Transport set

Transport sentences or transport theorems describe the rules for the time derivative of integrals with time-dependent integration limits. Such time derivatives occur in continuum and fluid mechanics , where the integrals represent, for example, a circulation , a volume flow through a surface or the momentum of a moving and deforming mass. The integration area can accordingly be a line, an area or a volume. The transport theorem for volumes is called Reynolds' transport theorem or Reynolds transport theorem (after Osborne Reynolds ). The transport theorems are used to derive basic conservation laws of continuum mechanics . From a mathematical point of view, these are generalizations of the Leibniz rule for parameter integrals .

All fields under consideration must be continuously differentiable both in terms of time and location and integrable in the area under consideration . However, points of discontinuity in the form of areas that can be continuously differentiated according to location, at which, for example, the density changes abruptly, can be taken into account. When deriving the transport theorems, the integration limits are shown independent of time, the integrand is derived over time and the integration area is brought back into the time-dependent form.

Line, surface and volume elements for areas transported by mass

If the areas move with the mass, i.e. if they have material limits, then the spatial differentiability of the movement function of the mass can be used. The substantial derivation of the line, area and volume elements are then given in accordance with the following table.

Line element	${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\,\mathrm {d} {\vec {x}}=\operatorname {grad} ({\vec {v}})\cdot \,\mathrm {d} {\vec {x}}}$
Vectorial surface element	${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\,\mathrm {d} {\vec {a}}=[\operatorname {div} ({\vec {v}})\mathbf {I} -\operatorname {grad} ({\vec {v}})^{\top }]\cdot \,\mathrm {d} {\vec {a}}}$
Volume element	${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\,\mathrm {d} V=\operatorname {div} ({\vec {v}})\,\mathrm {d} V}$

The operator or a high point as in denotes the substantial time derivative, the velocity field of the mass depending on the place and time , grad the gradient and div the divergence of a vector field , the unit tensor and the superscript the transposition . ${\displaystyle {\tfrac {\mathrm {D} }{\mathrm {D} t}}}$ ${\displaystyle {\dot {f}}}$ ${\displaystyle {\vec {v}}({\vec {x}},t)}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle t}$ ${\displaystyle \mathbf {I} }$ ${\displaystyle \top }$

If the integration area moves relative to the mass, then the line and surface elements indicated above cannot be calculated because the environment required for the gradient and divergence formation is missing in the areas . Instead of resorting to the Lagrangian approach to integrals, which allows material integration limits to be defined independently of time, the area is described with parameters from a fixed interval - here depending on the dimensions of the area. These fixed limits also allow the time derivative to be shifted in the integrand. ${\displaystyle [0,1]^{n},n=1,2,3}$ ${\displaystyle n}$

Transport set for line integrals

Given is a curve with a vectorial line element moving through the mass. For the curve, there is a parameter representation for the points on the curve with curve parameters in the interval at any time . The following then applies to the line element: The time derivative of the curve integral of a field size that is dependent on location and time over the path is then: ${\displaystyle b}$ ${\displaystyle \mathrm {d} {\vec {b}}\,.}$ ${\displaystyle t}$ ${\displaystyle {\vec {b}}(\xi ,t)}$ ${\displaystyle \xi }$ ${\displaystyle [0,1]}$ ${\displaystyle \mathrm {d} {\vec {b}}={\tfrac {\partial }{\partial \xi }}{\vec {b}}(\xi ,t)\,\mathrm {d} \xi \,.}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle t}$ ${\displaystyle {\vec {f}}({\vec {x}},t)}$ ${\displaystyle b}$

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{b}{\vec {f}}({\vec {b}},t)\cdot \,\mathrm {d} {\vec {b}}=&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{0}^{1}{\vec {f}}({\vec {b}}(\xi ,t),t)\cdot {\frac {\partial }{\partial \xi }}{\vec {b}}(\xi ,t)\,\mathrm {d} \xi =\int _{0}^{1}\left[\left({\frac {\partial {\vec {f}}}{\partial {\vec {x}}}}\cdot {\frac {\partial {\vec {b}}}{\partial t}}\right)\cdot {\frac {\partial {\vec {b}}}{\partial \xi }}+{\frac {\partial {\vec {f}}}{\partial t}}\cdot {\frac {\partial {\vec {b}}}{\partial \xi }}+{\vec {f}}\cdot {\frac {\partial ^{2}{\vec {b}}}{\partial \xi \partial t}}\right]\,\mathrm {d} \xi \\=&\int _{b}\left[{\frac {\partial {\vec {f}}}{\partial t}}+\operatorname {grad} ({\vec {f}})\cdot {\vec {v}}_{b}\right]\cdot \,\mathrm {d} {\vec {b}}+\int _{0}^{1}{\vec {f}}\cdot {\frac {\partial }{\partial \xi }}{\vec {v}}_{b}(\xi ,t)\,\mathrm {d} \xi \,.\end{aligned}}}$

