thin film fluid equation derivation for flat...

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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 will be 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 Dt =0 Where D Dt represents the material derivative. Plugging in the value for φ yields so called ”Kinetic Equation”: h t + uh x = 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 that it is the Surface Curvature is stressed. In the simple setting of a flat 2D plane we formulate this condition as: p(h)= p 0 - γκ (2) where p 0 is a constant. * This is for making an acknowledgement. 1

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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.

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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

which when you further simplify

ht =1

µ(1

3pxxh

3 + pxhxh2) =

1

∂x(pxh

3) =1

∂x(ρghxh

3 − γκxh3)

so that in the end you are left with the Thin Film Fluid Equation

ht =1

∂x(ρghxh

3 − γhxxxh3) (11)

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