material derivative

There are many other names for the material derivative, including:

• advective derivative{{cite book |title=Introduction to PDEs and Waves for the Atmosphere and Ocean | last=Majda |first=A. |author-link=Andrew Majda |isbn=0-8218-2954-8 |series=Courant Lecture Notes in Mathematics |volume=9 |year=2003 |publisher=American Mathematical Society |page=1 }}
• convective derivative{{cite book| first1=H. |last1=Ockendon |author1-link= Hilary Ockendon |last2=Ockendon |first2=J.R.|author2-link= John Ockendon | title=Waves and Compressible Flow | publisher=Springer | year=2004 | isbn=0-387-40399-X | page=6 }}
• derivative following the motion
• hydrodynamic derivative
• Lagrangian derivative{{cite book | first=G.L. | last=Mellor | title=Introduction to Physical Oceanography | publisher=Springer | year=1996 | isbn=1-56396-210-1 |page=19 }}
• particle derivative{{cite book |title=Water Waves: The Mathematical Theory with Applications |last=Stoker |first=J.J. |author-link=James J. Stoker |isbn=0-471-57034-6 |page=5 |year=1992 |publisher=Wiley }}
• substantial derivative{{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 |edition=Revised Second |publisher=John Wiley & Sons |year=2007 |isbn=978-0-470-11539-8 |page=83}}
• substantive derivative{{cite book| first=R.A. |last=Granger| title=Fluid Mechanics | publisher=Courier Dover Publications | year=1995 | isbn=0-486-68356-7 | page=30}}
• Stokes derivative
• total derivative,{{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 }} although the material derivative is actually a special case of the total derivative

The material derivative is defined for any tensor field y that is macroscopic, with the sense that it depends only on position and time coordinates, {{math|1=y = y(x, t)}}:

$$\backslash frac\{\backslash mathrm\{D\}\; y\}\{\backslash mathrm\{D\}t\}\; \backslash equiv\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; t\}\; +\; \backslash mathbf\{u\}\backslash cdot\backslash nabla\; y,$$

where {{math|∇y}} is the covariant derivative of the tensor, and {{math|u(x, t)}} is the flow velocity. Generally the convective derivative of the field {{math|u·∇y}}, the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field {{math|u·(∇y)}}, or as involving the streamline directional derivative of the field {{math|(u·∇) y}}, leading to the same result.{{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 }}

Only this spatial term containing the flow velocity describes the transport of the field in the flow, while the other describes the intrinsic variation of the field, independent of the presence of any flow. Confusingly, sometimes the name "convective derivative" is used for the whole material derivative {{math|D/Dt}}, instead for only the spatial term {{math|u·∇}}. The effect of the time-independent terms in the definitions are for the scalar and tensor case respectively known as advection and convection.

For example, for a macroscopic scalar field {{math|φ(x, t)}} and a macroscopic vector field {{math|A(x, t)}} the definition becomes:

$$\backslash begin\{align\}$$

\frac{\mathrm{D}\varphi}{\mathrm{D}t} &\equiv \frac{\partial \varphi}{\partial t} + \mathbf{u}\cdot\nabla \varphi, \\[3pt]

\frac{\mathrm{D}\mathbf{A}}{\mathrm{D}t} &\equiv \frac{\partial \mathbf{A}}{\partial t} + \mathbf{u}\cdot\nabla \mathbf{A}.

\end{align}

In the scalar case {{math|∇φ}} is simply the gradient of a scalar, while {{math|∇A}} is the covariant derivative of the macroscopic vector (which can also be thought of as the Jacobian matrix of {{math|A}} as a function of {{math|x}}).

In particular for a scalar field in a three-dimensional Cartesian coordinate system {{math|(x₁, x₂, x₃)}}, the components of the velocity {{math|u}} are {{math|u₁, u₂, u₃}}, and the convective term is then:

$$\backslash mathbf\{u\}\backslash cdot\; \backslash nabla\; \backslash varphi\; =\; u\_1\; \backslash frac\; \{\backslash partial\; \backslash varphi\}\; \{\backslash partial\; x\_1\}\; +\; u\_2\; \backslash frac\; \{\backslash partial\; \backslash varphi\}\; \{\backslash partial\; x\_2\}\; +\; u\_3\; \backslash frac\; \{\backslash partial\; \backslash varphi\}\; \{\backslash partial\; x\_3\}.$$

Consider a scalar quantity {{math|1=φ = φ(x, t)}}, where {{mvar|t}} is time and {{math|x}} is position. Here {{math|φ}} may be some physical variable such as temperature or chemical concentration. The physical quantity, whose scalar quantity is {{math|φ}}, exists in a continuum, and whose macroscopic velocity is represented by the vector field {{math|u(x, t)}}.

The (total) derivative with respect to time of {{math|φ}} is expanded using the multivariate chain rule:

$$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}\; t\}\backslash varphi(\backslash mathbf\; x(t),\; t)\; =\; \backslash frac\{\backslash partial\; \backslash varphi\}\{\backslash partial\; t\}\; +\; \backslash dot\; \backslash mathbf\; x\; \backslash cdot\; \backslash nabla\; \backslash varphi.$$

It is apparent that this derivative is dependent on the vector

$$\backslash dot\; \backslash mathbf\; x\; \backslash equiv\; \backslash frac\{\backslash mathrm\{d\}\; \backslash mathbf\; x\}\{\backslash mathrm\{d\}\; t\},$$

which describes a chosen path {{math|x(t)}} in space. For example, if $\backslash dot\; \backslash mathbf\; x=\; \backslash mathbf\; 0$ 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 $\backslash dot\; \backslash mathbf\; x\; =\; 0$, 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. In which case the term $\{\backslash partial\; \backslash varphi\}/\{\backslash partial\; t\}$ is sufficient to describe the rate of change of temperature.

