The Dynamics of Water Bells

Transcription

1 Department of Mathematics and Statistics, The University of Melbourne The Dynamics of Water Bells Eleanor C. Button Supervisor: A/Prof. John E. Sader Honours Thesis November 2005

2 Abstract When a vertically aimed liquid jet impacts on the underside of a horizontal plate, it spreads radially to an abrupt point, and then suddenly falls of its own accord. The falling film may coalesce to form a water bell, and by changing the flow rate of the impinging jet, many beautiful shapes are attained. We present an original formulation for the critical radius where the fluid departs the plate. This solution agrees remarkably well with experiments. We also give an approximation for the evolving water bell shape under a changing flow regime.

3 Contents 1 Introduction 2 2 Theory of Water Bells 5 3 Investigation of the Top Flow Inviscid Theory Solution of the Viscous Flow Problem Similarity Solution for Fully Developed Flow The Growing Boundary Layer A Composite Solution The Critical Radius Energy Balance Argument Surface Energy Solution of the Energy Equation Inviscid Case Incorporating Viscosity Asymptotic Analysis The Falling Sheet Derivation of Governing Equations High Gravity Approximation Scaling Analysis Initial Conditions Solution for α = Changing Flow Rate Volume Conservation The Evolving Bell Shape Solution for Arbitrary α ii

4 Contents iii Departure Angle Experimental Results The Critical Radius Volume Conservation The Evolving Bell Shape Departure Angle Conclusions 39 A Mathematica Code 41

5 List of Figures 1.1 Savart Water Bells The impinging jet The changing bell shape The inviscid jet impinging on the glass plate Regions of flow in the viscous problem Velocity profile Film and boundary layer thickness Strip used in energy balance analysis Comparison of exact and asymptotic solutions A fluid element Schematic of water bell surface Experimental results for critical radius: Experimental results for critical radius: Inviscid and viscous results Experimental bell volume Increasing flow rate results Decreasing flow rate results Departure angle results iv

6 Acknowledgements Sincere appreciation to John Sader, for never failing to inspire, and for endless time and enthusiasm. Thanks to Graeme Jameson, Claire Jenkins, Peter Ireland and Ben Dwyer for their experimental work, and to Anthony Van Eysden and Lila Warszawski for their contribution. * Thanks also to the dashpot. 1

7 Chapter 1 Introduction Little attention has been paid so far to the phenomena involved in the impact of a liquid jet on a solid surface with which it forms a defined angle: the modifications introduced in the shape and in the state of the jet, as well as in the motion of the molecules after the impact, are still completely unknown, even in the simplest case one can imagine. However, it was easy to foresee that the study of such phenomena could shed light on some peculiarities of jets, and even on the properties of liquids themselves. - Savart 1833 In 1833 Savart [20] set out to ensure the mysteries surrounding impacting liquid jets indeed received the attention they deserved. He published four papers that year on the subject, the second containing the above excerpt. Since that time, questions have continued to be posed. The most interesting by account of their natural beauty are those involving water bells. Typically, a liquid jet is fired downwards at a small circular horizontal plate. The fluid spreads radially, and upon reaching the edge of this disk, it forms a moving free film of liquid. When this film contacts a surface, it produces stable 12 and interesting shapes, such as below. Much C.Clanet time has been spent studying these Figure 6. Figure 11 of F.Savart Planche 5: D 0 =4.7 mm, U 0 =5.5 m.s 1, Re = 25850, Figure 1.1: Standard water bells, as formed by Savart. From Pl. 5 [20] We = 2031 and N 1: D i =0.65 cm, N 2: D i =4.05 cm, N 3: D i =5.40 cm, N 4: D i =10.80 cm, N 5: D i =16.30 cm, N 6: D i = cm. liquid film remains attached to the impactor. At this point, the critical diameter ratio X +,ispassed and we deduce that for these conditions, 8.04 <X + < In (j) and

8 Chapter 1. Introduction 3 bells, from Boussinesq [4] who derived the original governing equations, to more recent accomplishments such as transonic water bells [6]. Several new water bell phenomena were observed by Jameson [13] in The bells in question were unlike any studied previously. Notably, in this case, the liquid jet (water with a surfactant) is fired upwards at the underside of a horizontal plate. After impact the radial spread also occurs, but with a difference. In previous water bell experiments, the fluid leaves the plate at its edge - this point is predetermined. For Jameson s bell, the plate is effectively infinite. We see the radial flow continues to an abrupt point, and then the film suddenly falls of its own accord, see Fig Figure 1.2: The impinging jet, and the angle formed by the sheet on departure. [14] Amazing shapes are produced when the liquid sheet stabilises to form a bell. By altering the flow rate of the impinging jet, not only can the point of departure from the plate change, the sheets take on a large number of shapes. As the impingement radius increases, the bottom of the film gradually creeps up the central pipe. Eventually the sheet breaks, after which it can form even stranger shapes. As the flow rate is decreased, the bottom of the bell is seen to expand radially. The shapes that are formed are beautiful and seductive - they can give hours of innocent enjoyment. - Jameson [13]. Questions abound on the behaviour of these water bells. Where does the fluid leave the plate? What angle is formed at the departure point? Can we predict the shape of the liquid film? What are the underlying physical processes? All these problems remain unsolved. In this thesis we aim to investigate the physics behind these unexplained water bell phenomena, and develop mathematical models describing them.

9 4 Figure 1.3: The changing bell shape [14] Overview In Chapter 2 we review past work on water bells, and consider their possible application to our problem. Chapter 3 is spent detailing work done by Watson [24] on the impinging jet, as these results are crucial for the theory developed in this thesis. Chapter 4 is dedicated to the top flow along the glass plate, in particular determining its departure point. An energy balance argument is used to derive an expression for this radius. The method of Taylor [23] is applied to the falling sheet in Chapter 5, after which we make a high gravity simplification and find an analytic expression approximating the bell surface shape. Finally, in Chapter 6 we test the validity of these theories, where the derived results are compared to experimental data. Chapters 4, 5 and 6 contain original research.