For a scalar field the following applies accordingly:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{b}f({\vec {b}},t)\,\mathrm {d} {\vec {b}}=\int _{b}\left[{\frac {\partial f}{\partial t}}+\operatorname {grad} (f)\cdot {\vec {v}}_{b}\right]\,\mathrm {d} {\vec {b}}+\int _{0}^{1}f{\frac {\partial }{\partial \xi }}{\vec {v}}_{b}(\xi ,t)\,\mathrm {d} \xi \,.}$

If the speed of the curve is equal to the speed of the mass, then the substantial time derivative of the field size is in the square brackets and the derivative of the speed in the direction of the curve can be calculated with the speed gradient: ${\displaystyle {\vec {v}}_{b}}$ ${\displaystyle {\vec {v}}}$

${\displaystyle {\frac {\partial {\vec {v}}_{b}(\xi ,t)}{\partial \xi }}\,\mathrm {d} \xi ={\frac {\partial {\vec {v}}({\vec {b}}(\xi ,t),t)}{\partial \xi }}\,\mathrm {d} \xi ={\frac {\partial {\vec {v}}({\vec {x}},t)}{\partial {\vec {x}}}}\cdot {\frac {\partial {\vec {b}}(\xi ,t)}{\partial \xi }}\,\mathrm {d} \xi =\operatorname {grad} ({\vec {v}})\cdot \,\mathrm {d} {\vec {b}}={\frac {\mathrm {D} }{\mathrm {D} t}}\mathrm {d} {\vec {b}}({\vec {x}},t)\,.}$

Then this transport record goes over to the one for lines transported by the mass from the table.

Scalar field	${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\oint _{b}f\,\mathrm {d} {\vec {b}}=\oint _{b}\left({\frac {\mathrm {D} }{\mathrm {D} t}}f\,\mathrm {d} {\vec {b}}+f{\frac {\mathrm {D} }{\mathrm {D} t}}\,\mathrm {d} {\vec {b}}\right)=\oint _{b}[{\dot {f}}\mathbf {I} +f\operatorname {grad} ({\vec {v}})]\cdot \,\mathrm {d} {\vec {b}}}$
Vector field	${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\oint _{b}{\vec {f}}\cdot \,\mathrm {d} {\vec {b}}=\oint _{b}\left({\frac {\mathrm {D} }{\mathrm {D} t}}{\vec {f}}\cdot \,\mathrm {d} {\vec {b}}+{\vec {f}}\cdot {\frac {\mathrm {D} }{\mathrm {D} t}}\,\mathrm {d} {\vec {b}}\right)=\oint _{b}[{\dot {\vec {f}}}+{\vec {f}}\cdot \operatorname {grad} ({\vec {v}})]\cdot \,\mathrm {d} {\vec {b}}}$

Transport set for surface integrals

Given is a surface with a vectorial surface element that moves through the mass. For the area there is always a parametric representation for the points on the area with area parameters ( ) from the unit square . The vectorial surface element is then calculated with the cross product : ${\displaystyle a}$ ${\displaystyle \mathrm {d} {\vec {a}}\,,}$ ${\displaystyle t}$ ${\displaystyle {\vec {a}}(\xi ,\eta ,t)}$ ${\displaystyle \xi ,\eta }$ ${\displaystyle [0,1]^{2}}$

${\displaystyle \mathrm {d} {\vec {a}}={\frac {\partial {\vec {a}}}{\partial \xi }}\times {\frac {\partial {\vec {a}}}{\partial \eta }}\,\mathrm {d} \xi \,\mathrm {d} \eta \,.}$

The time derivative of the area integral of a field size that is dependent on location and time over the area is then: ${\displaystyle {\vec {x}}}$ ${\displaystyle t}$ ${\displaystyle {\vec {f}}({\vec {x}},t)}$ ${\displaystyle a}$

