Lecture.1.1: AMES 101 A / AMES 103 A; July 1,2 1996

Introduction to fluid mechanics.
Fluid mechanics is a study of fluids either in motion (fluid
dynamics) or at rest (fluid statics). Most of the matter in the
universe is fluid. Fluids may be liquids or gases. Fluids differ
from solids in that they are unable to withstand stresses without
moving. Stresses are forces acting on areas, and are tensors. The
force on an area is the dot product of a stress tensor times the area
by Cauchy's rule. The area vector is directed outward and normal to
a surface with magnitude equal to the area of the surface. An
example of a stress tensor is the pressure stress tensor, equal to
-pd_ij, where p is the pressure, d_ij is the kronecker delta tensor
equal to 1 if i=j and zero otherwise, and i and j are 1,2 or 3. The
pressure on a submarine porthole of area A is inward with
magnitude pA, which is equal to the dot product of -pd_ij with A
vector, as advertised.
Another example of a stress tensor is the inertial stress tensor
-rho v_i v_j, where rho is the fluid density and v_i and v_j are the i
and j components of the velocity vector v. You can find the force on
the nozzle area of a rocket by taking the dot product of this tensor
with the nozzle area. Suppose the area points in the x direction.
The force has magnitude rho v^2 A, which seems reasonable, and
the direction of the force is opposite to the direction of the velocity
as expected.
The final example of stress tensors is the viscous stress
tensor tau. Again, viscous forces on a surface area A equal tau dot
A. We will see later that tau = 2 mu e, where e is the rate of strain
tensor. The components of e are (vi,j + vj,i)/2, where commas
denote partial derivatives.
Fluid mechanics problems are solved by applying
conservation laws of physics to fluids. The conservation laws are:
1. The conservation of mass; 2. The conservation of momentum;
3. The conservation of energy. In Chapter 3 these laws are applied
to various control volumes using the so-called Reynolds Transport
Theorem to relate the conservation laws that apply to systems to
these volumes. As in thermodynamics, a system is defined as a
specified quantity of matter. Partial differential equations that reflect
the conservation laws are derived by applying the laws to control
volumes for systems. These can be solved under certain
circumstances, as shown in Chapter 4. Generally the equations
cannot be solved, so the engineering or experimental approach is
taken, as in Chapter 5. Dimensional analysis is used to organize the
relevant dimensional parameters into suitable dimensionless groups.
If the right set has been identified, the cost of the experiments is
greatly reduced, since the number of groups is less than the number
of dimensional parameters. All of these methods are combined in
Chapter 6, Viscous Flow in Ducts. A brief introduction to turbulent
flows will be given.
Consider a control volume enclosing a fluid system. The
volume surface moves with the fluid velocity v, so that no fluid
escapes the volume. By the conservation of mass, the total mass
contained by the control volume is constant. The total mass equals
the integral of the density over the volume: M = int [ rho ] dV. The
derivative with respect to time is dM/dt. Integrals may be
differentiated using Leibnitz' rule, so dM/dt = 0 = int [ partial ( rho
)/ partial t dV ] + int [ rho v_S dot dA ], where the surface velocity
v_S equals the fluid velocity v because the control volume is about a
system. Converting the surface integral to a volume integral using
the divergence theorem and combining integrals gives int [ partial (
rho )/ partial t + divergence ( rho v ) ] dV = 0. If the size of the
control volume becomes small enough the derivatives become
constant , so the entire integrand can be removed from the integral.
Thus the integrand partial ( rho )/ partial t + divergence ( rho v ) =
0, which is the "continuity equation": one of the basic differential
equations of fluid mechanics.
The same procedure gives the conservation of momentum
equations, or "equations of motion". The rate of change of the total
momentum P for a system is the sum of the forces according to
Newton's law of motion. Differentiate the integral P = int [ rho v ]
dV and convert the surface integral to a volume integral. The forces
are exerted at the surface by the pressure and viscous stress tensors,
and within the volume by gravity. Each surface integral is converted
to a volume integral, the volume integrals are combined, and the size
of the control volume shrunk until the integrand can be removed.
Again we have a combination of terms that must be identically zero
in continuous fluids, because their product with the finite volume is
zero. The equations of motion are: part ( rho v ) / part t + div ( rho
v v ) = - grad p + rho g + div tau . We use a special case of these
equations in Chapter 2 where v = 0, giving grad p = rho g: the
equations of hydrostatics.

