THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING EFFECT OF VARIATION OF ACCELERATION ON FREE-SURFACE INSTABILITY Chia-Shun Yih S. P. Lin April, 1963 IP-614

LIST OF CONTENTS Page LIST OF FIGURES AND PLATES.....................*..........o.. iii 1. INTRODUCTION......................,.,...,.,.......o...... 1 2. A DESCRIPTIVE EXPLANATION OF THE CAUSE OF POST-ROLL INSTABILITY............................................... 2 3. FORMULATION OF THE DIFFERENTIAL EQUATION GOVERNING FREE-SURFACE INSTABILITY.........................oo...oo. 4 4. FREE-SURFACE INSTABILITY DUE TO A SUDDEN CHANGE OF ACCELERATION...................................o... 11 5. FREE-SURFACE INSTABILITY DUE TO A GRADUAL CHANGE OF ACCELERATION......................o................. 16 6. EXPERIMENTAL VERIFICATION.......................... 18 7. APPLICATION TO POST-ROLL INSTABILITY OF STOCK ON FOURDRINIER WIRES.....................................oo 25 REFERENCES..................................................... 27 ii

LIST OF FIGURES AND PLATES Figure Page 1 (a) a = a1. Wave Attains Maximum Heighto (b) a Changes to a2, Which is Less Than al. Surface is Flat. (c) a = a2o Wave Attains Next Maximum Height... 28 2 (a) a = al. Wave Attains Maximum Height. (b) a Changes to a2, Which is Greater Than al. Surface is Flat. (c) a = a2. Wave Attains Next Maximum Height......... 28 3 An Idealized Acceleration Schedule.................... 29 4 A More Realistic Acceleration Schedule................ 29 5 An Acceleration Schedule Representing the Actual Situation On a Fourdrinier Wire............ o... o....o 30 6 That Portion of Figure 5 Used for Experimental Verification of the Theory..........,, O................ 30 Plates I Apparatus.............................................. 31 II Acceleration Graph.................................... 31 III Wave Form Soon After a2 (Nearly Equal to -g) is Reached by a, After Decreasing From a Large Positive Value o o o o o o o........................o o o o o..... o. 32 IV A Few Milli-Seconds After the Condition in Plate III was Reached............................, o.......... 32 iii

1. INTRODUCTION The instability of stock on a Fourdrinier wire occurs as it issues from the slice, as it passes over the table rolls, and especially immediately after it leaves the table rolls. There are many causes for the instability. On top of the table rolls the cause of instability is the centripetal and downward acceleration of the fluid, which can be many times as great as the gravitational acceleration. A paper dealing with this was published in the Proceedings of the Royal Society by Yih(l,2) with supporting experimental data. At the places where the streamlines are curved and the fluid is slowed down away from the center of the curvature, instability may take the form of the formation of Taylor-Gortler vortices. A paper dealing with this possibility was published by Yih and Debler in 1961. (2) Measurements of the growth of disturbances after the table roll were made by Spengos,(3) who also obtained some preliminary measurements on the rate of growth of the disturbances over the table roll. But the rather violent instability of the stock after the table rolls has so far not been satisfactorily explained. This paper provides a theory, with some supporting experimental data, which explains the main cause of instability after the table rolls, and constitutes a final report to the Technical Association of Pulp and Paper Industries which has generously sponsored the study of free-surface instability at the University of Michigan during the years 1957-59 and 1960-62. -1

2. A DESCRIPTIVE EXPLANATION OF THE CAUSE OF POST-ROLL INSTABILITY Consider a layer of liquid in a container and in wave motion, as shown in Figure 1. Suppose that at time t = 0 the wave attains a maximum height, when the acceleration of the container is al, directed upward. This acceleration is maintained until the free surface is flat, at which time all the energy of the wave motion is in the form of kinetic energy. If at this time the acceleration is suddenly changed to a2, which is less than al, then as the next maximum wave height is attained it would have to be greater than the previous maximum, because the potential energy relative to the container will remain the same, whereas the effective gravitational acceleration has been decreased from g + al to g + a2. The reverse is true if a2 is greater than al, as shown in Figure 2. It is also evident that the phase of the waves is important at the moment of change of the acceleration, whether it is decreased or increased. Since the waves on the stock as it moves past the table rolls are in all possible phases, a change of acceleration will make certain components of the waves unstable. The change of acceleration is experienced by the stock as the wire carrying it leaves the table rolls, above which it has undergone a downward acceleration, and goes through a region of reverse curvature, in which it is subjected to an upward acceleration. The first author owes this idea to a stimulating discussion with Mr. JTo Justus, who explained the instability in terms of pressure at the bottom of the container. A more precise explanation will be given by a mathematical analysis, which will be presented in the following two sections. -2

