|
|
| Line 1: |
Line 1: |
| In [[continuum mechanics]], the '''material derivative'''<ref name="BSLr2"/><ref name=Batchelor>{{cite book | first=G.K. | last=Batchelor | authorlink=George Batchelor | title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=0-521-66396-2 | pages=72–73}}</ref> describes the time [[rate of change]] of some physical quantity (like [[heat]] or [[momentum]]) for a [[material element]] subjected to a space-and-time-dependent [[flow velocity|velocity field]]. The material derivative can serve as a link between [[Continuum mechanics#Eulerian description|Eulerian]] and [[Continuum mechanics#Lagrangian description|Lagrangian]] descriptions of continuum deformation.<ref name=Trenberth>{{cite book | first=K. E. | last=Trenberth | authorlink=Kevin_Trenberth | title=Climate System Modeling | year=1993 | publisher=Cambridge University Press | isbn=0-521-43231-6 | page=99 }}</ref>
| | Historical past of the of the author is Gabrielle Lattimer. For years she's been working as a library assistant. For a while she's yet been in Massachusetts. As a woman what your woman really likes is mah jongg but she have not made a dime utilizing it. She could be described as running and maintaining a meaningful blog here: http://prometeu.net<br><br>Also visit my homepage :: [http://prometeu.net clash of clans cheats] |
| | |
| For example, in [[fluid dynamics]], take the case that the velocity field under consideration is the [[flow velocity]] itself, and the quantity of interest is the [[temperature]] of the fluid. Then the material derivative describes the temperature evolution of a certain [[fluid parcel]] in time, as it is being moved along its [[Streamlines, streaklines, and pathlines|pathline]] (trajectory) while following the fluid flow.
| |
| {{TOCright}}
| |
| | |
| ==Names== <!-- in bold, since the redirects lead to here -->
| |
| | |
| There are many other names for the material derivative, including:
| |
| *'''convective derivative'''<ref name=Ockendon>{{cite book| first=H. |last=Ockendon | coauthors=Ockendon, J.R. | title=Waves and Compressible Flow | publisher=Springer | year=2004 | isbn=0-387-40399-X | page=6 }}</ref>
| |
| *'''advective derivative'''
| |
| *'''substantive derivative'''<ref name=Granger>{{cite book| first=R.A. |last=Granger| title=Fluid Mechanics | publisher=Courier Dover Publications | year=1995 | isbn=0-486-68356-7 | page=30}}</ref>
| |
| *'''substantial derivative'''<ref name="BSLr2">{{cite book|last1=Bird |first1=R.B. |last2=Stewart |first2=W.E. | last3=Lightfoot |first3=E.N. |author3-link=Edwin N. Lightfoot |title=[[Transport Phenomena (book)|Transport Phenomena]] |edition=Revised Second Edition |publisher=John Wiley & Sons |year=2007 |isbn=978-0-470-11539-8 |page=83}}</ref>
| |
| *'''Lagrangian derivative'''<ref name=Mellor>{{cite book | first=G.L. | last=Mellor | title=Introduction to Physical Oceanography | publisher=Springer | year=1996 | isbn=1-56396-210-1 |page=19 }}</ref>
| |
| *'''Stokes derivative'''<ref name=Granger/>
| |
| *'''particle derivative'''
| |
| *'''hydrodynamic derivative'''<ref name="BSLr2"/>
| |
| *'''derivative following the motion'''<ref name="BSLr2"/>
| |
| *'''total derivative'''<ref name="BSLr2"/><ref>{{Cite book | publisher = Butterworth-Heinemann | isbn = 0-7506-2767-0
| |
| | first1 = L.D. | last1 = Landau | author1-link = Lev Landau | first2 = E.M. | last2 = Lifshitz | author2-link = Evgeny Lifshitz | title = Fluid Mechanics | edition = 2nd | series = Course of Theoretical Physics | volume = 6 | year = 1987 | pages = 3–4 & 227 }}</ref>
| |
| | |
| ==Definition==
| |
| | |