If the sun is not warming the water (i.e. $\{\backslash partial\; \backslash varphi\}/\{\backslash partial\; t\}\; =\; 0$), but the path {{math|x(t)}} is not a standstill, the time derivative of {{math|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 at a constant high temperature and the other end at a constant low 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 and the second term on the right $\backslash dot\; \backslash mathbf\; x\; \backslash cdot\; \backslash nabla\; \backslash varphi$ is sufficient to describe the rate of change of temperature. A temperature sensor attached to the swimmer would show temperature varying with time, simply due to the temperature variation from one end of the pool to the other.

The material derivative finally is obtained when the path {{math|x(t)}} is chosen to have a velocity equal to the fluid velocity

$$\backslash dot\; \backslash mathbf\; x\; =\; \backslash mathbf\; u.$$

That is, the path follows the fluid current described by the fluid's velocity field {{math|u}}. So, the material derivative of the scalar {{math|φ}} is

$$\backslash frac\{\backslash mathrm\{D\}\; \backslash varphi\}\{\backslash mathrm\{D\}\; t\}\; =\; \backslash frac\{\backslash partial\; \backslash varphi\}\{\backslash partial\; t\}\; +\; \backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash varphi.$$

An example of this case is a lightweight, neutrally buoyant particle swept along a flowing river and experiencing temperature changes as it does so. The temperature of the water locally may be increasing due to one portion of the river being sunny and the other in a shadow, or 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 a fluid current; however, no laws of physics were invoked (for example, it was assumed that a lightweight particle in a river will follow the velocity of the water), but it turns out that many physical concepts can be described concisely using the material derivative. The general case of advection, however, relies on conservation of mass of 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.

It may be shown that, in orthogonal coordinates, the {{math|j}}-th component of the convection term of the material derivative of a vector field $\backslash mathbf\{A\}$ is given by{{cite web

| url = http://mathworld.wolfram.com/ConvectiveOperator.html

| title = Convective Operator

| author = Eric W. Weisstein

| author-link = Eric W. Weisstein

| publisher = MathWorld

| access-date = 2008-07-22

}}

$$[\backslash left(\backslash mathbf\{u\}\; \backslash cdot\; \backslash nabla\; \backslash right)\backslash mathbf\{A\}]\_j\; =$$

\sum_i \frac{u_i}{h_i} \frac{\partial A_j}{\partial q^i} + \frac{A_i}{h_i h_j}\left(u_j \frac{\partial h_j}{\partial q^i} - u_i \frac{\partial h_i}{\partial q^j}\right),

where the {{math|h_i}} are related to the metric tensors by $h\_i\; =\; \backslash sqrt\{g\_\{ii\}\}.$

In the special case of a three-dimensional Cartesian coordinate system (x, y, z), and {{math|A}} being a 1-tensor (a vector with three components), this is just:

$$(\backslash mathbf\{u\}\backslash cdot\backslash nabla)\; \backslash mathbf\{A\}\; =$$

\begin{pmatrix}

\displaystyle

u_x \frac{\partial A_x}{\partial x} + u_y \frac{\partial A_x}{\partial y}+u_z \frac{\partial A_x}{\partial z}

\\

\displaystyle

u_x \frac{\partial A_y}{\partial x} + u_y \frac{\partial A_y}{\partial y}+u_z \frac{\partial A_y}{\partial z}

\\

\displaystyle

u_x \frac{\partial A_z}{\partial x} + u_y \frac{\partial A_z}{\partial y}+u_z \frac{\partial A_z}{\partial z}

\end{pmatrix} =

\frac{\partial (A_x, A_y, A_z)}{\partial (x, y, z)}\mathbf{u}

where $\backslash frac\{\backslash partial(A\_x,\; A\_y,\; A\_z)\}\{\backslash partial(x,\; y,\; z)\}$ is a Jacobian matrix.

There is also a vector-dot-del identity and the material derivative for a vector field $\backslash mathbf\; A$ can be expressed as:

:$\{\backslash displaystyle\; (\backslash mathbf\; \{A\}\; \backslash cdot\; \backslash nabla\; )\backslash mathbf\; \{A\}\; =\; \{\backslash frac\; \{1\}\{2\}\}\backslash nabla\; |\backslash mathbf\; \{A\}\; |^\{2\}-\backslash mathbf\; \{A\}\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf\; \{A\}\; )=\{\backslash frac\; \{1\}\{2\}\}\backslash nabla\; |\backslash mathbf\; \{A\}\; |^\{2\}+(\backslash nabla\; \backslash times\; \backslash mathbf\; \{A\}\; )\backslash times\; \backslash mathbf\; \{A\}\; \}.$

{{Div col|colwidth=25em}}

• Navier–Stokes equations
• Euler equations (fluid dynamics)
• Derivative (generalizations)
• Lagrangian and Eulerian specification of the flow field
• Lie derivative
• Levi-Civita connection
• Spatial acceleration
• Spatial gradient

{{Div col end}}

{{Reflist}}

• {{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=4th |year=2008 }}

• {{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=4th |year=2010 }}

Category:Fluid dynamics

Category:Multivariable calculus

Category:Rates

Category:Generalizations of the derivative