The Cauchy momentum equation is a vector
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
put forth by
Cauchy that describes the non-relativistic
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 ...
transport
Transport (in British English), or transportation (in American English), is the intentional movement of humans, animals, and goods from one location to another. Modes of transport include air, land ( rail and road), water, cable, pipelin ...
in any
continuum.
Main equation
In convective (or
Lagrangian) form the Cauchy momentum equation is written as:
:
where
*
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 ...
vector field, which depends on time and space, (unit:
)
*
is
time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
, (unit:
)
*
is the
material derivative of
, equal to
, (unit:
)
*
is the
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
at a given point of the continuum (for which the
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...
holds), (unit:
)
*
is the
stress tensor, (unit:
)
*
is a vector containing all of the accelerations caused by
body force
In physics, a body force is a force that acts throughout the volume of a body.
Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref>
Forces due to gravity, electric fields and magnetic fields are examples of body forces. ...
s (sometimes simply
gravitational acceleration
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodie ...
), (unit:
)
*
is the
divergence
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 ...
of stress tensor.
(unit:
)
Commonly used SI units are given in parentheses although the equations are general in nature and other units can be entered into them or units can be removed at all by
nondimensionalization
Nondimensionalization is the partial or full removal of dimensional analysis, physical dimensions from an mathematical equation, equation involving physical quantity, physical quantities by a suitable substitution of variables. This technique can ...
.
Note that only we use column vectors (in the
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 ...
) above for clarity, but the equation is written using physical components (which are neither
covariants ("column") nor
contravariants ("row") ).
However, if we chose a non-orthogonal
curvilinear coordinate system
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
, then we should calculate and write equations in covariant ("row vectors") or contravariant ("column vectors") form.
After an appropriate change of variables, it can also be written in
conservation form:
:
where is the
momentum density at a given space-time point, is the flux associated to the momentum density, and contains all of the
body force
In physics, a body force is a force that acts throughout the volume of a body.
Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref>
Forces due to gravity, electric fields and magnetic fields are examples of body forces. ...
s per unit volume.
Differential derivation
Let us start with the
generalized momentum conservation principle which can be written as follows: "The change in system momentum is proportional to the resulting force acting on this system". It is expressed by the formula:
:
where
is momentum in time ,
is force averaged over
. After dividing by
and passing to the limit
we get (
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
):
:
Let us analyse each side of the equation above.
Right side
We split the forces into
body force
In physics, a body force is a force that acts throughout the volume of a body.
Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref>
Forces due to gravity, electric fields and magnetic fields are examples of body forces. ...
s
and
surface forces
:
Surface forces act on walls of the cubic fluid element. For each wall, the ''X'' component of these forces was marked in the figure with a cubic element (in the form of a product of stress and surface area e.g.
with units
).
:
Adding forces (their ''X'' components) acting on each of the cube walls, we get:
:
After ordering
and performing similar reasoning for components
(they have not been shown in the figure, but these would be vectors parallel to the Y and Z axes, respectively) we get:
:
:
:
We can then write it in the symbolic operational form:
:
There are mass forces acting on the inside of the control volume. We can write them using the acceleration field
(e.g. gravitational acceleration):
:
Left side
Let us calculate momentum of the cube:
:
Because we assume that tested mass (cube)
is constant in time, so
:
Left and Right side comparison
We have
:
then
:
then
:
Divide both sides by
, and because
we get:
:
which finishes the derivation.
Integral derivation
Applying
Newton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in mo ...
(th component) to a
control volume
In continuum mechanics and thermodynamics, a control volume (CV) is a mathematical abstraction employed in the process of creating mathematical models of physical processes. In an inertial frame of reference, it is a fictitious region of a given v ...
in the continuum being modeled gives:
:
Then, based on the
Reynolds transport theorem and using
material derivative notation, one can write
:
where represents the control volume. Since this equation must hold for any control volume, it must be true that the integrand is zero, from this the Cauchy momentum equation follows. The main step (not done above) in deriving this equation is establishing that the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the stress tensor is one of the forces that constitutes .
