082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

27.7 Some simple volume and surface integrals 891
In Figure 27.19(b) we have placed an imaginary surface,S, within the flow and
it is reasonable to ask ‘what is the volume of fluid which crosses this surface in
any given time?’ Suppose we select a very small portion of the surface having area
δS. Note that δS is a scalar. We can also define an associated vector δS, which has
magnitude δS and whose direction is normal to this portion of the surface. The
component of fluid velocity in the direction of δS is given by the scalar product
v· ̂δS where ̂δS is a unit vector in the direction of δS. The volume of fluid crossing
this portion each second is v·δS. This is also known as the flux of v. If we
wishtocalculatethetotalfluxthroughthesurfaceSwemustsumallsuchcontributions
over the entire surface. This is found by integrating over the surface. We write
this as
∫
volume flowper second =total flux = v·dS
This is an example of a surface integral, so called because we must integrate over
a surface. It will usually take the form of a double integral such as those discussed
inSection27.6.1.Technically,therearetwointegralsigns,butforbrevitythesehave
been replaced by the single ∫ where it is to be understood that the integral is to be
S
performed over the surfaceS. In any specific problem care must be taken to ensure
that the entire surface is included when the integration is performed. The double
integralsevaluatedinSection27.6arespecialcasesofsurfaceintegrals,inwhichthe
surface isaplane region (thex--y plane).
Forexample,supposethevelocityfieldisgivenbyv = (x+2y)k.Thisrepresents
a flow in the z direction whose magnitude varies with x and y. Suppose S is the
plane surface defined by 1 x 4, 0 y 2,z = 0. This surface is shown in
Figure 27.20. Note that the normal to this surface is parallel to thezaxis and so we
can writedS = dxdyk. Then
v·dS=(x+2y)k·dxdyk
=(x+2y)dxdy
S
y
S
y= 2
dS = dxdyk
v= (x+2y)k
x
y= 0
0
x = 1
z
x = 4
Figure27.20
The vectorfieldvrepresents
fluid flowing acrossthe
surfaceS.Theintegral over
the surface ofv ·dSgives the
volume flow persecond
acrossS.
➔