Material derivative method

In the study of motion in continuum mechanics one deals with the time rates of changes of quantities that vary from one particle to the other. Such quantities include displacement, velocity and acceleration. These quantities may be expressed as functions described in the material form or the spatial form, and the meaning of the time rate of their change depends on the nature of the description.

Material time derivative.

Consider a real-valued function $f = f ( \mathbf{x} ^ { 0 } , t )$ that represents a scalar or a component of a vector or tensor. The point $\mathbf{x} ^ { 0 }$ determines a continuum particle uniquely, namely the one located at $\mathbf{x} ^ { 0 }$. With this notation, the function $f = f ( \mathbf{x} ^ { 0 } , t )$ can be interpreted as the value of $f$ experienced at time $t$ by the particle $\mathbf{x} ^ { 0 }$. The time derivative of $f$ with respect to time $t$, with $\mathbf{x} ^ { 0 }$ held fixed, is interpreted as the time rate of change of $f$ at the particle $\mathbf{x} ^ { 0 }$. This derivative is usually called the particle or material time derivative of $f$, denoted by $D f / D t$ and defined by

\begin{equation} \tag{a1} \frac { D f } { D t } = \left( \frac { \partial f ( \mathbf{x} ^ { 0 } , t ) } { \partial t } \right) | _ { \mathbf{x}^0 }, \end{equation}

where the subscript $\mathbf{x} ^ { 0 }$ accompanying the vertical line indicates that $\mathbf{x} ^ { 0 }$ is kept constant in the differentiation of $f$. Note that, like $f$, $D f / D t$ is a function of $\mathbf{x} ^ { 0 }$ and $t$ by definition. In other words, $D f / D t$ defined above is a function in the material form.

Local time derivative.

In order to define the local time derivative, one considers a real-valued function $\phi = \phi ( x , t )$ that represents a scalar or a component of a vector or tensor. Since $\mathbf{x}$ is point in the current configuration of a continuum, $\phi ( x , t )$ can be interpreted as the value of $\phi$ at the point $\mathbf{x}$ at time $t$. The partial derivative of $\phi$ with respect to time $t$, with $\mathbf{x}$ held fixed, is interpreted as the time rate of change of $\phi$ at the particle located at $\mathbf{x}$. This derivative is called the local time derivative of $\phi$, denoted by the usual partial derivative symbol $\partial \phi / \partial t$ and defined by

\begin{equation} \tag{a2} \frac { \partial \phi } { \partial t } = \left( \frac { \partial \phi ( \mathbf x , t ) } { \partial t } \right) | _ { \mathbf x }. \end{equation}

It is noted that, like $\phi$, $\partial \phi / \partial t$ is a function of $\mathbf{x}$ and $t$, and is a function in the spatial form.

The distinction between the material time derivative and the local time derivative should be emphasized. While both are partial derivatives with respect to $t$, the former is defined for a function of $\mathbf{x} ^ { 0 }$ and $t$ whereas the latter is defined for a function of $\mathbf{x}$ and $t$. Physically, the local time derivative of a function represents the rate at which the function changes with time as seen by an observer currently (momentarily) stationed at a point, whereas the material time derivative represents the rate at which the function changes with time as seen by an observer stationed at a particle and moving with it. The material time derivative is therefore also called the mobile time derivative or the derivative following a particle. For brevity, the material time derivative will be referred to as the material derivative or material rate, and the local time derivative as the local derivative or local rate.

Velocity and acceleration.

Since $\mathbf{x}$ is a function of $\mathbf{x} ^ { 0 }$ and $t$ in the material description of motion, the material derivative $f$ is denoted by $\mathbf{v}$ and is defined by

\begin{equation} \tag{a3} \mathbf {v}= \frac { D \mathbf{x} } { D t } = \left( \frac { \partial \mathbf{x} } { \partial t } \right) | _ { \mathbf{x} ^ { 0 } }. \end{equation}