10 Chapter 2 Theory of Water Bells Since Savart [20] revealed the existence of water bells, there have been many contributions to the development of their theory. In 1869, Boussinesq [4] balanced inertia with the forces of surface tension, gravity and pressure to produce the first governing equation for the shape of such a bell. This has been the basis for many works since. In the case of negligible gravity and pressure, Boussinesq showed the bell shape reduced to a simple catenary. Hopwood [10] renewed wide interest in 1952, when he produced the most amazing shapes to date. By slightly changing the experimental set up, Hopwood created dome shaped bubbles, the size of which increased with the jet flow rate. He observed that if the bell was punctured, it contracted slightly. This was the first solid evidence of a pressure difference across the surface of a water bell. He performed further experiments including an increase of the internal pressure by a stream of small air bubbles. These bells were seen to increase in diameter at the base, but contract higher up. Notably, the shapes he described were new, with extra features of beauty. Inspired by Hopwood s shapes, Lance and Perry [15] searched for a mathematical formulation. They rewrote the Boussinesq equation to give an expression for the local radius of curvature - a second order nonlinear differential equation. This was solved numerically, for particular parameter values, using an extension of Euler s method. It was found the shapes closely resembled the experiments. In 1959, Taylor [23] wrote a series of seminal papers. In the first of these, he nondimensionalised the differential equation found by Lance and Perry. Taylor noticed this governing equation for the bell shape depended upon only two parameters. One of these terms contained gravity, and the other pressure. Taylor presented an analytic solution in the case of both forces being negligible. He used an experimental result to justify 5

11 6 this assumption. By constructing bells with a horizontally aimed jet, and noting the symmetry, he concluded it was legitimate to ignore gravity. More recently, Benedetto and Caglioti [2] derived an identical solution by means of stationary action. The Lagrangian equations for a fluid sheet were found by taking the formal variation of the action. Using an appropriate parametrisation of the surface, this became the Euler-Lagrange equation of a found functional. This method was applied to water bells, and yielded Taylor s result for the zero pressure difference and gravity case. The effect of gravity on the shape of a water bell was further considered by Dumbleton [9]. The Lance and Perry equation was solved numerically for differing values of gravity. Dumbleton found that when gravity was included, the meridian sections of the bells were no longer symmetric, and the shapes were vastly different to the zero gravity solutions. He reported the new shapes were in good agreement with experimentally observed surfaces. In 2001, Clanet [8] varied the flow rate of the jet forming his water bells. He saw increases in the flow rate did not drastically change the shape of the bell, but there was a large increase in the radial dimension. He also tackled the water bell formation problem. Clanet wrote time evolution equations for the creation of the bell, by considering the pushing action of the feeding liquid, and the retracting capillary force. Although the construction of these previously studied water bells is different to that described by Jameson [13], the falling liquid sheet is a feature common to all problems. For this reason, the results mentioned above will be useful in constructing our theory. In particular, the governing equation set forward by Boussinesq [4], refined first by Lance and Perry [15] and even further by Taylor [23], is applied to the surface of our water bell. Another topic to receive considerable coverage in the literature is that of stability of the water bell. Liquid instability was investigated as early as 1878, when Rayleigh [19] considered the breaking of a fluid jet. He described the splitting of the jet into smaller pieces whose total surface area is less than that of the original cylindrical shape. Rayleigh attributed instability to the small disturbances a system is subject to. Squire [22] discussed the stability of a film of uniform thickness moving through stationary air. He reported the film may break due to the formation of transverse waves. Squire concluded instability occurs when the Weber number (the ratio of inertial to surface tension forces), W, is greater than one. However, Taylor [23] presented experimental results showing stable flow in a radially expanding liquid sheet can exist for W > 1. In 1970, Huang [11] considered the axisymmetric liquid sheets formed by impingement of two co-axial jets. He found two different instability regimes, separated according

12 Chapter 2. Theory of Water Bells 7 to Weber number. The first Huang attributed to growing waves on the sheet, and in a similar fashion to Squire [22], gave expressions for the wavelength and wavenumber of these thin film waves. This lead to an approximation for the break-up radius. By balancing the inertial force exerted radially to the outer edge surface forces, Huang found the equivalent radius for the second instability regime. He described the break up as being caused by liquid beads forming at the periphery. This second radius had in fact been derived previously by Bond [3], who by noting the point at which instability occurred was dependent on surface tension, introduced these liquid films as a new method for measuring this quantity. Instablities such as these have been observed in water bells. Clanet [8] included photos of breakages in the liquid film at the base of the bell. Similar phenomena are seen in Jameson s [13] water bells. As the bottom expands radially, the liquid sheet breaks several times. Buchwald and König [7] suggested that water bells could be used to measure surface tension by a similar method to that of Bond [3]. This was done by means of the Boussinesq equation, which contains a surface tension term. However they found results more than 10% higher than expected. This was incorrectly believed to be a new phenomenon, dynamic surface tension. Wegener and Parlange [25] realised this was not the case - the error was in the Boussinesq equation. It did not account for movement of air inside the bell. Hence the shape it predicted was not accurate, and the calculated surface tension would always be wrong. Parlange [18] continued with this idea in a second paper. He noted the boundary condition at the surface of the film required the gas and liquid to travel at the same speed. This resulted in circulation of the internal air, and hence variation in pressure, which through the Boussinesq equation changed the bell shape. This problem soon became complicated as the pressure difference was determined by the internal air motion, itself determined by the bell shape. By using a stream function for the internal velocity, Parlange presented an iterative method for solution, beginning with a convenient choice of water bell shape. As we have the same internal boundary condition in the current problem, a similar method could be applied to Jameson s bell [13].

