F. FLUID STATICS – THE STRESS TENSOR FOR A STATIONARY FLUID

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Basic Principles



the linear momentum equation, (2–32), reduces to a balance between body and surface

forces,

∇ · T + ρg = 0,



(2–58)



whereas the thermal energy equation reduces to the form (2–57). Although the equations are

thus considerably simplified for a stationary fluid, the basic problem of requiring constitutive

equations for T and q remains.

In this section, we consider an isothermal, stationary fluid. In this case, from thermodynamics, we know that the only surface force is the normal thermodynamic pressure, p.

The pressure at a point P acts normal to any surface through P with a magnitude that is

independent of the orientation of the surface. That is, for a surface with orientation denoted

by the unit normal vector n, the surface-force vector t(n) takes the form

t(n) = −n p.



(2–59)



The minus sign in this equation is a matter of convention: t(n) is considered positive

when it acts inward on a surface whereas n is the outwardly directed normal, and p is taken

as always positive. The fact that the magnitude of the pressure (or surface force) is independent of n is “self-evident” from its molecular origin but also can be proven on purely

continuum mechanical grounds, because otherwise the principle of stress equilibrium,

(2–25), cannot be satisfied for an arbitrary material volume element in the fluid. The form

for the stress tensor T in a stationary fluid follows immediately from (2–59) and the general

relationship (2–29) between the stress vector and the stress tensor:

T = − pI.



(2–60)



In other words, in this case T is strictly diagonal:

⎛



−p

T=⎝ 0

0



0

−p

0



⎞

0

0⎠ .

−p



Equation (2–60) is the constitutive equation for the stress in a stationary fluid.

Substituting (2–60) into the force balance (2–58), and noting that

∇ · T = ∇ · (− pI) = −∇ p,

we obtain the fundamental equation of fluid statics:

ρg − ∇ p = 0.



(2–61)



It follows that the presence of a body force leads to a nonzero gradient of pressure parallel to

the body force even in a stationary fluid. Indeed, it is well know that the pressure increases

with depth under the action of gravity. Provided the fluid density remains constant, the

pressure increases linearly with depth

p(z) = p0 + ρgz



(2–62)



where p0 is a reference pressure at the vertical position, z = 0, and z increases with depth.

If we consider any arbitrary volume element from within a larger body of stationary fluid, it,

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F. Fluid Statics – The Stress Tensor for a Stationary Fluid



is evident that there must be an exact balance between the total body force and the pressure

forces acting on the volume element. This balance can be easily demonstrated for a volume

element in the shape of a vertical cylinder of fluid, say between z = z 1 and z = z 2 . The net

pressure force acting on the ends acts upward and has a magnitude

[ p(z 2 ) − p(z 1 )]π R 2 .

This is precisely equal to the total downward body force (ρg times the volume), namely,

ρg(π R 2 )(z 2 − z 1 ).

It is the increase of the hydrostatic pressure with depth that is responsible for the buoyant

force on any body that is immersed within the fluid. If we consider a cylindrical body of

radius R and length L that is oriented vertically, there is a net upward force that is due to

the increase of the hydrostatic pressure with depth, which is the same as that acting on the

column of fluid above, i.e.,

ρg(π R 2 )L .

The net downward force that is due to gravity is

ρs g(π R 2 )L ,

where ρs is the density of the body. Hence, the net force on the cylinder is

F = (ρs − ρ)g(π R 2 )L ,

which may be either up or down depending on whether the density ρs is smaller or larger,

respectively, than the density of the fluid. Although this result was derived for a circular

cylinder that is oriented vertically, it can be generalized to bodies of arbitrary shape.

For any arbitrarily shaped body of density ρs immersed in a fluid of density ρ, there is

a net upward force that is due to the increase of hydrostatic pressure with depth, which is

ρg (volume of the body).

When combined with the direct body force on the body that is due to gravity, this leads to

a net “buoyancy” force,

(ρ − ρs ) g (volume of body),



(2–63)



for a body of arbitrary shape. This is known as Archimedes’ principle. It is important to note

that the presence of a nonzero force on a body immersed in a fluid means that the body (and

thus the fluid around it) will move unless the body is acted on by some additional force.

The condition for zero motion is that the density of the body must equal that of the fluid,

ρ = ρs .