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{a}{\vec {f}}({\vec {a}},t)\cdot \,\mathrm {d} {\vec {a}}=&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{0}^{1}\int _{0}^{1}{\vec {f}}({\vec {a}}(\xi ,\eta ,t),t)\cdot \left({\frac {\partial {\vec {a}}}{\partial \xi }}\times {\frac {\partial {\vec {a}}}{\partial \eta }}\right)\,\mathrm {d} \xi \,\mathrm {d} \eta \\=&\int _{0}^{1}\int _{0}^{1}\left[\left({\frac {\partial {\vec {f}}}{\partial {\vec {x}}}}\cdot {\frac {\partial {\vec {a}}}{\partial t}}\right)\cdot \left({\frac {\partial {\vec {a}}}{\partial \xi }}\times {\frac {\partial {\vec {a}}}{\partial \eta }}\right)+{\frac {\partial {\vec {f}}}{\partial t}}\cdot \left({\frac {\partial {\vec {a}}}{\partial \xi }}\times {\frac {\partial {\vec {a}}}{\partial \eta }}\right)+{\vec {f}}\cdot {\frac {\partial }{\partial t}}\left({\frac {\partial {\vec {a}}}{\partial \xi }}\times {\frac {\partial {\vec {a}}}{\partial \eta }}\right)\right]\,\mathrm {d} \xi \,\mathrm {d} \eta \\=&\int _{a}\left[{\frac {\partial {\vec {f}}}{\partial t}}+\operatorname {grad} ({\vec {f}})\cdot {\vec {v}}_{a}\right]\cdot \,\mathrm {d} {\vec {a}}+\int _{0}^{1}\int _{0}^{1}{\vec {f}}\cdot \left({\frac {\partial {\vec {v}}_{a}}{\partial \xi }}\times {\frac {\partial {\vec {a}}}{\partial \eta }}+{\frac {\partial {\vec {a}}}{\partial \xi }}\times {\frac {\partial {\vec {v}}_{a}}{\partial \eta }}\right)\,\mathrm {d} \xi \,\mathrm {d} \eta \,.\end{aligned}}}$

For a scalar field the following applies accordingly:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{a}f({\vec {a}},t)\cdot \,\mathrm {d} {\vec {a}}=\int _{a}\left[{\frac {\partial f}{\partial t}}+\operatorname {grad} (f)\cdot {\vec {v}}_{a}\right]\,\mathrm {d} {\vec {a}}+\int _{0}^{1}\int _{0}^{1}f\cdot \left({\frac {\partial {\vec {v}}_{a}}{\partial \xi }}\times {\frac {\partial {\vec {a}}}{\partial \eta }}+{\frac {\partial {\vec {a}}}{\partial \xi }}\times {\frac {\partial {\vec {v}}_{a}}{\partial \eta }}\right)\,\mathrm {d} \xi \,\mathrm {d} \eta \,.}$

If the speed of the surface is the same as the speed of the mass, then the substantial time derivative of the field size is in the square brackets and this transport sentence is converted into the table for surfaces transported by the mass. ${\displaystyle {\vec {v}}_{a}}$ ${\displaystyle {\vec {v}}}$

Scalar field

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\int _{a}f\,\mathrm {d} {\vec {a}}=\int _{a}\left({\frac {\mathrm {D} }{\mathrm {D} t}}f\,\mathrm {d} {\vec {a}}+f{\frac {\mathrm {D} }{\mathrm {D} t}}\,\mathrm {d} {\vec {a}}\right)=\int _{a}[{\dot {f}}\mathbf {I} +f\operatorname {div} ({\vec {v}})\mathbf {I} -f\operatorname {grad} ({\vec {v}})^{\top }]\cdot \,\mathrm {d} {\vec {a}}}$

Vector field

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\int _{a}{\vec {f}}\cdot \,\mathrm {d} {\vec {a}}=\int _{a}\left({\frac {\mathrm {D} }{\mathrm {D} t}}{\vec {f}}\cdot \,\mathrm {d} {\vec {a}}+{\vec {f}}\cdot {\frac {\mathrm {D} }{\mathrm {D} t}}\,\mathrm {d} {\vec {a}}\right)=\int _{a}[{\dot {\vec {f}}}+{\vec {f}}\operatorname {div} ({\vec {v}})-{\vec {f}}\cdot \operatorname {grad} ({\vec {v}})^{\top }]\cdot \,\mathrm {d} {\vec {a}}}$