13 Chapter 3 Investigation of the Top Flow U 0 z Figure 3.1: The inviscid jet impinging on the glass plate. When the vertically aimed jet impacts on the underside of the horizontal plate, the fluid spreads radially in a thin layer. At some point it turns, and begins to fall. As it leaves the plate, the fluid may form a complete sheet, or fall in broken threads. Experiments show there is little difference in departure radius for these two cases. Thus we say, to leading order, the departure radius is independent of what happens to the falling fluid. We then ask two distinct questions: 1. Where does the fluid begin to fall? 2. What is the path taken by the fluid after its departure from the plate? The former question is considered in this chapter, where we aim to understand the behaviour of the fluid in this thin layer, before it falls. This is a necessary step before considering the mechanism controlling the point of departure from the plate. Following 8

14 Chapter 3. Investigation of the Top Flow 9 the analysis of Watson [24], the fluid is first assumed inviscid, and then a solution of the boundary layer equations is provided. 3.1 Inviscid Theory We initially consider cases where viscous effects may be neglected. Since the fluid in the jet is unidirectional, and travelling at a constant velocity, the vorticity vanishes. The persistence of irrotationality then implies the entire problem is one of potential flow. Thus p ρ q2 + χ = constant throughout the flow, where p is pressure, ρ the density, q 2 = u u, u the velocity, and χ is the body force potential. If we assume the film is thin enough to ignore any effect on velocity or free surface profiles caused by the gravitational pressure gradient, the χ term inthe above expression may be neglected. We then note the fluid is bounded by free streamlines; At the surface, the pressure is constant, and hence the velocity also remains at a constant value of U 0, the speed of the impinging jet. There is a stagnation point at the centre of impact of the jet. From here, the fluid accelerates radially at the surface of the plate, and the pressure drops, causing an increase in velocity. This increase in velocity must be bounded, and for such a flow the maximum speed will be attained on the free surface [17]. Hence the speed approaches U 0 as r. Then for r a (where a is the jet radius), the inviscid flow is radial, with a constant speed U 0. By conservation of mass, we must have Q = πa 2 U 0 = 2πrh(r)U 0 for r a, where h(r) is the thickness of the film at radius r. We then see the bounding streamline of the liquid film is given by h(r) = a2 2r (3.1) 3.2 Solution of the Viscous Flow Problem We now include the effects of viscosity using the approach of Watson [24]. The fluid impinging on the plate is irrotational. From the stagnation point, the no-slip boundary condition at the plate s surface generates vorticity. As this vorticity is swept radially downstream, a boundary layer forms. Initially this layer is thin, and the undisturbed potential flow continues near the free surface. Eventually, the layer grows to contain the

15 10 entire flow. To simplify matters, we split the flow into three distinct regions. These are defined as follows, and shown in Fig The incoming inviscid flow, continuing near the free surface. 2. The growing boundary layer near the solid surface, from the stagnation point, to r 0,where it first contains the whole flow. 3. The fully developed region, in which the boundary layer equations describe flow right up to the free surface. The transition between regions is in fact continuous, however by solving the three problems separately, and forming a composite solution, we hope to produce a good approximation to this complicated problem. 1 2 r 0 3 z Figure 3.2: Regions of flow in the viscous problem Similarity Solution for Fully Developed Flow (Region 3) When viscous effects are present all the way to the free surface, we may describe the flow using the boundary layer equations (see, for example, [16]). (ru) r + (rw) z u u r + w u z = 0 (3.2) = ν 2 u z 2 (3.3) Here u and w are, respectively, the r and z components of velocity, and ν the kinematic viscosity. Since we neglect the body force term in these equations, the problem

16 Chapter 3. Investigation of the Top Flow 11 is equivalent to that dealt with by Watson [24], who considered a vertically falling jet impacting on a flat plate. We have used a cylindrical coordinate system (r, ϕ, z), noting the problem is axisymmetric. We consider the shear stress at the free surface to vanish, as the viscosity µ air µ water. Eqns. (3.2) and (3.3) are then to be solved subject to u = w = 0 at z = 0 (3.4) u = 0 z at z = h(r) (3.5) Q = 2πr h(r) 0 udz (3.6) where h(r) is the free surface shape, or equivalently the film thickness, Q the volumetric flow rate, and conservation of mass leads to the final condition. similarity solution of the form u = U(r)f(η) where η = z h(r) Here we look for a (3.7) and U(r) is the free surface velocity. Applying the no-slip boundary condition leads to the following: Substitution of Eqn. (3.7) into Eqn. (3.6) gives f(0) = 0 f(1) = 1 f (1) = 0 (3.8) Q = 2πr h(r) 0 Ufdz = 2πrUh 1 0 f(η)dη (3.9) Hence we may conclude ruh is a constant for constant Q. By using this fact, and writing ru = ruh( f(r) h(r) ), (ru) r Then from the continuity equation (3.2), = ruh r = ruf h η h ( ) f(r) h(r) rufh h whose solution is w z = Uf h η h + Ufh h w = Uh ηf (3.10) With these expressions for u and w, the equation of motion (3.3) becomes νf = h 2 U f 2

17 12 Dividing through by f 2, we see all z dependence appears on the left hand side. Hence h 2 U is also constant. Physically, the shear stress is greatest at the plate, and decreases with z. Now the shear stress T rz = µ u z = µ U h f (η). Then f (η) is greatest at η = 0, and f (η) 0 as η 1. Hence f (η) < 0. Following Watson [24], we set h 2 U = 3 2 c2 ν, where c is a constant. The equation of motion now reduces to from which we obtain 2f = 3c 2 f 2 (f ) 2 = c 2 f 3 + d where d is a constant. The boundary conditions in Eqn. (3.8) give a final differential equation for f: (f ) 2 = c 2 (1 f 3 ) (3.11) Now noting that f (η) > 0, we choose positive c when taking the square root of Eqn. (3.11) f = c(1 f 3 ) 1/2 or alternatively, cη = Application of the boundary condition f(1) = 1 gives a value for c: ) c = 1 Now using the expression for f (η) given above, 1 0 f(η)dη = 1 Substitution of this into Eqn. (3.9) gives 0 0 f 0 (1 x 3 ) 1/2 dx ( π Γ 4 (1 x 3 ) 1/2 3 dx = Γ ( ) 5 (3.12) 6 1 c f(1 f 3 ) 1/2 df = 1 ( π Γ 2 3) c Γ ( ) 1 = 2π 6 3 (3.13) 3c 2 h(r) = 3 3c 2 Q 4π 2 1 ru(r) (3.14) and recalling h 2 U = 3 2 c2 ν, Eqn. (3.14) becomes whose solution is 3 2 c2 ν = 27c4 Q 2 16π 4 U (r) r 2 U(r) 2 1 U(r) = 8π4 ν 9c 2 Q 2 ( r 3 + l 3)