Evidently, $\mathbf{v}$ represents the time rate of change of position of the particle $\mathbf{x} ^ { 0 }$ at time $t$. This is called the velocity of the particle $\mathbf{x} ^ { 0 }$ at time $t$. If $v_i$ are the components of $\mathbf{v}$, then the velocity components of the particle $\mathbf{x} ^ { 0 }$ at time $t$ take the form

\begin{equation} \tag{a4} v _ { i } = - \frac { D x _ { i } } { D t } = \left( \frac { \partial x _ { i } } { \partial t } \right) | _ { x _ { k }^{ 0} }. \end{equation}

The displacement vector $\mathbf{u}$ of the particle $\mathbf{x} ^ { 0 }$ is defined as $\mathbf{u} = \mathbf{x} - \mathbf{x} ^ { 0 }$. Thus, $\mathbf{u}$ may be regarded as a function of $\mathbf{x} ^ { 0 }$ and $t$, or of $\mathbf{x}$ and $t$. Treating $\mathbf{u}$ as a function of $\mathbf{x} ^ { 0 }$ and $t$, it follows from the above that

\begin{equation} \tag{a5} \mathbf{v} = \frac { \partial } { \partial t } ( \mathbf{x} ^ { 0 } + \mathbf{u} ) | _ { \mathbf{x} ^ { 0 } } = \left( \frac { \partial \mathbf{u} } { \partial t } \right) | _ { \mathbf{x} ^ { 0 } } = \frac { D u } { D t }. \end{equation}

Thus, the velocity of a particle at time $t$ is precisely the rate of change of displacement of that particle at time $t$. The above definition of velocity $\mathbf{v}$ assumes the component form

\begin{equation} \tag{a6} v _ { i } = \frac { D u _ { i } } { D t }. \end{equation}

It may be pointed out that, in solid mechanics, the deformation and motion are generally described in terms of the displacement vector. In fluid mechanics, the motion is generally described in terms of the velocity vector. When a motion is described in terms of velocity, it is commonly referred to as a flow.

Since $\mathbf{v}$ is a function of $\mathbf{x} ^ { 0 }$ and $t$ by definition, the material derivative of $\mathbf{v}$, namely, $D \mathbf{v} / D t$, can be defined. This derivative is called the acceleration of the particle $\mathbf{x} ^ { 0 }$ at time $t$. One often writes $\mathbf{v}{c}$ for $D \mathbf{v} / D t$. Thus, the acceleration of a particle at time $t$ is the rate of change of velocity of that particle at time $t$. The components of the acceleration are denoted by $D v _ { i } / D t$ or $\dot { v }_i$.

It is to be emphasized that the velocity and acceleration are defined with reference to a particle and are basically functions of $\mathbf{x} ^ { 0 }$ and $t$. In the spatial description of motion, $\mathbf{x} ^ { 0 }$ is a function of $\mathbf{x}$ and $t$. Hence, like the displacement, velocity and acceleration can also be expressed as functions of $\mathbf{x}$ and $t$. When $\mathbf{v}$ is expressed as a function of $\mathbf{x}$ and $t$, $\mathbf v ( \mathbf x , t )$ is referred to as the instantaneous velocity at the point $\mathbf{x}$. This actually means that $\mathbf v ( \mathbf x , t )$ is the velocity at time $t$ of the particle currently located at the point $\mathbf{x}$. Similar terminology is used in respect of acceleration also.

Next, one can deduce a formula enabling one to compute the instantaneous acceleration from the instantaneous velocity.

Material derivative in spatial form.

Consider again the function $\phi = \phi ( x , t )$ for which the local derivative was defined by (a2). This function can be expressed as a function of $x _ { i } ^ { 0 }$ and $t$, as explicitly indicated in the following:

\begin{equation} \tag{a7} \phi = \phi ( x _ { i } , t ) = \phi ( x _ { i } ( x _ { k } ^ { 0 } , t ) , t ). \end{equation}

Consequently, the material derivative of $\phi$ can also be defined. By the chain rule of partial differentiation, we obtain from (a7)

\begin{equation} \tag{a8} \left( \frac { \partial \phi } { \partial t } \right) | _ { x _ { k }^0 } = \left( \frac { \partial \phi } { \partial t } \right) | _ { x _ { i } } + \left( \frac { \partial \phi } { \partial x _ { i } } \right) | _ { t } \left( \frac { \partial x _ { i } } { \partial t } \right) | _ { x _ { k }^ 0 }. \end{equation}

