Substantial derivative

The substantial derivative (also material derivative or local derivative plus convective derivative ) describes the rate at which a given physical field changes at the location of a fluid particle while it is carried by a flow through the field.

From a mathematical point of view it is the total derivative of the field along the path of the particle. The change perceived by the particle on its path is made up of two components: the change due to different field strengths at locations that the particle passes through one after the other, and a possible time dependence of the field at the location through which the particle passes.

The field can be an externally predetermined field through which the fluid flows, e.g. B. an electric or magnetic field, a gravitational field, but also a gravitational potential or any other physical or mathematical variable, as long as their derivatives can be formed. The field can also describe a property of the flowing fluid, e.g. B. its temperature, its density, its pressure or its enthalpy density . In these cases, the field is usually described from the standpoint of an observer at rest.

In particular, the field under consideration can be the velocity field of the flow itself. In this case, the substantial derivative describes the change in the speed of the particle while it is following the flow, i.e. its acceleration in (and through) the flow. The determination of this acceleration as a function of the forces acting on the particle is the starting point for fluid dynamics .

In the following, the movement of the observed particle through the field is considered to be due to the flow; however, it can also more generally be the movement of a volume element during the deformation of an elastic or inelastic medium. The continuum mechanics treats all these cases on a common basis.

definition

The substantial derivative of a scalar or vector field quantity is written as or and is defined as: ${\displaystyle \Phi ({\vec {x}},t)}$ ${\displaystyle {\frac {{\text{D}}\Phi }{{\text{D}}t}}}$ ${\displaystyle {\frac {{\text{d}}\Phi }{{\text{d}}t}}}$

${\displaystyle {\frac {{\text{D}}\Phi ({\vec {x}},t)}{{\text{D}}t}}:={\frac {\partial \Phi }{\partial t}}+({\vec {v}}\cdot {\vec {\nabla }})\Phi }$.

in which

${\displaystyle \Phi }$	: scalar or vector field.
${\displaystyle \textstyle {\frac {\partial }{\partial t}}}$	: partial derivative with respect to time also called local derivative or local change, ${\displaystyle t}$
${\displaystyle {\vec {v}}}$	: Velocity vector of the flow at the location and at the time , ${\displaystyle {\vec {x}}}$ ${\displaystyle t}$
${\displaystyle {\vec {\nabla }}}$	: Nabla operator
${\displaystyle \cdot }$	: Dot product

The first summand is called the local change. It describes the explicit time dependence of the field and therefore indicates how the fixed location , i.e. H. local, changed. ${\displaystyle {\frac {\partial \Phi }{\partial t}}}$ ${\displaystyle \Phi }$ ${\displaystyle {\vec {x}}}$

The second term is the convective change. It describes the additional change that occurs due to the movement of the fluid particle. ${\displaystyle ({\vec {v}}\cdot {\vec {\nabla }})\Phi }$

If it is a scalar field quantity, then the convective change is equal to the scalar product of the velocity vector and the gradient of . ${\displaystyle \Phi }$ ${\displaystyle ({\vec {v}}\cdot {\vec {\nabla }})\Phi ={\vec {v}}\cdot {\vec {\nabla }}\Phi }$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {\nabla }}\Phi }$ ${\displaystyle \Phi }$

If it is a vectorial field quantity , then the convective change is a vector with the components . ${\displaystyle {\vec {\Phi }}}$ ${\displaystyle ({\vec {v}}\cdot {\vec {\nabla }}){\vec {\Phi }}}$ ${\displaystyle \left({\vec {v}}\cdot {\vec {\nabla }}\Phi _{x},\ {\vec {v}}\cdot {\vec {\nabla }}\Phi _{y},\ {\vec {v}}\cdot {\vec {\nabla }}\Phi _{z}\right)}$

The convective change can be clearly interpreted as follows. Let be the unit vector pointing in the direction of the velocity . Then is , and holds for the convective change of a scalar ${\displaystyle {\vec {e_{v}}}}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {v}}=|{\vec {v}}|{\vec {e_{v}}}}$ ${\displaystyle \Phi }$

${\displaystyle {\vec {v}}\cdot {\vec {\nabla }}\Phi =|{\vec {v}}|{\vec {e_{v}}}\cdot {\vec {\nabla }}\Phi =|{\vec {v}}|{\frac {\partial \Phi }{\partial s}}}$

(where is a position coordinate counted in the direction of the unit vector), because the scalar product of a unit vector and the gradient of a function is the spatial rate of change of this function in the direction described by the unit vector (see derivation of direction ). Multiplying the spatial rate of change by the amount of flow velocity gives the temporal rate of change to which the fluid element is subjected while it is moving with the flow. ${\displaystyle s}$