| The material derivatives of a [[scalar field]] ''φ''( '''x''', ''t'' ) and a [[vector field]] '''u'''( '''x''', ''t'' ) are defined respectively as:
| |
| | |
| :<math>\frac{\mathrm{D}\varphi}{\mathrm{D}t} = \frac{\partial \varphi}{\partial t} + \mathbf{v}\cdot\nabla \varphi,</math>
| |
| | |
| :<math>\frac{\mathrm{D}\mathbf{u}}{\mathrm{D}t} = \frac{\partial \mathbf{u}}{\partial t} + \mathbf{v}\cdot\nabla \mathbf{u},</math>
| |
| | |
| where the distinction is that <math>\nabla \varphi</math> is the [[gradient]] of a scalar, while <math>\nabla \mathbf{u}</math> is the [[covariant derivative]] of a vector. In case of the material derivative of a vector field, the term '''v'''•∇'''u''' can both be interpreted as '''v'''•(∇'''u''') involving the [[tensor derivative (continuum mechanics)|tensor derivative]] of '''u''', or as ('''v'''•∇)'''u''', leading to the same result.<ref>{{Cite book | last=Emanuel | first=G. | title=Analytical fluid dynamics | publisher=CRC Press | year=2001 | edition=second | isbn=0-8493-9114-8 |pages=6–7 }}</ref> As an example: in a three-dimensional [[Cartesian coordinate system]] (''x''<sub>1</sub>,''x''<sub>2</sub>,''x''<sub>3</sub>), '''v'''•∇''φ'' is equal to {{nowrap|''v''<sub>1</sub> ∂''φ''/∂''x''<sub>1</sub> + ''v''<sub>2</sub> ∂''φ''/∂x<sub>2</sub> + ''v''<sub>3</sub> ∂''φ''/∂x<sub>3</sub>}}.
| |
| | |
| Confusingly, the term convective derivative is both used for the whole material derivative ''Dφ/Dt'' or ''D'''''u'''/''Dt'', and for only the spatial rate of change part, '''v'''•∇''φ'' or '''v'''•∇'''u''' respectively.<ref name=Batchelor/> For that case, the convective derivative only equals ''D/Dt'' for time independent flows.
| |
|
| |
| These derivatives are physical in nature and describe the transport of a scalar or vector quantity in a velocity field '''v'''( '''x''', ''t'' ). The effect of the time independent terms in the definitions are for the scalar and vector case respectively known as [[advection]] and convection.
| |
| | |
| ==Development==
| |
| Consider a scalar quantity ''φ'' = ''φ''( '''x''', ''t'' ), where ''t'' is understood as time and '''x''' as position. This may be some physical variable such as temperature or chemical concentration. The physical quantity exists in a fluid, whose velocity is represented by the vector field '''v'''( '''x''', ''t'' ).
| |
| | |
| The (total) derivative with respect to time of ''φ'' is expanded through the multivariate [[chain rule]]:
| |
| | |
| :<math>\frac{\mathrm{d}}{\mathrm{d} t}(\varphi(\mathbf x, t)) = \frac{\partial \varphi}{\partial t} + \nabla \varphi \cdot \frac{\mathrm{d} \mathbf x}{\mathrm{d} t}.</math>
| |
| | |
| It is apparent that this derivative is dependent on the vector
| |
| | |
| :<math>\frac{\mathrm{d} \mathbf x}{\mathrm{d} t} = \left(\frac{\mathrm{d} x}{ \mathrm{d}t}, \frac{\mathrm{d} y}{\mathrm{d} t}, \frac{\mathrm{d} z}{\mathrm{d} t}\right)^T</math>
| |
| | |
| which describes a ''chosen'' path '''x'''(''t'') in space. For example, if <math>\mathrm{d} \mathbf x/\mathrm{d} t = 0</math> is chosen, the time derivative becomes equal to the partial time derivative, which agrees with the definition of a [[partial derivative]]: a derivative taken with respect to some variable (time in this case) holding other variables constant (space in this case). This makes sense because if <math>\mathrm{d} \mathbf x/\mathrm{d} t = 0</math>, then the derivative is taken at some ''constant'' position. This static position derivative is called the Eulerian derivative.
| |
| | |
| An example of this case is a swimmer standing still and sensing temperature change in a lake early in the morning: the water gradually becomes warmer due to heating from the sun.
