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Experiment Goal Model Solution Results

Viscous jet falling onto a moving surface

A. Hlod, M.A. Peletier, A.C.T. Aarts, and A.A.F. van de Ven

CASACenter for Analysis, Scientific Computing and Applications

Department of Mathematics and Computer Science

9-May-2006



Experimental setup

Pour syrup onto the moving surface



Experiment (curved stream)

Bottle is close to the surface (small L)

Surface moves fast (large vbelt)



Experiment (vertical stream)

Bottle is high above the surface (large L)

Surface moves slowly (small vbelt)



Outcome of experiment

ObservationsStream becomes vertical when:

1. Distance between the bottle and the surface increases L ↑

2. Surface velocity decreases vbelt ↓

QuestionsWhat happens if:

3. Viscosity is lower (e.g. water instead of syrup) η ↓?

4. Syrup flows faster from the bottle vnozzle ↑?



Goal

1. Model falling of syrup onto the moving surface

2. Determine when the flow is vertical/curved

3. Understand why the flow is vertical/curved



Modeling strategy

1. Stream of syrup is modeled as a jet

2. Jet is stationary

3. Fluid is Newtonian

4. Model curved flow



Model

η - viscosity, ρ - density



Model equations

Mass conservation

(A(s)v(s))′ = 0

A(s) - cross-sectional area, v(s) - flow velocity

Momentum conservation

(A(s)v(s)v(s))′ =1ρ(P(s)es(s))′ + gA(s)

v(s) = v(s)es(s), g - acceleration of gravity, P(s) - longitudinalforce

P(s) = 3ηA(s)v ′(s)



Equations for v(s) and Θ(s)

v ′(s) =g sin(Θ(s))

v(s)+ µ

(v ′(s)

v(s)

)′

v(s)Θ′(s) =g cos(Θ(s))

v(s)+ µ

(v ′(s)

v(s)

)Θ′(s)

where µ = 3η/ρ



Conditions

Boundary conditions

v(0) = vnozzle

v(send) = vbelt

Θ(send) = 0

send - unknown jet’s length

Integral condition

L =

∫ send

0sin(Θ(s))ds



Solution (1)

Dimensionalize

Get 3 dimensionless parameters A, B (Reynolds number),and vnozzle

Move onset of s to send, s → send − s

Introduce

ξ(s) = v(s) +v ′(s)

v(s)

(inertia force - viscous force)



Solution ξ, Θ (2)

Replace s by t (ds = vdt , send → tend)

Findξ(t) = −

√A2t2 + w

Θ(t) = arcsin(

At√A2t2 + w

)ξ(0) = −

√w , w ≥ 0

w is unknown



System for v(t), tend, w

v ′(t) = −v2(t)(√

A2t2 + w + v(t)) (1)

v(0) = 1 (2)

v(tend) = vnozzle∫ tend

0

Atv(t)√A2t2 + w

dt = B

Define v(t ; w) - solution of (1)-(2) for given w ≥ 0



Lemma (Properties of v(t ; w))

v(t ; w) exists and is unique

v(·, w) : [0,∞) → (0, 1]

v(·, w) ∈ C1([0,∞))

v(·, w) - strictly decreasing

v(t , ·) - strictly decreasing

v(t ; w) < 22+t√

A2t2+w

w 7→ v(·, w) - continuous from [0,∞) to L∞(0,∞)



Function I(w)

Definition tend(w) and I(w)

v(tend(w); w) = vnozzle

and

I(w) =

∫ tend(w)

0

Atv(t ; w)√A2t2 + w

dt

Algebraic equation for w

I(w) = B



Lemma (Properties of I(w))

Strictly decreasing

Continuous

limw→∞ I(w) = 0.



Main result

Theorem (Existence & Uniqueness)

I(0; A, vnozzle) > B

Existence region ((A, vnozzle, B) below I(0; A, vnozzle))



Different flow regimes (curved/vertical) for B = 1



Different L

Jet becomes vertical when L ↑



Different µ

Jet becomes vertical when µ ↓



Different vbelt

Jet becomes vertical when vbelt ↓



Different vnozzle

Jet becomes vertical when vnozzle ↑



Conclusions

We modeled curved jet

Jet is vertical when I(0; A, vnozzle) ≤ B

Jet is curved when I(0; A, vnozzle) > B

Sign of ξ(0) determines jet’s shape

Results from the model correspond to the experimentalresults

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