Topical Problems of Fluid Mechanics 2014 Computationally
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Topical Problems of Fluid Mechanics 2014 Computationally inexpensive method to determine size of g-l interfacial area of rivulet spreading on inclined wetted plate
Martin Isoz ICT Prague
19.– 21. 2. 2014
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Outline 1 Introduction – Why to study rivulet interface – Basic principle
– Coordinate system
2 Simplifications 3 Proposed method 4 Comparison with experiment 5 Conclusions 6 Discussion Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Why to study rivulets Numerous applications in mass transfer and reaction engineering
Hydrodynamics • Fuel cells – water management inside PEMFC fuel cells
• Aerospace engineering – in flight formation of rivulets on plane wings
Gas-liquid interface • Packed columns – wetting performance – mass transfer coefficients
• Catalytic reactors – wetting of the catalyst [Sulzer ChemTech] Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Used coordinate system Cartesian coordinate system and basic notations
Notations a . . . . half-width of the rivulet, [m] h . . . . . . . . . . . . . . . . . . . . . height, [m] l . . . . . . intermediate region length scale, [m]
x, y, z . . . . coordinate system, [m] α . . . . . . plate inclination angle, [◦ ] β . . . . . dynamic contact angle, [◦ ]
Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Proposed method principle Parallel between spreading of trickle in time and along the plate
Duffy and Moffat[4] Description of an uniform rivulet flowing down an inclined plate Cox-Voinov Law[3] Description of spreading of the 2D symmetric object in time Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Outline 1 Introduction 2 Simplifications – Assumptions – Equations 3 Proposed method 4 Comparison with experiment 5 Conclusions 6 Discussion Martin Isoz – ICT Prague
Conclusions
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Cox-Voinov law Solution of thin film governing equation for spreading of symmetric object[3]
Thin film governing equation - outer and inner γ ∂ 3 1 ∂ 3 ht + (h haaa ) = 0, ht + h + 3λh2 γhaaa = 0 3µ ∂a 3µ ∂a
Cox-Voinov law[3] da(t) µ β(t) = 9 ln dt γ 3
Martin Isoz – ICT Prague
a(t) 2e2 l
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Navier-Stokes equations for shallow uniform rivulet Published by Duffy and Moffat in [4]
Simplified NS 0 = −px + ρg sin α + µuzz 0 = −py ,
0 = −pz − ρg cos α
Used boundary conditions z= 0: z= h: y = ±a :
u = u(x, y) = 0 p − pA = −γh00 h=0
and and
uz = 0 h0 = ± tan β
Solution p(z) = pA + a =
γ tan β, a
η tan3/4 β
,
tan β 2 h(y) = (a − y 2 ) 2a 1 105µQ 4 η= 4ρg sin α
Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Outline 1 Introduction 2 Simplifications 3 Proposed method – Derivation – Calculation algorithm 4 Comparison with experiment 5 Conclusions 6 Discussion Martin Isoz – ICT Prague
Conclusions
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Basic principle remainder Combination of Cox-Voinov law with research of Duffy and Moffat
Spreading of a trickle in time a(t) da(t) µ 3 β(t) = 9 ln dt γ 2e2 l Uniform rivulet (g) − (l) interface h(y) =
Martin Isoz – ICT Prague
tan β 2 (a − y 2 ), 2a
a≈η
1 β 3/4
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Spreading of flowing rivulet in time Substituting for a(t) = η/β(t)3/4 into Cox-Voinov law and solving arising ODE
First order ODE with separable variables dβ B 27 ηµ 19/4 β = −A ln , β = β(t), A = , 3/4 dt 4 γ β
B=
η 2e2 l
Implicit dependence of rivulet (g) − (l) interface shape on time B 1 4 A ln − t− +C =0 (1) 15 β 15/4 5 β 3/4 Introduction of initial condition, β(0) = β0 " ! # 4 A B 1 C= ln − 3/4 15 β 15/4 5 β 0
0
Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Transformation from t to x Presence of falling thin liquid film, l, on all the plate
From t to x using uτ uτ =
ρg sin α 2 2µ l =⇒ t = x = $x 2 2µ ρgl sin α
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Spreading of flowing rivulet along the plate Substituting for t into the equation (1)
Implicit dependence of rivulet (g) − (l) interface shape on x A¯ B 1 x − 15/4 ln + C¯ = 0 (2) − 5 β β 3/4 4 A A¯ = 15 $
4 C C¯ = 15 $
Notes on the equation (2)
Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Spreading of flowing rivulet along the plate Substituting for t into the equation (1)
Implicit dependence of rivulet (g) − (l) interface shape on x A¯ B 1 x − 15/4 ln + C¯ = 0 (2) − 5 β β 3/4 4 A A¯ = 15 $
4 C C¯ = 15 $
Notes on the equation (2) • The obtained profiles will be all of the shape of circle segments.
Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Spreading of flowing rivulet along the plate Substituting for t into the equation (1)
Implicit dependence of rivulet (g) − (l) interface shape on x A¯ B 1 x − 15/4 ln + C¯ = 0 (2) − 5 β β 3/4 4 A A¯ = 15 $
4 C C¯ = 15 $
Notes on the equation (2) • The obtained profiles will be all of the shape of circle segments. • The problem of finding the shape of the rivulet’s interface was reduced to specifying the right intermediate region length scale, l. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Derived method for Sg−l calculation Equation (2) defines the shape of rivulet interface in implicit dependence of x
Proposed algorithm: • Discretize domain in x to N subdomains • For all subdomains solve (2) with x = xi and obtain βi , i = 1, 2, . . . , N • From βi calculate shape of each (g) − (l) interface, hi (x, y) • Evaluate integral Z
L Z a(x)
s
1+
Sg−l = 0
−a(x)
∂h(x, y) ∂y
2
and obtain Sg−l
Martin Isoz – ICT Prague
dy dx
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Outline 1 Introduction 2 Simplifications 3 Proposed method 4 Comparison with experiment – Measurements principle – Data 5 Conclusions 6 Discussion Martin Isoz – ICT Prague
Conclusions
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Measurements principle and data origin Light Induced Fluorescence
Measurements principle – LIF[5, 6]
Camera
• Illumination of marked liquid by monochromatic light • Measurements of emitted light intensities • Conversion of measured light intensities in local film thicknesses
Emitted light Calibration cell
Light source Rivulet
Plate
Output of measurements Image of (g) − (l) interface in a form of grayscale photography Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Comparison of calculated and measured Sg−l Silicon oil spreading on steel, α = 60 ◦
0.016 0.015
Sg−l , [m2 ]
0.014 0.013 0.012 0.011 0.01 0.009
fit.: l = 2.81 · 10−5 m exp.: l = 3.00 · 10−5 m experimental data
0.008 0.007 0
1
2
3 −1
Q, [ml s
Martin Isoz – ICT Prague
4 ]
5
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Conclusions Simple yet accurate enough method to determine size of (g) − (l) interface of rivulet
Practical advantages of proposed algorithm • Suitable for parallelization • Usable for qualitative study of dependence of Sg−l on process parameters Accuracy • Deviation < 5 % in comparison with experimental data. Outlook • Study of spreading on dry plate • Study of transition states between dry and wetted plate
Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
References In order of appearance [1] Cooke, J. J. Gu, S. Armstrong, L. M., Luo K. H.: Gas-liqud flow on smooth and textured inclined planes, World Ac. of Sc., Eng. and Technol., vol. 68: (2012) pp. 1712–1719. [2] Ataki, A. Kolb, P. Buhlman, U. Bart, H. J.: Wetting performance and ¨ pressure drop of structured packing: CFD and experiment, I. Chem. E. – symp. series, vol. 152: (2009) pp. 534–543. [3] Bonn, D. Eggers, J. Indekeu, J. Meunier, J. Rolley, E.: Wetting and Spreading. Rev. Mod. Phys., vol. 81: (2009) pp. 739–805. [4] Duffy, B. R. Moffat, H. K.: Flow of a viscous trickle on a slowly varying incline, Chem. Eng. J., vol. 60: (1995) pp. 141–146. [5] S. V. Alekseenko, V. A. Antipin, A. V. Bobylev, D. M. Markovich: Application of PIV to velocity measurements in a liquid film flowing down an inclined cylinder, In: Exp. Fluids, 2008, 43 pp. 197 – 207. ´ ¨ [6] T. Hagemeier, M. Hartmann, M. Kuhle, D. Thevenin, K. Zahringer: ¨ Experimental characterization of thin films, droplets and rivulets using LED fluorescence, In: Exp. Fluids, 2011, 52, pp. 361–374. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Thank you for your attention
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Rivulet distinction Find rivulet edges on the plate
Rivulet edges identification aiL ⇐⇒ first value of hi < Q25 (hi ) + ˜l left from max |hi | Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Experimental data Example of data output from measurements of (g)-(l) interface of rivulets
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Experimental data Example of data output from measurements of (g)-(l) interface of rivulets
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Experimental data Example of data output from measurements of (g)-(l) interface of rivulets
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Experimental data Example of data output from measurements of (g)-(l) interface of rivulets
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Experimental data Example of data output from measurements of (g)-(l) interface of rivulets
Sg−l (aL + aR ), hm , . . .
