Material derivative

Template:Short description In continuum mechanics, the material derivative<ref name="BSLr2"/><ref name=Batchelor>Template:Cite book</ref> describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation.<ref name=Trenberth>Template:Cite book</ref>

For example, in fluid dynamics, the velocity field is the flow velocity, and the quantity of interest might be the temperature of the fluid. In this case, the material derivative then describes the temperature change of a certain fluid parcel with time, as it flows along its pathline (trajectory).

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Other names

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

• advective derivative<ref>Template:Cite book</ref>
• convective derivative<ref name=Ockendon>Template:Cite book</ref>
• derivative following the motion<ref name="BSLr2"/>
• hydrodynamic derivative<ref name="BSLr2"/>
• Lagrangian derivative<ref name=Mellor>Template:Cite book</ref>
• particle derivative<ref>Template:Cite book</ref>
• substantial derivative<ref name="BSLr2">Template:Cite book</ref>
• substantive derivative<ref name=Granger>Template:Cite book</ref>
• Stokes derivative<ref name=Granger/>
• total derivative,<ref name="BSLr2"/><ref name="LandauLifshitz">Template:Cite book</ref> although the material derivative is actually a special case of the total derivative<ref name="LandauLifshitz"/>

Definition

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, Template:Math: <math display="block">\frac{\mathrm{D} y}{\mathrm{D}t} \equiv \frac{\partial y}{\partial t} + \mathbf{u}\cdot\nabla y,</math> where Template:Math is the covariant derivative of the tensor, and Template:Math is the flow velocity. Generally the convective derivative of the field Template:Math, the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field Template:Math, or as involving the streamline directional derivative of the field Template:Math, leading to the same result.<ref>Template:Cite book</ref> 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 Template:Math, instead for only the spatial term Template:Math.<ref name=Batchelor/> The effect of the time-independent terms in the definitions are for the scalar and tensor case respectively known as advection and convection.

Scalar and vector fields

For example, for a macroscopic scalar field Template:Math and a macroscopic vector field Template:Math the definition becomes: <math display="block">\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}</math>

In the scalar case Template:Math is simply the gradient of a scalar, while Template:Math is the covariant derivative of the macroscopic vector (which can also be thought of as the Jacobian matrix of Template:Math as a function of Template:Math). In particular for a scalar field in a three-dimensional Cartesian coordinate system Template:Math, the components of the velocity Template:Math are Template:Math, and the convective term is then: <math display="block"> \mathbf{u}\cdot \nabla \varphi = u_1 \frac {\partial \varphi} {\partial x_1} + u_2 \frac {\partial \varphi} {\partial x_2} + u_3 \frac {\partial \varphi} {\partial x_3}.</math>

Development

Consider a scalar quantity Template:Math, where Template:Mvar is time and Template:Math is position. Here Template:Math may be some physical variable such as temperature or chemical concentration. The physical quantity, whose scalar quantity is Template:Math, exists in a continuum, and whose macroscopic velocity is represented by the vector field Template:Math.

The (total) derivative with respect to time of Template:Math is expanded using the multivariate chain rule: <math display="block">\frac{\mathrm{d}}{\mathrm{d} t}\varphi(\mathbf x(t), t) = \frac{\partial \varphi}{\partial t} + \dot \mathbf x \cdot \nabla \varphi.</math>

It is apparent that this derivative is dependent on the vector <math display="block">\dot \mathbf x \equiv \frac{\mathrm{d} \mathbf x}{\mathrm{d} t},</math> which describes a chosen path Template:Math in space. For example, if <math> \dot \mathbf x= \mathbf 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>\dot \mathbf x = 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. In which case the term <math> {\partial \varphi}/{\partial t}</math> is sufficient to describe the rate of change of temperature.

If the sun is not warming the water (i.e. <math> {\partial \varphi}/{\partial t} = 0</math>), but the path Template:Math is not a standstill, the time derivative of Template:Math 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 <math> \dot \mathbf x \cdot \nabla \varphi </math> 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 Template:Math is chosen to have a velocity equal to the fluid velocity <math display="block">\dot \mathbf x = \mathbf u.</math>

That is, the path follows the fluid current described by the fluid's velocity field Template:Math. So, the material derivative of the scalar Template:Math is <math display="block">\frac{\mathrm{D} \varphi}{\mathrm{D} t} = \frac{\partial \varphi}{\partial t} + \mathbf u \cdot \nabla \varphi.</math>

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.

Orthogonal coordinates

It may be shown that, in orthogonal coordinates, the Template:Math-th component of the convection term of the material derivative of a vector field <math>\mathbf{A}</math> is given by<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">[\left(\mathbf{u} \cdot \nabla \right)\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), </math>

where the Template:Math are related to the metric tensors by <math>h_i = \sqrt{g_{ii}}.</math>

In the special case of a three-dimensional Cartesian coordinate system (x, y, z), and Template:Math being a 1-tensor (a vector with three components), this is just: <math display="block">(\mathbf{u}\cdot\nabla) \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} </math>

where <math>\frac{\partial(A_x, A_y, A_z)}{\partial(x, y, z)}</math> is a Jacobian matrix.

There is also a vector-dot-del identity and the material derivative for a vector field <math>\mathbf A</math> can be expressed as:

<math> {\displaystyle (\mathbf {A} \cdot \nabla )\mathbf {A} = {\frac {1}{2}}\nabla |\mathbf {A} |^{2}-\mathbf {A} \times (\nabla \times \mathbf {A} )={\frac {1}{2}}\nabla |\mathbf {A} |^{2}+(\nabla \times \mathbf {A} )\times \mathbf {A} }.</math>

See also

• 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

Further reading

• Template:Cite book
• Template:Cite book