18 Chapter 3. Investigation of the Top Flow 13 where l is a length constant, to be determined by the matching of solutions between Regions 1, 2 and 3 in Fig Using Eqn. (3.14), the required solutions for the velocity and height profiles in the fully developed region of flow are U(r) = 27c2 Q 2 8π 4 ν 1 (r 3 + l 3 ) h(r) = 2π2 ν 3 (r 3 + l 3 ) 3Q r (3.15) (3.16) A power series approximation to the solution of Eqn. (3.11) on [0, 1] is found in the following way: We note that f(0) = 0, and so set f(η) = a 1 η + a 2 η 2 + a 3 η 3 + (3.17) Substituting this into Eqn. (3.11), and equating terms by order of η: [η 0 ] : a 2 1 c 2 = 0 a 1 = c [η 1 ] : 4a 2 = 0 [η 2 ] : 6a 3 = 0 [η 3 ] : c 5 + 8ca 4 = 0 a 4 = 1 8 c4 Continuing on in this fashion, we reach a series representation for f: f(η) = cη (cη)4 8 + (cη)7 112 (cη) (cη) (cη) O( η 19) (3.18) The velocity profile given by f as a function of η is shown in Fig f Η Η Figure 3.3: Velocity profile

19 The Growing Boundary Layer (Regions 1 and 2) For small r, the fluid velocity rises from zero at the plate surface to its maximum value, U 0, in the potential flow region. This change occurs rapidly in a thin layer along the plate, while the layer thickness, δ(r), increases radially. Here the velocity is given by the Blasius flat plate profile [1], obtained by solving the boundary layer equations (3.2) and (3.3) subject to the conditions u = w = 0 at z = 0, and u U 0 as z. In order to make a simple match between fluid velocity profiles, we here present an approximation (see [21]) to the Blasius solution. Integration of Eqn. (3.3) across the thickness of the film gives h 0 ( u u r + w u ) dz = ν u z z (3.19) z=0 After representing w via the continuity equation (3.2), w = 1 (ru(r,z )) r r dz, and integrating the second term by parts, this becomes ( r + 1 ) δ ( U0 u u 2) dz = ν u r 0 z (3.20) z=0 Note the upper terminal has been changed, as the integrand vanishes outside the boundary layer. This is the momentum integral equation, valid for r < r 0. Here since the flow is constant for z δ(r), we notice u z 0 at z = δ(r). This condition at the edge of the boundary layer is identical in form to the no shear stress condition at the free surface of region 3. Hence the solution to Eqns. (3.2) and (3.3) in the growing boundary layer will take an identical form to that in Region 3. In this case, the similarity solution is z 0 u = U 0 f(η ) where η = z δ(r) (3.21) Substituting this into Eqn. (3.20) gives U 2 0 ( r + 1 ) δ ( f + f 2 ) dz = νu 0 f (0) 1 r 0 δ which leads to U 2 0 2(π c 3) 3 3c 2 ( dδ dr + δ ) r = νu 0c δ since f satisfies Eqn. (3.11). Now this can be written 2δr 2 dδ dr + 2rδ2 = 3 3νc 3 r 2 2(π c 3)U 0

20 Chapter 3. Investigation of the Top Flow 15 from which we obtain r 2 δ 2 = 3νc 3 r 3 2(π c 3)U 0 + C where C is a constant. Clearly δ(r) must be finite as r 0, and so we require C = 0. Thus the thickness of the growing boundary layer is given by [ 3νπa 2 c 3 ] 1/2 r (3.22) δ(r) = Q(π c 3) Conservation of mass provides the total film thickness, H(r), for r < r 0. Since any fluid flux must be either in the boundary layer or in the inviscid region above it, Q = [ U 0 δ(r) 1 0 ] f(η )dη + U 0 (H(r) δ(r)) 2πr Eqn. (3.13) and the volumetric flow condition Q = πa 2 U 0 consequently give the total height in Regions 1 and 2: ( H(r) = a2 2r + δ(r) 1 2π ) 3 3c 2 (3.23) A Composite Solution (Regions 1, 2 and 3) The film thickness and fluid velocity are now known either side of r 0. In matching these solutions, we must determine at what radius this transition occurs. By definition, it is where the boundary layer first contains the entire flow. As the growing layer flow takes the same form as that in the fully developed region, we have the analogy to Eqn. (3.14) r 0 U 0 δ(r 0 ) = 3 3c 2 Q. Thus 4π 2 r 0 = [ 9 3c(π c ] 1/3 3)a 2 Q 16νπ 3 (3.24) For the dimensions of our problem, r 0 is typically much less than the departure point where the film leaves the solid surface, and hence viscous effects are important in nearly the entire flow. The final undetermined quantity, the length constant l, is found by requiring continuity of the free surface velocity at r 0. That is, Rearranging this leads to U(r 0 ) = 27c 2 Q 2 8νπ 4 (r l3 ) = U 0 l 3 = 9 3c(3 3c π)a 2 Q 16νπ 3 (3.25)

21 16 Hence the problem is now complete. For r < r 0 the boundary layer thickness is given by Eqn. (3.22), and the total film thickness by Eqn. (3.23). For r > r 0, this thickness varies according to Eqn. (3.16). Fig. 3.4 shows these height profiles. The velocity in the inviscid region (r < r 0, z > δ(r)) is U 0, and is everywhere else given by u = U(r)f(η). Here U(r) = U 0 for r < r 0, and is otherwise defined in Eqn. (3.15). Recall that f is given by Eqn. (3.18), and finally the transition point r 0 by Eqn. (3.24). z H(r) r) r 0 h(r) r Figure 3.4: Film and Boundary Layer Thickness for Q = 8 L/min

22 Chapter 4 The Critical Radius The behaviour of the fluid film on the underside of the plate is now solved, and so we return to the question regarding the point of its departure from the solid surface. Experiments show the departure radius depends heavily, among other things, on the flow rate, Q. In this chapter, we aim to investigate the process causing the film to fall. This is a previously unsolved problem, and we propose the following energy balance as a mechanism for determining the critical radius, R. 4.1 Energy Balance Argument As the fluid spreads radially for the first time, new surface is being created. This creation requires energy. We consider the energy contained in a fluid element such as the one described by Fig. 4.1; due to the axisymmetry of the problem, we in fact take the volume obtained by rotating this area around the z axis. dr direction of flow z Figure 4.1: Strip used in energy balance analysis. 17

