Hydrodynamics and Elasticity: Class 3
Ostrogradski-Gauss theorem For a volume V enclosed by a surface Σ and a vector field A Z
Σ
A · dS = Z
V
(∇ · A)dV. (1)
Stokes theorem For a surface area Σ spanned on a closed contour γ I
γ
A · d` = Z
Σ
(∇ × A) · dS. (2)
1. Prove the following identities for a scalar field φ and a vector field A I
Σ
φdS = Z
V
∇φdV, (3)
I
Σ
dS × A = Z
V
(∇ × A)dV. (4)
2. Prove the basic properties of material elements and hydrodynamic balance equations:
(a) Show that the deformation (stretch) of a material line in a velocity gradient is D
Dtd` = (d` · ∇)u. (5)
Apply this to show that an integral along a material curve with endpoints P (t) and Q(t) of a scalar function θ(r, t) satisfies
d dt
Z Q(t) P (t)
θ d` = Z Q
P
Dθ Dtd` +
Z Q P
θ d` · ∇u. (6)
Then show that an integral over a closed material loop C(t) satisfies d
dt I
C(t)
u · d` = I
C
Du
Dt · d`. (7)
(b) Find the deformation of a volume element to be D
Dt(dV ) = (∇ · u)dV. (8)
Check that for an incompressible flow Dt(dV ) = 0. Apply this result to show that if ρ(r, t) is the fluid density, the mass balance equation takes the form
d dt
Z
V (t)
ρ(r, t)dV = 0. (9)
Extend this to show that for an arbitrary scalar function θ(r, t), the transport equation is d
dt Z
V (t)
ρ(r, t)θ(r, t)dr = Z
V
ρDθ
DtdV, (10)
(c) Find the deformation of a surface element D
Dt(dS) = (∇ · u)dS − ∇u · dS (11)
and apply it to show that an integral over a material surface S(t) of a scalar function θ(r, t) takes the form
d dt
Z
Σ(t)
θ ndS = Z
Σ
Dθ
Dt ndS + θ(∇ · u) ndS − θ∇u · ndS
, (12)
where n is the outward oriented normal unit vector of the surface S.
