Convective derivative
Encyclopedia
The material derivative is a derivative
taken along a path moving with velocity v, and is often used in fluid mechanics
and classical mechanics
. It describes the time rate of change of some quantity (such as heat
or momentum
) by following it, while moving with a space- and time-dependent velocity field
.
For example, in fluid dynamics, take the case that the velocity field under consideration is the flow velocity
itself, and the quantity of interest is the temperature
of the fluid. Then the material derivative describes the temperature evolution of a certain fluid parcel in time, as it is being moved along its pathline (trajectory) while following the fluid flow.
The material derivative can serve as a link between Eulerian and Lagrangian descriptions of fluid motion.
There are many other names for this operator, including:
φ( x, t ) and a vector field
u( x, t ) are defined respectively as:
where the distinction is that
is the gradient
of a scalar, while
is the covariant derivative
of a vector. In case of the material derivative of a vector field, the term v•∇u can both be interpreted as v•(∇u) involving the tensor derivative
of u, or as (v•∇)u, leading to the same result.
Confusingly, the term convective derivative is both used for the whole material derivative Dφ/Dt or Du/Dt, and for only the spatial rate of change part, v•∇φ or v•∇u respectively. For that case, the convective derivative only equals D/Dt for time independent flows.
These derivatives are physical in nature and describe the transport of a scalar or vector quantity in a velocity field v( x, t ). The effect of the time independent terms in the definitions are for the scalar and vector case respectively known as advection
and convection.
The (total) derivative with respect to time of φ is expanded through the multivariate chain rule
:
It is apparent that this derivative is dependent on the vector
which describes a chosen path x(t) in space. For example, if
is chosen, the time derivative becomes equal to the partial derivative, which agrees with the definition of a partial derivative
: a derivative taken with respect to some variable (time in this case) holding other variables constant (space in this case). This makes sense because if
, then the derivative is taken at some constant position. This static position derivative is called the Eulerian derivative.
An example of this case is a swimmer standing still and sensing temperature change in a lake early in the morning: the water gradually becomes warmer due to heating from the sun.
If, instead, the path x(t) is not a standstill, the (total) time derivative of φ may change due to the path. For example, imagine the swimmer is in a motionless pool of water, indoors and unaffected by the sun. One end happens to be a constant hot temperature and the other end a constant cold temperature. By swimming from one end to the other the swimmer senses a change of temperature with respect to time, even though the temperature at any given (static) point is a constant. This is because the derivative is taken at the swimmer's changing location. A temperature sensor attached to the swimmer would show temperature varying in time, even though the pool is held at a steady temperature distribution.
The material derivative finally is obtained when the path x(t) is chosen to have a velocity equal to the fluid velocity:
That is, the path follows the fluid current described by the fluid's velocity field v. So, the material derivative of the scalar φ is:
An example of this case is a lightweight, neutrally buoyant particle swept around in a flowing river undergoing temperature changes, maybe due to one portion of the river being sunny and the other in a shadow. The water as a whole may be heating as the day progresses. The changes due to the particle's motion (itself caused by fluid motion) is called advection
(or convection if a vector is being transported).
The definition above relied on the physical nature of fluid current; however no laws of physics were invoked (for example, it hasn't been shown that a lightweight particle in a river will follow the velocity of the water). It turns out, however, that many physical concepts can be written concisely with the material derivative. The general case of advection, however, relies on conservation of mass in the fluid stream; the situation becomes slightly different if advection happens in a non-conservative medium.
Only a path was considered for the scalar above. For a vector, the gradient becomes a tensor derivative; for tensor
fields we may want to take into account not only translation of the coordinate system due to the fluid movement but also its rotation and stretching. This is achieved by the upper convected time derivative.
, the
component of convection is given by:
where the his are related to the metric tensor
s by
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
taken along a path moving with velocity v, and is often used in fluid mechanics
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
and classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
. It describes the time rate of change of some quantity (such as heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
or momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
) by following it, while moving with a space- and time-dependent velocity field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
.
For example, in fluid dynamics, take the case that the velocity field under consideration is the flow velocity
Flow velocity
In fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of a fluid...
itself, and the quantity of interest is the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
of the fluid. Then the material derivative describes the temperature evolution of a certain fluid parcel in time, as it is being moved along its pathline (trajectory) while following the fluid flow.
The material derivative can serve as a link between Eulerian and Lagrangian descriptions of fluid motion.