23 18 For this element to move from a radial position r, to position r + dr, extra surface of area 2π(r + dr)dr must be created. The energy for this to happen must come from somewhere, and the only source is the kinetic energy in the flow. The kinetic energy in such an element is fixed if we consider the fluid velocity to be given by the steady flow solutions derived in Ch. 3. If there is enough kinetic energy in the element at radius r to create the surface of an element at radius r + dr, the radial spreading continues. If there is not, however, this fluid surface can not be created. The fluid must then depart from the plate. For small r, the flow is still mostly inviscid, and so a fluid element will possess its greatest kinetic energy. In addition to this, the new surface created is small. Hence, for small r, KE(r) SE(r + dr), where KE and SE are the kinetic and surface energies of a fluid element, respectively. As r increases, viscous dissipation leads to a decrease in kinetic energy, while the area of surface being created is increasing. Thus, eventually we reach a radius where KE(r) < SE(r + dr), and no more surface can be created. The critical point occurs when these two quantities are equal. If we let dr 0, the surface area 2π(r + dr)dr, is to leading order 2πrdr. Since surface energy must be proportional to surface area, we can also say, to leading order, SE(r) = SE(r + dr). This simplification is used, and we conclude that the critical radius, R, satisfies KE(R) = SE(R) (4.1) The kinetic energy of a fluid element at r described by Fig. 4.1 is given by 1 KE(r) = 2 ρu2 dv V = πρrdr h(r) 0 u 2 dz (4.2) ere, ρ is the density, u the fluid velocity, and h(r) the thickness of the film at r Surface Energy It is the difference in energies between molecules at the surface and molecules in the bulk of the material that leads to surface tension [12]. This surface tension has dimensions of force per unit length, or equivalently, energy per unit area. It is the latter interpretation we use here. There are three types of interface present in this problem: solid-air, solidliquid and liquid-air. We denote the corresponding surface energies (tensions) by γ SA, γ SL, γ LA.

24 Chapter 4. The Critical Radius 19 Consider the radial spread of a fluid, creating a new surface of total area A. The solidair interface is effectively replaced by the combined solid-liquid and liquid-air interface system of the liquid film. The energy required to make this change is A(γ SL +γ LA γ SA ). If a drop of this liquid sits on the plate in equilibrium, these surface energies can be related to each other and the contact angle of the drop, via Young s equation [12]: γ SA = γ SL + γ LA cos θ eq (4.3) where θ eq is the equilibrium contact angle. Thus γ SL + γ LA γ SA = γ LA (1 cos θ eq ), and since in this case A = 2πrdr, we have the surface energy of our fluid element at r SE(r) = 2πγ LA (1 cos θ eq )rdr (4.4) Comparing Eqns. (4.2) and (4.4) gives a final equation for the departure radius: ρ h(r) 4.2 Solution of the Energy Equation 0 u 2 dz = 2γ LA (1 cos θ eq ) (4.5) We now solve Eqn. (4.5) using the results of Ch. 3. As with previous work, the flow is first considered inviscid, and then the more complicated solution incorporating viscosity is presented Inviscid Case Recall from the inviscid theory detailed in Sec. 3.1, flow is radial with constant velocity U 0 = Q, and the thickness of the film is given by Eqn. (3.1). Substituting these into πa 2 Eqn. (4.5) gives R = ρq 2 4π 2 a 2 γ LA (1 cos θ eq ) (4.6) Thus, inviscid theory predicts the departure radius R Q 2. This relationship is not supported by experimental observations, see Ch. 6, with Eqn. (4.6) dramatically over estimating the critical point for all values of Q. Since we have neglected viscous dissipation, the kinetic energy in any fluid element is incorrectly large. Hence, more surface can be created, leading to an over estimate of the true critical radius. Consequently, inviscid theory is insufficient and the effects of viscous dissipation must be taken into account.

25 Incorporating Viscosity Here we solve Eqn. (4.5) using the similarity solution to the boundary layer equations presented in Ch. 3. We noted the point of transition into fully developed flow, r 0, is typically much less than the departure radius, R. Hence we assume it is possible to approach this equation using only the solution valid for r > r 0. The left hand side of Eqn. (4.5) becomes h(r) ρu(r) 2 f(η) 2 dz = ρu(r) 2 h(r) where U(r), h(r) and f(η) are given by Eqns f(η) 2 dη (3.15), (3.16) and (3.18) respectively. Using the differential equation (3.11), the integral may be evaluated as 1 0 f(η) 2 dη = 1 c 1 0 f 2 1 f 3 df = 2 3c Thus, substitution of this equation into Eqn. (4.5) yields the required equation for the departure radius: R ( R 3 + l 3) = 27 3c 3 ρq 3 32νπ 6 γ LA (1 cos θ eq ) λ (4.7) where l is given by Eqn. (3.25). 4.3 Asymptotic Analysis While an analytic solution exists for Eqn. (4.7), it is complex in nature, and gives little information on the importance of the many parameters. For this reason, we do not present it here. Instead, we consider two asymptotic limits. If R is defined by R = λ 1/4 R, Eqn. (4.7) becomes R ( R 3 + ε 3) = 1 where ε = l λ 1/4 (4.8) This equation now depends on only one parameter, and we solve this for the limiting cases of ε. For small ε, we expand the analytic real solution to Eqn. (4.8) as a power series in ε: R = ε ε6 + O ( ε 12) (4.9) For ε 1, we expand the same solution to Eqn. (4.8) as a power series in ε 1 : R = ε 3 ε 15 + O ( ε 27) (4.10)

26 Chapter 4. The Critical Radius 21 Since both the parameters we expand in are small, we make an approximation by here truncating the two series to order 3. In combining these equations, we note the two solutions for R intersect when ε3 = ε 3, or ε = 2 1/3. Thus we form the composite solution R = { ε3 ε < 2 1/3 ε 3 ε 2 1/3 (4.11) Fig. 4.2 shows this composite solution compared with the exact solution to Eqn. (4.8). Further analysis shows the error in choosing the composite solution is no more than 6.8% for all values of ε. The error attains a maximum for ε around 2 1/3, and is otherwise negligible. R* Exact Composite ε Figure 4.2: Comparison of exact and asymptotic solutions This accuracy is sufficient for the current purpose, and so we form a general composite solution for the departure radius: ( ) l l R = λ 1/4 4 < 2 1/3 λ 1/4 λ ( ) 1/4 λ 1/4 3 (4.12) l l 2 1/3 λ 1/4 Typically for our problem, λ 1/4 l, and so the first solution is chosen. To force the flow into the other profile, for similar parameter values, we would require a very small flow rate. While this may be possible, we concentrate here on the first profile.