A body satisfying this condition is called “neutrally buoyant.”

We can also interpret Eq. (2–61) as a necessary condition for the fluid to remain motionless. In particular, the hydrostatic balance between ρg and ∇ p can be satisfied only if the

pressure gradient is colinear with (or in the same direction as) the body force, ρg. If there is

a component of ∇ p in any other direction, the fluid must flow. This observation allows us

to examine certain fluid configurations to determine whether they are stable or will evolve

by means of motion of the fluid to some other configuration. To see how this works, we can

begin with an example for which the result will be obvious. Let us consider a horizontal layer

of liquid, within an open cylindrical container, which we assume to initially have a surface

elevation that is raised at the edges and lower in the middle, as sketched in Fig. 2–6.15 Our

intuition probably tells us that such a configuration cannot exist as a stable, stationary state

with no fluid motion, but that fluid will flow from the outer edge toward the middle until

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Basic Principles

Interface

g



Hydrostatic

pressure

increases

with distance

from interface



Contours of

constant

hydrostatic

pressure



B



A



B



Figure 2–6. A sketch of the contours of hydrostatic pressure in a body of liquid within a cylindrical

container when the upper interface is depressed in the center and raised at the walls as shown. As discussed

in the text, the fact that the horizontal component of the pressure gradient is nonzero implies that this fluid

configuration is unstable and will undergo spontaneous motion from the walls toward the center (i.e., in

the direction of decreasing hydrostatic pressure in any horizontal plane) until the contours of hydrostatic

pressure are all horizontal and the upper interface is also flat and horizontal.



the depth of the fluid is uniform everywhere in the container. What makes this happen? The

hydrostatic pressures in the fluid layer create a horizontal pressure gradient, which cannot be

balanced by ρg, and thus the fluid flows in such a way as to produce a fluid layer of uniform

depth. Let us suppose that the pressure at all points on the upper free surface of the fluid

layer is the ambient pressure of air at the conditions of the experiment16 and that the fluid is

motionless. Then, because the depth of the liquid is larger near the outer edge and smaller in

the middle, it is clear that the hydrostatic pressure near the bottom of the container at point

B (near the outer edge) would be larger than the hydrostatic pressure at point A. Hence,

the fluid could not remain motionless in the configuration shown in Fig. 2–6, because there

would then be a horizontal pressure gradient that would drive fluid motion from B toward

A. This motion will continue until the depth is the same at every point, at which stage the

horizontal hydrostatic pressure gradient will completely vanish. It may be noted that the

hydrostatic pressure gradient in the horizontal direction will diminish as the surface flattens,

and thus we may expect the flow rate to decrease with time. It should also be emphasized that

we cannot calculate the actual flow by using the hydrostatic pressure gradients. The flow will

actually cause changes in the pressure distribution, and thus the flow too will be different

in detail from what we would calculate from the hydrostatic pressure gradient alone.

Another example of the qualitative usefulness of thinking about hydrostatic pressure

distributions occurs for a drop that is initially placed on a flat surface, as illustrated in

Fig. 2–7. We shall later see that interfacial tension at the interface between the drop and

surrounding air, as well as line tension at the solid/liquid/gas contact line can play a role

in the dynamics of such a drop. For now we assume that these factors play a negligible

role17 and concentrate on the role of the gravitational force on the drop, which we initially

imagine to be motionless. As we have seen, this leads to a hydrostatic pressure distribution

within the drop. With an initial configuration, such as that shown in Fig. 2–7, there would

be a horizontal pressure gradient within the drop, with the largest hydrostatic pressure at the

center where the drop is highest and the lowest pressures at the outer rim where the height

goes to zero. The drop cannot therefore be motionless because the horizontal pressure

gradient will produce motion of the fluid within the drop, causing the drop to spread on the

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F. Fluid Statics – The Stress Tensor for a Stationary Fluid



Interface



g



Air



Contours of constant

hydrostatic pressure

Drop



Solid substrate



Figure 2–7. A sketch of the contours of constant hydrostatic pressure within a drop that is initially placed on

a flat surface in air. As discussed in the text, the fact that the horizontal component of the pressure gradient

is nonzero implies that this fluid configuration is unstable and will undergo a spontaneous “spreading”

motion to decrease the hydrostatic pressure gradients in any horizontal plane. In this case, this process will

continue indefinitely but at a decreasing rate as the contours of hydrostatic pressure become increasingly

horizontal and flat. If the edge of the drop encounters container walls these will stop further spread and the

drop will continue to move only until it has filled the container to a constant depth.



