Flow, deformation, and transport - Physical Processes in Earth and Environmental Sciences - page 193

Before collision
u 1 = u x + u y

2D elastic collision between a molecule and wall

Signs and coordinates

- u x After collision

u 2 = - u x + u y

u y

+ y

u 2

Momentum change is thus
∆ P = mu 2 - mu 1 = m (- u x + u y ) - m ( u x + u y )
or
∆ P = - mu x - mu x = -2 mu x

u 1

- x + x

u y

- y

u x

And Momentum transfer is
∆ P = -(-2 mu x ) = 2 mu x

The overall pressure, force per unit area, acting on any surface is given by the contribution of all molecules colliding with
the wall in unit time. This number will be half of the total molecules, N , in any volume, V (the other half traveling away from the
wall over the same time interval). The pressure is 0.5( N/V )(2 mu x ). An N is given by u x d t and p = mu x 2 N/V . Finally, since
u x 2 = 1/3 u rms 2 and u rms 2 = 2 E/mN , we have the important result that:

pV = 2/3( E ).

Fig. 4.144 Origin of molecular pressure and its relation to internal thermal energy: link between mechanics and thermodynamics.

Atomic vibration

Hotter Cooler

Heat flow

Fig. 4.145 Conductive heat flow in solids is movement of heat
energy in the form of atomic vibrations from hot areas to cool areas
so as to reduce temperature.

A steady-state condition of heat flow occurs when the
quantity of heat arriving and leaving is equal. Many natu-
ral systems are not in steady state, for example, the cooling
of molten magma that has risen up into or onto the crust
(Fig. 4.146; Section 5.1) and in such cases the physics is a
little more complicated (Cookie 20).
The rate of movement of heat by conduction across unit
area, Q , is controlled by a bulk thermal property of the
substance in question, the thermal conductivity, k , so that
overall, for steady-state conditions when all temperatures
are constant with time, Q k d T /d x (Fig. 4.147).

Conductivity relates to spatial rate of transfer, the effi-
ciency of a substance to transfer its internal heat energy
from one point to another. Heat transfer may also be
expressed via a quantity known as the thermal diffusivity ,

Fig. 4.146 Bodies of molten magma intruded into the crust like the
dyke shown here (see Section 5.1) or extruded as lava flows cool by
conduction of heat energy outward into adjacent cooler rocks (or
the atmosphere in the case of lava). The rate of cooling and the
gradual decay of temperature with time may be calculated from
variants of Fourier's law of heat conduction (see Cookie 20).

(kappa; dimensions L 2 T 1 ), defined as k / is the

density and c is the specific heat (Section 2.2). It indicates
the time rate of heat energy dissemination, being the ratio

c , where