**Lecture.1.1: MAE 101 A / MAE
103 A; July 3, 2000 **

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Introduction to fluid mechanics.

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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.

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Conservation of mass equation by differential element method.

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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 ] =

0.

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Conservation of momentum equations.

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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

example

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

using

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Cauchy's Rule

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******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.

Similarly,

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

Rule

F_inertial = - rho V V dot A,

but

V dot A = VA

so

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.

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Hydrostatic equations.

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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).

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Various time derivatives.

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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.

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Incompressible flows.

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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.