In continuum mechanics, the focus of study is the behavior of a body on a length scale large compared with the size of molecules or the distances between them. The assumption therefore is that the body has a continuous structure. This is the continuum hypothesis.
All variables are then continuous functions of position in the fluid and a definite meaning for the notion of properties at a point.
Deformations are defined only relative to a reference frame. A reference frame is a body whose component parts are fixed relative to each other.
There is an infinite set of reference frames, of which a subset are inertial reference frames. An inertial reference frame is a reference frame in which linear and angular momentum are conserved. Alternatively, they can be defined as reference frames in which a particle moving without action from any forces travels in a straight line with a constant velocity.
For some purposes, the Earth can be regarded as an inertial reference frame whereas a rotating object cannot.
Given a reference frame, a three-dimensional Euclidean space can be fixed to it and a three-dimensional Coordinate System selected to span the space. A three-dimensional coordinate system is defined by an origin O and three noncoplanar directions, ε_{1}, ε_{2} and ε_{3}. There exists an infinite set of coordinate systems. Always, an orthogonal right-handed system is chosen. An orthogonal system is one for which the directions associated with ε_{1}, ε_{2} and ε_{3} are mutually perpendicular or normal. A right-handed system is one such that, if ε_{1} and ε_{2} are orientated in the plane of the page, ε_{3} is orientated out of the page (i.e., toward the viewer). The three quantities i_{z1}, i_{z2} and i_{z3} denote unit vectors (i.e., vectors of unit magnitude) aligned in the ε_{1}, ε_{2} and ε_{3} directions, respectively.
The two most common coordinate systems are:
The directions of unit vectors i_{r} and θ in cylindrical polar coordinates change with position in space. Cylindrical polar coordinates are related to rectangular ones as follows:
Since the choice of coordinate systems is motivated by convenience, it follows, for example, that rectangular coordinates would reasonably be used in problems involving flows past flat plates and cylindrical polar coordinates in problems involving flows through pipes of circular cross-section.
A scalar (or 0^{th} order tensor) field is a quantity with which only magnitude can be associated. An example is the density field, ρ. Scalar fields have one component, s.
A vector (or 1^{st} order tensor) field is a quantity with which magnitude and direction are associated. An example is the velocity field, u. Vector fields have three components.
A tensor (or 2^{nd} order tensor) is a quantity with which magnitude and two directions are associated. An example is the stress field, τ, since this is a force (a vector with one associated direction) per unit area (another vector with another associated direction). Tensor fields have nine components corresponding to the nine possible combinations of the two base vectors, i.e.,
There is no simple physical interpretation of the entity called a tensor; it is a mathematical entity that represents a physical entity and is defined such that its components transform in a certain way during a change of reference frame.
Vectors and tensors are written in boldface.
Operations with scalars, vectors and tensors. A tensor, T may be written as:
the transpose of T is then
For all scalar fields, s, vector fields, v, a and b and tensor fields, T, A and B:
Scalar products:
Trace:
Vector product:
Unit tensor:
Tensor symmetry:
Dyadic product:
Vector Laplacian:
Identities:
Divergence theorem:
For a volume V enclosed by an orientable (two-sided) surface, A,
where n is the unit outer normal to an element dA of the surface.
Kinematics is the study of motion without considering how it is caused.
Langrangian and Eulerian descriptions. There are two possible descriptions of fluid motion:
To identify a material point (fluid element), X such that X is at x_{0} at time t_{0}, then at time t, X is at x_{0} + Δx (see Figure 3). The velocity of X could be written
(25)this is the Lagrangian description and gives the velocity of material points as a function of time.
The velocity of the fluid at a particular point in space and at a particular time is specified by u(x, t), which, if specified at all points x in a fluid, would give a complete picture of fluid motion. This is the Eulerian description.
In practice, the Lagrangian description is usually cumbersome and the. Eulerian description is almost invariably adopted.
Differentiation following the motion. In an Eulerian description, if the velocity at x over a period of time is observed velocity changes can be noted, but (∂u/∂)_{x} is not the acceleration of the fluid because particles at x change with time. However, in a Lagrangian description for a particular fluid particle X, (∂u/∂)_{x} is the acceleration of fluid.