The substantial derivation uses the model of the moving observer, which is also known as Lagrange's approach . In addition, there is Euler's approach , which uses a fixed observer and is linked to the local change (the convective term falls out without movement). Strictly speaking, the size itself is defined by Euler, while its material derivation is Lagrangian. ${\displaystyle \Phi }$

Example: Movement in the temperature field

As an example, consider a lake surface, the temperature distribution of which in the fixed coordinate system is due to the two-dimensional time-dependent temperature field ${\displaystyle (x,y)}$

${\displaystyle \Theta (x,y,t)=300\,{\text{K}}+(1\,{\text{K/m}})\,x+(2\,{\text{K/m}})\,y+(3\,{\text{K/s}})\,t}$

is described. The water becomes warmer in the direction of the positive x and y axes, i.e. from southwest to northeast (e.g. because of a series of warm tributaries). In addition, the entire lake is continuously heated by the supply of heat (e.g. by solar radiation). The water flows with the speed

${\displaystyle {\vec {v}}={\begin{pmatrix}u\\v\end{pmatrix}}={\begin{pmatrix}3\\1\end{pmatrix}}{\text{m/s}}}$

from southwest to northeast through the lake.

The partial derivative with respect to time describes the temperature change for a stationary observer who is standing in the water with respect to the shore. This observer only perceives the time dependence of the temperature field at the fixed observation location. In this example the time dependency is the same for all locations and is:

${\displaystyle {\frac {\partial \Theta }{\partial t}}=3\,{\text{K/s}}}$.

The substantial derivative describes the change in temperature for an observer who moves with the water in a boat. she is

${\displaystyle {\frac {{\text{D}}\Theta }{{\text{D}}t}}={\frac {\partial \Theta }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}\Theta =3\,{\text{K/s}}+{\begin{pmatrix}3\\1\end{pmatrix}}{\text{m/s}}\cdot {\begin{pmatrix}1\\2\end{pmatrix}}{\text{K/m}}=8\,{\text{K/s}}}$

and thus by the convective part of greater, because the boat is also moving in the direction of the warmer area. ${\displaystyle 5\,{\text{K/s}}}$

Derivation

The derivation for the case of a scalar field in a Cartesian coordinate system is outlined below . The position and time dependence of the field are given by the function . ${\displaystyle \Phi }$ ${\displaystyle \Phi (x,y,z,t)}$

An observer who is currently at the location is exposed to the value of the field there. If the observer moves along a space curve, which is described by the time-dependent coordinates , he is exposed to changing values . For the temporal change of the field that the observer perceives, the following applies due to the generalized chain rule : ${\displaystyle t}$ ${\displaystyle (x,y,z)}$ ${\displaystyle \Phi (x,y,z,t)}$ ${\displaystyle (x(t),y(t),z(t))}$ ${\displaystyle \Phi (x(t),y(t),z(t),t)}$

${\displaystyle {\frac {{\text{d}}\Phi (x(t),y(t),z(t),t)}{{\text{d}}t}}={\frac {\partial \Phi }{\partial x}}{\frac {{\text{d}}x(t)}{{\text{d}}t}}+{\frac {\partial \Phi }{\partial y}}{\frac {{\text{d}}y(t)}{{\text{d}}t}}+{\frac {\partial \Phi }{\partial z}}{\frac {{\text{d}}z(t)}{{\text{d}}t}}+{\frac {\partial \Phi }{\partial t}}}$

Where , and are the speed components of the observer. ${\displaystyle {\frac {{\text{d}}x}{{\text{d}}t}}}$ ${\displaystyle {\frac {{\text{d}}y}{{\text{d}}t}}}$ ${\displaystyle {\frac {{\text{d}}z}{{\text{d}}t}}}$

Referring now in place of any observer who moves along an arbitrary space curve, especially a fluid element of a flow with the velocity component , and is carried through the field, so it is ${\displaystyle u={\frac {{\text{d}}x}{{\text{d}}t}}}$ ${\displaystyle v={\frac {{\text{d}}y}{{\text{d}}t}}}$ ${\displaystyle w={\frac {{\text{d}}z}{{\text{d}}t}}}$

