Material derivative

In continuum mechanics, the material derivative^[1]^[2] 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 macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation.^[3]

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 pathline (trajectory) while following the fluid flow.

Names

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

advective derivative^[4]
convective derivative^[5]
derivative following the motion^[1]
hydrodynamic derivative^[1]
Lagrangian derivative^[6]
particle derivative^[7]
substantial derivative^[1]
substantive derivative^[8]
Stokes derivative^[8]
total derivative^[1]^[9]

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, $y=y(\mathbf {x} ,t)$ and is given by

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

where $\nabla y$ is the covariant derivative of the tensor, and $\mathbf {u} (\mathbf {x} ,t)$ is the flow velocity. Generally the convective derivative of the field $\mathbf {u} \cdot \nabla y$ , the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field $\mathbf {u} \cdot (\nabla y)$ , or as involving the streamline directional derivative of the field $(\mathbf {u} \cdot \nabla )y$ , leading to the same result.^[10] 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 by the presence of any flow. Confusingly, sometimes the name "convective derivative" is used for the whole material derivative D/Dt, instead for only the spatial term, u•∇.,^[2] which is also a redundant nomenclature. In the nonredundant nomenclature the material derivative only equals the convective derivative for absent flows. The effect of the time independent terms in the definitions are for the scalar and tensor case respectively known as advection and convection.

Scalar & vector fields

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

{\frac {\mathrm {D} \varphi }{\mathrm {D} t}}\equiv {\frac {\partial \varphi }{\partial t}}+\mathbf {u} \cdot \nabla \varphi ,

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

In the scalar case ∇φ is simply the gradient of a scalar, while ∇A is the covariant derivative of the macroscopic vector. In particular for a scalar field in a three-dimensional Cartesian coordinate system(x₁,x₂,x₃), the convective term is:

\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}}}.

Development

Consider a scalar quantity $\phi$ 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 continuum, whose macroscopic velocity is represented by the vector field u( x, t ).

The (total) derivative with respect to time of φ is expanded through the multivariate chain rule:

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

It is apparent that this derivative is dependent on the vector:

{\dot {\mathbf {x} }}\equiv {\frac {\mathrm {d} \mathbf {x} }{\mathrm {d} t}}

which describes a chosen path x(t) in space. For example, if ${\dot {\mathbf {x} }}=\mathbf {0}$ 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 ${\dot {\mathbf {x} }}=0$ , 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.

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 frame of reference path x(t) is solidal with the local stream in the continuum (lagrangian reference system) so the reference velocity is equal to the macroscopic velocity in the continuum:

{\dot {\mathbf {x} }}=\mathbf {u} .

So, the material derivative of the scalar φ is:

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

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 j-th component of convection is given by:^[11]

[\mathbf {u} \cdot \nabla \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 h_i's are related to the metric tensors by

h_{i}={\sqrt {g_{ii}}}.

In the special case of a three-dimensional Cartesian coordinate system (x,y,z) this is just

\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}}\\[2ex]\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}}\\[2ex]\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}}.

References

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↑ Landau, L.D.; Lifshitz, E.M. (1987). Fluid Mechanics. Course of Theoretical Physics. 6 (2nd ed.). Butterworth-Heinemann. pp. 3–4 & 227. ISBN 0-7506-2767-0.
↑ Emanuel, G. (2001). Analytical fluid dynamics (second ed.). CRC Press. pp. 6–7. ISBN 0-8493-9114-8.
↑ Eric W. Weisstein. "Convective Operator". MathWorld. Retrieved 2008-07-22.