In order to use the Eulerian description, the acceleration of the fluid must be known. For small Δt, and hence small Δx, a Taylor series expansion in space and time leads to
and so
Taking the limit as Δt → 0,
where D/Dt is the substantial (material) derivative written as:
for any scalar, vector or tensor quantity, f.
Rate of strain and vorticity. The rate of strain (or strain rate, or rate of deformation) tensor field e is defined as:
and the vorticity (or spin) tensor field, w, as:
Evidently e is symmetric (e = e^{T}) and w is antisymmetric (w = - w^{T}), also
and so, to within a factor of two, e is the symmetric part and w, the antisymmetric part of u.
It can be shown that there exists a vector field, ω, also called the vorticity such that, for an arbitrary vector field, v
and equivalently,
In rectangular coordinates, with velocity components u_{x}, u_{y} and u_{z} in the x, y and z directions, respectively,
In cylindrical polar coordinates with velocity components u_{r}, u_{θ} and u_{z} in the r, θ and z directions, respectively,
Irrotational and solenoidal fields. An irrotational flow is one in which the angular velocity and hence the vorticity vector, ω (which is a measure of the local rotation in a flow) vanishes everywhere, i.e.,
For any scalar field, s, so a function φ is defined such that
which satisfies
so that u is irrotational.
Conversely, it can be shown that if u is irrotational, a scalar field exists such that Eq. (44) is true. The scalar field, φ, is called a scalar velocity potential.
A solenoidal flow is one for which
It will be shown later (in conservation equations) that any incompressible flow is solenoidal.
Since for any vector field, v, defining
will have
So if the flow is solenoidal, a vector velocity potential, ψ, exists such that Eq. (47) is true. Note that the definition is not unique, but this does not cause any problems in practice.
There are two main types of force which may act on a body: long range, e.g., gravitational, and short range, e.g., intermolecular forces.
Long range forces, often called body or volume forces, vary slowly over the deforming body and are relatively easy to deal with.
Short range forces of molecular origin are generally negligible unless there is direct contact between interacting parts of a body, since molecular forces only extend over very small distances. Short range forces are viewed as surface forces and are the focus here.
Consider an elementary tetrahedron of the body, as shown in Figure 4, in which t is the stress tensor and n is the unit outer normal to face ABC. Stress is the force per unit area on the surface. The convention is that t is exerted by material on the side from which n points on the material to which n points and is positive when in the direction of n.
Thus t_{1}, t_{2} and t_{3} can be written in component form as:
Denoting the surface area of ABC by A, then the areas of PCB, PAC and PAB are A(n·i_{z1}), A(n·i_{z2}), and A(n·i_{z3}), respectively. If the tetrahedron is small enough, then the force acting on a given face is the product of the component of the stress vector and the area of the surface.
A force balance on the material inside the tetrahedron leads to
as surface area tends to zero. There are no inertial or body force terms in this expression since these vary as volume (i.e., l^{3}, where l is a typical linear dimension of the tetrahedron), whereas the surface forces vary as area (l^{2}), and so the former becomes negligible when l is very small. Thus,
where τ is a second order tensor known as the stress tensor. This is Cauchy’s fundamental theorem for stress.
The diagonal elements of τ (τ _{11}, τ _{22} and τ _{33}) are referred to as the normal stress components and the remainder, as the shear stress components.
Having assumed the continuum hypothesis, it can now be supposed that mass, momentum and energy are conserved in any motion or flow. To mathematically represent the conservation of these quantities, the notion of control volumes must be introduced.
A control volume is a volume located in the body which is fixed with respect to a frame of reference (see Figure 5). Any convenient frame of reference may be chosen for the conservation of mass and energy, but an inertial frame of reference is required for momentum conservation. Note that control volumes should not be confused with material volumes; the latter contains a fixed amount of material, but are not fixed with respect to a reference frame and moves with the material.
The volume, V, is enclosed by the surface. A, where A is orientable (i.e., two-sided).
Considering the flow of mass through the control volume, V;
or
and since V is independent of time,
Applying the divergence theorem. Eq. (23), to the second term leads to
Since V is arbitrary, the only way for this expression to be always true is if the integrand itself is zero, i.e.,
This expression of the conservation of mass is often referred to as the continuity equation. Another form of this expression may be derived by applying the mathematical identity, Eq. (20), to ·(ρu) giving
which, upon substitution into Eq. (57), results in
Recalling the definition of the substantial derivative. Eq. (29), this may be written as:
Equations (57) and (60) are alternative forms of the continuity equation.
