Brief verview of Vector Calculus
Vector Calculus
Denition of a Vector
-i . .
A vector F 18 on ordered trlple (a, b, C) where a, b, c are real num-
bers.
Denition of 3. Norm of a Vector
. ~ > .
A norm (or magnltude) of the vector F 18 a real number denoted

Compression and Expansion of Gases
ISOTHERMAL PROCESS - temperature is constant:
B = constant
. P
ISENTROPIC PROCESS - no heat exchange:
3; = constant
0
where k is the ratio of specic heat at constant pressure to the
specic heat at constant volume and p i

Basic Equations for Fluid Pressure
Pressure and gravity forces: Pressure Forces
Pressure force in x direction for volume dxdy dz:
6 dx 6]: dx
derwx = "(D + 537Mde +( ~- 5;?)dydz
6p
6p dy 6]) Eu
der_y = (D + 51:7)dxdz + (I) w "a; 2i)dXdZ
ab
Pressure force

Problem Identication: Desks are needed by students, businessmen, and many others
Conventional desks take up take up oor space and occupy an area at all times. People living in
small res1dences, lack the space required for a desk.
Description: A wall-mount

Fluid Mass and Weight
DENSITY p
J Mass of a, unit volume (mass per volume).
BC: slugs/ft3
SI: kg / m3
Density of gasses is strongly inuenced by temperature and pressure -
Water at 60degrees F: p :2 1.9431ugs/ft3
SPECIFIC VOLUME v = 1/9
Volume of a unit. (

DIMENSIONALLY HOMOGENEOUS EQUATIONS
All additive terms have the same dimensions.
Example: ow through an orice:
Q = 0.61A\/ 29h
vaolume/time [£73]
[12]
gmaccelerat ion of gravity [:5]
mheight [U FORCE
WEIGHT
F 2 ma
F i MLT2
11b 213111.91
ft
E7 ' . :- - "

DEFINITION
A uid is a substance that defomls continuously when acted on by a
shearing stress of any magnitude
TYPES. OF FLUIDS
Liquids (Water, Oil, .)
GasseS(Air, Methane.)
CONTINUUM CONCEPT
'Ireats uids as hypothetical substances with properties which ar

ty
O
lSCOSl
V
VELOCITY PROFILE
u 11
FOR A SMALL TIME INCREMENT
tancSB «- 66 *-
moving plate
fixed plate RATE OF SHEARING STRAIN
. 53
m L' .
y 5913(1) 5t
*HE_§E
.owdy
I SHEARING STRESS OF A NEWTONIAN FLUID
WherepWTUZ] iSd37n31nie ViSCOSity _ v visc

Gravity Force
Horizontal forces are zero, vertical force is the weight of
the uid element and points downwards:
fgrwx r":
fary =
fgrwz m
-.->
fgr : alance of Forces
p' = Vp - p"? + viscous forces
(
6p . . ' ._ ._ 1
pax = a; + 0 + VISCOUS' force per mm. v

Surface Tension
. . .- u . . .-. I . cf ._. . . . r. . I .-_._ . . ._. J.
.J r.- _._ _. a. far. .r. ._U. Ins. w.:.:.-.:. a. J. .3.Ju.lrm. aw. Par 33.
a.
J . . .
. J. . x
. .
771?th = ZWRO cos 0:
Where 0'[FL] is SURFACE T ENSION
Surface tension: molecula