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In
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such mo ...
, the material derivative describes the time rate of change of some physical quantity (like
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
or
momentum In Newtonian mechanics, momentum (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. If is an object's mass ...
) of a
material element In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constant, ...
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 and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, 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 expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
of the fluid. In which case, the material derivative then describes the temperature change of a certain
fluid parcel In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constant, ...
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


Definition

The material derivative is defined for any
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
''y'' that is ''macroscopic'', with the sense that it depends only on position and time coordinates, : :\frac \equiv \frac + \mathbf\cdot\nabla y, where ∇''y'' is the
covariant 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 differe ...
of the tensor, and u(x, ''t'') 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 u·∇''y'', the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline
tensor 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 different ...
of the field u·(∇''y''), or as involving the streamline
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
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 ''D/Dt'', instead for only the spatial term u·∇. The effect of the time-independent terms in the definitions are for the scalar and tensor case respectively known as
advection In the field 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 al ...
and convection.


Scalar & vector fields

For example, for a macroscopic
scalar field In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity ...
and a macroscopic vector field the definition becomes: :\begin \frac &\equiv \frac + \mathbf\cdot\nabla \varphi, \\ pt \frac &\equiv \frac + \mathbf\cdot\nabla \mathbf. \end In the scalar case ∇''φ'' is simply the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of a scalar, while ∇A 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. When this matrix is square, that is, when the function takes the same number of variable ...
of A as a function of x). In particular for a scalar field in a three-dimensional
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
(''x''1, ''x''2, ''x''3), the components of the velocity u are ''u''1, ''u''2, ''u''3, and the convective term is then: : \mathbf\cdot\nabla \varphi = u_1 \frac + u_2 \frac + u_3 \frac .


Development

Consider a scalar quantity ''φ'' = ''φ''(x, ''t''), where ''t'' is time and x is position. Here ''φ'' may be some physical variable such as temperature or chemical concentration. The physical quantity, whose scalar quantity is ''φ'', exists in a continuum, and whose macroscopic velocity is represented by the vector field u(x, ''t''). The (total) derivative with respect to time of ''φ'' is expanded using the multivariate
chain rule In calculus, the chain rule is a formula that expresses the derivative of the 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(x)=f(g(x)) for every , ...
: :\frac\varphi(\mathbf x, 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 x(''t'') 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). Pa ...
: 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 \frac is sufficient to describe the rate of change of temperature. If the sun is not warming the water (i.e. \frac=0 ), but the path x(''t'') is not a standstill, the 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 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 x(''t'') 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 u. So, the material derivative of the scalar ''φ'' 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 field 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 al ...
'' (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 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 different ...
; for
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other 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.


Orthogonal coordinates

It may be shown that, in
orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
, the ''j''-th component of the convection term of the material derivative is given by : left(\mathbf\cdot\nabla \right)\mathbfj = \sum_i \frac \frac + \frac\left(u_j \frac - u_i \frac\right), where the ''h''''i'' 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 allow ...
s by :h_i = \sqrt. In the special case of a three-dimensional
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
(''x'', ''y'', ''z''), and A 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. When this matrix is square, that is, when the function takes the same number of variable ...
.


See also

*
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
*
Euler equations (fluid dynamics) In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations wit ...
* Derivative (generalizations) *
Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
*
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 (i.e. affine connection) that preserves ...
*
Spatial acceleration In physics, the study of rigid body motion allows for several ways to define the acceleration of a body. The usual definition of acceleration entails following a single particle/point of a rigid body and observing its changes in velocity. Spatial a ...
* 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