The foregoing consideration of the instability phenomenon, for the first time, also explains in physical terms the observation of Faraday in 1831 (Phil. Trans, p. 319) that the frequency of liquid oscillations was only half that of an oscillating vessel containing the liquid, since, as the liquid accomplishes half an oscillation (Figure 1), the vessel has already accomplished one oscillation, ready to start the next cycle by changing a2 to al again. (The change in acceleration occurs both at extreme wave heights and at zero wave heights.)

3. FORMULATION OF THE DIFFERENTIAL EQUATION GOVERNING FREE-SURFACE INSTABILITY Wave motion in a layer of liquid of uniform depth will be considered. The formulation of the problem is identical to that of Benjamin and Ursell(5) for instability due to periodic acceleration. In fact, the qualitative explanation given in the previous section makes it possible to understand in physical terms the mathematical results obtained by Benjamin and Ursell. The hydrodynamic equations of motion with reference to the container are, with viscous effects neglected, D (u,v,w) = l(a a a ) p + (0,0, -g-a), (1) in wi an Dt p x by oz in which a is the acceleration, p is the density, p is the pressure, g is the gravitational acceleration, and u, v, and w are the velocity components in the directions of increasing x, y, and z respectively, The coordinates x, y, and z are Cartesian coordinates, and the symbol D stands for Dt +u +a vva +w - at ax by az and is the operator for substantial differentiation. From Equation (1), it can already be seen that when the frame of reference is taken to be the container, the body force g per unit mass in the negative z-direction is replaced by g + a, so that the potential energy with respect to the container is based on g + a instead of g. Although the concept of potential energy is meaningful only if a is constant, the explanation based -4

-5 on constant a given in the last section is qualitatively applicable to the case of variable acceleration. The unknowns in (1) are u, v, w, and p. Since there are only three equations in (1),a fourth one is needed. This is the equation of continuity ~u + _V + _w = O. 6u+1+6w`=0. (2) ax ay az Now Equations (1) are nonlinear, and are very difficult to solve although the number of equations is now sufficient with the addition of (2). Fortunately, the theorem of Helmholtz and Kelvin on the persistence of irrotationality is valid here because the acceleration a of the container can only be time-dependent, so that the effective body force with components (0, 0, -g-a) is conservative, i.e., its curl is zero. This can be demonstrated quite simply. With the vorticity components denoted by a _w _v au _w 6v au b = - ~ -^ 4 - T- ~ — ^ ^ - "~~ ~ T-^^(3) ay Tz az dx =x a =y equations governing the vorticity of the fluid can be obtained from (1) by cross-differentiation. For instance, with the second and third equation in (1) written as at ay p 2 W - uT + vi =- ( + u +2+2- (g+a), (5) dt 0z p 2 (5) can be differentiated with respect to y and (4) differentiated with respect to z. Utilizing (2) and the identity ~5 + y + az =0, a y z

-6 the result can be reduced to the form D_ eu= + u + + u (6) Dt ax by az Similarly, D = a+ 2 +; av (7) Dt ax by az D = e aww+ + aw + w (8) Dt ax by az Now if the flow starts from an irrotational state, 0 = T = 0 everywhere at t = O. Then (6), (7), and (8) state that the substantial derivatives of A, i, and ~ are zero, or that the true derivatives of A, r, and ~ of each particle as we follow its motion are zero. This being ture of every particle originating from an irrotational state, the vorticity of every particle will remain zero, and the persistence of irrotationality is demonstrated. Since the motion under consideration can be considered to have started from rest, which is an irrotational state, the subsequent motion is irrotational. For irrotational motion, Equation (3) states that the curl of the velocity is zero. This implies that the velocity can be expressed as the gradient of a potential (: (Uv,,) = - (Y,, E) (9) Combining (2) and (9), we have 2) = 0 (10)' "2' + "" (10)

