Reynolds Transport Theorem Derivation
From charlesreid1
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^{M}_{B}(\boldsymbol{\xi},t) $, or more simply $ V^{M}_{B} $) 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 $ 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^{M}_{B} $
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^{M}_{B} $ is a volume whose boundaries move such that there is no flux of $ B $ across the boundaries
Derivative Frames of Reference
Frames of References
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 $ V^{M}_{B}(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 $.
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 $ B $ with respect to time, keeping $ \boldsymbol{x} $ constant
This derivative corresponds to the rate of change of $ B $ in a control volume (which is a fixed point in space; see nomenclature above)
Material Derivative
The material derivative is denoted by
$ \displaystyle{\frac{dB}{dt}} $ (alternatively, $ \displaystyle{\frac{DB}{Dt}} $)
The material derivative derivative of B with respect to time, keeping $ \boldsymbol{\xi} $ constant
This derivative corresponds to the rate of change of $ B $ in a material volume (which is a volume whose boundaries are moving with time such that the flux of $ B $ across the boundaries is zero)
Keep in mind that the material derivatives are not partial derivatives because $ \boldsymbol{\xi} $ (the initial particle position) is a constant for a given fluid particle
Position and Velocity
For a material volume $ V^{M}_{B} $, the material volume moves at some rate.
Let the position of the material volume be denoted $ \boldsymbol{x}_{B} $.
Then the velocity of the material volume $ V^{M}_{B} $ is denoted $ \boldsymbol{v}_{B} $
The material derivative of the position of the fluid particle is the velocity:
$ 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 $ \boldsymbol{\xi} $ constant (that is, considering a material volume $ V^{M}_{B} $ with the initial position $ \boldsymbol{\xi} $), the rate of change of the fluid particle's current position $ x_{i,B} $ is the velocity of the material volume $ V^{M}_{B} $.
$ \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 $ \left( \frac{ \partial B }{ \partial t } \right)_{\boldsymbol{x}} $ - rate of change of $ B $ for a fixed control volume $ V(\boldsymbol{x}) $
Material derivative $ \left( \frac{ dB }{ dt } \right)_{\boldsymbol{\xi}} $ - the rate of change of $ B $ for a material volume $ V^{M}_{B}(\boldsymbol{\xi},t) $
Next, the derivatives of $ 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:
$ \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
$ \frac{ dB }{ dt } = \frac{ \partial B }{ \partial t } + ( \boldsymbol{v}_{B} \cdot \nabla ) B $
The operator $ \frac{d}{dt} $ can be expanded as:
$ \frac{d}{dt} = \frac{\partial}{\partial t} + (\boldsymbol{v} \cdot \nabla) $
Dilation and the Euler Expansion Formula
Changing coordinates from $ \boldsymbol{\xi} $ to $ \boldsymbol{x} $ is likely to cause a change in the volume of the fluid particle:
$ 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 $ J $ is the Jacobian.
The Jacobian is equal to:
$ J = \frac{ dV }{ dV_0 } $
This quantity is called the dilation.
The Jacobian is a matrix that looks like this:
$ 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:
$ J = \left| \frac{\partial x_i}{\partial \xi_j} \right| $
or as:
$ 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 $ \epsilon_{ijk} $ is the Levi-Civita symbol (wikipedia:Levi-Civita symbol).
The initial volume of the fluid particle is $ d \xi_1 d \xi_2 d \xi_3 = dV_0 $ at $ t=0 $
Motion is continuous, so the volume cannot break up.
Another way to state that is, $ 0 < J < \infty $ (required so that neither $ J $ nor $ J^{-1} $ vanish, and the mapping from $ \xi $ to $ x $ and vice-versa are continuous and smooth).
The natural question to ask is how the volume dilation changes with time - mathematically, $ \frac{ dJ }{ dt } $
Aris Approach
First, the time derivative of the terms in the Jacobian matrix can be simplified:
$ \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 $ \frac{d}{dt} $ holds $ \boldsymbol{\xi} $ constant.
It was specified above that the velocity is a function of location, $ \boldsymbol{v} = \boldsymbol{v}( x_1, x_2, x_3 ) $. This can be plugged into the relation $ \frac{ \partial v_i }{ \partial \xi_j } $, and the chain rule used, to yield:
$ \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:
$ \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,
$ \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 $ \frac{dJ}{dt} $ above) has a value of
$ \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:
$ \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,
$ \displaystyle{ \frac{1}{J} \frac{dJ}{dt} = div (\boldsymbol{v}) } $
or,
$ \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,
$ div (\boldsymbol{v}) = 0 $
Milhas Approach
From the compact form of the Jacobian, the time derivative can be written:
$ \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:
$ \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 $ \frac{\partial v_i}{\partial \xi_j} $ can be expanded as
$ \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:
$ \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:
$ \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 $ \epsilon_{ijk} $. This makes the term substantially simpler:
$ \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 $ \frac{dJ}{dt} $ can be treated the same way, to get $ \frac{\partial v_2}{\partial x_2} J $ and $ \frac{\partial v_3}{\partial x_3} J $, respectively, which leads to the final conclusion that
$ \frac{dJ}{dt} = (\nabla \cdot \boldsymbol{v} ) J $
or,
$ \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 $ V_{B}^{M} $, and an initial volume of $ V_{B,0} $.
If we have a material volume $ V_{B}^{M} $ with a spatial location $ \boldsymbol{x}_{B}(\boldsymbol{\xi},t) $ and velocity $ \boldsymbol{v}_{B} = \frac{d \boldsymbol{x}_{B} }{dt} $, then the Jacobian is defined as:
$ J = \left| \frac{ \partial x_{i,B} }{ \partial \xi_j } \right| $
such that
$ V_{B}^{M} = J V_{0,B} $
and Euler's Expansion Formula can be written for that material volume as:
$ \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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