Reynolds Transport Theorem Derivation

Reynolds Transport Theorem can be derived using some mathematical relations. The derivation yields an equation governing the balance of a given quantity in a given control volume.

Nomenclature

Intensive vs. Extensive Properties

For any given amount of fluid, the fluid will have some properties associated with it

Extensive properties $B$ - change with the mass of the fluid (e.g. concentration of species $i$)

Intensive properties $b={\frac {\partial B}{\partial t}}$ - are independent of the mass of the fluid (e.g. mass fraction of species $i$)

Material Volume vs. Control Volume

A material volume (denoted $V_{B}^{M}({\boldsymbol {\xi }},t)$, or more simply $V_{B}^{M}$) is a volume of fluid whose boundaries are defined such that there is zero flux of a specified extensive property $B$ (the typical case is $B=m$, mass, so that Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle b=1} and there is zero flux of mass through the material volume's boundaries). The material volume is a function of time $t$, since the boundaries of the volume will change, with the fluid, in time. It is also a function of initial position ${\boldsymbol {\xi }}$.

A control volume (denoted $V({\boldsymbol {x}})$) is a fixed and constant volume, whose location is fixed. It is not a function of time, only of location ${\boldsymbol {x}}$.

(See illustrations below)

Fluid particle

A fluid particle is a particle, in the macroscopic (as opposed to the molecular) sense, that consists of a mass of fluid

This mass of fluid has some properties associated with it, an extensive property $B$ and an intensive property $b$

The volume of the fluid is a material volume $V_{B}^{M}$

The boundaries of a material volume are such that there is zero flux of the property of interest $B$

That is, for a given fluid particle, with an extensive property $B$, a material volume $V_{B}^{M}$ is a volume whose boundaries move such that there is no flux of $B$ across the boundaries

Derivative Frames of Reference

Frames of References

Illustration of the changes undergone by a single fluid particle $\xi$ with extensive property $B$; the volume depicted is a material volume $V_{B}^{M}$ such that the flux of $B$ across the surface of the material volume is zero. The rate of change of a fluid property $B$ within the material volume is equal to the substantial (material) derivative ${\frac {dB}{dt}}$.

To begin, the initial position of a fluid particle (not a particle in the molecular sense, but a particle in a macroscopic, continuum sense) may be written as a random variable ${\boldsymbol {\xi }}$.

As specified above, a fluid particle volume containing some extensive property $B$ is contained within a material volume Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{B}^{M}(t)} such that the flux of $B$ across the boundaries of the material volume are zero. The material volume is a function of time.

The location of this particular fluid particle at a later time is determined, first, by its initial location ${\boldsymbol {\xi }}$, and second, by the time that has passed, $t$.

For a volume with a fixed position ${\boldsymbol {x}}$, referred to as a control volume, multiple fluid particles, each with a different initial position ${\boldsymbol {\xi }}_{n}$, pass through ${\boldsymbol {x}}$ at different times $t_{n}$. The rate of change, keeping ${\boldsymbol {x}}$ constant, of a fluid property $B$ is the partial derivative ${\frac {\partial B}{\partial t}}$.

This later position can be written as ${\boldsymbol {x}}({\boldsymbol {\xi }},t)$

We will assume that, given a location ${\boldsymbol {x}}$, we can backtrack and find the fluid particle's initial location ${\boldsymbol {\xi }}$

This means the function ${\boldsymbol {x}}({\boldsymbol {\xi }},t)$ is assumed to be invertible.

In other words, we assume we can find a function ${\boldsymbol {\xi }}={\boldsymbol {\xi }}({\boldsymbol {x}},t)$ that is continuous and single-valued (i.e. invertible).