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Measured system Rivulet spreading down a steel wetted plate
4 liquids • Silicon oils DC 5 a DC 10 • Water and water with surfactants
4 plate inclination angles • 45, 52, 60 a 75◦
Different liquid flow rates • 0.2 – 13 ml/s Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Something is rotten in the state of Denmark Liquid spreading and no slip boundary condition
Derived equation – assumption of no slip at z = 0 1 ∂ 3 h (ρghx − γκx ) = 0 ⇐⇒ u(0) = w(0) = 0 ht − 3µ ∂x
solid
Huh and Scriven paradox[?] Even Heracles could not sink a solid.
no slip u(0) = 0 ⇐⇒ F → ∞
liquid Martin Isoz – ICT Prague
Discussion
Simplifications
Proposed method
Comparison with experiment
Conclusions
Topical Problems of Fluid Mechanics 2014 Computationally inexpensive method to determine size of g-l interfacial area of rivulet spreading on inclined wetted plate
Martin Isoz ICT Prague
19.– 21. 2. 2014
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Outline 1 Introduction – Why to study rivulet interface – Basic principle
– Coordinate system
2 Simplifications 3 Proposed method 4 Comparison with experiment 5 Conclusions 6 Discussion Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Why to study rivulets Numerous applications in mass transfer and reaction engineering
Hydrodynamics • Fuel cells – water management inside PEMFC fuel cells
• Aerospace engineering – in flight formation of rivulets on plane wings
Gas-liquid interface • Packed columns – wetting performance – mass transfer coefficients
• Catalytic reactors – wetting of the catalyst [Sulzer ChemTech] Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Used coordinate system Cartesian coordinate system and basic notations
Notations a . . . . half-width of the rivulet, [m] h . . . . . . . . . . . . . . . . . . . . . height, [m] l . . . . . . intermediate region length scale, [m]
x, y, z . . . . coordinate system, [m] α . . . . . . plate inclination angle, [◦ ] β . . . . . dynamic contact angle, [◦ ]
Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Proposed method principle Parallel between spreading of trickle in time and along the plate
Duffy and Moffat[4] Description of an uniform rivulet flowing down an inclined plate Cox-Voinov Law[3] Description of spreading of the 2D symmetric object in time Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Outline 1 Introduction 2 Simplifications – Assumptions – Equations 3 Proposed method 4 Comparison with experiment 5 Conclusions 6 Discussion Martin Isoz – ICT Prague
Conclusions
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Symplifying assumptions Reduce problem to one spatial and one time coordinate
• Newtonian liquid, ρ, µ and γ are constant • ht (t, x, y) = 0, Q is constant • u = (u, v, w), u v ∼ w • z = h : ux = vy = 0 • Gravity is the only acting body force. • Gravity effects on (g) − (l) interface shape can be neglected and β = β(x) 1 • There is a thin precursor film of height l on the whole studied surface. Thus there is no contact angle hysteresis and βm = 0. The height of the precursor film, l, can also be taken as the intermediate region length scale well separating the inner and outer solution for the profile shape[3]. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Cox-Voinov law Solution of thin film governing equation for spreading of symmetric object[3]
Thin film governing equation - outer and inner γ ∂ 3 1 ∂ 3 ht + (h haaa ) = 0, ht + h + 3λh2 γhaaa = 0 3µ ∂a 3µ ∂a
Cox-Voinov law[3] da(t) µ β(t) = 9 ln dt γ 3
Martin Isoz – ICT Prague
a(t) 2e2 l
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Navier-Stokes equations for shallow uniform rivulet Published by Duffy and Moffat in [4]
Simplified NS 0 = −px + ρg sin α + µuzz 0 = −py ,
0 = −pz − ρg cos α
Used boundary conditions z= 0: z= h: y = ±a :
u = u(x, y) = 0 p − pA = −γh00 h=0
