Material derivative

Last updated December 08, 2024

In continuum mechanics, the material derivative^[1]^[2] describes the time rate of change of some physical quantity (like heat or momentum) 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.^[3]

Other names

There are many other names for the material derivative, including:

advective derivative^[4]
convective derivative^[5]
derivative following the motion^[1]
hydrodynamic derivative^[1]
Lagrangian derivative^[6]
particle derivative^[7]
substantial derivative^[1]
substantive derivative^[8]
Stokes derivative^[8]
total derivative,^[1]^[9] although the material derivative is actually a special case of the total derivative ^[9]

Definition

The material derivative is defined for any tensor field y that is macroscopic, with the sense that it depends only on position and time coordinates, $y = y (x, t)$ : ${\frac {\mathrm {D} y}{\mathrm {D} t}}\equiv {\frac {\partial y}{\partial t}}+\mathbf {u} \cdot \nabla y,$ where $\nabla y$ is the covariant derivative of the tensor, and $u (x, t)$ is the flow velocity. Generally the convective derivative of the field $u \cdot\nabla y$ , the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field $u \cdot(\nabla y)$ , or as involving the streamline directional derivative of the field $(u \cdot\nabla) y$ , leading to the same result.^[10] 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 $u \cdot\nabla$ .^[2] The effect of the time-independent terms in the definitions are for the scalar and tensor case respectively known as advection and convection.

Scalar and vector fields

For example, for a macroscopic scalar field $φ (x, t)$ and a macroscopic vector field $A (x, t)$ the definition becomes: ${\begin{aligned}{\frac {\mathrm {D} \varphi }{\mathrm {D} t}}&\equiv {\frac {\partial \varphi }{\partial t}}+\mathbf {u} \cdot \nabla \varphi ,\\[3pt]{\frac {\mathrm {D} \mathbf {A} }{\mathrm {D} t}}&\equiv {\frac {\partial \mathbf {A} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {A} .\end{aligned}}$

In the scalar case $\nabla φ$ is simply the gradient of a scalar, while $\nabla A$ is the covariant derivative of the macroscopic vector (which can also be thought of as the Jacobian matrix of $A$ as a function of $x$ ). In particular for a scalar field in a three-dimensional Cartesian coordinate system $(x 1, x 2, x 3)$ , the components of the velocity $u$ are $u 1, u 2, u 3$ , and the convective term is then: $\mathbf {u} \cdot \nabla \varphi =u_{1}{\frac {\partial \varphi }{\partial x_{1}}}+u_{2}{\frac {\partial \varphi }{\partial x_{2}}}+u_{3}{\frac {\partial \varphi }{\partial x_{3}}}.$

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: ${\frac {\mathrm {d} }{\mathrm {d} t}}\varphi (\mathbf {x} ,t)={\frac {\partial \varphi }{\partial t}}+{\dot {\mathbf {x} }}\cdot \nabla \varphi .$

It is apparent that this derivative is dependent on the vector ${\dot {\mathbf {x} }}\equiv {\frac {\mathrm {d} \mathbf {x} }{\mathrm {d} t}},$ 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: 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 ${\partial \varphi }/{\partial t}$ is sufficient to describe the rate of change of temperature.

If the sun is not warming the water (i.e. ${\partial \varphi }/{\partial t}=0$ ), 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 ${\frac {\mathrm {D} \varphi }{\mathrm {D} t}}={\frac {\partial \varphi }{\partial t}}+\mathbf {u} \cdot \nabla \varphi .$

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 (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 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, the $j$ -th component of the convection term of the material derivative of a vector field $\mathbf {A}$ is given by^[11] $[\left(\mathbf {u} \cdot \nabla \right)\mathbf {A} ]_{j}=\sum _{i}{\frac {u_{i}}{h_{i}}}{\frac {\partial A_{j}}{\partial q^{i}}}+{\frac {A_{i}}{h_{i}h_{j}}}\left(u_{j}{\frac {\partial h_{j}}{\partial q^{i}}}-u_{i}{\frac {\partial h_{i}}{\partial q^{j}}}\right),$