The continuity equation can be simplified if the material is incompressible (i.e., if the density of each material point is constant with respect to time), to give
Recalling Eq. (46), it can be seen that the velocity field of an incompressible material is solenoidal. Note that when reference is made to an incompressible flow it really means an isochoric (volume preserving) flow.
The mass conservation equation for a fluid of constant density, ρ, in rectangular coordinates with velocity components u_{x}, u_{y} and u_{z} in the x, y and z directions, respectively, is
and in cylindrical polar coordinates with velocity components u_{r}, u_{θ} and u_{z} in the r, θ and z directions, respectively,
Considering the flow of momentum (linear as opposed to angular) through control volume V;
which is written mathematically as:
where it has been assumed that all body forces are gravitational in origin. Since V is independent of time, the time derivative inside the integral sign in the first term can again be taken. Also, by using Cauchy’s fundamental theorem for stress, Eq. (53), the expression becomes
using the divergence theorem. Eq. (24):
and
Thus,
and since V is arbitrary
Applying the mathematical identity. Eq. (19), to the ·ρuu term results in
and substituting this expression into Eq. (69) leads to
Substitution of the conservation of mass, Eq. (57), into this expression gives
which is often referred to as the equation of motion and is an expression of Cauchy’s first law of motion.
A scalar field, p, can now be defined which is conceptually similar to the static pressure and which reduces to the static pressure when the fluid is at rest:
it is thus the mean normal stress in a fluid. It is now possible to decompose the total stress, τ, thus:
where I is the unit tensor (see above section on scalar, vector and tensor fields) and τ_{D} is the deviatoric stress tensor field. Note that the isotropic components from the total stress tensor have been stripped out and these are identified with the pressure (the negative sign accounts for the fact that pressure is positive). Henceforth only deviatoric stress is referred to unless otherwise stated, so the subscript D is dropped.
Substituting this decomposition into Eq. (72) gives
where the mathematical identity given by Eq. (22) has been applied to the –pI term.
If a material is incompressible, gravitational acceleration can be eliminated by defining a modified pressure, , such that
thus
In an inertial frame of reference, angular momentum is conserved. By following the same procedure for mass and momentum conservation, the expression for a nonpolar material becomes
i.e., the stress tensor is symmetric (see section on Scalar, vector and tensor fields). A nonpolar material is one in which the torques within it arise only as the moments of direct forces. Most materials, and certainly most fluids, are nonpolar; so it is generally assumed that conservation of angular momentum implies symmetry of the stress tensor.
Taking the dot product of each term in the linear momentum conservation, Eq. (75) results in
or
or
where (l/2)|u|^{2} is the kinetic energy per unit mass. These equations express the conservation of mechanical energy.
The energy of a material comprises internal, kinetic and potential energy. Defining u as the internal energy per unit mass and −g·x as the potential energy per unit mass, in the absence of heat generation by chemical or nuclear reaction, conservation of energy requires that:
Mathematically,
where q is the heat flux vector and τ is the total (not deviatoric) stress. The last term is the rate at which surface stresses do work on the material inside A, i.e., force × velocity (∫_{A}t·udA) which is ∫_{A}n·(τ·u)dA by Cauchy’s fundamental theorem for stress [Eq. (53)].
Applying the divergence theorem and noting that V is independent of time leads to
Considering the left-hand side of this equation; expanding the internal and kinetic energy terms gives
which, upon substitution of mass conservation [Eq. (57)] becomes
Similarly for the potential energy terms,
which, upon substitution of mass conservation and noting that g is independent of time and position, results in
Substituting Eqs. (85) and (87) into Eq. (83) and decomposing the stress tensor (as described in linear momentum conservation) gives
where τ is now the deviatoric stress.
Substitution of the mechanical energy [Eq. (81)] into this result gives
where Eq. (21) has been used and the double dot product of two tensors is defined in the section on scalar, vector and tensor fields. This is the equation of conservation of energy, it is an expression of the first law of thermodynamics. The physical meanings of the terms in this equation are as follows:
Term (a) is the rate of gain of internal energy.