proof

If the surface is a material surface, then there is a reference configuration with a time-independent Lagrangian description of the surface that merges into the current surface with a motion function : The tangent vectors on the surface then transform into one another with the deformation gradient , e.g. B .: ${\displaystyle {\vec {A}}(\xi ,\eta )\,,}$ ${\displaystyle {\vec {\chi }}}$ ${\displaystyle {\vec {a}}(\xi ,\eta ,t)={\vec {\chi }}({\vec {A}}(\xi ,\eta ),t)\,.}$ ${\displaystyle \mathbf {F} }$ ${\displaystyle {\frac {\partial {\vec {a}}(\xi ,\eta ,t)}{\partial \xi }}={\frac {\partial {\vec {\chi }}({\vec {A}}(\xi ,\eta ),t)}{\partial \xi }}={\frac {\partial {\vec {\chi }}({\vec {X}},t)}{\partial {\vec {X}}}}\cdot {\frac {\partial {\vec {A}}(\xi ,\eta )}{\partial \xi }}=\mathbf {F} \cdot {\frac {\partial {\vec {A}}(\xi ,\eta )}{\partial \xi }}\,.}$ The term appearing in the transport theorem in the second integrand is simplified to: ${\displaystyle {\begin{aligned}\left({\frac {\partial {\vec {v}}_{a}}{\partial \xi }}\times {\frac {\partial {\vec {a}}}{\partial \eta }}+{\frac {\partial {\vec {a}}}{\partial \xi }}\times {\frac {\partial {\vec {v}}_{a}}{\partial \eta }}\right)\,\mathrm {d} \xi \,\mathrm {d} \eta =&{\frac {\partial }{\partial t}}\left({\frac {\partial {\vec {a}}}{\partial \xi }}\times {\frac {\partial {\vec {a}}}{\partial \eta }}\right)\,\mathrm {d} \xi \,\mathrm {d} \eta ={\frac {\partial }{\partial t}}\left[\left(\mathbf {F} \cdot {\frac {\partial {\vec {A}}}{\partial \xi }}\right)\times \left(\mathbf {F} \cdot {\frac {\partial {\vec {A}}}{\partial \eta }}\right)\right]\,\mathrm {d} \xi \,\mathrm {d} \eta \\=&{\frac {\partial }{\partial t}}\operatorname {cof} (\mathbf {F} )\cdot \left({\frac {\partial {\vec {A}}}{\partial \xi }}\times {\frac {\partial {\vec {A}}}{\partial \eta }}\right)\,\mathrm {d} \xi \,\mathrm {d} \eta ={\frac {\partial }{\partial t}}[\operatorname {det} (\mathbf {F} )\mathbf {F} ^{\top -1}]\cdot \,\mathrm {d} {\vec {A}}={\frac {\mathrm {D} }{\mathrm {D} t}}\,\mathrm {d} {\vec {a}}\,,\end{aligned}}}$ see the calculation of the cross product and the cofactor cof with the outer tensor product and the time derivative of line, area and volume elements . It was used that in Lagrange's approach the partial time derivative is equal to the substantial time derivative. With this result, the above transport rate goes over to that for areas transported by the mass.

Reynolds' transport theorem or transport theorem for volume integrals

The Reynolds transport theorem is used to derive basic conservation laws of continuum mechanics. Is z. If, for example, the density is used for the scalar field , then a formulation for the conservation of mass results . ${\displaystyle \rho }$ ${\displaystyle f}$

Given is a control volume with a volume element and a surface with an outwardly directed vectorial surface element that moves through the mass. Then the time derivative of the volume integral of a field variable that is dependent on location and time over the control volume , or Reynolds' transport theorem for short, is: ${\displaystyle V_{k}}$ ${\displaystyle \mathrm {d} V_{k}}$ ${\displaystyle a_{k}}$ ${\displaystyle \mathrm {d} {\vec {a}}_{k}\,,}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle t}$ ${\displaystyle {\vec {f}}({\vec {x}},t)}$

Scalar field	${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V_{k}}f\,\mathrm {d} V_{k}=\int _{V_{k}}{\frac {\partial f}{\partial t}}\,\mathrm {d} V_{k}+\int _{a_{k}}f\,({\vec {v}}_{k}\cdot \,\mathrm {d} {\vec {a}}_{k})}$
Vector field	${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V_{k}}{\vec {f}}\,\mathrm {d} V_{k}=\int _{V_{k}}{\frac {\partial {\vec {f}}}{\partial t}}\,\mathrm {d} V_{k}+\int _{a_{k}}{\vec {f}}\,({\vec {v}}_{k}\cdot \,\mathrm {d} {\vec {a}}_{k})}$

