Basics of Fluid Mechanics, 2014a

258 CHAPTER 8. DIFFERENTIAL ANALYSIS
that integration constant is unknown). In same instances, this constant is determined
from the volume conservation. In index notation equation (8.121) is written 21 as
τ (1)
ij n j + σn i
( 1
R 1
+ 1 R 2
)
= τ (2)
ij n j (8.123)
where 1 is the upper surface and 2 is the lower surface. For example in one dimensional 22
̂n =
̂t =
(−f ′ (x), 1)
√
1+(f ′ (x)) 2
(1,f ′ (8.124)
(x))
√
1+(f ′ (x)) 2
the unit vector is given as two vectors in x and y and the radius is given by equation
(1.57). The equation is given by
∂f
∂t + U ∂f
x
∂x = U y (8.125)
The Pressure Condition
The second condition that commonality prescribed at the interface is the static
pressure at a specific location. The static pressure is measured perpendicular to the
flow direction. The last condition is similar to the pressure condition of prescribed shear
stress or a relationship to it. In this category include the boundary conditions with issues
of surface tension which were discussed earlier. It can be noticed that the boundary
conditions that involve the surface tension are of the kind where the condition is given
on boundary but no at a specific location.
Gravity as Driving Force
The body forces, in general and gravity in a particular, are the condition that
given on the flow beside the velocity, shear stress (including the surface tension) and
the pressure. The gravity is a common body force which is considered in many fluid
mechanics problems. The gravity can be considered as a constant force in most cases
(see for dimensional analysis for the reasons).
Shear Stress and Surface Tension as Driving Force
21 There is no additional benefit in this writing, it just for completeness and can be ignored for most
purposes.
22 A one example of a reference not in particularly important or significant just a random example.
Jean, M. Free surface of the steady flow of a Newtonian fluid in a finite channel. Arch. Rational Mech.
Anal. 74 (1980), no. 3, 197–217.