Fluid mechanics
Fluid is matter that flows under influence of
external forces (?)
Fluids flow under shear!
While solids experience
some finite deformation.
Do Newton’s laws apply in fluid mechanics?
Of course they do, since they apply everywhere unless the
things get relativistic or quantum.
What is the problem with the fluid mechanics then?
Why do not we study it together with the regular mechanics?
The trouble is we have got to deal with a continuum…
where the notion of a “body” is difficult to define and it is not
always straightforward where the forces come from.
What do we do?
For appropriate description of
continuum mechanics we have
got to introduce new notions
and parameters.
Density measures the mass per unit volume.
SI units are kg/m3
! = m /V
3
[ kg/m ]
There are compressible and incompressible fluids
Pressure measures the normal force per unit area exerted by a fluid.
SI units are N/m2 or Pa, Pascal.
F dF
P= =
A dA
2
[ N/m ], Pa
Abrahams tank
Mass – 63 tons ( 63,000 kg)
Caterpillar area – 6 m2
Ground pressure:
mosquito
Proboscis – about 0.1 mm in
diameter.
Pressure of 105 Pa corresponds to
F mg 63,000 kg ! 9.8 m/s 2
P= =
=
=
2
A
A
6m
= 105 Pa " 3.14 " (10 !4 m ) 2 / 4 = 8 " 10 !4 N
= 105 Pa = 1 atm
Equivalent mass is
F = P " A = P " #d 2 / 4 =
m = F / g ! 10 "4 kg = 0.1 g
Abrahams tank
F mg 63,000 kg ! 9.8 m/s 2
P= =
=
=
2
A
A
6m
5
= 10 Pa = 1 atm
Equivalent mass is
m = F / g ! 10 "4 kg = 0.1 g
A mosquito weighs about 2 mg, 0.002 g.
An ant can lift 50× its weight.
If the same is true for a mosquito, it can produce a force equivalent to
50 ! 0.002 g = 0.1g and exert the same pressure as an Abrahams tank
Can pressure be NOT the same for different
directions?
No net force!
A solid would be in equilibrium.
What about a fluid?
Pressure is a scalar quantity.
Hydrostatic equilibrium
(no net force on any element of liquid)
Condition for hydrostatic
equilibrium – constant
pressure throughout the
fluid volume.
Variation of pressure
creates net force in the
direction of decreasing
pressure.
Hydrostatic equilibrium with
external forces: Gravity
Force from above
(pushing down)
Force from below
(pushing up)
PA
( P + dP) A
Gravitational force (pulling down)
dFg = mg = !gA dh
Balance (equilibrium) equation
PA + !gA dh = ( P + dP) A
!gA dh = AdP
dP
= !g
dh
Hydrostatic equilibrium with
external forces: Gravity
dP
= !g
dh
We have got a differential
equation…
Where do we go from here?
We need to integrate it. Can we?
If both ρ and g are constant, certainly yes.
P = !gh + P0
What is P0? And BTW, what is h?
P0 is the constant of integration – the value of pressure at h = 0.
It is natural to count h downwards from the fluid surface (h = depth).
Then P0 is the pressure at the surface – the atmospheric pressure.
If the atmospheric pressure changes, does the pressure at a given
depth change?
Yep!