and and
uz = 0 h0 = ± tan β
Solution p(z) = pA + a =
γ tan β, a
η tan3/4 β
,
tan β 2 h(y) = (a − y 2 ) 2a 1 105µQ 4 η= 4ρg sin α
Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Outline 1 Introduction 2 Simplifications 3 Proposed method – Derivation – Calculation algorithm 4 Comparison with experiment 5 Conclusions 6 Discussion Martin Isoz – ICT Prague
Conclusions
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Basic principle remainder Combination of Cox-Voinov law with research of Duffy and Moffat
Spreading of a trickle in time a(t) da(t) µ 3 β(t) = 9 ln dt γ 2e2 l Uniform rivulet (g) − (l) interface h(y) =
Martin Isoz – ICT Prague
tan β 2 (a − y 2 ), 2a
a≈η
1 β 3/4
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Spreading of flowing rivulet in time Substituting for a(t) = η/β(t)3/4 into Cox-Voinov law and solving arising ODE
First order ODE with separable variables dβ B 27 ηµ 19/4 β = −A ln , β = β(t), A = , 3/4 dt 4 γ β
B=
η 2e2 l
Implicit dependence of rivulet (g) − (l) interface shape on time B 1 4 A ln − t− +C =0 (1) 15 β 15/4 5 β 3/4 Introduction of initial condition, β(0) = β0 " ! # 4 A B 1 C= ln − 3/4 15 β 15/4 5 β 0
0
Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Transformation from t to x Presence of falling thin liquid film, l, on all the plate
From t to x using uτ uτ =
ρg sin α 2 2µ l =⇒ t = x = $x 2 2µ ρgl sin α
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Spreading of flowing rivulet along the plate Substituting for t into the equation (1)
Implicit dependence of rivulet (g) − (l) interface shape on x A¯ B 1 x − 15/4 ln + C¯ = 0 (2) − 5 β β 3/4 4 A A¯ = 15 $
4 C C¯ = 15 $
Notes on the equation (2)
Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Spreading of flowing rivulet along the plate Substituting for t into the equation (1)
Implicit dependence of rivulet (g) − (l) interface shape on x A¯ B 1 x − 15/4 ln + C¯ = 0 (2) − 5 β β 3/4 4 A A¯ = 15 $
4 C C¯ = 15 $
Notes on the equation (2) • The obtained profiles will be all of the shape of circle segments.
Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Spreading of flowing rivulet along the plate Substituting for t into the equation (1)
Implicit dependence of rivulet (g) − (l) interface shape on x A¯ B 1 x − 15/4 ln + C¯ = 0 (2) − 5 β β 3/4 4 A A¯ = 15 $
4 C C¯ = 15 $
Notes on the equation (2) • The obtained profiles will be all of the shape of circle segments. • The problem of finding the shape of the rivulet’s interface was reduced to specifying the right intermediate region length scale, l. Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Derived method for Sg−l calculation Equation (2) defines the shape of rivulet interface in implicit dependence of x
Proposed algorithm: • Discretize domain in x to N subdomains • For all subdomains solve (2) with x = xi and obtain βi , i = 1, 2, . . . , N • From βi calculate shape of each (g) − (l) interface, hi (x, y) • Evaluate integral Z
L Z a(x)
s
1+
Sg−l = 0
−a(x)
∂h(x, y) ∂y
2
and obtain Sg−l
Martin Isoz – ICT Prague
dy dx
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Outline 1 Introduction 2 Simplifications 3 Proposed method 4 Comparison with experiment – Measurements principle – Data 5 Conclusions 6 Discussion Martin Isoz – ICT Prague
Conclusions
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Measurements principle and data origin Light Induced Fluorescence
Measurements principle – LIF[5, 6]
Camera
• Illumination of marked liquid by monochromatic light • Measurements of emitted light intensities • Conversion of measured light intensities in local film thicknesses
Emitted light Calibration cell
Light source Rivulet
Plate
Output of measurements Image of (g) − (l) interface in a form of grayscale photography Martin Isoz – ICT Prague
Introduction
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Proposed method
Comparison with experiment