The necessary and sufficient condition for invertability is for a non-vanishing Jacobian to exist:

$J={\frac {\partial {\boldsymbol {x}}}{\partial {\boldsymbol {\xi }}}}$

Now, let's consider some extensive fluid property $B$. This property field evolves with the state of the fluid, and can be written one of two ways, each with a unique interpretation:

$B({\boldsymbol {x}},t)=B({\boldsymbol {\xi }}({\boldsymbol {x}},t),t)$

This way of writing $B$ can be interpreted as follows: The value of the property $B$ at the spatial/temporal location $({\boldsymbol {x}},t)$ is the value appropriate to the fluid particle located at $({\boldsymbol {x}},t)$

Alternatively,

$B({\boldsymbol {\xi }},t)=B({\boldsymbol {x}}({\boldsymbol {\xi }},t),t)$

This way of writing $B$ can be interpreted as: The value seen by the particle ${\boldsymbol {\xi }}$ at time $t$ is the value of the property $B$ at the position the particle occupies at that time $t$

In keeping with these interpretations, two different temporal derivatives can be written.

Derivatives

Partial Derivative

The partial derivative is denoted by

$\displaystyle {\frac {\partial B}{\partial t}}$

The partial derivative is the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} with respect to time, keeping Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{x}} constant

This derivative corresponds to the rate of change of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} in a control volume (which is a fixed point in space; see nomenclature above)

Material Derivative

The material derivative is denoted by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\frac{dB}{dt}}} (alternatively, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\frac{DB}{Dt}}} )

The material derivative derivative of B with respect to time, keeping Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\xi}} constant

This derivative corresponds to the rate of change of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} in a material volume (which is a volume whose boundaries are moving with time such that the flux of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} across the boundaries is zero)

Keep in mind that the material derivatives are not partial derivatives because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\xi}} (the initial particle position) is a constant for a given fluid particle

Position and Velocity

For a material volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V^{M}_{B}} , the material volume moves at some rate.

Let the position of the material volume be denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{x}_{B}} .

Then the velocity of the material volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V^{M}_{B}} is denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{v}_{B}}

The material derivative of the position of the fluid particle is the velocity:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_{i,B} = \frac{d x_{i,B}}{dt} = \left( \frac{ \partial x_{i,B}(\boldsymbol{\xi},t)}{\partial t}\right)_{\boldsymbol{\xi}} }

In other words, holding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\xi}} constant (that is, considering a material volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V^{M}_{B}} with the initial position Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\xi}} ), the rate of change of the fluid particle's current position Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_{i,B}} is the velocity of the material volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V^{M}_{B}} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{v}_{B} = \frac{ d \boldsymbol{x}_{B} }{ dt } }

Derivative Relationships

The two derivatives, partial and material, can be related.

First, to review what they mean:

Partial derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \left( \frac{ \partial B }{ \partial t } \right)_{\boldsymbol{x}}} - rate of change of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} for a fixed control volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V(\boldsymbol{x})}

Material derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \left( \frac{ dB }{ dt } \right)_{\boldsymbol{\xi}}} - the rate of change of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} for a material volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V^{M}_{B}(\boldsymbol{\xi},t)}

Next, the derivatives of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} with respect to time can be equated at a particular spatial and temporal location, and the chain rule used, to get the relationship between these two derivatives:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{ \frac{dB}{dt} } &=& \displaystyle{ \frac{\partial}{\partial t} \left( B(\boldsymbol{\xi},t) \right) } \\ &=& \displaystyle { \frac{\partial}{\partial t} \left( B( \boldsymbol{x}( \boldsymbol{\xi},t ), t ) \right) } \\ &=& \displaystyle{ \frac{ \partial B }{\partial x_i} \left( \frac{d x_i}{dt} \right)_{\boldsymbol{\xi}} + \left( \frac{ \partial B }{\partial t} \right)_{\boldsymbol{x}} } \\ &=& \displaystyle{ \frac{ \partial B }{ \partial t } + v_i \frac{ \partial B }{ \partial x_i } } \end{array} }

This can be written conveniently as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ dB }{ dt } = \frac{ \partial B }{ \partial t } + ( \boldsymbol{v}_{B} \cdot \nabla ) B }

The operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}} can be expanded as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt} = \frac{\partial}{\partial t} + (\boldsymbol{v} \cdot \nabla) }

Dilation and the Euler Expansion Formula

Changing coordinates from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\xi}} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{x}} is likely to cause a change in the volume of the fluid particle:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle dV = \frac{ \partial (x_1, x_2, x_3) }{ \partial (\xi_1, \xi_2, \xi_3) } d \xi_1 d \xi_2 d \xi_3 = J dV_0 }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J} is the Jacobian.