Names
There are many other names for this operator, including:
- convective derivative
- advective derivative
- substantive derivative
- substantial derivative
- Lagrangian derivative
- Stokes derivative
- particle derivative
- hydrodynamic derivative
- derivative following the motion
- total derivative
Definition
The material derivatives of a scalar fieldScalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
φ( x, t ) and a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
u( x, t ) are defined respectively as:
where the distinction is that
is the gradientGradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of a scalar, while
is the covariant derivativeCovariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
of a vector. In case of the material derivative of a vector field, the term v•∇u can both be interpreted as v•(∇u) involving the tensor derivative
Tensor derivative (continuum mechanics)
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical...
of u, or as (v•∇)u, leading to the same result.
Confusingly, the term convective derivative is both used for the whole material derivative Dφ/Dt or Du/Dt, and for only the spatial rate of change part, v•∇φ or v•∇u respectively. For that case, the convective derivative only equals D/Dt for time independent flows.
These derivatives are physical in nature and describe the transport of a scalar or vector quantity in a velocity field v( x, t ). The effect of the time independent terms in the definitions are for the scalar and vector case respectively known as advection
Advection
Advection, in chemistry, engineering and earth sciences, is a transport mechanism of a substance, or a conserved property, by a fluid, due to the fluid's bulk motion in a particular direction. An example of advection is the transport of pollutants or silt in a river. The motion of the water carries...
and convection.
Development
Consider a scalar quantity φ = φ( x, t ), where t is understood as time and x as position. This may be some physical variable such as temperature or chemical concentration. The physical quantity exists in a fluid, whose velocity is represented by the vector field v( x, t ).The (total) derivative with respect to time of φ is expanded through the multivariate chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
:
It is apparent that this derivative is dependent on the vector
which describes a chosen path x(t) in space. For example, if
is chosen, the time derivative becomes equal to the partial derivative, which agrees with the definition of a partial derivativePartial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
: a derivative taken with respect to some variable (time in this case) holding other variables constant (space in this case). This makes sense because if
, then the derivative is taken at some constant position. This static position derivative is called the Eulerian derivative.An example of this case is a swimmer standing still and sensing temperature change in a lake early in the morning: the water gradually becomes warmer due to heating from the sun.
If, instead, the path x(t) is not a standstill, the (total) time derivative of φ may change due to the path. For example, imagine the swimmer is in a motionless pool of water, indoors and unaffected by the sun. One end happens to be a constant hot temperature and the other end a constant cold temperature. By swimming from one end to the other the swimmer senses a change of temperature with respect to time, even though the temperature at any given (static) point is a constant. This is because the derivative is taken at the swimmer's changing location. A temperature sensor attached to the swimmer would show temperature varying in time, even though the pool is held at a steady temperature distribution.
The material derivative finally is obtained when the path x(t) is chosen to have a velocity equal to the fluid velocity:
That is, the path follows the fluid current described by the fluid's velocity field v. So, the material derivative of the scalar φ is:
An example of this case is a lightweight, neutrally buoyant particle swept around in a flowing river undergoing temperature changes, maybe due to one portion of the river being sunny and the other in a shadow. The water as a whole may be heating as the day progresses. The changes due to the particle's motion (itself caused by fluid motion) is called advection
Advection
Advection, in chemistry, engineering and earth sciences, is a transport mechanism of a substance, or a conserved property, by a fluid, due to the fluid's bulk motion in a particular direction. An example of advection is the transport of pollutants or silt in a river. The motion of the water carries...
(or convection if a vector is being transported).
The definition above relied on the physical nature of fluid current; however no laws of physics were invoked (for example, it hasn't been shown that a lightweight particle in a river will follow the velocity of the water). It turns out, however, that many physical concepts can be written concisely with the material derivative. The general case of advection, however, relies on conservation of mass in the fluid stream; the situation becomes slightly different if advection happens in a non-conservative medium.
Only a path was considered for the scalar above. For a vector, the gradient becomes a tensor derivative; for tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
fields we may want to take into account not only translation of the coordinate system due to the fluid movement but also its rotation and stretching. This is achieved by the upper convected time derivative.
Orthogonal coordinates
It may be shown that, in orthogonal coordinatesOrthogonal coordinates
In mathematics, orthogonal coordinates are defined as a set of d coordinates q = in which the coordinate surfaces all meet at right angles . A coordinate surface for a particular coordinate qk is the curve, surface, or hypersurface on which qk is a constant...
, the
component of convection is given by:where the his are related to the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
s by
See also
- Navier–Stokes equations
- Euler equationsEuler equationsIn fluid dynamics, the Euler equations are a set of equations governing inviscid flow. They are named after Leonhard Euler. The equations represent conservation of mass , momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and heat conduction terms. Historically,...
- Derivative (generalizations)Derivative (generalizations)The derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.- Derivatives in analysis :...
- Lie derivativeLie derivativeIn mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...