${\displaystyle {\begin{aligned}{\frac {{\text{d}}\Phi (x(t),y(t),z(t),t)}{{\text{d}}t}}&={\frac {\partial \Phi }{\partial x}}u+{\frac {\partial \Phi }{\partial y}}v+{\frac {\partial \Phi }{\partial z}}w+{\frac {\partial \Phi }{\partial t}}\\&={\frac {\partial \Phi }{\partial t}}+u{\frac {\partial \Phi }{\partial x}}+v{\frac {\partial \Phi }{\partial y}}+w{\frac {\partial \Phi }{\partial z}}\\&=\left({\frac {\partial }{\partial t}}+u{\frac {\partial }{\partial x}}+v{\frac {\partial }{\partial y}}+w{\frac {\partial }{\partial z}}\right)\Phi \\&=:{\frac {\text{D}}{{\text{D}}t}}\Phi \end{aligned}}}$

With the introduction of the position vector with the components , , and the velocity vector with the components , , could write the substantial derivative as ${\displaystyle {\vec {x}}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle w}$

${\displaystyle {\begin{aligned}{\frac {{\text{d}}\Phi ({\vec {x}},t)}{{\text{d}}t}}={\frac {{\text{D}}\Phi ({\vec {x}},t)}{{\text{D}}t}}&={\frac {\partial \Phi }{\partial t}}+u{\frac {\partial \Phi }{\partial x}}+v{\frac {\partial \Phi }{\partial y}}+w{\frac {\partial \Phi }{\partial z}}\\&=\underbrace {\frac {\partial \Phi }{\partial t}} _{\mbox{lokal}}+\underbrace {({\vec {v}}\cdot {\vec {\nabla )}}\Phi } _{\mbox{konvektiv}}\end{aligned}}}$

For a vector-valued function , the substantial derivative is written out component-wise ${\displaystyle {\vec {\Phi }}({\vec {x}},t)=(\Phi _{x}({\vec {x}},t),\Phi _{y}({\vec {x}},t),\Phi _{z}({\vec {x}},t))}$

${\displaystyle {\begin{aligned}{\frac {{\text{D}}{\vec {\Phi }}({\vec {x}},t)}{{\text{D}}t}}={\frac {\partial {\vec {\Phi }}}{\partial t}}+({\vec {v}}\cdot {\vec {\nabla )}}{\vec {\Phi }}={\bigg (}&{\frac {\partial \Phi _{x}}{\partial t}}+u{\frac {\partial \Phi _{x}}{\partial x}}+v{\frac {\partial \Phi _{x}}{\partial y}}+w{\frac {\partial \Phi _{x}}{\partial z}},\\&{\frac {\partial \Phi _{y}}{\partial t}}+u{\frac {\partial \Phi _{y}}{\partial x}}+v{\frac {\partial \Phi _{y}}{\partial y}}+w{\frac {\partial \Phi _{y}}{\partial z}},\\&{\frac {\partial \Phi _{z}}{\partial t}}+u{\frac {\partial \Phi _{z}}{\partial x}}+v{\frac {\partial \Phi _{z}}{\partial y}}+w{\frac {\partial \Phi _{z}}{\partial z}}{\bigg )}\\\end{aligned}}}$

Illustrative special cases

Consider a fluid flow with the (possibly time-dependent) velocity field . A streamline is a curve that has the same direction at every point it traverses as the flow velocity at that point and at that time. ${\displaystyle {\vec {v}}({\vec {x}},t)=(u(x,y,z,t),v(x,y,z,t),w(x,y,z,t))}$

Stationary flow

Streamlines that describe the steady flow around a wing profile.

A flow is stationary if the velocity field does not explicitly depend on time:

${\displaystyle {\vec {v}}={\vec {v}}({\vec {x}})=(u(x,y,z),\ v(x,y,z),\ w(x,y,z))}$,

${\displaystyle {\frac {\partial {\vec {v}}}{\partial t}}=0}$

In this case, the streamline pattern remains unchanged over time and a fluid particle that is located on a certain streamline at a given point in time will continue to follow this streamline over time.

If it can be determined for a stationary problem through suitable considerations that

${\displaystyle {\frac {{\text{D}}\Phi }{{\text{D}}t}}=0}$

is (because it is sufficient to prove that ) then it follows that along a streamline always has the same value. (Nothing is said about whether it has the same value or different values on different streamlines.) ${\displaystyle {\frac {\partial {\vec {v}}}{\partial t}}=0}$ ${\displaystyle ({\vec {v}}\cdot {\vec {\nabla }})\Phi =0}$ ${\displaystyle \Phi }$ ${\displaystyle \Phi }$

Unsteady flow

If the flow is unsteady, the velocity field depends explicitly on time. If for a given problem it holds that

${\displaystyle {\frac {{\text{D}}\Phi }{{\text{D}}t}}=0}$

then it follows that for each fluid element remains constant on its path . (Nothing is said about whether the value has the same or different values ​​for different fluid elements. However, the value once assumed remains unchanged for any given fluid element.) ${\displaystyle \Phi }$ ${\displaystyle \Phi }$

In this case, local and convective changes always cancel each other out on the path of the fluid element.