The Jacobian is equal to:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J = \frac{ dV }{ dV_0 } }

This quantity is called the dilation.

The Jacobian is a matrix that looks like this:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J = \displaystyle{ \left| \begin{array}{ccc} \frac{ \partial x_1 }{ \partial \xi_1 } & \frac{ \partial x_1 }{ \partial \xi_2 } & \frac{ \partial x_1 }{ \partial \xi_3 } \\ \frac{ \partial x_2 }{ \partial \xi_1 } & \frac{ \partial x_2 }{ \partial \xi_2 } & \frac{ \partial x_2 }{ \partial \xi_3 } \\ \frac{ \partial x_3 }{ \partial \xi_1 } & \frac{ \partial x_3 }{ \partial \xi_2 } & \frac{ \partial x_3 }{ \partial \xi_3 } \end{array} \right| } }

which can be written compactly as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J = \left| \frac{\partial x_i}{\partial \xi_j} \right| }

or as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J = \epsilon_{ijk} \frac{\partial x_1}{\partial \xi_i} \frac{\partial x_2}{\partial \xi_j} \frac{\partial x_3}{\partial \xi_k} }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon_{ijk}} is the Levi-Civita symbol (wikipedia:Levi-Civita symbol).

The initial volume of the fluid particle is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d \xi_1 d \xi_2 d \xi_3 = dV_0} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t=0}

Motion is continuous, so the volume cannot break up.

Another way to state that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0 < J < \infty} (required so that neither Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J} nor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J^{-1}} vanish, and the mapping from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \xi} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x} and vice-versa are continuous and smooth).

The natural question to ask is how the volume dilation changes with time - mathematically, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ dJ }{ dt }}

Aris Approach

First, the time derivative of the terms in the Jacobian matrix can be simplified:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt} \left( \frac{\partial x_i}{\partial \xi_j} \right) = \frac{\partial}{\partial \xi_j} \frac{d x_i}{d t} = \frac{ \partial v_i }{ \partial \xi_j } }

where the second step is possible because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}} holds Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\xi}} constant.

It was specified above that the velocity is a function of location, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{v} = \boldsymbol{v}( x_1, x_2, x_3 )} . This can be plugged into the relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ \partial v_i }{ \partial \xi_j }} , and the chain rule used, to yield:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ \partial v_i }{ \partial \xi_j } = \frac{ \partial v_i }{ \partial x_1 } \frac{ \partial x_1 }{ \partial \xi_j } + \frac{ \partial v_i }{ \partial x_2 } \frac{ \partial x_2 }{ \partial \xi_j } + \frac{ \partial v_i }{ \partial x_3 } \frac{ \partial x_3 }{ \partial \xi_j } }

which can be generalized to the result:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ d v_i }{ dt } = \frac{ d v_i }{ d x_k } \frac{ d x_k }{ dt } }

This expression gives a way to write the time derivative of the terms in the Jacobian matrix.