27 22 We observe the zeroth order term in the theoretical departure radius, which we call R 0, can be expressed as R 0 = [ 27 ] 1/4 3c 3 ρq 3 32π 6 (4.13) νγ LA (1 cos θ eq ) So we have predicted, to leading order, the departure radius Q 3/4. The validity of this relationship is supported by experiments in Ch. 6. Eqn. (4.13) also indicates that the critical point is independent of the size of the impinging jet, an experimental observation made by Jameson [13]. We now have an analytic expression for the radius of departure, R, that is experimentally testable. For further comparison with experiments, see Ch. 6. While the leading order solution given by Eqn. (4.13) is generally a good approximation, we propose the following result for the departure radius R in this problem: ( R = λ 1/4 1 1 ( ) ) l 3 4 λ 1/4 (4.14) where λ and l are given by Eqns. (4.7) and (3.25) respectively. This leads to a very accurate solution to the critical radius for the flow regime considered by Jameson [13].

28 Chapter 5 The Falling Sheet What path is taken by the fluid after its departure from the plate? We now investigate the second of the two questions posed earlier. In this chapter, we derive equations governing the behaviour of the falling film, and solve them to find the shape of the bell surface. The results of Ch. 4 are used as initial conditions for this part of the problem. We say the fluid begins to fall at position R, this being determined by means of Eqn. (4.14). Following the work of Taylor [23], we further assume the flow in the film to be completely inviscid, and construct a force balance at right angles to the stream. 5.1 Derivation of Governing Equations The problem is axisymmetric, and we retain the cylindrical coordinate system used in Ch. 3. We also define local coordinates, ŝ, ˆn, such that u = uŝ. These coordinate systems, and the meaning of the variables φ and t are defined in Fig dy dx Figure 5.1: A fluid element. 23

29 24 Consider an element of fluid surface of length dx and width dy, as shown in Fig Note this element has two principal radii of curvature, and assume it is small enough to approximate the thickness, t, as constant throughout. The element is acted upon by several forces: Gravitational force Pressure force caused by difference between internal and external pressures, which we assume to be constant and uniform. Force due to surface tension The force due to gravity acting on such a fluid element is F g = ρgt(cos φŝ + sin φˆn)dxdy, where ρ is the fluid density, and g acceleration due to gravity. If we define the pressure difference p = p in p out, the force due to pressure will be F p = pdxdy ˆn. t r n s z Figure 5.2: Schematic of Water Bell Surface. Note that positive φ corresponds to the bell pointing away from the z axis. Without loss of generality, we may describe the curvature of the fluid element by R A and R M, respectively the axisymmetric and meridian radii of curvature. We take the surface tension γ to be constant along the surface. Due to symmetry, the tangential component of surface force vanishes. If we note there are in fact two surfaces present, the force due the axisymmetric direction of curvature then reduces to F sa = 2γdxdy R A ˆn. An identical argument from the other centre of curvature gives the total force due to surface tension: ( 1 F s = 2γdxdy + 1 ) ˆn R A R M

30 Chapter 5. The Falling Sheet 25 From the action of the above forces, the fluid element experiences a centripetal acceleration. This is directed towards the centre of an osculating circle at any point on a meridian section. The acceleration takes the value u2 R M ˆn where u is the speed of fluid in that element. Newton s second law then gives us the centripetal force. Equating the normal components of this and previously mentioned forces we find ( 1 2γ + 1 ) + ρgt sin φ p = ρtu2 (5.1) R A R M R M Substituting the geometric identities R A = r cos φ ; 1 = dφ R M ds, into Eqn. (5.1) gives the force balance equation ( cos φ 2γ dφ ) + ρgt sin φ p + ρtu 2 dφ r ds ds = 0 (5.2) Now consider a cross-section of the bell, looking down the z-axis. Since the flow is axisymmetric, these cross-sections are circular. Define the position of the bell by r = 1 2 (r out + r in ), and note the thickness is given by t(z) = r out r in. Then if the flow rate remains constant at Q, by continuity we must have for all z: Eqn. (5.2) now beomes ( cos φ 2γ r We here define dimensionless variables Q = ( πr 2 out(z) πr 2 in(z) ) u(z) = 2πrt(z)u(z) (5.3) dφ ) ρgq sin φ p + ds 2πru + ρqu dφ 2πr ds = 0 ẑ = z L, ˆr = r L, ŝ = s L, û = u u 0, (5.4) where u 0 is the fluid velocity at the point of departure from the plate, and Under this scaling regime, we obtain cos φ ˆr L = ρqu 0 4πγ dφ dŝ α + β sin φ ûˆr (5.5) + û dφ ˆr dŝ = 0 (5.6) α = ρqu 0 p 8πγ 2 (5.7) β = ρgq 4πγu 0 (5.8)

