thin film fluid equation derivation for flat...
TRANSCRIPT
Thin Film Fluid Equation Derivation For Flat
Surfaces
A. Uribe∗
Columbia University
12/3/2011
Abstract
The Thin Film Fluid Equation is derived from the Navier Stokes Equa-tion assuming fluid motion on a flat substrate.
1 Derivation of the Thin Film Equation
The thin film equation will be derived. First, the boundary conditions that willbe used in the derivation will be listed.
1.1 Boundary Conditions
1.1.1 Kinetic Equation
Let φ = h(x, t) − y so that the 0 level set of φ describes the thin film surface.We want this statement to be valid for all times, which happens iff Dφ
Dt = 0
Where DDt represents the material derivative. Plugging in the value for φ yields
so called ”Kinetic Equation”:
ht + uhx = v (1)
1.1.2 Surface Tension Pressure
At the surface should be proportional to the negative of the curvature (p ∝ −κ)of the fluid surface in order to drive the surface to minimize area. The fact thatit is the Surface Curvature is stressed. In the simple setting of a flat 2D planewe formulate this condition as:
p(h) = p0 − γκ (2)
where p0 is a constant.
∗This is for making an acknowledgement.
1
1.1.3 Free Surface
On the surface boundary, the fluid is free to flow:
uy(h) = 0 (3)
1.1.4 No Slip
A no-slip condition where the fluid is touching the substrate is also imposed.
u(0) = v(0) = 0 (4)
1.1.5 Continuity Equation
Since we will be dealing with a Newtonian fluid, the divergence of the velocityfield has to be zero. This is equivalent to saying that the volume of the fluidwill be conserved.
∇ · ~u = ux + vy = 0
Another form of this equation which is will be useful at a later step is
v(h)− v(0) = v(h) =
∫ h
0
−uxdy
which can be combined with the kinetic equation to obtain
ht + uhx =
∫ h
0
−uxdy (5)
1.2 Non-dimensionalizing Navier-Stokes
Writing the Navier-Stokes equations for each dimension, for a fluid includingviscosity forces results in the following pair of equations:
ut + uux + vuy = −1
ρpx +
µ
ρ(uxx + uyy) (6)
vt + uvx + vvy = −1
ρpy +
µ
ρ(vxx + vyy) + g (7)
For non-dimesionalizing purposes we will use x = Lx, y = εLy, u = Uu,
v = εUv, and p = µU2
ε2L p. Replacing all these values into 6 and 7 one obtains:
U2
L(ut + uux + vuy) =
µU
ε2ρL2(−px + ε2uxx + uyy)
εU2
L(vt + uvx + vvy) =
µU
ε3ρL2(−py + ε4vxx + ε2vyy) + g
2
Assuming that the Reynolds number Re = ρLUµ << 1 one can rewrite the
previous equations as:
ε2Re(ut + uux + vuy) = −px + ε2uxx + uyy
ε4Re(vt + uvx + vvy) = −py + ε4vxx + ε2vyy + g
From the previous equations, one can argue that an approximation in theThin Film scenario where Reε2 << 1 and ε << 1 of the N-S equations is thefollowing so called “Thin Film Approximation”:
px = µuyy (8)
py = ρg (9)
1.3 Thin Film Equations
In this section, 8 and 9 will be manipulated and all the boundary conditionswill be used to come up with a single PDE that is equivalent.
Equation 9 can be integrated:∫ h
y
pydy =
∫ h
y
ρgdy
p(h)− p(y) = ρgh− ρgyCombining this with 2 gives:
p(y) = −ρgh+ ρgy + p0 − γκ
This expression for the pressure can be plugged into 8 so that:
−ρghx + γκx = px = µuyy
Notice that px does not depend on y, so it can easily be integrated. Integratingthe last equation yields
pxh− pxy =
∫ h
y
pxdy = µ
∫ h
y
uyydy = µ(uy(h)− uy(y)) = −µuy(y)
where we used the Free Surface Flow boundary condition (3) to drop the lastterm. We can integrate this equation again, which gives
pxhy −pxy
2
2=
∫ y
0
pxh− pxydy = −µ∫ y
0
uydy = −µ(u(y)− u(0)) = −µu(y)
where we used the No-slip boundary condition 4. This equation is known as the“Quadratic velocity profile” equation
u(y) =1
µ(pxy
2
2− pxhy) (10)
3
At this point, the only pair of boundary conditions we haven’t used are theContinuity and Kinetic equations. Previously, we had combined both of themto obatin Eq. 5. Plugging the Quadratic Velocity Profile into 5 and using thefact that:
ux =1
µ(pxxy
2
2− pxxhy − pxhxy)
gives rise to
ht +hxµ
(pxh
2
2− pxh2) =
∫ h
0
−uxdy
=
∫ h
0
(1
µ(pxxhy + pxhxy −
pxxy2
2))dy
=pxxh
3
2µ+pxhxh
2
2µ− pxxh
3
6µ
which when you further simplify
ht =1
µ(1
3pxxh
3 + pxhxh2) =
1
3µ
∂
∂x(pxh
3) =1
3µ
∂
∂x(ρghxh
3 − γκxh3)
so that in the end you are left with the Thin Film Fluid Equation
ht =1
3µ
∂
∂x(ρghxh
3 − γhxxxh3) (11)
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