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| {{dablink| This article is about fluid mechanics. For the use of generalized coordinates in classical mechanics, see [[generalized coordinates]], [[Lagrangian mechanics]] and [[Hamiltonian mechanics]]}}
| | The person who wrote the article is known as Jayson Hirano and he totally digs that title. One of the things she loves most is canoeing and she's been doing it for quite a while. Mississippi is exactly where his house is. Office supervising is my occupation.<br><br>my weblog - [http://afeen.fbho.net/v2/index.php?do=/profile-210/info/ psychic readers] |
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| In [[fluid dynamics]] and finite-deformation [[plasticity (physics)|plasticity]] the '''Lagrangian specification of the flow field''' is a way of looking at fluid motion where the observer follows an individual [[fluid parcel]] as it moves through space and time.<ref name=Batchelor>Batchelor (1973) pp. 71–73.</ref><ref name=Lamb>Lamb (1994) §3–§7 and §13–§16.</ref> Plotting the position of an individual parcel through time gives the [[pathline]] of the parcel. This can be visualized as sitting in a boat and drifting down a river.
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| The '''Eulerian specification of the flow field''' is a way of looking at fluid motion that focuses on specific locations in the space through which the fluid flows as time passes.<ref name=Batchelor/><ref name=Lamb/> This can be visualized by sitting on the bank of a river and watching the water pass the fixed location.
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| The Lagrangian and Eulerian specifications of the flow field are sometimes loosely denoted as the '''Lagrangian and Eulerian frame of reference'''. However, in general both the Lagrangian and Eulerian specification of the flow field can be applied in any observer's [[frame of reference]], and in any [[coordinate system]] used within the chosen frame of reference.
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| ==Description==
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| In the ''Eulerian specification'' of the flow field, the flow quantities are depicted as a function of position '''x''' and time ''t''. Specifically, the flow is described by a function
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| :<math>\mathbf{v}\left(\mathbf{x},t\right)</math>
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| giving the flow [[velocity]] at position '''x''' at time ''t''.
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| On the other hand, in the ''Lagrangian specification'', individual fluid parcels are followed through time. The fluid parcels are labelled by some (time-independent) vector field '''a'''. (Often, '''a''' is chosen to be the center of mass of the parcels at some initial time ''t<sub>0</sub>''. It is chosen in this particular manner to account for the possible changes of the shape over time. Therefore the center of mass is a good parametrization of the velocity '''v''' of the parcel.)<ref name=Batchelor/> In the Lagrangian description, the flow is described by a function
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| :<math>\mathbf{X}\left(\mathbf{a},t\right)</math>
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| giving the position of the parcel labeled '''a''' at time ''t''.
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| The two specifications are related as follows:<ref name=Lamb/>
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| :<math>\mathbf{v}\left(\mathbf{X}(\mathbf{a},t),t \right) = \frac{\partial \mathbf{X}}{\partial t}\left(\mathbf{a},t \right)</math>
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| because both sides describe the velocity of the parcel labeled '''a''' at time ''t''.
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| Within a chosen coordinate system, '''a''' and '''x''' are referred to as the '''Lagrangian coordinates''' and '''Eulerian coordinates''' of the flow.
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| ==Substantial derivative==
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| {{Main|Material derivative}}
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| The Lagrangian and Eulerian specifications of the [[kinematics]] and [[dynamics (physics)|dynamics]] of the flow field are related by the [[substantial derivative]] (also called the Lagrangian derivative, convective derivative, material derivative, or particle derivative).<ref name=Batchelor/>
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| Suppose we have a flow field with Eulerian specification '''v''', and we are also given some function '''F'''('''x''',''t'') defined for every position '''x''' and every time ''t''. (For instance, '''F''' could be an external force field, or temperature.) Now one might ask about the total rate of change of '''F''' experienced by a specific flow parcel. This can be computed as
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| :<math>\frac{\mathrm{D}\mathbf{F}}{\mathrm{D}t} = \frac{\partial \mathbf{F}}{\partial t} + (\mathbf{v}\cdot \nabla)\mathbf{F}</math>
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| (where ∇ denotes the [[gradient]] with respect to '''x''', and the operator '''v'''⋅∇ is to be applied to each component of '''F'''.) This tells us that the total rate of change of the function '''F''' as the fluid parcels moves through a flow field described by its Eulerian specification '''v''' is equal to the sum of the local rate of change and the convective rate of change of '''F'''. This is a consequence of the [[chain rule]] since we are differentiating the function '''F'''('''X'''('''a''',''t''),''t'') with respect to ''t''.
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| Conservation laws for a unit mass have a Lagrangian form, which together with mass conservation produce Eulerian conservation; on the contrary when fluid particle can exchange the quantity (like energy or momentum) only Eulerian conservation law exists, see Falkovich.
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| ==See also==
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| * [[Contour advection]]
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| * [[Coordinate system]]
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| * [[Equivalent latitude]]
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| * [[Fluid dynamics]]
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| * [[Frame of reference]]
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| * [[Generalized Lagrangian mean]]
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| * [[Lagrangian particle tracking]]
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| * [[Semi-Lagrangian scheme]]
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| * [[Streamlines, streaklines, and pathlines]]
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| * [[Trajectory (fluid mechanics)]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation
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| | first=G.K. | last=Batchelor | authorlink=George Batchelor
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| | title=An introduction to fluid dynamics
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| | publisher=Cambridge University Press
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| | year=1973
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| | isbn=0-521-09817-3
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| }}
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| *{{citation
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| | first=H. | last=Lamb | authorlink=Horace Lamb
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| | title=Hydrodynamics
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| | edition=6th
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| | publisher=Cambridge University Press
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| | year=1994
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| | origyear=1932
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| | isbn=978-0-521-45868-9
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| }}
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| * {{citation | last=Falkovich | first=Gregory | year=2011 | title=Fluid Mechanics (A short course for physicists)|url=http://www.cambridge.org/gb/knowledge/isbn/item6173728/?site_locale=en_GB | publisher=Cambridge University Press | isbn=978-1-107-00575-4 }}
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| [[Category:Fluid dynamics]]
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| [[Category:Aerodynamics]]
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The person who wrote the article is known as Jayson Hirano and he totally digs that title. One of the things she loves most is canoeing and she's been doing it for quite a while. Mississippi is exactly where his house is. Office supervising is my occupation.
my weblog - psychic readers