Term (b) is the rate of internal energy input by conduction.
Term (c) is the reversible rate of internal energy conversion.
Term (d) is the irreversible rate of internal energy increase by surface forces.
The conservation equations themselves are not sufficient to solve problems in fluid mechanics. This is because they describe the behavior of a completely general material which may be a fluid, but could equally well be a solid. What are required then, are additional relationships that represent the intrinsic response of a particular material. Such relationships are known as constitutive equations or, equivalently, Equations of State, because they describe the constitution or state of a material. Specifically, relationships expressing p, and τ in terms of ρ, u and u or T (and, perhaps, each other) are required. Then, there will be sufficient equations to determine all the unknowns and the equation set is said to be closed. Constitutive equations cannot be obtained from continuum mechanics alone, instead they are generally based on phenomenological arguments and experiment.
Pressure. For compressible materials, the following constitutive equation for pressure applies:
An example is that for a perfect gas:
where R is the universal gas constant and M, the molecular weight or relative molar mass of the gas.
For incompressible material,
and p, which is called the ‘pressure’ but is not necessarily the thermodynamic pressure, is determined as part of the solution of the equations of motion.
Heat flux. An example of a constitutive equation for heat flux is Fourier’s law:
where λ denotes the thermal conductivity of the fluid. This experimental law states that heat flux is proportional to the temperature gradient, with the minus sign indicating that heat flows down a temperature gradient from a region of higher temperature to one of lower temperature. Most materials, and certainly most fluids, obey this law under normal conditions. Note however that in the energy balance equation includes heat transmission due to radiation as well as conduction, whereas Fourier’s law incorporates only conduction. The justification for using Fourier’s law is that, since radiation is proportional to T^{4}, where T is the absolute temperature, heat flux due to radiation can be neglected where modest temperature differences are involved.
The energy equation [Eq. (89)] for a Fourier fluid becomes
It is easy to show that
and furthermore, heat capacity can be defined as:
so
where c is assumed constant. Generally, c ≡ c_{p}, the heat capacity at constant pressure, is used. Equation 94 thus becomes
which expresses conservation of energy for a Fourier fluid of constant density, ρ, constant viscosity, η and constant thermal conductivity, λ. This equation is, in fact, unchanged if the fluid is of variable viscosity. In rectangular coordinates with velocity components u_{x}, u_{y} and u_{z} in the x, y and z directions, respectively, it becomes
and in cylindrical polar coordinates with velocity components u_{r}, u_{θ} and u_{z} in the r, θ and z directions, respectively,
Stress. The simplest constitutive equation for stress is that of the inviscid fluid, in which there is no stress so that
Obviously, no such fluid exists in practice; however this idealization is very useful due to its simplicity and is often utilized, most notably in the theory of gas dynamics (where viscous effects are generally small). It is also used for theories of rapid liquid flows. (See Flow of Fluids.)
The next simplest constitutive equation for stress is where the stress is linearly related to the rate of strain, thus
where η is the (dynamic) viscosity of the fluid. A fluid obeying this constitutive equation is said to be Newtonian; many common fluids are Newtonian. It is apparent that a Newtonian fluid is analogous to a Fourier material, for which it will be recalled that heat flux, , is linearly related to temperature gradient, T.
If this result is used in the linear momentum conservation equation [Eq. (75)], the following expression is obtained:
Using mass conservation for an incompressible fluid [Eq. (61)] along with Eqs. (15), (30), (14) and (18), it can be shown that
and so
which are the Navier-Stokes equations. They cannot be solved exactly for the general case owing to the nonlinearity in u and thus, recourse is usually made to the following two methods:
Obtain approximate solutions to the exact equations by numerical simulation (computer methods).
Obtain exact solutions to simplified forms of the equations.
A basis for the latter approach is discussed briefly below in the section on Euler Equation.
The Navier-Stokes equations for a fluid of constant density, ρ, and constant viscosity, η, may be written in rectangular coordinates as:
for each direction, i (i = x, y, z), where u_{x}, u_{y} and u_{z} are the velocity components in the x, y and z directions, respectively.
In cylindrical polar coordinates with velocity components u_{r}, u_{θ} and u_{z} in the r, θ and z directions, respectively,
In this section, order of magnitude arguments based on a dimensionless formulation are used to obtain approximate forms of the Navier-Stokes equations.