solid substrate. Assuming that interfacial tension and contact line forces can be neglected,

as stated previously, the drop will continue to spread until it either encounters the bounding

walls of a container (at which point it will again flow until it achieves a fluid layer of uniform

depth) or until it thins to molecular dimensions when other nonhydrodynamic effects will

come into play. Again, the rate of spread is expected to diminish with time as the horizontal

component of the hydrostatic pressure gradient decreases.

A similar dynamics will occur at the “nose” or front of any fluid layer that has a

density ρ1 that exceeds the density ρ2 of a surrounding fluid. An important example from

geophysics is the so-called “gravity current” that occurs with a fluid configuration such as

that sketched in Fig. 2–8. In the geophysical examples, fluid 1 may contain a heavier solid

phase, but in any case, its density exceeds that of surrounding fluid 2. As a consequence

there is a horizontal hydrostatic pressure gradient near the nose, from a larger pressure

at A to a smaller pressure at B, and the heavy fluid, 1, must move from left to right,

displacing the lighter fluid, 2, as it does. In this case, the fact that flow is accompanied by

modifications of the pressure distribution in the fluid is very obvious, because the hydrostatic

pressure distribution in fluid 2 has no horizontal component, even though it must clearly

undergo motion in the horizontal direction as the gravity current propagates from left to

right.

Upper free

surface



g



ρ2 < ρ 1



Interface

ρ1

A



B



Figure 2–8. A sketch of the profiles of constant hydrostatic pressure near the “nose” of a gravity current.

Because of the horizontal gradients of hydrostatic pressure within the nose region, it will propagate to the

right, displacing the exterior fluid as it goes.



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Basic Principles



To emphasize again, the preceding arguments are useful only in identifying systems in

which flow will occur as a consequence of variations in the hydrostatic pressure that cannot

be balanced by a body force. To analyze any details of the motion, we would have to solve

the equations of motion to determine the velocity distribution, u(x, t), and the pressure

p(x, t), which would be modified from the hydrostatic form because of the motion of the

fluid.



G. THE CONSTITUTIVE EQUATION FOR THE HEAT FLUX

VECTOR – FOURIER’S LAW



The simplicity of the constitutive equation for stress in a stationary fluid is due to the fact

that the only surface force in this case is the thermodynamic pressure. The constitutive

equation for the heat flux vector q is not so easy to obtain, even if we assume that the fluid

is stationary, i.e., u ≡ 0. In fact, we shall see later that, for many fluids, the form of the

constitutive equation for q is expected to be the same whether the fluid is moving or not.

However, here, for simplicity of presentation, we imagine u ≡ 0. The governing thermal

energy equation is then (2–57). To determine the constitutive equation for q, we adopt a

continuum approach in the sense that we essentially attempt to guess the relevant form.

However, we first consider the process responsible for q from a qualitative, molecular point

of view, with the goal of providing as much insight as possible for this guess.

Let us begin by recalling that q represents the flux of mean molecular kinetic energy (or

heat) that is due to the purely random component of the motions of individual molecules. In a

system with a nonzero continuum (average) velocity, heat may also be transported because

of the mean motion – and this is called transport of heat by convection. However, here,

we have assumed u = 0. Let us consider the simplest molecular (or kinetic) fluid model,

namely, that of a hard-sphere (or billiard-ball) gas in which the only molecular interactions

are due to collisions and the random molecular motions are purely translational in character.

If we focus our attention on an arbitrary surface within this fluid, it is clear that the only

possible mechanism for a flux of heat is the random interchange of hard-sphere molecules

across the surface and that this will lead to heat transfer only if there is a nonzero gradient of

temperature (or average molecular kinetic energy) with a component normal to the surface.