The volume is capitalized here as below to avoid confusion with the speed . ${\displaystyle V}$ ${\displaystyle {\vec {v}}}$

Reynolds' transport law can be interpreted as follows: The temporal change in the content of a field size in a variable control volume is made up of a local and a convective part. The local component consists of the integral over the local time derivative, which is formed with the partial derivative , and the convective component is determined from the transport of the field size over the limit a _{k of} the control volume. With is the transition amount per unit of time and area. ${\displaystyle \mathrm {d} {\vec {a}}_{k}=|\,\mathrm {d} {\vec {a}}_{k}|{\vec {n}}}$ ${\displaystyle {\vec {f}}({\vec {v}}_{k}\cdot {\vec {n}})}$

proof

For the spatial points in the volume there is a one-to-one parameter representation with parameters at any time t . The volume element is then calculated with the late product to: ${\displaystyle {\vec {x}}({\vec {\Theta }},t)}$ ${\displaystyle {\vec {\Theta }}\in [0,1]^{3}}$ ${\displaystyle {\begin{aligned}\mathrm {d} V_{k}=&{\frac {\partial {\vec {x}}}{\partial \Theta _{1}}}\cdot \left({\frac {\partial {\vec {x}}}{\partial \Theta _{2}}}\times {\frac {\partial {\vec {x}}}{\partial \Theta _{3}}}\right)\,\underbrace {\mathrm {d} \Theta _{1}\,\mathrm {d} \Theta _{2}\,\mathrm {d} \Theta _{3}} _{\mathrm {d} \Omega }=\operatorname {det} {\begin{pmatrix}{\frac {\partial {\vec {x}}}{\partial \Theta _{1}}}&{\frac {\partial {\vec {x}}}{\partial \Theta _{2}}}&{\frac {\partial {\vec {x}}}{\partial \Theta _{3}}}\end{pmatrix}}\,\mathrm {d} \Omega \\=&\operatorname {det} \left({\frac {\partial {\vec {x}}}{\partial {\vec {\Theta }}}}\right)\,\mathrm {d} \Omega =:\operatorname {det} (\mathbf {J} )\,\mathrm {d} \Omega \,.\end{aligned}}}$ The derivation of the location according to the coordinates is calculated like a Jacobi matrix . The time derivative of the volume integral of a on the location and the time -dependent field size over the volume is then: because the derivative of the determinant of a tensor according to the tensor is calculated as follows and the operator forms the track of its argument and the speed in the control volume is the time derivative Furthermore, the Trace of the speed gradient the divergence of the speed field. For a scalar field, the following applies accordingly: With the product rule and the Gaussian integral theorem , these integrals can be further transformed and the results are obtained in the table. The arithmetic symbol " " forms the dyadic product . In the literature there is also a divergence operator for tensors that forms the divergence of the rows of the tensor and not - as here - the columns. With the operator, the following applies: so that the end result agrees again. ${\displaystyle \mathbf {J} }$ ${\displaystyle {\vec {\Theta }}}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle t}$ ${\displaystyle {\vec {f}}({\vec {x}},t)}$ ${\displaystyle V_{k}}$ ${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V_{k}}{\vec {f}}({\vec {x}},t)\,\mathrm {d} V_{k}=&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\vec {f}}({\vec {x}}({\vec {\Theta }},t),t)\cdot \operatorname {det} (\mathbf {J} )\,\mathrm {d} \Omega \\=&\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\left[{\frac {\partial {\vec {f}}}{\partial {\vec {x}}}}\cdot {\frac {\partial {\vec {x}}}{\partial t}}\operatorname {det} (\mathbf {J} )+{\frac {\partial {\vec {f}}}{\partial t}}\operatorname {det} (\mathbf {J} )+{\vec {f}}\cdot \operatorname {det} (\mathbf {J} )\left(\mathbf {J} ^{\top -1}:{\frac {\partial \mathbf {J} }{\partial t}}\right)\right]\,\mathrm {d} \Omega \\=&\int _{V_{k}}\left[{\frac {\partial {\vec {f}}}{\partial t}}+\operatorname {grad} ({\vec {f}})\cdot {\vec {v}}_{k}+{\vec {f}}\operatorname {div} ({\vec {v}}_{k})\right]\,\mathrm {d} V_{k}\,,\end{aligned}}}$ ${\displaystyle {\tfrac {\mathrm {d} }{\mathrm {d} \mathbf {T} }}\operatorname {det} \mathbf {T} =\operatorname {det} (\mathbf {T} )\mathbf {T} ^{\top -1}}$ ${\displaystyle {\begin{aligned}\mathbf {J} ^{\top -1}:{\frac {\partial \mathbf {J} }{\partial t}}:=&\operatorname {Sp} \left(\mathbf {J} ^{-1}\cdot {\frac {\partial \mathbf {J} }{\partial t}}\right)=\operatorname {Sp} \left({\frac {\partial \mathbf {J} }{\partial t}}\cdot \mathbf {J} ^{-1}\right)=\operatorname {Sp} \left({\frac {\partial {\vec {v}}_{k}}{\partial {\vec {\Theta }}}}\cdot {\frac {\mathrm {d} {\vec {\Theta }}}{\mathrm {d} {\vec {x}}}}\right)=\operatorname {Sp} \left({\frac {\partial {\vec {v}}_{k}}{\partial {\vec {x}}}}\right)=\operatorname {div} ({\vec {v}}_{k})\,.\end{aligned}}}$ ${\displaystyle \operatorname {Sp} }$ ${\displaystyle {\vec {v}}_{k}:={\tfrac {\partial }{\partial t}}{\vec {x}}({\vec {\Theta }},t)\,.}$ ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V_{k}}f({\vec {x}},t)\,\mathrm {d} V_{k}=\int _{V_{k}}\left[{\frac {\partial f}{\partial t}}+\operatorname {grad} (f)\cdot {\vec {v}}_{k}+f\operatorname {div} ({\vec {v}}_{k})\right]\,\mathrm {d} V_{k}\,.}$ ${\displaystyle \operatorname {div} (f{\vec {v}})=\operatorname {grad} (f)\cdot {\vec {v}}+f\operatorname {div} ({\vec {v}})\quad {\text{bzw.}}\quad \operatorname {div} ({\vec {v}}\otimes {\vec {f}})=\operatorname {grad} ({\vec {f}})\cdot {\vec {v}}+\operatorname {div} ({\vec {v}}){\vec {f}}}$ ${\displaystyle \int _{v}\operatorname {div} (f{\vec {v}})\,\mathrm {d} v=\int _{a}f{\vec {v}}\cdot \,\mathrm {d} {\vec {a}}\quad {\text{bzw.}}\quad \int _{v}\operatorname {div} ({\vec {v}}\otimes {\vec {f}})\,\mathrm {d} v=\int _{a}{\vec {f}}({\vec {v}}\cdot \,\mathrm {d} {\vec {a}})}$ ${\displaystyle \otimes }$ ${\displaystyle \operatorname {\tilde {div}} }$ ${\displaystyle \operatorname {\tilde {div}} ({\vec {f}}\otimes {\vec {v}})=\operatorname {grad} ({\vec {f}})\cdot {\vec {v}}+\operatorname {div} ({\vec {v}}){\vec {f}}\quad {\text{und}}\quad \int _{v}\operatorname {\tilde {div}} ({\vec {f}}\otimes {\vec {v}})\,\mathrm {d} v=\int _{a}{\vec {f}}({\vec {v}}\cdot \,\mathrm {d} {\vec {a}})\,,}$