-7 which is linear. The problem is then to solve (10) with the appropriate boundary conditions. But one of the boundary conditions, the one at the free surface, involves the pressure. It is therefore necessary to obtain an expression for p in terms of (. This is supplied by the Bernoulli equation for irrotational unsteady flows: + + +w + (g+a)z F(t) (11) at p 2 The derivation of (11) is quite simple. With the vorticity components in (4) and (5) equal to zero, and with v and w on the left-hand sides given by (9), the Equations (4) and (5) can be written as ax aX 0, 0 by az in which X =- u2++ + (g+a) z dt p 2 Similarly the first equation in (1) gives dX 0 dx Hence, by integration, (11) is obtained. The function F(t) is independent of x, y, and z, and can be absorbed in ( by adding to ( the function - f F(t)dt, without affecting either the velocity components or p. Hence, for convenience F(t) will be taken to be zero. The conditions at the rigid boundaries are X = 0 at the bottom, (12) az and X- = 0 at the walls. (13) an

-8 At the free surface, p = T(- + ), (14) R1 R2 in which T is the surface tension and R1 and R2 are the principal radii of the surface z = ~(xy,t) (15) with ~ now and henceforth denoting the surface displacement rather than the third vorticity component. Since (15) is valid for all values of time t, D [z - (x,y,t)] = 0, Dt or w - + u + v (16) at ax by This is the kinematic condition at the free surface, relating w to ~. The dynamic boundary condition is obtained from (11) and (14), and is T(1 + 1) - + (u2+v2+w2) + (g+a)t = 0 (17) p Ri R2 6t 2 If f and its derivatives with respect the x and y are everywhere small, u, v, and w will be everywhere small, and squares and products in u, v, w, and ~ can be neglected. Equations (16) and (17) can then be written as w = t at z =h, (18) t + +a. (19) Tp + + (g+a)z =0. (19) pdx < o z=h

-9 Following Benjamin and Ursell,(5) we shall take 0 5(xgyvt) = ~ am(t)Sm(x,y), (20) o and (yzt) = dam(t) cosh km Sm(x,y) + G(t), (21) 1 dt km sinh kmh in which Sm(x,y) satisfies (2 2 2( ( + ay2 + k) Sm(xy) = 0, (22) in which the k's are the eigenvalues that make m( X) equal to an zero, and of course depend only on the shape of the container. Note that (13) and (18) imply that a = 0 at the walls, an and that Equation (20) satisfies this condition. Also, ( given in (21) satisfies (12) and (13), as well as (18). The eigenvalue ko = 0 corresponds to So(x,y) = Oo As explained by Benjamin and Ursell, ao(t) is constant, since the total volume of the liquid is constant. If the origin of ~ is taken from the mean free surface, ao(t) = O Hence, it follows from (19) that G(t) can only be a constant, which can be taken to be zero without affecting anything. With (20) and (21) substituted into (19), the result is Xk Sm(x[,y) d + am tanh kh(mT ga)] (2 km tanh kmh [dt - kmh( + g+a)] = 0. (23) 1 km tanhkmh dt- P Since the functions Sm(xy) are linearly independent, d2am dam + (pm+qm) am = 0, (24) dt2

-10in which Pm km tanh kmh(T km + g), qm = a(t) km tanh kmho (25) pEquation (24) is the basis for the analyses in the subsequent se Equation (224) is the basis for the analyses in the subsequent sections.

4. FREE-SURFACE INSTABILITY DUE TO A SUDDEN CHANGE OF ACCELERATION To bring out the effect of variable acceleration on the amplitude of waves, consider the simplest case in which the acceleration is zero at first, then assumes a constant finite value a, and finally becomes zero again, as shown in Figure 3. For simplicity the quantity am, which is a time-dependent amplitude in the sense that it is proportional to the maximum surface height (with respect to x and y) at time t for the m-th mode, will be denoted by A. If attention is focused on the m-th mode, (24) reads dt2 d2A+ (pm+qm)A = 0. (26) Suppose that the acceleration a(t) is changed from zero to the constant a at time t = 0 Then for t < 0, A = B1 cos(slt + @1), B1 > 0. (27) in which sl =Pm, and Q1 is the angle specifying the phase of A at t = 0. For t > 0 the acceleration is a, so that the solution of (26) is A = B2 cos(s2t + 92), (28) in which 2 s2 = Pm+qm with a(t) = a in qm dA Now at t = 0 both A and d- must be continuous. Hence dt B1 cos1 = B2 cosG2, (29) slB1 sinGl = s2B2 sing2 (30) -11

