Fluid Basic Properties and Governing Equations
Density (): mass per unit volume (kg/m3 or slug/ft3) Specific Volume (v=1/): volume per unit mass Temperature (T): thermodynamic property that measures the molecular activity of an object. It is used to determine whether an object has reached thermal equilibrium. Pressure (p):pressure can be considered as an averaged normal force exerted on a unit surface area by impacting molecules. ( P = lim F , N/m2 or pascal; lb/in2 or psi)
A 0
A
Pascal law: (under static condition) pressure acts uniformly in all directions. It also acts perpendicular to the containing surface. If a fluid system is not in motion, then the fluid pressure is equal its thermodynamic pressure. Atomspheric pressure (patm): pressure measured at the earth's surface. 1 atm = 14.696 psi = 1.01325 x 105 N/m2 (pascal) Absolute pressure: pressure measured without reference to other pressures. Gage pressure: pgage= p absolute - patm
Atmospheric pressure can be measured using a barometer: p=0 Vacuum
p=0
Patm=1.01x105 Pa
L
p=patm
Force balance patm A = W = mg = ALg Patm = gL
is the density of the fluid, g is the gravitational constant
Similarly, this balance can be applied to a small fluid element as shown dp pA - ( p + dp) A = mg = Agdy, = - g, integrate from fluid element to dy Free surface, p=p the free surface p(h) = p + gh
y p+dp p x ax h Example: If a container of fluid is accelerating with an acceleration of ax to the right as shown below, what is the shape of the free surface of the fluid? p p+dp
dy
dp pA - ( p + dp) A = max = Adxax , - = a x dx dp dy a g a x tan( ) = = = , = tan -1 x dx dp g g a x
dx
FI GJ HK
Buoyancy of a submerged body
h2 h1 free surface p1=p+ Lgh1 Net force due to pressure difference dF=(p2-p1)dA= Lg(h2-h1)dA Total net force (buoyancy)
FB = dF = L g (h2 - h1 )dA = L gVdisplaced
p2=p+ Lgh2
z z
The principle of Archimedes: The buoyancy acting on a submerged object is equal to the weight of the displaced fluid due to the presence of the object. This law is valid for all fluid and regardless of the shape of the body. It can also be applied to both fully and partially submerged bodies.
Example: Titanic sank when it struck an iceberg on April 14, 1912. Five of its 16 watertight compartments were punctuated when it collides with the iceberg underwater. Can you estimate the percentage of the iceberg that is actually beneath the water surface? It is known that when water freezes at 0 C, it expands and its specific gravity changes from 1 to 0.917. When the iceberg floats, its weight balances the buoyancy force exerted on the iceberg by the displaced water. weight
W = FB
ice gVice - berg = water gVsubmerged
Vsubmerged
buoyancy
Vice - berg
=
ice 0.917 = = 91.7% water 1
Therefore, more than 90% of the iceberg is below the water surface.
Properties (cont.)
Viscosity: Due to interaction between fluid molecules, the fluid flow will resist a shearing motion. The viscosity is a measure of this resistance. Moving Plate constant force F constant speed U Stationary Plate From experimental observation, F A(U/H)=A(dV/dy)
H
F dV , where is shear stress A dy dV lb sec N sec = , where is dynamic viscosity, The unit of is or dy ft 2 m2
=
ft 2 m2 kinematic viscosity = , has unit of or sec sec
Boundary Layer Concept
Immediately adjacent to a solid surface, the fluid particles are slowed by the strong shear force between the fluid particles and the surface. This relatively slower moving layer of fluid is called a "boundary layer".
Laminar Turbulent
dV = dy
Question: which profile has larger wall shear stress? In words, other which profile produces more frictional drag against the motion of the solid surface?
Partial Differential Equations (PDE): Many physical phenomena are
governed by PDE since the physical functions involved usually depend on two or more independent variables (ex. Time, spatial coordinates). Their variation with respect to these variables need to be described by PDE not ODE (Ordinary Differential Equations). Example: In dynamics, we often track the change of the position of an object in time. Time is the only variable in this case. X=x(t), u=dx/dt, a=du/dt. In heat transfer, temperature inside an object can vary with both time and space. T=T(x,t). The temperature varies with time since it has not reach its thermal equilibrium.
T C p = qin - qout 0 t
The temperature can also vary in space as according to the Fourier's law:
T T q = - KA , if q 0, then 0 x x
Basic equations of Fluid Mechanics
Mass conservation (continuity equation):
The rate of mass stored = the rate of mass in - the rate of mass out
dm = min - mout dt
min
m= V
mout
Area A
L
Within a given time t, the fluid element with a cross-sectional area of A moves a distance of L as shown. The mass flow rate can be represented as
m L m= = A = AV t t
dm d ( V ) dV d = = +V = min - mout dt dt dt dt
Keep density constant, volume changes with time ex: blow up a soap bubble Keep volume constant, density changes with time ex: pump up a basketball
dm = ( AV ) in - ( AV ) out dt
For steady state condition: mass flow in = mass flow out
( AV )in = ( AV ) out
Examples: filling up an empty tank
Water is fed into an empty tank using a hose of cross-sectional area of 0.0005 m2. The flow speed out of the hose is measured to be 10 m/s. Determine the rate of increase for the water height inside the tank dh/dt. The cross-sectional area of the tank is 1 m2. 0, no mass out
h(t)
dm dV dh = = Atan k = min - mout = AhoseV dt dt dt
dh Ahose = dt Atan k
0.0005 V = (10) = 0.005(m / s ) 1
V1=20 m/s
Section 2, A2=1 m2
Section 1, A1=5 m2
V2=?
min = mout
1 A1V1 = 2 A2V2 , for constant density
A1 5 V2 = V1 = (20) = 100(m / s ) A 1 2
Momentum Conservation: (N...