In order to make the Navier-Stokes equations dimensionless, a characteristic length, l_{c}, and a characteristic velocity, u_{c}, need to be defined. These should be chosen such that the dimensionless variables are all of order one. The choice is usually straightforward; for flow in a pipe for example, l_{c} might be the pipe radius and u_{c}, the mean axial velocity. The following dimensionless variables are thus obtained:
where g is the magnitude of the gravitational acceleration.
Substitution into Eq. (105) yields
Defining the following dimensionless numbers (see Froude Number and Reynolds Number):
Eq. (105) can be written
Since Re may be interpreted as the ratio of inertial to viscous forces, the viscous terms in the Navier-Stokes equation could be neglected when Re is very high (and conversely, the inertial terms neglected when Re is very low). This can be seen from Eq. (113) since the inertial terms (first two) will be of order one and the viscous terms (last one) of order 1/Re, i.e., much less than one when Re is large. The Navier-Stokes equations become
which is the Euler equation. It describes the motion of an inviscid fluid.
From Eq. (18) comes
where ω is the vorticity (see section on Rate of Strain and vorticity) defined by
Substitution of Eq. (115) into (114) gives
where |u| is the speed, |u| = √(u·u).
If the flow is irrotational such that the vorticity vanishes, then from the above section on Irrotational and Solenoidal forms, u can be expressed in the form
where φ is a scalar potential field. Noting that g = (g·x), the equation becomes
which, since the operators ∂/∂t and are commutative
yields
This can be integrated to give
where ζ is, at most, a function of time. This is the Bernoulli equation. It can often be simplified; for example, if the flow is steady (independent of time) the
where ζ is a constant and u = φ has been used again.
The terms in this equation may be identified with kinetic, potential and pressure energy, respectively. The equation admits to the following physical interpretation: for an incompressible, inviscid fluid in steady flow, the sum of the kinetic, potential and pressure energies is constant.
Nomenclature
a vector field
A area
A tensor field
b vector field
B tensor field
c specific heat
e rate of strain
f arbitrary scalar, vector or tensor quantity
Fr Froude number
g gravitational acceleration
i unit vector
I unit tensor
l length
M molecular weight or relative molecular mass
n unit normal
p pressure
modified pressure
heat flux
r, θ, z cylindrical polar coordinates
R universal gas constant
Re Reynolds number
s scalar field
t time
t stress (vector)
T temperature
T tensor field
u internal energy per unit mass
u velocity
v vector field
V volume
w vorticity (tensor)
x, y, z rectangular coordinates
x position
X material point
ζ constant
λ thermal conductivity
η viscosity
ε_{1}, ε_{2}, ε_{3} right-handed orthogonal coordinates
ρ density
τ stress (tensor)
φ scalar potential
Ψ vector potential
ω vorticity (vector)
Subscripts
c characteristic value
Superscripts
ˆ Lagrangian specification
~ dimensionless variable
REFERENCES
Aris, R. (1962) Vectors, Tensors and the Basic Equations of Fluid Mechanics. Prentice-Hall, Englewood Cliffs, N. J.
Bird, R. B., Stewart, W. E., and Lightfoot, E. N. (1960) Transport Phenomena. Wiley. Kay, J. M. and Nedderman, R. M. (1985) Fluid Mechanics and Transfer Processes. Cambridge University Press.
Richardson, S. M. (1989) Fluid Mechanics. Hemisphere, New York.
Slattery, J. C. (1972) Momentum, Energy, and Mass Transfer Continua. McGraw-Hill.
References
- Aris, R. (1962) Vectors, Tensors and the Basic Equations of Fluid Mechanics. Prentice-Hall, Englewood Cliffs, N. J.
- Bird, R. B., Stewart, W. E., and Lightfoot, E. N. (1960) Transport Phenomena. Wiley. Kay, J. M. and Nedderman, R. M. (1985) Fluid Mechanics and Transfer Processes. Cambridge University Press.
- Richardson, S. M. (1989) Fluid Mechanics. Hemisphere, New York.
- Slattery, J. C. (1972) Momentum, Energy, and Mass Transfer Continua. McGraw-Hill.