| |
| | |
| If, instead, the path '''x'''(''t'') is not a standstill, the (total) time derivative of ''φ'' may change due to the path. For example, imagine the swimmer is in a motionless pool of water, indoors and unaffected by the sun. One end happens to be a constant hot temperature and the other end a constant cold temperature. By swimming from one end to the other the swimmer senses a change of temperature with respect to time, even though the temperature at any given (static) point is a constant. This is because the derivative is taken at the swimmer's changing location. A temperature sensor attached to the swimmer would show temperature varying in time, even though the pool is held at a steady temperature distribution.
| |
| | |
| The material derivative finally is obtained when the path '''x'''(''t'') is chosen to have a velocity equal to the fluid velocity:
| |
| | |
| :<math>\frac{\mathrm{d} \mathbf x}{\mathrm{d} t} = \mathbf v.</math> | |
| | |
| That is, the path follows the fluid current described by the fluid's velocity field '''v'''. So, the material derivative of the scalar ''φ'' is:
| |
| | |
| :<math>\frac{\mathrm{D} \varphi}{\mathrm{D} t} = \frac{\partial \varphi}{\partial t} + \nabla \varphi \cdot \mathbf v.</math>
| |
| | |
| An example of this case is a lightweight, neutrally buoyant particle swept around in a flowing river undergoing temperature changes, maybe due to one portion of the river being sunny and the other in a shadow. The water as a whole may be heating as the day progresses. The changes due to the particle's motion (itself caused by fluid motion) is called ''[[advection]]'' (or convection if a vector is being transported).
| |
| | |
| The definition above relied on the physical nature of fluid current; however no laws of physics were invoked (for example, it hasn't been shown that a lightweight particle in a river will follow the velocity of the water). It turns out, however, that many physical concepts can be written concisely with the material derivative. The general case of advection, however, relies on conservation of mass in the fluid stream; the situation becomes slightly different if advection happens in a non-conservative medium.
| |
| | |
| Only a path was considered for the scalar above. For a vector, the gradient becomes a [[tensor derivative]]; for [[tensor]] fields we may want to take into account not only translation of the coordinate system due to the fluid movement but also its rotation and stretching. This is achieved by the [[upper convected time derivative]].
| |
| | |
| ==Orthogonal coordinates==
| |
| It may be shown that, in [[orthogonal coordinates]], the <math>j^{th}</math> component of convection is given by:<ref>{{cite web
| |
| | url = http://mathworld.wolfram.com/ConvectiveOperator.html
| |
| | title = Convective Operator
| |
| | author = [[Eric W. Weisstein]]
| |
| | publisher = [[MathWorld]]
| |
| | accessdate = 2008-07-22
| |
| }}</ref>
| |
| | |
| :<math>[\mathbf{v}\cdot\nabla \mathbf{u}]_j = | |
| \sum_i \frac{v_i}{h_i} \frac{\partial u_j}{\partial q^i} + \frac{u_i}{h_i h_j}\left(v_j \frac{\partial h_j}{\partial q^i} - v_i \frac{\partial h_i}{\partial q^j}\right),
| |
| </math>
| |
| | |
| where the ''h<sub>i</sub>'''s are related to the [[metric tensor]]s by
| |
| | |
| :<math>h_i=\sqrt{g_{ii}}.</math>
| |
| | |
| ==See also==
| |
| | |
| * [[Navier–Stokes equations]]
| |
| * [[Euler equations (fluid dynamics)]]
| |
| * [[Derivative (generalizations)]]
| |
| * [[Lie derivative]]
| |
| * [[Spatial acceleration]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| ==Further reading==
| |
| * {{cite book
| |
| |first1=Ira M.|last1=Cohen|first2=Pijush K|last2=Kundu
| |
| |title=Fluid Mechanics|isbn=978-0-12-373735-9|publisher=[[Academic Press]]|edition=4}}
| |
| * {{cite book|first1=Michael|last1=Lai|first2=Erhard|last2=Krempl|first3=David|last3=Ruben
| |
| |title=Introduction to Continuum Mechanics|isbn=978-0-7506-8560-3|publisher=Elsevier|edition=4}}
| |
| | |
| [[Category:Fluid dynamics]]
| |
| [[Category:Multivariable calculus]]
| |
Historical past of the of the author is Gabrielle Lattimer. For years she's been working as a library assistant. For a while she's yet been in Massachusetts. As a woman what your woman really likes is mah jongg but she have not made a dime utilizing it. She could be described as running and maintaining a meaningful blog here: http://prometeu.net
Also visit my homepage :: clash of clans cheats