Conclusions
Comparison of calculated and measured Sg−l Silicon oil spreading on steel, α = 60 ◦
0.016 0.015
Sg−l , [m2 ]
0.014 0.013 0.012 0.011 0.01 0.009
fit.: l = 2.81 · 10−5 m exp.: l = 3.00 · 10−5 m experimental data
0.008 0.007 0
1
2
3 −1
Q, [ml s
Martin Isoz – ICT Prague
4 ]
5
Discussion
Introduction
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Proposed method
Comparison with experiment
Conclusions
Discussion
Conclusions Simple yet accurate enough method to determine size of (g) − (l) interface of rivulet
Practical advantages of proposed algorithm • Suitable for parallelization • Usable for qualitative study of dependence of Sg−l on process parameters Accuracy • Deviation < 5 % in comparison with experimental data. Outlook • Study of spreading on dry plate • Study of transition states between dry and wetted plate
Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
References In order of appearance [1] Cooke, J. J. Gu, S. Armstrong, L. M., Luo K. H.: Gas-liqud flow on smooth and textured inclined planes, World Ac. of Sc., Eng. and Technol., vol. 68: (2012) pp. 1712–1719. [2] Ataki, A. Kolb, P. Buhlman, U. Bart, H. J.: Wetting performance and ¨ pressure drop of structured packing: CFD and experiment, I. Chem. E. – symp. series, vol. 152: (2009) pp. 534–543. [3] Bonn, D. Eggers, J. Indekeu, J. Meunier, J. Rolley, E.: Wetting and Spreading. Rev. Mod. Phys., vol. 81: (2009) pp. 739–805. [4] Duffy, B. R. Moffat, H. K.: Flow of a viscous trickle on a slowly varying incline, Chem. Eng. J., vol. 60: (1995) pp. 141–146. [5] S. V. Alekseenko, V. A. Antipin, A. V. Bobylev, D. M. Markovich: Application of PIV to velocity measurements in a liquid film flowing down an inclined cylinder, In: Exp. Fluids, 2008, 43 pp. 197 – 207. ´ ¨ [6] T. Hagemeier, M. Hartmann, M. Kuhle, D. Thevenin, K. Zahringer: ¨ Experimental characterization of thin films, droplets and rivulets using LED fluorescence, In: Exp. Fluids, 2011, 52, pp. 361–374. Martin Isoz – ICT Prague
Introduction
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Proposed method
Comparison with experiment
Conclusions
Thank you for your attention
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Rivulet distinction Find rivulet edges on the plate
Rivulet edges identification aiL ⇐⇒ first value of hi < Q25 (hi ) + ˜l left from max |hi | Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Experimental data Example of data output from measurements of (g)-(l) interface of rivulets
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Experimental data Example of data output from measurements of (g)-(l) interface of rivulets
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Experimental data Example of data output from measurements of (g)-(l) interface of rivulets
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Experimental data Example of data output from measurements of (g)-(l) interface of rivulets
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Experimental data Example of data output from measurements of (g)-(l) interface of rivulets
Sg−l (aL + aR ), hm , . . .
Martin Isoz – ICT Prague
Discussion
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Discussion
Measured system Rivulet spreading down a steel wetted plate
4 liquids • Silicon oils DC 5 a DC 10 • Water and water with surfactants
4 plate inclination angles • 45, 52, 60 a 75◦
Different liquid flow rates • 0.2 – 13 ml/s Martin Isoz – ICT Prague
Introduction
Simplifications
Proposed method
Comparison with experiment
Conclusions
Something is rotten in the state of Denmark Liquid spreading and no slip boundary condition
Derived equation – assumption of no slip at z = 0 1 ∂ 3 h (ρghx − γκx ) = 0 ⇐⇒ u(0) = w(0) = 0 ht − 3µ ∂x
solid
Huh and Scriven paradox[?] Even Heracles could not sink a solid.
no slip u(0) = 0 ⇐⇒ F → ∞
liquid Martin Isoz – ICT Prague
Discussion