If the speed of the control volume is equal to the speed of the mass (no inflow and outflow of matter), then this transport rate is converted into the table for the volume transported by the mass. ${\displaystyle {\vec {v}}_{k}}$ ${\displaystyle {\vec {v}}}$

Scalar field

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\int _{V}f\,\mathrm {d} V=\int _{V}\left({\frac {\mathrm {D} }{\mathrm {D} t}}f\,\mathrm {d} V+f{\frac {\mathrm {D} }{\mathrm {D} t}}\,\mathrm {d} V\right)=\int _{V}[{\dot {f}}+f\operatorname {div} ({\vec {v}})]\,\mathrm {d} V=\int _{V}{\frac {\partial f}{\partial t}}\,\mathrm {d} V+\int _{a}f{\vec {v}}\cdot \,\mathrm {d} {\vec {a}}}$

Vector field

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\int _{V}{\vec {f}}\,\mathrm {d} V=\int _{V}\left({\frac {\mathrm {D} }{\mathrm {D} t}}{\vec {f}}\cdot \,\mathrm {d} V+{\vec {f}}\cdot {\frac {\mathrm {D} }{\mathrm {D} t}}\,\mathrm {d} V\right)=\int _{V}[{\dot {\vec {f}}}+{\vec {f}}\operatorname {div} ({\vec {v}})]\,\mathrm {d} V=\int _{V}{\frac {\partial {\vec {f}}}{\partial t}}\,\mathrm {d} V+\int _{a}{\vec {f}}({\vec {v}}\cdot \,\mathrm {d} {\vec {a}})}$

This is Reynolds' transport theorem specializing in volumes carried by masses . Here is the surface of the mass with an outwardly directed vectorial surface element ${\displaystyle V}$ ${\displaystyle a}$ ${\displaystyle \mathrm {d} {\vec {a}}\,.}$

If the limits of the control volume and the mass coincide at a point in time, then the local part can be eliminated from the general Reynolds' transport theorem and the special, latter version:

Scalar field	${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\int _{V}f\,\mathrm {d} V={\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V_{k}}f\,\mathrm {d} V_{k}+\int _{a_{k}}f\;({\vec {v}}-{\vec {v}}_{k})\cdot \,\mathrm {d} {\vec {a}}_{k}}$
Vector field	${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\int _{V}{\vec {f}}\,\mathrm {d} V={\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V_{k}}{\vec {f}}\,\mathrm {d} V_{k}+\int _{a_{k}}{\vec {f}}\;[({\vec {v}}-{\vec {v}}_{k})\cdot \,\mathrm {d} {\vec {a}}_{k}]}$

The material time derivative of the content of a field size in a volume is therefore equal to the temporal change in the time-dependent volume and the transport over the moving surface with a flow rate that is determined by the speed difference between the particles of the mass and the surface in the direction of the surface normal. ${\displaystyle V_{k}}$ ${\displaystyle a_{k}}$ ${\displaystyle {\vec {v}}-{\vec {v}}_{k}}$ ${\displaystyle a_{k}}$

Influence of cracks

A jump point on the surface a _s separates two spatial areas V ⁺ and V ^-

The constant local differentiability of the transported field required at the beginning is violated under real conditions if, for example, density jumps at material boundaries or shock waves occur. Such flat jumps can, however, be taken into account in the transport theorem if the area itself is locally continuously differentiable and thus has a normal vector in each of its points . The area - hereinafter referred jump point -. Has no material surface be, so can move at a different speed than the mass itself through this area is the mass into two pieces and divided and it is agreed that the normal vector of the jump point toward the speed of the jump point and the volume , see picture on the right. ${\displaystyle V^{+}}$ ${\displaystyle V^{-}}$ ${\displaystyle a_{s}}$ ${\displaystyle V^{+}}$

Then the transport record for cases with a jump is obtained from the table.

Scalar field	${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\int _{V}f\,\mathrm {d} V=\int _{V}{\frac {\partial f}{\partial t}}\,\mathrm {d} V+\int _{a}f\;({\vec {v}}\cdot \,\mathrm {d} {\vec {a}})-\int _{a_{s}}[[f\;(({\vec {v}}-{\vec {v}}_{s})\cdot {\vec {n}})]]\mathrm {d} a_{s}}$
Vector field	${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\int _{V}{\vec {f}}\,\mathrm {d} V=\int _{V}{\frac {\partial {\vec {f}}}{\partial t}}\,\mathrm {d} V+\int _{a}{\vec {f}}\;({\vec {v}}\cdot \,\mathrm {d} {\vec {a}})-\int _{a_{s}}[[{\vec {f}}\;(({\vec {v}}-{\vec {v}}_{s})\cdot {\vec {n}})]]\mathrm {d} a_{s}}$

The newly added last term integrates the jump function over the jump point, for example:

${\displaystyle [[f\;(({\vec {v}}-{\vec {v}}_{s})\cdot {\vec {n}})]]:=f^{+}\;(({\vec {v}}^{+}-{\vec {v}}_{s})\cdot {\vec {n}})-f^{-}\;(({\vec {v}}^{-}-{\vec {v}}_{s})\cdot {\vec {n}})\,.}$