31 26 Notice that Eqn. (5.6), and hence the path taken by the falling film, depend only upon two dimensionless parameters, α and β. The internal-external pressure difference enters the governing equation only through α, while the β term accounts for gravity. We seek an expression for ˆr, the distance from the bell surface to the z-axis, in terms of z, so we introduce a function: ˆr = Y (z). The hats are now omitted for convenience, and all variables are taken to be dimensionless. Eqn. (5.6) now becomes Taylor s [23] governing equation cos φ Y + dφ ( u ) ds Y 1 α + β sin φ uy = 0 (5.9) Brenner [5] made further simplification to this equation by noting Y (z) = tan φ, which leads to ( ( ) ) 1 1 dy 2 2 cos φ = 1 + dz and dφ ds = cos3 φ d2 Y dz 2 Substituting these results into Eqn. (5.9) yields the final governing equation for the bell shape: Y (u Y ) + (1 + ( Y ) ) ( β ) ( u Y αy 1 + ( Y ) ) This is to be solved with the initial conditions where φ 0 is the angle of departure from the plate. = 0 (5.10) Y (0) = R (5.11) Y (0) = tan φ 0 (5.12) The only remaining undetermined quantity is u, the velocity at height z. We assume the film is thin enough that we may approximate the velocity to be constant through a cross-section at given z. Note that the surface of the liquid film is itself a streamline. Hence by the Bernoulli equation in dimensional form 1 2 u2 + p ρ + χ = constant where u 2 = u u, and the body force gk = χ. The latter implies χ is a function of z only, and furthermore g = χ (z). Since only χ is a physical quantity, we are justified in setting χ = gz. As such a streamline is also a free surface, it is a surface of constant pressure. As such, the pressure term in Bernoulli s equation may be neglected. Thus u 2 2gz = κ for some

32 Chapter 5. The Falling Sheet 27 constant κ. The remaining initial condition requires the fluid leave the plane with a speed u 0. Hence u 2 = u gz Returning to the dimensionless variables defined in Eqn. (5.4), this becomes u 2 = 1 + 2βz (5.13) where β is given by Eqn. (5.8). This equation, together with Eqns. (5.10) to (5.12) provide the general governing equations for the falling sheet after it departs the plate. While an analytic solution to Eqns. (5.10) to (5.13) may be found for the case α = β = 0 (see [23] and [5]), this is of no value to the current problem. Using typical dimensions of the bell, the Froude number is O(1). This shows gravitational and inertial forces are comparable, and hence the β term must be retained. Experiments show the bell can be made to form a perfect cylindrical shape. In this case, all derivatives of Y with respect to z vanish, and Eqn. (5.10) reduces to αy = 1. Clearly, in this case α is non-zero. The inclusion of non-zero α and β terms in the governing equation allows only for numerical solutions. 5.2 High Gravity Approximation When the fluid reaches R, it acts as a reservoir for the falling film. The sheet simply falls under gravity, and so we may expect this force to dominate during determination of the bell shape Scaling Analysis Recall the original governing equation given by (5.1). If we define a function r = y(z), where y, r and z are unscaled quantities, Eqn. (5.1) can by written using previous results as 2γ y (1 + (y ) 2 ) 3/2 + 2γ y ρgt y (y ) (y ) p = ρtu2 y 2 (1 + (y ) 2 ) 3/2 (5.14) An appropriate scale for y is y R, where R is the radius of departure from the plate. We consider a gravity dominated flow. That is, gravitational forces are much larger than surface tension forces. In this limiting case, we expect the fluid to fall approximately vertically, and hence the length scale for z will be very large. We therefore scale z Λ,

33 28 where Λ R. A scaling analysis of Eqn. (5.14) reveals ( ) 2 2γ ( R Λ 1 + ( ) ) R 2 3/2 + Λ 2γ 1 + ( ) + ρgtr ( ) R ( ) Λ R 2 Λ 1 + ( ) + pr ρtu2 R 2 Λ R 2 ( Λ 1 + ( ) ) R 2 3/2 (5.15) Λ Now Λ R implies ( R Λ ) 2 1, and hence the first term on the left hand side, and the term on the right may be neglected in this high gravity limit. The non-linearities may also be removed. Eqn. (5.15) then reduces to Balancing the first and second terms gives 2γ R + ρgtr + p 0 (5.16) Λ 2γ R ρgtr Λ 2γ ρgtr R Λ 1 Thus the high gravity limit formally corresponds to this inequality. Scaling Eqn. (5.16) in the same fashion as Sec. 5.1, an approximation to the governing equation is obtained. Thus in the high gravity limit 1 + β u Y (z) αy (z) = 0 (5.17) Since the order has been reduced, we now need only one initial condition, the departure point R, which is specified. As we shall see, this is a valid simplification in this limit Initial Conditions From the results of Ch. 4, we determine the initial dimensions of the falling film. In the previous section the velocity was considered constant across the film at height z. The initial velocity was denoted u 0. Consider the bounding free streamline close to the departure point. The kinetic energy of a fluid particle must be conserved around the corner on this streamline, and hence we conclude u 0 = U(R) (5.18) where U and R are given by Eqns. (3.15) and (4.14) respectively. Eliminating Q from Eqns. (3.14) and (5.3) then gives the initial film thickness: t(0) = 2π 3 h(r) (5.19) 3c2 The shape of the falling sheet may now be calculated.

34 Chapter 5. The Falling Sheet Solution for α = 0 We first deal with the simple case of constant flow rate. When the fluid leaves the plate for the first time, the film has not yet coalesced to form a bell. No air is cut off from the atmosphere until the sheet is complete, and so in this case we may be sure there is no pressure difference across the surface of the bell. While we hold Q at the initial flow rate, p = 0. Hence α = 0 for the constant flow rate problem. Recalling Eqn. (5.13), in this case Eqn. (5.17) reduces to which gives Y (z) = 1 (1 + 2βz)1/2 β Y (z) = 1 3β 2 (1 + 2βz)3/2 + D where D is a constant. Finally, application of the initial condition Y (0) = R produces the required result. 5.3 Changing Flow Rate Y (z) = R + 1 3β 2 [ 1 (1 + 2βz) 3/2] (5.20) The most interesting shapes are produced when, after the bell has been formed, the flow rate is altered. For an increased flow rate, the radius of departure, R, occurs further from the impact point of the jet and the film is observed to creep up the central pipe. Decreasing Q will also decrease R, but causes the bottom of the bell to expand radially. In this section we aim to predict the evolving bell shape, given an initial flow rate Volume Conservation Clearly when the bell is formed, an amount of gas is trapped by the flowing film. Therefore, this air must obey conservation of mass. Recall that the bell can be made to form a perfect cylinder with vertical sides, see Fig In this case Eqn. (5.10) reduces to α = 1/R, or p = 8πγ2 ρqru 0 which for these bells is O(1). This is tiny compared to atmospheric pressure. The ideal gas law states any change in volume will result in a change in pressure. The pressure change induced here is negligible. Thus, to leading order, we conclude the volume of gas must be conserved.