If (29) is squared and (30) is divided by s2 and squared, and the results are added, the following equation is obtained: 2 2 2 ol 2 2 2 2 B2 = Bl[cos 1 + ( l- sinG1) = Bl[l - (1-r ) sin G1], (31) S2 in which Sl r -. S2 Thus, whatever the value of r, IB21 = JB11 if 91= 0. That is to say, the amplitude is unchanged if the change of acceleration from zero to a occurs at the time of maximum A. This is understandable, because at that moment the kinetic energy is zero, and a sudden change in acceleration only brings about a sudden change in potential energy in the proportion (g+a)/g. If the acceleration is maintained constant (= a), the subsequent maximum A will remain unchanged. If, on the other hand, 01 = t/2, then IB21 rIB11, (32) and the amplitude is reduced by the ratio r. We can, indeed, compute the maximum and minimum of (B2/B1)2 from (31). Thus d (B2) = -2(1-r2) singl cosGi = -(1-r2) sin 291, del B1 which is zero for G1 = 0, + t/2, + iT, + 3cr/2, etc. Since r < 1, it is easy to see that the values 0, and + nt (n = integer) correspond to the maximum values of IB2/Bll, and the values +(2n+l)ir/2 correspond to the minimum values. In other words, the most severe amplitudereduction ratio is simply sl/s2.

-13 When the acceleration suddenly drops from a to zero, the analysis is similar, but physically the situation is quite different, Since we are going to consider all possible phase angles, we can again take, without loss of generality, the moment of acceleration change to be the origin of time. For t < 0, we have A =B2 cos(s2t + 2), (5) in which s2 is as defined before, 1B21 maintains the same magnitude, but 02 is no longer 92, because we have changed the origin of time for convenience. For t > 0, A = B3 cos(slt + 03), (34) in which sl = \/p as before. Continuity in A and dA at t - 0 dt demands that B2 os2 = B3 cosQ g c s2B2 sinG2 = slB3 sin93, which produce B2 B2[1 + (r-2-1) sin2G], (35) in which r has the same meaning as before. A similar calculation gives the maximum value for IB3/B21 to be r-1, and the minimum value to be 1. The minimum value, corresponding to no change of amplitude, is again understandable. Since it corresponds to 92 = 0 or nT, it corresponds to the state of maximum potential energy and no kinetic energy.

-14 The maximum value of IB3/B21 correspond to 02 = or +(2n+l)r/2. For 0 < 92 < T/2, the value of IB3/B21 is between 1 and l/r. This brings out the importance of the phase angle 92. (The same is true of 91.) Since waves with all phases are assumed to exist, the maximum ratio of amplitude increase is s2/sl. Consider now the acceleration graph given in Figure 3. If to = 0 and @1 = 0, the amplitude suffers no change on passing through to. At t = tl the value of 92 is really s2(tl-to) + @2 = s2tl + @2. Depending on sl, s2, and t1 (or tl-to), it may or may not be one of the values + (2n+l)n/2. But as tl is varied, it can be one of these values. Thus the absolute maximum for IB3/Bll, or the greatest ratio of amplitude increase, is s2/sl. If the value a is very large, say 48 g, this ratio is exactly 7 if surface-tension effect is neglected, because (a+g)/g = 72. It is significant that it is the region of acceleration decrease that causes the increase in amplitude. At the post-roll region on a Fourdrinier wire, the velocity changes from a downward one along the roll to a slightly upward one along the wire a short distance after a reverse curvature (concave upward), then becomes horizontal again. There is a region where the acceleration changes from a large positive value (upward acceleration) to zero. This is the narrow region of dramatic increase in the amplitudes of the disturbances. The acceleration schedule described by Figure 4 is more realistic. The downward acceleration al corresponds to the region over the table roll. The upward acceleration, a, corresponds to the region of reverse curvature (concave upward). But the wire has to become horizontal

-15 finally. So there must be a region of ing to downward acceleration a2. At ratio is 1, whatever the values of a1 possible amplitude ratio is upward convexity again, correspondt = to the greatest amplitude and ao At t = t1 the maximum ( _1/2 (g+a ) g-a2 which is also the maximum ratio of the amplitude after t2 to the amplitude before ti. Since a2 may be quite near to g, this ratio can be very large. This explains the rather dramatic amplification of disturbances in the post-roll region.

