this essay followsfrom the question, How can we apply these conservationlaws to the analysis or prediction ﬂuid ﬂow?In principle the answer is straightforward;ﬁrst we erect a coordinate system that is suitable fordescribing a ﬂuid ﬂow, and then we derive the mathematical form of the conservation laws that correspond tothat system. The deﬁnition of a coordinate system is a matter of choice, and the issues to be considered aremore in the realm of kinematics than of physics. However, as we will describe in this introductorysection,the kinematics of a ﬂuid ﬂow are certainly dependent upon the physicalproperties of the ﬂuid (reviewed inSection 1.1), and, the kinematics of even the smallest and simplest ﬂuid ﬂow is likely to be quite complex;fully three-dimensional and time-dependent ﬂows are common rather than exceptional (Section 1.2). Thuskinematics is at the nub of what makes ﬂuid mechanics challenging, and speciﬁcally, requires that thedescription of ﬂuid ﬂows be in terms of ﬁelds (beginningin Section 1.3).
1
;
2
1.1 Physical properties of material; how are ﬂuids different from solids?
For most purposes of classical ﬂuid dynamics, ﬂuids such as air and water can be idealized as an inﬁnitelydivisiblecontinuum withinwhich the pressure,
P
, and the velocity,
V
, temperature,
T
, are in principledeﬁnable at every point in space.
3
The molecular makeup of the ﬂuid will be studiouslyignored, and thecrucially important physicalproperties of a ﬂuid, e.g., its mass density,
, its heat capacity,
Cp
, amongothers, must be provided from outside of a continuum theory, Table (1).The space occupied by the material will be called the domain. Solids are materials that have a more orless deﬁnite or intrinsicshape, and will not conform to their domain under normal conditions. Fluids (gasesand liquids)have no intrinsicshape or preferred conﬁguration. Gases are ﬂuids that will completely ﬁll theirdomain (or container) and liquidsare ﬂuids that form a free surface in the presence of an acceleration ﬁeld,i.e., gravity.An important property of any material is its response to an applied force, Fig. (1). If the force on the
1
Footnotes provide references, extensions or qualiﬁcations of material discussed in the main text, along with a few homework assignments. They may be skippedon ﬁrst reading.
2
An excellent web page that surveys the wide range of ﬂuid mechanicsis http://physics.about.com/cs/ﬂuiddynamics/
3
Readers are presumed to have a college-level background in physics and multivariable calculus and to be familiar with basicphysical concepts such as pressure and velocity, Newton’s laws of mechanics and the ideal gas laws. We will review the deﬁnitionswhen we require an especially sharp or distinct meaning.

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