where the $h i$ are related to the metric tensors by $h_{i}={\sqrt {g_{ii}}}.$

In the special case of a three-dimensional Cartesian coordinate system (x, y, z), and $A$ being a 1-tensor (a vector with three components), this is just: $(\mathbf {u} \cdot \nabla )\mathbf {A} ={\begin{pmatrix}\displaystyle u_{x}{\frac {\partial A_{x}}{\partial x}}+u_{y}{\frac {\partial A_{x}}{\partial y}}+u_{z}{\frac {\partial A_{x}}{\partial z}}\\\displaystyle u_{x}{\frac {\partial A_{y}}{\partial x}}+u_{y}{\frac {\partial A_{y}}{\partial y}}+u_{z}{\frac {\partial A_{y}}{\partial z}}\\\displaystyle u_{x}{\frac {\partial A_{z}}{\partial x}}+u_{y}{\frac {\partial A_{z}}{\partial y}}+u_{z}{\frac {\partial A_{z}}{\partial z}}\end{pmatrix}}={\frac {\partial (A_{x},A_{y},A_{z})}{\partial (x,y,z)}}\mathbf {u}$

where ${\frac {\partial (A_{x},A_{y},A_{z})}{\partial (x,y,z)}}$ is a Jacobian matrix.

There is also a vector-dot-del identity and the material derivative for a vector field $\mathbf {A}$ can be expressed as:

{\displaystyle (\mathbf {A} \cdot \nabla )\mathbf {A} ={\frac {1}{2}}\nabla |\mathbf {A} |^{2}-\mathbf {A} \times (\nabla \times \mathbf {A} )={\frac {1}{2}}\nabla |\mathbf {A} |^{2}+(\nabla \times \mathbf {A} )\times \mathbf {A} }.

Related Research Articles

In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field.

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

<span class="mw-page-title-main">Gradient</span> Multivariate derivative (mathematics)

In vector calculus, the gradient of a scalar-valued differentiable function $of several variables is the vector field whose value at a point gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of . If the gradient of a function is non-zero at a point, the direction of the gradient is the direction in which the function increases most quickly from, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function may be defined by:$

<span class="mw-page-title-main">Navier–Stokes equations</span> Equations describing the motion of viscous fluid substances

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 George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity present in the flow.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols $, (where is the nabla operator), or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δ f (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (p) .$

In fluid dynamics, Stokes' law gives the frictional force – also called drag force – exerted on spherical objects moving at very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of change of the fluid's velocity vector.

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 by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field under the line integral being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected.

In fluid mechanics, or more generally continuum mechanics, incompressible flow refers to a flow in which the material density of each fluid parcel — an infinitesimal volume that moves with the flow velocity — is time-invariant. An equivalent statement that implies incompressible flow is that the divergence of the flow velocity is zero.

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group $O(p, q)$ . As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold. The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.

The following are important identities involving derivatives and integrals in vector calculus.

The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space rather than just the real line.

There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation are listed below.

The derivation of the Navier–Stokes equations as well as their application and formulation for different families of fluids, is an important exercise in fluid dynamics with applications in mechanical engineering, physics, chemistry, heat transfer, and electrical engineering. A proof explaining the properties and bounds of the equations, such as Navier–Stokes existence and smoothness, is one of the important unsolved problems in mathematics.

The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

References

1 2 3 4 5 Bird, R.B.; Stewart, W.E.; Lightfoot, E.N. (2007). Transport Phenomena (Revised Second ed.). John Wiley & Sons. p. 83. ISBN 978-0-470-11539-8.
1 2 Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. pp. 72–73. ISBN 0-521-66396-2.
↑ Trenberth, K. E. (1993). Climate System Modeling. Cambridge University Press. p. 99. ISBN 0-521-43231-6.
↑ Majda, A. (2003). Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics. Vol. 9. American Mathematical Society. p. 1. ISBN 0-8218-2954-8.
↑ Ockendon, H.; Ockendon, J.R. (2004). Waves and Compressible Flow. Springer. p. 6. ISBN 0-387-40399-X.
↑ Mellor, G.L. (1996). Introduction to Physical Oceanography. Springer. p. 19. ISBN 1-56396-210-1.
↑ Stoker, J.J. (1992). Water Waves: The Mathematical Theory with Applications. Wiley. p. 5. ISBN 0-471-57034-6.
1 2 Granger, R.A. (1995). Fluid Mechanics. Courier Dover Publications. p. 30. ISBN 0-486-68356-7.
1 2 Landau, L.D.; Lifshitz, E.M. (1987). Fluid Mechanics. Course of Theoretical Physics. Vol. 6 (2nd ed.). Butterworth-Heinemann. pp. 3–4 & 227. ISBN 0-7506-2767-0.
↑ Emanuel, G. (2001). Analytical fluid dynamics (second ed.). CRC Press. pp. 6–7. ISBN 0-8493-9114-8.
↑ Eric W. Weisstein. "Convective Operator". MathWorld . Retrieved 2008-07-22.