The size is the value of the field of interest when approaching the jump point in is the size when approaching the jump point in and so the field on the surface makes the jump . The same applies to the speed, which can be different on both sides of the jump point , for example in the case of a shock wave . The jump point speed and the normal to the jump point - defined with - are identical on both sides of the jump point. The minus sign in front of the last integral results from the agreement that the normal point to the jump point and the jump point speed into the volume . ${\displaystyle f^{+}}$ ${\displaystyle V^{+},f^{-}}$ ${\displaystyle V^{-}}$ ${\displaystyle a_{s}}$ ${\displaystyle [[f]]:=f^{+}-f^{-}}$ ${\displaystyle {\vec {v}}_{s}}$ ${\displaystyle {\vec {n}}}$ ${\displaystyle \mathrm {d} {\vec {a}}_{s}=|\,\mathrm {d} {\vec {a}}_{s}|{\vec {n}}=:{\vec {n}}\,\mathrm {d} a_{s}}$ ${\displaystyle V^{+}}$

proof

There is a spatially continuously differentiable area a s , which moves through the mass at its own jump point speed , see picture. The surface of the entire control volume V (the inner surface of a s does not count them) consists of the volume V + belonging part a + and the complement of a - and moves with the mass, so that the surfaces of material surfaces represent. Only at the jump point does the surface of the control volume have its own jump point speed. Application of Reynold's transport theorem in the form ${\displaystyle {\vec {v}}_{s}}$ ${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\int _{V}{\vec {f}}\,\mathrm {d} V={\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V_{k}}{\vec {f}}\,\mathrm {d} V_{k}+\int _{a_{k}}{\vec {f}}[({\vec {v}}-{\vec {v}}_{k})\cdot \,\mathrm {d} {\vec {a}}_{k}]}$ supplies the two partial volumes because the surface element should always be directed outwards and is therefore included once with a negative and once with a positive sign on the jump point. The terms marked with the curly brackets disappear according to the assumption. Adding the terms on the left side of the two equations gives the material time derivative of the volume integral with a jump point. The transport rate for control volume is used for the sum of the first terms on the right. The result is the transport record for cases with a jump point from the table above. ${\displaystyle {\begin{aligned}{\frac {\mathrm {D} }{\mathrm {D} t}}\int _{V^{+}}{\vec {f}}\,\mathrm {d} {V^{+}}=&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V_{k}^{+}}{\vec {f}}\,\mathrm {d} V_{k}^{+}+\int _{a_{k}^{+}}{\vec {f}}[(\underbrace {{\vec {v}}-{\vec {v}}_{k}^{+}} _{={\vec {0}}})\cdot \,\mathrm {d} {\vec {a}}_{k}^{+}]-\int _{a_{s}}{\vec {f}}^{+}[({\vec {v}}^{+}-{\vec {v}}_{s})\cdot \,\mathrm {d} {\vec {a}}_{s}]\\{\frac {\mathrm {D} }{\mathrm {D} t}}\int _{V^{-}}{\vec {f}}\,\mathrm {d} {V^{-}}=&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V_{k}^{-}}{\vec {f}}\,\mathrm {d} V_{k}^{-}+\int _{a_{k}^{-}}{\vec {f}}[(\overbrace {{\vec {v}}-{\vec {v}}_{k}^{-}} )\cdot \,\mathrm {d} {\vec {a}}_{k}^{-}]+\int _{a_{s}}{\vec {f}}^{-}[({\vec {v}}^{-}-{\vec {v}}_{s})\cdot \,\mathrm {d} {\vec {a}}_{s}]\,,\end{aligned}}}$

Footnotes

1. ↑ ^{a b c d e f g} The Frechet derivative of a function according to the limited linear operator of the - if it exists - in all directions the Gâteaux derivative corresponds, so ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle h}$

   ${\displaystyle {\mathcal {A}}(h)=\left.{\frac {\mathrm {d} }{\mathrm {d} s}}f(x+sh)\right|_{s=0}=\lim _{s\rightarrow 0}{\frac {f(x+sh)-f(x)}{s}}\quad \forall \;h}$

   applies. In it is scalar, vector or tensor valued but and similar. Then will too ${\displaystyle s\in \mathbb {R} \,,f,x\,{\textsf {und}}\,h}$ ${\displaystyle x}$ ${\displaystyle h}$

   ${\displaystyle {\mathcal {A}}={\frac {\partial f}{\partial x}}}$

   written.

literature

• H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 .
• WH Müller: Forays through the continuum theory . Springer, 2011, ISBN 978-3-642-19869-4 .
• P. Haupt: Continuum Mechanics and Theory of Materials . Springer, 2010, ISBN 978-3-642-07718-0 .
• Pieter Wesseling: Principles of Computational Fluid Dynamics , Springer Verlag, 2001

Web links

• http://planetmath.org/reynoldstransporttheorem