Download 02_Basic biorheology and gemodynamics

Basic biorheology and
gemodynamics
Equation on Continuity
 Bernoullis equation
 Torricellis theorem
 Viscosity
 Poiseuille formula
 Reynolds number
 Stoke's law

Equation on Continuity
where A1 and A2 are the cross section areas of the tube and
v1 and v2 are the velocities of the liquid at these sections
respectively.
The volume of an incompressible fluid entering one part of a
tube or pipe must be matched by an equal volume leaving
downstream.
Flow is faster in narrower parts of a tube, slower in wider
parts
Volume Flow Rate –
Bernoulli's Equation
"The Bernoulli Equation can be considered to be a statement of the
conservation of energy principle appropriate for flowing fluids."
Torricelli’s Theorem:
Velocity of efflux (of liquid flowing out through a hole) =√(2gh)
where ‘h’ is the depth of the hole (fig).
(It is interesting to note that a particle dropped from a height ‘h’ will strike
the ground with the above velocity)
The expression for the efflux velocity follows from Bernoulli’s theorem by
considering the cases at the free surface of the liquid in the tank (fig)
and at the hole:
P + ρgH + 0 = P + ρg(H–h) + (½)ρv2 so that v =√(2gh),
where P is the atmospheric pressure. Note that the hole is open to the
atmosphere, as is the free liquid surface in the tank.
The horizontal range of liquid jet (fig), R= 2√[h(H–h)]
Note that the above expression is obtained by considering the horizontal
range (on the ground) of a particle projected horizontally from a height
(H–h). Do it as an exercise.
The range R will be maximum if h = H/2. [You may show this by putting
dR/dh = 0 when R is maximum]. The maximum range is H.
In the next post we will consider some typical questions in this section.
Of course, their solution also will be discussed.
Viscosity
The viscosity of a fluid is an important property in the
analysis of liquid behavior and fluid motion near solid
boundaries.
The viscosity is the fluid resistance to shear or flow and is a
measure of the adhesive/cohesive or frictional fluid property.
The resistance is caused by intermolecular friction exerted
when layers of fluids attempt to slide by one another.
Viscosity is a measure of a fluid's resistance to flow
The knowledge of viscosity is needed for proper design of
required temperatures for storage, pumping or injection of
fluids.
There are two related measures of fluid viscosity - known as
dynamic (or absolute) and kinematic viscosity.
Dynamic (absolute) Viscosity
is the tangential force per unit area required to move one
horizontal plane with respect to the other at unit velocity
when maintained a unit distance apart by the fluid.
The shearing stress between the layers of non turbulent fluid
moving in straight parallel lines can be defined for a
Newtonian fluid as:
The dynamic or absolute viscosity can be expressed like
τ = μ dc/dy
(1)
where
τ = shearing stress
μ = dynamic viscosity
Equation (1) is known as the Newtons Law of Friction.
Kinematic Viscosity is the ratio of absolute or dynamic
viscosity to density - a quantity in which no force is involved.
Kinematic viscosity can be obtained by dividing the absolute
viscosity of a fluid with it's mass density
ν=μ/ρ
(2)
where
ν = kinematic viscosity
μ = absolute or dynamic viscosity
ρ = density
Viscosity and
Temperature
Kinematic viscosity of
liquids like water, mercury,
oils SAE 10 and oil no. 3 and gases like air,
hydrogen and helium are
indicated below. Note that
•for liquids viscosity
decreases with
temperature
•for gases viscosity
increases with
temperature
Laminar and Turbulent Flow:
Part b of the figure shows the streamlines of simple flow.
These lines show the path a tiny particle in the fluid follows
as it moves along the pipe. Flow such as this is termed
laminar flow.
The fluid velocity is lower in the large cross-sectional
region. Part c of the figure shows what happens if the flow
becomes too swift past an obstruction. The smooth flow
lines no longer exist. As the fluid rushes past the obstacle, it
starts to swirl in erratic motion. No longer can one predict
the exact path a particle will follow. This region of constantly
changing flow lines is said to consist of turbulent flow.
Reynolds number
Re 
pvD

p
Newtons equation for fluids
dv
F  S
dt
A viscous material is gooey. Runny materials have low
viscosity. There are various ways of measuring viscosity,
the most common of which is to drop a ball-bearing into the
liquid, and measuring its terminal speed. Remember that
at terminal speed, the upwards forces of upthrust and drag
and the downwards force of the weight are balanced.
The upthrust is the same as the weight of fluid displaced by
Archimedes' principle. Therefore, if the weight is greater
than the upthrust, the object will accelerate downwards until
the drag balances the difference between the weight and the
upthrust. This is true of all fluids, for example air, or water, or
chocolate.
We can calculate the drag force by Stokes' Law:
The terms of the equation are:
F - the viscous drag (N);
r - the radius (m);
η - the coefficient of viscosity (N s m-2) [The strange looking
symbol η is 'eta' a Greek letter long 'ē'];
v - the terminal velocity (m/s);
Bernoullis equation
1
1
P
v12  P2  v 22
21
2
Torricellis theorem
v2  2 g h1  h2 
Newtons equation for fluids
F  S
dv
dt
Poiseuille formula
Q
 p r
1 p2
4
8

Hydraulic resistence
X 
Reynolds number
Re 
Stake's law
F=6r
pvD

8l
r 4