35 The Evolving Bell Shape As the flow rate changes, so too does R. For all flow rates, the correct bell shape must satisfy Eqns. (5.17) and (5.13), with the initial condition Y (0) = R, together with internal volume conservation. The following method is used to determine the evolving bell shape: 1. The bell is first formed: following the method of Sec. 5.2, set α = 0, and solve Eqn. (5.17) for the initial shape, Y (z). This is given by Eqn. (5.20). 2. Calculate the internal volume V = zc 0 πy (z) 2 dz where z c is defined to be the point the film intersects the pipe, if it does, and the total height of the bell if it does not. 3. Alter the flow rate. This results in a new initial condition Y (0) = R. Let Yᾱ(z) be the result obtained by setting α = ᾱ in Eqn. (5.17) and solving. 4. Define the volume as a function of ᾱ: V (ᾱ) = zc ᾱ 0 πyᾱ(z) 2 dz, where z cᾱ is defined in the same way as z c. 5. Use Newton s method to find ᾱ such that V (ᾱ) = V. 6. This ᾱ is the required α for volume conservation. The correct bell shape is then given by Y α (z). The resultant pressure difference may now be calculated from Eqn. (5.7). For the Mathematica code implementing this method, see Appendix A Solution for Arbitrary α We now find the general solution to Eqn. (5.17). This can be written [ d dz exp α ] (1 + 2βz)3/2 Y = 1 [ 3β2 β (1 + 2βz)1/2 exp α ] (1 + 2βz)3/2 3β2 and so the required solution is given by Y (z) = 1 [ α exp α 3β 2 (1 + 2βz)3/2 ] + b

36 Chapter 5. The Falling Sheet 31 where b is a constant. The initial condition Y (0) = R gives a value for b, and we finally obtain Y (z) = 1 ( α + R 1 ) [ exp α ] (1 + 2βz)3/2 α 3β2 (5.21) In the limit α 0, this solution reduces to Eqn. (5.20) as required. Eqn. (5.21), together with the algorithm described above, allows us to determine the bell shape as it changes with flow rate. In the high gravity limit, numerical solutions to Eqn. (5.10) show the existence of a boundary layer near z = 0. The solution given by Eqn. (5.21) corresponds to the outer solution, giving the natural boundary condition at z = 0 in this limit. Investigation of this boundary layer is a topic for further work Departure Angle We now find an expression for the departure angle, consistent with our current theory. This is possible via Eqn. (5.12). Evaluating the derivative of Eqn. (5.21) at z = 0 gives us this angle: [ ] 1 φ 0 = arctan (αr 1) β (5.22)

37 Chapter 6 Experimental Results We have now developed a model for several aspects of the water bell problem. In this chapter, we test our theories against experimental results. The following experiments were conducted by Graeme Jameson and Claire Jenkins at the Centre for Multiphase Processes, the University of Newcastle. 6.1 The Critical Radius Two similar experiments were performed, first by Jameson (2001) [13], and then by Jenkins (2005). The bell was formed and then the flow rate changed, to induce different points of departure. Photographs and movies of these experiments were taken, and these later analysed to find the critical radii. Figs. 6.1 and 6.2 compare experimental results with the theory of Ch. 4 for the critical radius. The theoretical curves shown are calculated using Eqn. (4.14). Unfortunately the surface tension of the fluid used in these experiments is not known. The lines shown correspond to three values of the correct order expected in the experiments. Further experiments with controlled surface tension must be conducted to test just how accurate this theory is. These two independent experiments offer great support to Eqn. (4.14) and the energy balance theory we present here. In particular, the relationship between departure radius and flow rate, R Q 3/4, is accurately captured, and the insignificance of the impinging jet size is also predicted by the theory. 32

38 Chapter 6. Experimental Results 33 R (m) Experiment: Increasing surface tension (0.034, , N/m) Q (L/min) Figure 6.1: Comparison of theory and experiment: Feb Triton X-100 solution impinging with jet radius a = 4mm. θ eq = 44.9, 49.3, 60.1 in the direction of increasing surface tension. R (m) Experiment: Increasing surface tension (0.034, , N/m) Q (L/min) Figure 6.2: Comparison of theory and experiment: Triton X-100 solution impinging with jet radius a = 5.5mm. θ eq = 44.9, 49.3, 57.3 in the direction of increasing surface tension.

39 34 For completion, Fig. 6.3 compares the viscous and inviscid predictions for critical radius. R (m) 1 Inviscid Theory Increasing Surface Tension Viscous Theory Q (L/min) Figure 6.3: Comparison of inviscid and viscous theories.

40 Chapter 6. Experimental Results Volume Conservation A series of photos of the evolving bell were analysed by Jenkins in a CAD program. The shape of the bells was accurately measured, allowing the volume to be determined. The volume was found for a sequence of flow rates. Fig. 6.4 shows this data. Volume 3 (cm ) Increasing Flow Rate Q (L/min) Figure 6.4: Experimental bell volume Observe the sharp jump in volume at Q 7.5 L/min corresponds to a break in the falling sheet. This experiment supports the theoretical assumption that the bell conserves volume until the film breaks, causing the pressure to equalise. 6.3 The Evolving Bell Shape Using the model presented above, we produced a series of curves representing the bell shape subject to a changing flow rate. This was done by means of Eqn. (5.21), and the algorithm described in Sec Figs. 6.5 and 6.6 allow visual comparison to stills taken from a movie of the Jenkins experiment. Note the good agreement between theory and experiment in all cases. We emphasise,

41 36 that the theory contains no fitting parameters, and only the flow rate was changed between figures. It is interesting to observe the decreasing flow rate results appear more accurate. This is expected, as in the high gravity approximation, we require Thus, the solution will be more accurate for large R, as in Fig γ ρgtr 1. As our solution relies on conservation of volume, if the initial volume is incorrect, such an error will propagate through to the other flow rate solutions. In any case, the leading order high gravity solution still manages to capture the important features of the falling film. Flow Rate (L/min) Figure 6.5: Comparison of Theory and Experiment: Increasing Flow Rate [14]