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In
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mec ...
, the material derivative describes the time rate of change of some physical quantity (like
heat In thermodynamics, heat is energy in transfer between a thermodynamic system and its surroundings by such mechanisms as thermal conduction, electromagnetic radiation, and friction, which are microscopic in nature, involving sub-atomic, ato ...
or
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation. For example, in
fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
, the velocity field is the
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
, and the quantity of interest might be the
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
of the fluid. In this case, the material derivative then describes the temperature change of a certain
fluid parcel In fluid dynamics, a fluid parcel, also known as a fluid element or material element, is an infinitesimal volume of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel rema ...
with time, as it flows along its pathline (trajectory).


Other names

There are many other names for the material derivative, including: *advective derivative *convective derivative *derivative following the motion *hydrodynamic derivative *Lagrangian derivative *particle derivative *substantial derivative *substantive derivative *Stokes derivative *total derivative, although the material derivative is actually a special case of the
total derivative In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with res ...


Definition

The material derivative is defined for any
tensor field In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...
''y'' that is ''macroscopic'', with the sense that it depends only on position and time coordinates, : \frac \equiv \frac + \mathbf\cdot\nabla y, where is the
covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
of the tensor, and is the
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
. Generally the convective derivative of the field , the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field , or as involving the streamline
directional derivative In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. The directional derivative of a multivariable differentiable (scalar) function along a given vect ...
of the field , leading to the same result. Only this spatial term containing the flow velocity describes the transport of the field in the flow, while the other describes the intrinsic variation of the field, independent of the presence of any flow. Confusingly, sometimes the name "convective derivative" is used for the whole material derivative , instead for only the spatial term . The effect of the time-independent terms in the definitions are for the scalar and tensor case respectively known as
advection In the fields of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is a ...
and convection.


Scalar and vector fields

For example, for a macroscopic
scalar field In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
and a macroscopic
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
the definition becomes: \begin \frac &\equiv \frac + \mathbf\cdot\nabla \varphi, \\ pt \frac &\equiv \frac + \mathbf\cdot\nabla \mathbf. \end In the scalar case is simply the
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
of a scalar, while is the covariant derivative of the macroscopic vector (which can also be thought of as the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...
of as a function of ). In particular for a scalar field in a three-dimensional
Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
, the components of the velocity are , and the convective term is then: \mathbf\cdot \nabla \varphi = u_1 \frac + u_2 \frac + u_3 \frac .


Development

Consider a scalar quantity , where is time and is position. Here may be some physical variable such as temperature or chemical concentration. The physical quantity, whose scalar quantity is , exists in a continuum, and whose macroscopic velocity is represented by the vector field . The (total) derivative with respect to time of is expanded using the multivariate
chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...
: \frac\varphi(\mathbf x(t), t) = \frac + \dot \mathbf x \cdot \nabla \varphi. It is apparent that this derivative is dependent on the vector \dot \mathbf x \equiv \frac, which describes a ''chosen'' path in space. For example, if \dot \mathbf x= \mathbf 0 is chosen, the time derivative becomes equal to the partial time derivative, which agrees with the definition of a
partial 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 (as opposed to the total derivative, in which all variables are allowed to vary). P ...
: 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 \dot \mathbf x = 0, 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. In which case the term / is sufficient to describe the rate of change of temperature. If the sun is not warming the water (i.e. / = 0), but the path is not a standstill, the time derivative of 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 at a constant high temperature and the other end at a constant low 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 and the second term on the right \dot \mathbf x \cdot \nabla \varphi is sufficient to describe the rate of change of temperature. A temperature sensor attached to the swimmer would show temperature varying with time, simply due to the temperature variation from one end of the pool to the other. The material derivative finally is obtained when the path is chosen to have a velocity equal to the fluid velocity \dot \mathbf x = \mathbf u. That is, the path follows the fluid current described by the fluid's velocity field . So, the material derivative of the scalar is \frac = \frac + \mathbf u \cdot \nabla \varphi. An example of this case is a lightweight, neutrally buoyant particle swept along a flowing river and experiencing temperature changes as it does so. The temperature of the water locally may be increasing due to one portion of the river being sunny and the other in a shadow, or 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 In the fields of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is a ...
'' (or convection if a vector is being transported). The definition above relied on the physical nature of a fluid current; however, no laws of physics were invoked (for example, it was assumed that a lightweight particle in a river will follow the velocity of the water), but it turns out that many physical concepts can be described concisely using the material derivative. The general case of advection, however, relies on conservation of mass of 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 In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
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 coordinates In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...
, the -th component of the convection term of the material derivative of a
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
\mathbf is given by left(\mathbf \cdot \nabla \right)\mathbfj = \sum_i \frac \frac + \frac\left(u_j \frac - u_i \frac\right), where the are related to the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
s by h_i = \sqrt. In the special case of a three-dimensional
Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
(''x'', ''y'', ''z''), and being a 1-tensor (a vector with three components), this is just: (\mathbf\cdot\nabla) \mathbf = \begin \displaystyle u_x \frac + u_y \frac+u_z \frac \\ \displaystyle u_x \frac + u_y \frac+u_z \frac \\ \displaystyle u_x \frac + u_y \frac+u_z \frac \end = \frac\mathbf where \frac is a
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...
. There is also a vector-dot-del identity and the material derivative for a vector field \mathbf A can be expressed as: : .


See also

*
Navier–Stokes equations The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
*
Euler equations (fluid dynamics) In fluid dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity ...
* Derivative (generalizations) *
Lagrangian and Eulerian specification of the flow field Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem wit ...
* Lie derivative *
Levi-Civita connection In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ( pseudo-) Riemannian ...
* Spatial acceleration * Spatial gradient


References


Further reading

* * {{cite book, first1=Michael, last1=Lai, first2=Erhard, last2=Krempl, first3=David, last3=Ruben , title=Introduction to Continuum Mechanics, isbn=978-0-7506-8560-3, publisher=Elsevier, edition=4th , year=2010 Fluid dynamics Multivariable calculus Rates Generalizations of the derivative