Download Introduction Outline: The fundamental forces Pressure gradient

Introduction
Outline:
The fundamental forces
Pressure gradient
Gravitational
Viscosity
Noninertial frame
Centrifugal force
Coriolis
Forces:
If there is a NET force acting on an inertial system –the system will
experience acceleration. Conversely, if there are observed accelerations, in
an inertial system, then a NET force is acting upon the system.
Why do I emphasize ‘NET’?
What do I mean by inertial?
Newtonian laws hold only in an inertial reference frame, i.e. one that is at
rest with respect to the ‘fixed stars’. This means that the coordinate system
cannot be rotating if Newtonian laws are to apply.
A rotating reference frame gives rise to additional ‘apparent’ forces that will
appear in our equations. Note however, that these additional terms can and
do appear in non-rotating reference frames – resulting from coordinate
transformations only.
We will return to these concepts as part of our discussion of the Coriolis
‘force’ and a rotating reference frame. However, for now we consider only
the fundamental forces that impact (i.e., accelerate) fluid flow.
In an inertial (non-rotating) frame, we have:
r
r
F = ma , where
r
F = the sum of forces acting on an object
m = mass of object, a = acceleration of object
We could also write Newton’s 2nd law as:
r
r
r
r
r DV
DV
, where V = the velocity of the object, and a ≡
F =m
Dt
Dt
r
r
D
D2
r
=
mV = 2 (mr ), where r is a position vector (position of object)
Dt
Dt
( )
In classical mechanics, the object is typically a rigid solid body
In continuum Mechanics (e.g., Atmospheric Dynamics), the object is a tiny
(infinitesimal) chunk o’ fluid (liquid or gas). For meteorology, we usually
refer to this chunk as a ‘fluid element’ or ‘air parcel’. We do NOT consider
the microscopic aspects of the fluid in continuum mechanics – only the
macroscopic.
We will be applying Newton’s 2nd law to the atmosphere – the final result
won’t look anything like what we started with!
Fluid forces are broken down into two different types of forces”
“body” force: Force on an object that is proportional to its mass,
acting from a distance. The objects interacting are not in physical
contact with each other, yet are able to exert a push or pull despite a
physical separation (e.g. gravity, electrical, magnetic).
“surface” force: Also known as a ‘contact’ force as it involves objects
that are physically interacting with one another. The Force on an
object that is proportional to the area of an object. Force exerted on a
surface fluid element by an outside fluid. (e.g., pressure gradient,
frictional, tensional, air resistance forces, spring force, applied forces
etc.)
Pressure Gradient Force (PGF):
Many examples of this in our every day lives (tire pressure, soda
cans/bottles, aerosols, etc.)
Start with a really really really small box (infinitesimally small) with
dimensions δx, δy, and δz in the x, y, and z directions respectively.
δz
z
y
x
δx
δy
box volume =δV
= δxδyδz
What is the net pressure force acting on this fluid element?
Pressure is a compressive force and acts perpindicular to surface element
(compare with another surface force ‘stress’ which acts parallel to the
surface).
Let’s first consider the x-component of the pressure force, thus we consider
only two faces of our cube – those that are perpindicular to the x-axis.
O
FAx
FBx
z
Face B
y
Face A
δx
XB
XA
x
where, point ‘O’ is at the box center denoted by (xo,yo,zo), and FAx is the
force on face A in the ‘-x’ direction, and FBx is the force on face B in the
‘+x’direction. Also,
δx
δx
x A = x0 + ,
x B = x0 −
2
2
Pressure at ‘O’ is given as p0.
What is the pressure on face A (pA) for this infinitesimal fluid element?
USE Taylor Series expansion!
p A = p0 +
∂p
( x A − x0 ) + Higher order terms (HOT)
∂x
x0 y0 , z0
δx
∂p 

 x0 + − x0  + HOT
∂x 
2

valid at center of box
HOT vanish for tiny box
∂p δx
= p0 +
(' =' valid in the limit as δx → 0)
∂x 2
= p0 +
We recall that pressure is defined as the force/area, thus the ‘force’ exerted
on face A due to the pressure is
∂p δx 

FAx = − p0 +
δyδz , and
∂x 2 

similarly for face B we have:
∂p δx 

FBx = p Bδyδz =  p0 −
δyδz
∂x 2 

Thus, the NET force acting on the box due to the pressure is given as
Fx ≡ FAx + FBx
∂p δx 
∂p δx 


= − p0 +
δyδz +  p0 −
δyδz
∂x 2 
∂x 2 


∂p
= − δxδyδz
∂x
SO, net pressure force acting on our fluid element is proportional to the
gradient of the pressure – hence we call it the pressure gradient force (PGF).
We’re not quite done however, as the equations of motion are in terms of per
unit mass. The mass of the box is m = ρδxδyδz. Thus, we have
Fx
1 ∂p
, and for the y and z components, we have
=−
m
ρ ∂x
r
Fy
Fz
1 ∂p
1 ∂p
F
1
1 ∂p
, and
, or in vector form = − ∇p = −
=−
=−
m
m
ρ ∂y
m
ρ ∂z
ρ
ρ ∂xi
The PGF acts in direction opposite of the gradient in p. This makes sense
because the net force is ‘down gradient’.
Therefore, the PGF is in toward the center of a low pressure system, and out
away from the center of a high pressure system.
L
PGF
∇p
H