Conservation of mass equation by differential element method.
We have derived the continuity equation by the
"shrinking control volume method". A control volume for a
system (specified quantity of matter) surrounding a point in space-
time was allowed to shrink around the point until the partial
derivatives in the control volume became constants so that they
could be taken out of the integral, giving the continuity equation.
The same equation can be derived by considering a control
volume dx, dy, dz for a time interval dt. The mass budget is:
accumulation = in - out. Accumulation is the difference between
the mass in the finite control volume at time t+dt and that at t. In is
the x velocity times the x face area times the density at x, plus the
mass flows through the y and z faces; all times dt. Out is the sum
of the mass flows through the x, y, and z faces and x+dx, y+dy,
z+dz, respectively; all times the time interval dt.
Dividing by the finite volume dxdydz and dt, and taking
the limits of the terms as dx, dy, dz, and dt approach zero ...
using the definitions of the various partial derivatives ... gives the
continuity equation: partial rho / partial t + divergence [ rho v ] =

Conservation of momentum equations.
Newton's second law of motion says that the rate of
change of momentum for a system is the sum of the forces exerted
on it. Call the momentum of a system of fluid P, where P is the
integral over the system control volume CVsyst of the momentum
per unit volume rho v. This equals the sum of the forces exerted
on the fluid in CVsyst. The forces are the integrated pressure and
viscous forces over the surface and the integrated gravity forces
over the volume.
The time derivative of the momentum is obtained using
Leibnitz' Rule for differentiating integrals: dP/dt = integral [ part
rho v /part t ] dV + integral [ rho v dot ] dA, where v_S of the
surface equals the fluid velocity v for a system control volume (no
leaks are allowed through the surface).
The pressure force on the CV surface is the integral of the
pressure stress tensor dotted with the differential surface areas dA
(outward normals) to give the local surface forces by Cauchy's
Rule for stress tensors. This surface integral is converted to a
volume integral of divergence [-p d_ij ] = - grad p using the
divergence theorem (prove this by summing del_j [p d_ij] over j
values 1,2,3 setting d_ij = 0 if i is not equal to j).
The viscous force is the integral of the viscous stress
tensor tau dotted with dA, giving a volume integral of div [ tau ].
The gravity force is integral [ rho g ] dV.
Combining all integrals gives integral { partial rho v /
partial t + div ( rho v v ) + grad p - div [ tau ] - rho g } dV = 0.
Now shrink the system control volume around the point x,
t. By the continuum hypothesis the terms in {.} all become
constant for dV less than L_K^3, where L_K is the size of the
smallest velocity fluctuation (smaller fluctuations are damped by
viscous forces). L_K is assumed to be much larger than the
intermolecular distance L_M for the continuum hypothesis to be
true. Thus, {.} may be factored out of the momentum integral
giving {.} CVsyst = 0. Since the small volume [L_M^3] <<
CVsyst << [L_K^3] is not zero, then {.} must be zero. Since the
same argument can be made for every point in the continuous
fluid, the quantity {.} is identically zero; that is, the equations of
motion are true for every point x, t in the fluid.

partial rho v / partial t + div ( rho v v ) + grad p
- div [ tau ] - rho g = 0

The momentum equations can be written in a variety of
forms that illustrate various fluid mechanical phenomena. For
partial rho v = div [ - rho v v - p d + tau ] + rho g

is in the form of Newton's law dP/dt = sum forces, where the forces
consist of the body force per unit volume rho g, and the divergence
of the total stress tensor in the bracket [], where
tau_T = [total stress tensor] = [ - rho v v - p d + tau ],
pressure stress tensor - p d (d is the identity tensor), the viscous
stress tensor tau (=2 mu e, where e is the rate of strain tensor), and
- rho v v is the inertial stress tensor.
Forces on surfaces are obtained
Cauchy's Rule

******Force = [Stress Tensor] dot [Area Vector]******

as illustrated by pressure forces on submersible windows:

^^^^^pressure forces
F_pressure = - p d dot A = - p A
which confirms our intuition that the pressure force on the
window will be inward, proportional to the pressure times
the area of the window.