Example: fluid dynamics

In this example, the substantial derivative of the velocity field of the flow itself is used. So it describes the change in the speed of the particle while it follows the flow, and thus its acceleration. ${\displaystyle {\vec {v}}}$

Consider an element of fluid with volume and constant density in an incompressible flow . The acting forces are caused by the location-dependent hydrostatic pressure in the fluid and the gravitation with the gravitational acceleration . ${\displaystyle \Delta V}$ ${\displaystyle \rho }$ ${\displaystyle p}$ ${\displaystyle {\vec {g}}=(0,0,-g)}$

If the fluid element is a cuboid with side lengths , and , then the pressure exerts the force on the left side and the force on the right side , so that the net force acts in the x direction . Generalized to all three dimensions results in the force caused by the pressure ${\displaystyle \Delta x}$ ${\displaystyle \Delta y}$ ${\displaystyle \Delta z}$ ${\displaystyle p}$ ${\displaystyle p(x)\Delta y\Delta z}$ ${\displaystyle p(x+\Delta x)\Delta y\Delta z=\left[p(x)+{\frac {{\text{d}}p}{{\text{d}}x}}\Delta x\right]\Delta y\Delta z}$ ${\displaystyle -{\frac {{\text{d}}p}{{\text{d}}x}}\Delta x\Delta y\Delta z}$

${\displaystyle {\vec {F_{p}}}=-{\vec {\nabla }}p\ \Delta V}$

The weight force acting on the fluid element of the mass in the gravitational field is ${\displaystyle m}$

${\displaystyle {\vec {F_{g}}}=m\cdot {\vec {g}}=\rho \ \Delta V\cdot {\vec {g}}}$

According to Newton's second law , the product of the mass and acceleration of the fluid element is equal to the total force acting: ${\displaystyle \rho \ \Delta V}$ ${\displaystyle {\frac {{\text{D}}{\vec {v}}}{{\text{D}}t}}}$

${\displaystyle \rho \ \Delta V\ {\frac {{\text{D}}{\vec {v}}}{{\text{D}}t}}={\vec {F_{p}}}+{\vec {F_{g}}}=(-{\vec {\nabla }}p+\rho \ {\vec {g}})\ \Delta V}$

The Euler equation provides for shortening :

${\displaystyle {\frac {{\text{D}}{\vec {v}}}{{\text{D}}t}}=-{\frac {1}{\rho }}{\vec {\nabla }}p+{\vec {g}}}$

Written out in components, it reads:

${\displaystyle {\begin{aligned}{\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}+w{\frac {\partial u}{\partial z}}=&-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}}\\{\frac {\partial v}{\partial t}}+u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}+w{\frac {\partial v}{\partial z}}=&-{\frac {1}{\rho }}{\frac {\partial p}{\partial y}}\\{\frac {\partial w}{\partial t}}+u{\frac {\partial w}{\partial x}}+v{\frac {\partial w}{\partial y}}+w{\frac {\partial w}{\partial z}}=&-{\frac {1}{\rho }}{\frac {\partial p}{\partial z}}-g\\\end{aligned}}}$

If the shear force occurring in a viscous fluid is added as an additional force , the Navier-Stokes equation results .

application

Substantial derivation is used especially in continuum mechanics . These include, for example, the balance equations of fluid mechanics or solid mechanics in engineering. It often appears there when the behavior of the physical system is caused by the conservation of mass , energy, etc. Ä. Is described.

The limits of the term and its application arise when the definition of material points or associated speeds fails. This is the case, for example, in mixture theory with several phases or at the atomic level when there is no longer a continuum. The concept of a material point or its associated speed may be adapted.

literature

• DJ Acheson: Elementary Fluid Dynamics. Oxford University Press, Oxford 1990, ISBN 0-19-859679-0 , chap. 1.
• P. Haupt: Continuum Mechanics and Theory of Materials. Springer-Verlag, 2000, p. 21.
• Gerhard A. Holzapfel: Nonlinear Solid Mechanics: A Continuum Approach for Engineering. John Wiley & Sons, Chichester 2005, ISBN 0-471-82319-8 , pp. 90 ff.
• Horst Parisch: Solid-state continuum mechanics. Teubner Verlag, 2003, p. 90.
• C. Eck: Mathematical Modeling. Springer-Verlag, 2011, ISBN 978-3-642-18423-9 , p. 210 f.

Individual evidence

1. ^ DJ Acheson: Elementary Fluid Dynamics. Oxford University Press, Oxford 1990, ISBN 0-19-859679-0 , p. 5.