Conservation form
The Cauchy momentum equation can also be put in the following form:
simply by defining:
:
where is the
momentum density at the point considered in the continuum (for which the
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...
holds), is the flux associated to the momentum density, and contains all of the
body force
In physics, a body force is a force that acts throughout the volume of a body.
Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref>
Forces due to gravity, electric fields and magnetic fields are examples of body forces. ...
s per unit volume. is the
dyad of the velocity.
Here and have same number of dimensions as the flow speed and the body acceleration, while , being a
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 ...
, has .
[In 3D for example, with respect to some coordinate system, the vector has 3 components, while the tensors and have 9 (3×3), so the explicit forms written as matrices would be:
:
Note, however, that if symmetrical, will only contain 6 '']degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
''. And 's symmetry is equivalent to 's symmetry (which will be present for the most common Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
s), since dyads of vectors with themselves are always symmetrical.
In the Eulerian forms it is apparent that the assumption of no deviatoric stress brings Cauchy equations to the
Euler equations.
Convective acceleration
A significant feature of the Navier–Stokes equations is the presence of convective acceleration: the effect of time-independent acceleration of a flow with respect to space. While individual continuum particles indeed experience time dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle.
Regardless of what kind of continuum is being dealt with, convective acceleration is a
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
effect. Convective acceleration is present in most flows (exceptions include one-dimensional incompressible flow), but its dynamic effect is disregarded in
creeping flow (also called Stokes flow). Convective acceleration is represented by the
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
quantity , which may be interpreted either as or as , with the
tensor derivative of the velocity vector . Both interpretations give the same result.
Advection operator vs tensor derivative
The convection term
can be written as , where is the
advection operator. This representation can be contrasted to the one in terms of the tensor derivative.
[ The tensor derivative is the component-by-component derivative of the velocity vector, defined by , so that
:
]
Lamb form
The vector calculus identity of the cross product of a curl holds:
:
where the Feynman subscript notation is used, which means the subscripted gradient operates only on the factor .
Lamb in his famous classical book Hydrodynamics (1895), used this identity to change the convective term of the flow velocity in rotational form, i.e. without a tensor derivative:
:
where the vector is called the Lamb vector In fluid dynamics, Lamb vector is the cross product of vorticity vector and velocity vector of the flow field, named after the physicist Horace Lamb.Truesdell, C. (1954). The kinematics of vorticity (Vol. 954). Bloomington: Indiana University Pr ...
. The Cauchy momentum equation becomes:
:
Using the identity:
:
the Cauchy equation becomes:
:
In fact, in case of an external conservative field, by defining its potential :
:
In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes:
:
And by projecting the momentum equation on the flow direction, i.e. along a '' streamline'', the cross product disappears due to a vector calculus identity of the triple scalar product:
:
If the stress tensor is isotropic, then only the pressure enters: (where is the identity tensor), and the Euler momentum equation in the steady incompressible case becomes:
:
In the steady incompressible case the mass equation is simply:
:
that is, ''the mass conservation for a steady incompressible flow states that the density along a streamline is constant''. This leads to a considerable simplification of the Euler momentum equation:
:
The convenience of defining the total head for an inviscid liquid flow is now apparent:
:
in fact, the above equation can be simply written as:
:
That is, ''the momentum balance for a steady inviscid and incompressible flow in an external conservative field states that the total head along a streamline is constant''.
Irrotational flows
The Lamb form is also useful in irrotational flow, where the curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was ...
of the velocity (called vorticity) is equal to zero. In that case, the convection term in reduces to
:
Stresses
The effect of stress in the continuum flow is represented by the and terms; these are 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 ...
s of surface forces, analogous to stresses in a solid. Here is the pressure gradient and arises from the isotropic part of the Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
. This part is given by the normal stresses that occur in almost all situations. The anisotropic part of the stress tensor gives rise to , which usually describes viscous forces; for incompressible flow, this is only a shear effect. Thus, is the deviatoric stress tensor, and the stress tensor is equal to:
:
where is the identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
in the space considered and the shear tensor.