^^^^^viscous forces
The viscous force on a surface is best illustrated by considering
the force on the surface of a fixed flat plate, with a plate above
it by a distance h in the y direction, moving at velocity V in the x direction.
From Newton's viscosity law we know that the force on the lower plate
is in the x direction, with value mu V/h A, where mu is the viscosity
of the fluid between the plates. We can confirm this using Cauchy's Rule
and the viscous stress tensor tau = 2 mu e because the rate of strain tensor
in the gap between the plates is easy to evaluate. The only non zero elements
in the array e_ij are e_12 = du/dy /2. Therefore
F_1 = tau_11 A_1 + tau_12 A_2 + tau_13 A_3
= tau_11 0 + tau_12 A_2 + tau_13 0
= 0 + 2 mu e_12 A + 0 = mu du/dy A, as expected,
and F_2 = tau_21 A_1 + tau_22 A_2 + tau_23 A_3
= F_2 = tau_21 0 + tau_22 A + tau_23 0 = 0.
F_3 = tau_31 A_1 + tau_32 A_2 + tau_33 A_3
= tau_31 0 + 0 A_2 + tau_33 0 = 0. QED.

^^^^^inertial forces
Inertial forces can be illustrated by considering the force on
the area A of a rocket engine in a test stand, were the velocity of the
rocket gases V are in the direction of the nozzle area A. From Cauchy's
F_inertial = - rho V V dot A,
V dot A = VA
F_inertial = - rho V^2 A,
as expected; that is, the force is in a direction opposite to the direction
of the nozzle gas flow.

Hydrostatic equations.
A special case of the equations of motion is when v = 0.
The first two terms on the left are zero, and so is the viscous term
div [ tau ] because tau = 2 mu e. The rate of strain tensor e
consists of velocity derivatives: e_ij = ( vi,j + vj,i ) / 2. Since v =
0 everywhere, all these derivatives go to zero too, leaving

grad p = rho g.

Now g = (0,0,-gz) if the z direction is up. Therefore grad
p = (0,0, dp/dz). Pressure does not vary in the x or y directions:
only in the z direction. Integrating dp/dz = - rho gz for constant
rho and g gives p = - rho gz + constant. If p = p_o at z = 0, then
the constant is p_o. Thus for constant density fluids, p = p_o -
rho gz, where g is the acceleration of gravity (9.81 m/s^2 on the
earth's surface).

####### Example: At the bottom of the ocean (z = -11,000 m) the
pressure p = p_atm - rho gz = 101,000 Pa + 1030 kg/m^3 x 9.81
m/s^2 x 11,000 = 1.112x10^8 Pascals (1101 atmospheres).

Various time derivatives (done last time).
Consider some function of space and time C. The change
in C for an arbitrary change in space and time is dC = [part C / part
xi]dxi + [part C / part t]dt by the chain rule. Dividing by dt gives:
dC/dt = [part C / part xi]v'i + [part C / part t] .

Three cases are interesting in fluid mechanics: 1. v'i = 0,
dC/dt = part C / part t in "laboratory coordinates"; 2. dC/dt =
[part C / part xi]v'i + [part C / part t], the general case; 3. dC/dt =
[part C / part xi]vi + [part C / part t] = DC/Dt, where v'i = vi
following the fluid particle. In case 3. a special derivative symbol
DC/Dt is used, called the "substantive derivative". The text does
not always use this symbol, so be careful.

Incompressible flows.
The continuity equation can be rearranged to show that the
fractional rate of change in density of a fluid particle equals the
negative divergence of the velocity field. The positive divergence
of the velocity equals the fractional substantive derivative of the
specific volume 1/ rho. Thus the criterion for incompressible flow
is that div v = 0. Incompressible flows may have variable density.
However, the densities and specific volumes of fluid particles in
incompressible flows do not change.