The derivative of the determinant of the Jacobian is the sum of three terms; each term is the Jacobian matrix, with only one row differentiated. Thus,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dJ}{dt} = \displaystyle{ \left| \begin{array}{ccc} \frac{d}{dt} \frac{ \partial x_1 }{ \partial \xi_1 } & \frac{d}{dt} \frac{ \partial x_1 }{ \partial \xi_2 } & \frac{d}{dt} \frac{ \partial x_1 }{ \partial \xi_3 } \\ \frac{ \partial x_2 }{ \partial \xi_1 } & \frac{ \partial x_2 }{ \partial \xi_2 } & \frac{ \partial x_2 }{ \partial \xi_3 } \\ \frac{ \partial x_3 }{ \partial \xi_1 } & \frac{ \partial x_3 }{ \partial \xi_2 } & \frac{ \partial x_3 }{ \partial \xi_3 } \end{array} \right| } + \displaystyle{ \left| \begin{array}{ccc} \frac{ \partial x_1 }{ \partial \xi_1 } & \frac{ \partial x_1 }{ \partial \xi_2 } & \frac{ \partial x_1 }{ \partial \xi_3 } \\ \frac{d}{dt} \frac{ \partial x_2 }{ \partial \xi_1 } & \frac{d}{dt} \frac{ \partial x_2 }{ \partial \xi_2 } & \frac{d}{dt} \frac{ \partial x_2 }{ \partial \xi_3 } \\ \frac{ \partial x_3 }{ \partial \xi_1 } & \frac{ \partial x_3 }{ \partial \xi_2 } & \frac{ \partial x_3 }{ \partial \xi_3 } \end{array} \right| } + \displaystyle{ \left| \begin{array}{ccc} \frac{ \partial x_1 }{ \partial \xi_1 } & \frac{ \partial x_1 }{ \partial \xi_2 } & \frac{ \partial x_1 }{ \partial \xi_3 } \\ \frac{ \partial x_2 }{ \partial \xi_1 } & \frac{ \partial x_2 }{ \partial \xi_2 } & \frac{ \partial x_2 }{ \partial \xi_3 } \\ \frac{d}{dt} \frac{ \partial x_3 }{ \partial \xi_1 } & \frac{d}{dt} \frac{ \partial x_3 }{ \partial \xi_2 } & \frac{d}{dt} \frac{ \partial x_3 }{ \partial \xi_3 } \end{array} \right| } }

Using the identity derived above, it can be shown that in the derivative of the first row, only the terms with k=1 survive, since the coefficients of the other terms have a coefficient that is a determinant of a matrix with two rows the same.

For this reason, the determinant of the first matrix (in the expression for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dJ}{dt}} above) has a value of

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ \partial v_1 }{ \partial x_1 } J }

and the others have similar values. This makes the final value of the time-derivative of the Jacobian:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dJ}{dt} = \left( \frac{d v_1}{d x_1} + \frac{ d v_2 }{d x_2} + \frac{ d v_3 }{ d x_3 } \right) J }

which can also be written,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \displaystyle{ \frac{1}{J} \frac{dJ}{dt} = div (\boldsymbol{v}) } }

or,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \displaystyle{ \frac{ d \ln{J} }{ dt } = div (\boldsymbol{v}) } }

This is also called the Euler Expansion Formula.<ref name="Aris">Aris, Rutherford (1962). "4". Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Prentice-Hall. </ref>

NOTE: For incompressible fluids, the fluid particle's volume will not change due to compression or dilution, so the Jacobian is zero (that is, the fluid particle volume is always equal to the initial fluid particle volume)

This means that, for a fluid particle,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle div (\boldsymbol{v}) = 0 }

Milhas Approach

From the compact form of the Jacobian, the time derivative can be written:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dJ}{dt} = \frac{d}{dt} \left( \epsilon_{ijk} \frac{\partial x_1}{\partial \xi_i} \frac{\partial x_2}{\partial \xi_j} \frac{\partial x_3}{\partial \xi_k} \right) }