In view of (a1), (a2) and (a4), it follows that

\begin{equation} \tag{a9} \left( \frac { \partial \phi } { \partial t } \right) | _ { x _ { k } ^ { 0 } } = \frac { D \phi } { D t } , \left( \frac { \partial \phi } { \partial t } \right) | _ { x _ { i } } = \frac { \partial \phi } { \partial t } , \left( \frac { \partial x _ { i } } { \partial t } \right) | _ { x _ { k } ^ { 0 } } = v _ { i }. \end{equation}

Hence, denoting $( \partial \phi / \partial x _ { i } ) | _ { t }$ as just $\partial \phi / \partial x _ { i } = \phi _ { ,i }$, (a8) can be rewritten as

\begin{equation} \tag{a10} \frac { D \phi } { D t } = \frac { \partial \phi } { \partial t } + v _ { i } \phi _ { , i } = \frac { \partial \phi } { \partial t } + ( {\bf v} . \nabla ) \phi . \end{equation}

When $\mathbf{v}$ is known as a function of $\mathbf{x}$ and $t$, expression (a10) enables one to compute $D \phi / D t$ as a function of $\mathbf{x}$ and $t$. As such, (a10) serves as a formula for the material derivative in the spatial form. Note that the first term on the right-hand side of this formula, namely $\partial \phi / \partial t$, represents the local rate of change of $\phi$, and the second term, namely $v _ { i } \phi _ { , i } = ( {\bf v} . \nabla ) \phi$, is the contribution due to the motion. The second term is referred to as the convective rate of change of $\phi$.

It can be easily verified that the material derivative operator

\begin{equation} \tag{a11} \frac { D } { D t } = \frac { \partial } { \partial t } + v _ { i } ( . ) , _ { i } = \frac { \partial } { \partial t } + {\bf v} . \nabla \end{equation}

which operates on functions represented in spatial form, satisfies all the rules of partial differentiation.

The concept of the material derivative and formula (a11) are attributed to L. Euler (1770) and J. Lagrange (1783).

Acceleration in spatial form.

Taking $\phi = v _ { i }$ in (a10) gives the following expression for the acceleration:

\begin{equation} \tag{a12} \frac { D v _ { i } } { D t } = \frac { \partial v _ { i } } { \partial t } + v _ { k } v _ { i , k} \end{equation}

or, equivalently,

\begin{equation} \tag{a13} \frac { D \mathbf{v} } { D t } = \frac { \partial \mathbf{v} } { \partial t } + ( \mathbf{v} \cdot \nabla ) \mathbf v . \end{equation}

When $\mathbf{v}$ is known as a function of $\mathbf{x}$ and $t$, expression (a13) determines $D \mathbf{v} / D t$ directly in terms of $\mathbf{x}$ and $t$; this expression therefore serves as a formula for acceleration in the spatial form.

By using the standard vector identity, (a13) can be put in the following useful form:

\begin{equation} \tag{a14} \frac { D \mathbf{v} } { D t } = \frac { \partial \mathbf{v} } { \partial t } + \frac { 1 } { 2 } \nabla v ^ { 2 } + ( \operatorname { curl } \mathbf{v} ) \times \mathbf{v}. \end{equation}

From (a13) and (a14), one notes that the acceleration vector is made up of two parts, namely,

\begin{equation*} ( {\bf v} . \nabla ) {\bf v} = \frac { 1 } { 2 } \nabla v ^ { 2 } + ( \operatorname { curl } {\bf v} ) \times {\bf v}. \end{equation*}

Evidently, the second part is quadratically non-linear in nature. Thus, the acceleration depends quadratically on the velocity field, and a given motion cannot be viewed as a superposition of two independent motions in general.

References

[a1]	D.S. Chandrasekhariah, L. Debnath, "Continuum mechanics" , Acad. Press (1994)
[a2]	Y.C. Fung, "Foundations of solid mechanics" , Prentice-Hall (1965)