All non-relativistic momentum conservation equations, such as the Navier–Stokes equation, can be derived by beginning with the Cauchy momentum equation and specifying the stress tensor through a constitutive relation. By expressing the shear tensor in terms of viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
and fluid velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
, and assuming constant density and viscosity, the Cauchy momentum equation will lead to the 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 ...
. By assuming inviscid flow
In fluid dynamics, inviscid flow is the flow of an inviscid (zero-viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, suc ...
, the Navier–Stokes equations can further simplify to the Euler equations.
The divergence of the stress tensor can be written as
:
The effect of the pressure gradient on the flow is to accelerate the flow in the direction from high pressure to low pressure.
As written in the Cauchy momentum equation, the stress terms and are yet unknown, so this equation alone cannot be used to solve problems. Besides the equations of motion—Newton's second law—a force model is needed relating the stresses to the flow motion. For this reason, assumptions based on natural observations are often applied to specify the stresses in terms of the other flow variables, such as velocity and density.
External forces
The vector field represents body force
In physics, a body force is a force that acts throughout the volume of a body.
Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref>
Forces due to gravity, electric fields and magnetic fields are examples of body forces. ...
s per unit mass. Typically, these consist of only gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
acceleration, but may include others, such as electromagnetic forces. In non-inertial coordinate frames, other "inertial accelerations" associated with rotating coordinates may arise.
Often, these forces may be represented as the gradient of some scalar quantity , with in which case they are called conservative force
In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done (the sum ...
s. Gravity in the direction, for example, is the gradient of . Because pressure from such gravitation arises only as a gradient, we may include it in the pressure term as a body force . The pressure and force terms on the right-hand side of the Navier–Stokes equation become
:
It is also possible to include external influences into the stress term rather than the body force term. This may even include antisymmetric stresses (inputs of angular momentum), in contrast to the usually symmetrical internal contributions to the stress tensor.
Nondimensionalisation
In order to make the equations dimensionless, a characteristic length and a characteristic velocity need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained:
:
Substitution of these inverted relations in the Euler momentum equations yields:
:
and by dividing for the first coefficient:
:
Now defining the Froude number
In continuum mechanics, the Froude number (, after William Froude, ) is a dimensionless number defined as the ratio of the flow inertia to the external field (the latter in many applications simply due to gravity). The Froude number is based on ...
:
:
the Euler number:
:
and the coefficient of skin-friction or the one usually referred as 'drag' co-efficient in the field of aerodynamics:
:
by passing respectively to the conservative variables
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization ...
, i.e. the momentum density and the force density
In fluid mechanics, the force density is the negative gradient of pressure. It has the physical dimensions of force per unit volume. Force density is a vector field representing the flux density of the hydrostatic force within the bulk of a ...
:
:
the equations are finally expressed (now omitting the indexes):
Cauchy equations in the Froude limit (corresponding to negligible external field) are named free Cauchy equations:
and can be eventually conservation equations
Conservation is the preservation or efficient use of resources, or the conservation of various quantities under physical laws.
Conservation may also refer to:
Environment and natural resources
* Nature conservation, the protection and manageme ...
. The limit of high Froude numbers (low external field) is thus notable for such equations and is studied with perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
.
Finally in convective form the equations are:
3D explicit convective forms
Cartesian 3D coordinates
For asymmetric stress tensors, equations in general take the following forms:
:
Cylindrical 3D coordinates
Below, we write the main equation in pressure-tau form assuming that the stress tensor is symmetrical ():
:
See also
*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 ...
*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 ...
*Burnett equations Burnett may refer to:
Places
;Antarctica
* Burnett Island, an island in the Swain Islands
;Australia
*Burnett County, New South Wales, a cadastral division
* The Burnett River in Queensland
* Burnett Heads, Queensland
* Shire of Burnett, a forme ...
* Chapman–Enskog expansion
Notes
References
{{reflist
Continuum mechanics
Equations of physics
Momentum
Partial differential equations