The fact that a molecule passing across the surface in one direction is accompanied on

average by a second molecule going in the opposite direction is guaranteed by the continuum

approximation and conservation of mass. Furthermore, in this case of a hard-sphere gas, it

is evident that the net flux of heat across a surface will be proportional to the product of the

magnitude of the temperature gradient normal to the surface, the mean free path between

successive molecular collisions, and the frequency of the molecular exchange process. Of

course, the molecular transport process for real fluids will be more complicated than for a

billiard-ball gas. Nevertheless, we conclude from our considerations of that simple model

fluid that

q = q(∇θ, terms involving higher-order spatial derivatives of θ)



(2–64)



for all real fluids; that is, the rate of heat transport by means of molecular motions is

dependent on the magnitude of temperature gradients in the fluid (and quite possibly on

higher-order spatial derivatives of θ as well). The right-hand side of (2–64) represents,

in this case, either a function of ∇θ or, possibly, a functional over the past “history” of

∇θ for the fluid point of interest. Except for simple model materials like the hard-sphere

gas, this is as much as we can deduce from our understanding of the molecular origins

of q. From this point, we must guess the constitutive form for q and ultimately judge

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G. The Constitutive Equation for the Heat Flux Vector – Fourier’s Law



the success of our guess by comparing measured and predicted temperature fields in real

fluids.

A reasonable initial guess is the simplest assumption that is consistent with the relationship (2–64); namely, that the heat flux vector, q(x, t), depends linearly on∇θ(x, t):

q = −K · ∇θ.



(2–65)



Here, K is a second-order tensor that is known as the thermal conductivity tensor, and

the constitutive equation is known as the generalized Fourier heat conduction model for

the surface heat flux vector q. The minus sign in (2–65) is a matter of convention; the

components of K are assumed to be positive whereas a positive heat flux is defined as going

from regions of high temperature toward regions of low temperature (that is, in the direction

of −∇θ).

The reader may well be curious why the particular linear form of (2–65) was chosen

because at least one other vector function is linear in ∇θ , namely β ∧ (∇θ ), where β is a

constant vector. To provide a complete answer, it is necessary to introduce two important

principles that all constitutive relations are expected to obey. The first, which is frequently

taken for granted, may be called coordinate invariance. This principle states simply that

the form of a constitutive equation must be invariant under orthogonal coordinate transformations. Underlying this principle is the obvious fact that a change in orientation or sense

of the coordinate system cannot influence the relevant physical processes and thus should

not influence the form of the constitutive equation. The second invariance requirement of

a constitutive equation is that it must also remain unchanged under a transformation in the

frame of reference of the observer, even if the frame of the observer (or the fluid) is accelerating with respect to an inertial frame. This is usually thought of as being a consequence of

the intuitive notion that the mechanical or thermal properties of a material element cannot

depend on any motion of the person observing the material and is called the principle of material objectivity. Material objectivity is a stronger requirement than coordinate invariance,

but is relevant only for constitutive equations that involve dynamical variables, such as u.

Returning to the form of the constitutive equation for q, we have seen that there are two

distinct possibilities that are linear in∇θ , namely (2–65) and β ∧ (∇θ ). In this case, the

principle of coordinate invariance is sufficient to distinguish between these two possibilities.

The reader who is experienced with vector and tensor analysis may immediately recognize

that β ∧ (∇θ ) is not an acceptable form because it consists of the vector product (or cross

product) of two vectors and is thus a pseudo-vector. A key property of a pseudo-vector is

that it changes sign if we invert the coordinate axes from a right- to a left-handed coordinate

system whereas a true vector is invariant to this transformation. In particular, if we define L as

the coordinate transformation matrix (L · LT = I when the transformation is orthogonal),

then a pseudo-vector transforms according to the rule B = (det L)L · B whereas a true

vector transforms according to A = L · A. The vector formed as the cross product of the

two vectors β ∧ (∇θ ) changes sign on inversion of coordinates and it is thus a pseudo-vector.

The heat flux vector, on the other hand, is a true vector that is invariant to such changes of

coordinate systems. One condition for satisfying coordinate invariance is that all terms in

any equation involving vectors or tensors must have the same “parity” – that is, they must

all be either true vectors or they must all be pseudo-vectors. Because q is a true vector, the

only choice for the form of a constitutive equation that is linear in ∇θ and involves ∇θ only

at the present moment in time and the same point in space as q, is (2–65). Although the

same result can be obtained formally by application of a coordinate transformation to the

terms, q, K · ∇θ , and β ∧ (∇θ ), we will be content here to accept the conclusion based on

the qualitative arguments previously outlined.

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