This can be expanded as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dJ}{dt} = \epsilon_{ijk} \frac{\partial v_1}{\partial \xi_i} \frac{\partial x_2}{\partial \xi_j} \frac{\partial x_3}{\partial \xi_k} + \epsilon_{ijk} \frac{\partial x_1}{\partial \xi_i} \frac{\partial v_2}{\partial \xi_j} \frac{\partial x_3}{\partial \xi_k} + \epsilon_{ijk} \frac{\partial x_1}{\partial \xi_i} \frac{\partial x_2}{\partial \xi_j} \frac{\partial v_3}{\partial \xi_k} }

and the term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial v_i}{\partial \xi_j}} can be expanded as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial v_i}{\partial \xi_j} = \frac{\partial v_i}{\partial x_m} \frac{\partial x_m}{\partial \xi_j} }

which makes the time derivative of the Jacobian:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dJ}{dt} = \epsilon_{ijk} \frac{\partial v_1}{\partial x_m} \frac{\partial x_m}{\partial \xi_i} \frac{\partial x_2}{\partial \xi_j} \frac{\partial x_3}{\partial \xi_k} + \epsilon_{ijk} \frac{\partial x_1}{\partial \xi_i} \frac{\partial v_2}{\partial x_m} \frac{\partial x_m}{\partial \xi_j} \frac{\partial x_3}{\partial \xi_k} + \epsilon_{ijk} \frac{\partial x_1}{\partial \xi_i} \frac{\partial x_2}{\partial \xi_j} \frac{\partial v_3}{\partial x_m} \frac{\partial x_m}{\partial \xi_k} }

The first term, expanded in all of its glory, is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon_{ijk} \frac{\partial v_1}{\partial x_m} \frac{\partial x_m}{\partial \xi_i} \frac{\partial x_2}{\partial \xi_j} \frac{\partial x_3}{\partial \xi_k} = \frac{\partial v_1}{\partial x_1} \left( \epsilon_{ijk} \frac{\partial x_1}{\partial \xi_i} \frac{\partial x_2}{\partial \xi_j} \frac{\partial x_3}{\partial \xi_k} \right) + \frac{\partial v_1}{\partial x_2} \left( \epsilon_{ijk} \frac{\partial x_2}{\partial \xi_i} \frac{\partial x_2}{\partial \xi_j} \frac{\partial x_3}{\partial \xi_k} \right) + \frac{\partial v_1}{\partial x_3} \left( \epsilon_{ijk} \frac{\partial x_3}{\partial \xi_i} \frac{\partial x_2}{\partial \xi_j} \frac{\partial x_3}{\partial \xi_k} \right) }

The second and third terms are zero, due to the definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon_{ijk}} . This makes the term substantially simpler:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon_{ijk} \frac{\partial v_1}{\partial x_m} \frac{\partial x_m}{\partial \xi_i} \frac{\partial x_2}{\partial \xi_j} \frac{\partial x_3}{\partial \xi_k} = \frac{\partial v_1}{\partial x_1} J }

Next, the second and third terms in the expression for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dJ}{dt}} can be treated the same way, to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial v_2}{\partial x_2} J} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial v_3}{\partial x_3} J} , respectively, which leads to the final conclusion that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dJ}{dt} = (\nabla \cdot \boldsymbol{v} ) J }

or,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d \ln{(J)}}{dt} = \nabla \cdot \boldsymbol{v} }

which is Euler's Expansion Formula.<ref>Milhaus, Dmitri and Barbara Milhaus (1984). Foundations of Radiation Hydrodynamics. Oxford University Press. </ref>

Euler's Expansion Formula for a Material Volume

The expression for Euler's expansion formula derived above can be written in a slightly more general way, for a material volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{B}^{M}} , and an initial volume of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{B,0}} .

If we have a material volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{B}^{M}} with a spatial location Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{x}_{B}(\boldsymbol{\xi},t)} and velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{v}_{B} = \frac{d \boldsymbol{x}_{B} }{dt}} , then the Jacobian is defined as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J = \left| \frac{ \partial x_{i,B} }{ \partial \xi_j } \right| }

such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{B}^{M} = J V_{0,B} }

and Euler's Expansion Formula can be written for that material volume as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{J} \frac{dJ}{dt} = div( \boldsymbol{v}_{B} ) }

This result can now be used to obtain Reynolds Transport Theorem.

References

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