5. FREE-SURFACE INSTABILITY DUE TO A GRADUAL CHANGE OF ACCELERATION An acceleration schedule representing the actual situation on a Fourdrinier wire is given in Figure 5. The region t < to represents the region of downward acceleration over the table rolls. The region to < t < t1 represents the region of increasing upward acceleration. The region t1 < t < t2 represents the region of decreasing acceleration. For t > t2, a region of negative (downward) acceleraagain exists, which corresponds to a region of upward-convexity of the Fourdrinier wire, following the region of reverse curvature (concave upward) where the acceleration is positive. The governing equation is still (26), in which Pm and qm retain their forms in (25), so that -ao for t < to, ao + al+a(t-to) for to < t < t1 tl-to ^Jn a (+ ^ a al+a. ( tanh h = a(t) = al2 (t-tl) for t1 < t < t2K km tanh kmh t2t(t l -a2 for t > t2 Thus it is easy to see that (26) has in all four regions the general form d2A + (a + t)A =. (36) dt2 If P is zero, as for t < to or t > t2, then the solution is of the form A = B cos (la t + 0). (37)

If P is not zero, as in the region of increasing or decreasing acceleration, the solution of (36) is A = (a+t) /2B J1/3(2 (t + )3/) + N1( (t + ) )] (38) Matching of the solutions of types (37) and (38) at t = to and t = t2, and of two solutions of the type (38) at t - tl, can be made on the demand that A and dA/dt be continuous at the moments to, t19 and t2. Since the possibility of tremendous increase in amplitude has already been demonstrated in the last section for acceleration schedules described by Figures 3 and 4, and the demonstration for the present case would differ only in some detail, it will not be given hereo Instead, (37) and (38) will be used to correlate the experimental data, because the actual acceleration schedule in the experiments is not far from that given in Figure 5.

6. EXPERIMENTAL VERIFICATION The theory has been advanced that variable acceleration in a direction normal to the free surface can bring about a tremendous increase in amplitude of surface disturbanceso To test the validity of this theory, an apparatus was constructed (See Plate I) which allowed a plastic 5-inch-square box containing a layer of water to fall about 2 to 3 inches onto a pad of foam-rubber layerso Surface disturbances were made by either one line-jet of air blowing at the middle of the water surface or two such jets parallel to two sides of the box and blowing at two symmetric quarter positions (hence not at the center line). Thus the wave lengths were controlled. When only one jet was used, the wave length X is equal to the inner measure of the side of the box (5 inches). When two jets were used, the wave length was one half of that. As the water-bearing box was released, it fell with very nearly the downward acceleration g. When it hit the pad, this acceleration was first reduced to zero, then became positive, reaching a large positive value after some fluctuations, then decreased to a downward acceleration a2, say, for a length of time before finally becoming zero as the box was brought to rest. The magnitude of a2 is less than g. The acceleration graph is shown in Plate II, in which the value of time t increases from right to left, and positive acceleration is registered below the time axis. The tremendous increase in amplitude occurred in the interval of time during which the acceleration decreased to a2 and -18

-19 and was maintained at a2, as described schematically in Figure 60 Attention will be focused on this period. The magnitude of a2 is near go Hence a2 is assumed to be -g in the calculations. A movie camera took the motion pictures at 250 frames per secondo The surface waves had the form and magnitude shown in Plate III when the acceleration was a2, very shortly after t = t2o The surface form a few milli-seconds after a2 was reached is shown in Plate IV, from which the tremendous increase in amplitude is very evident indeed. A more detailed check of the theory was provided by the following procedure. Two values of A were taken from the motion pictures for two values of t very near t2 but less than t2o Then from the solution (38), in which, as in (36), a and p can be calculated from the experimental data, B and C are determined. Then the value of A calculated for a value of t less than t2 is compared with the experimental value. Similarly, with A and dA/dt known at t = t2 (the latter from two values of A at two instants near t = t2)9 A for t > t2 can be calculated. The values so calculated can be compared with the measured oneso The comparison is shown on Tables 1 to 12. Due to the smallness of A for t < t2, and the consequent difficulty of measurement, the calculated and experimental values of A for t < t2 agreed only in order of magnitudeo The agreement between the calculated and experimental values is much better for t > t2. The values of /2 given in Tables 1 and 2 are for that in (37) for t > t2o Both the pictorial description provided by Plates II and III and the more detailed record provided by Tables 1 and 12 demonstrate

-20very vividly the striking effect of acceleration variation on amplitude increase. The comparison of calculated and experimental values given in these tables also indicates the general validity of the theory given in Sections 3 and 4.

-21 TABLE 1 Max. acceleration = 10.7g X = 3.6 t2 = 16 millisec. h = 0.25" Max. (2A) = 3.19" \a = 3.06 rad/seco t (millisec.) 5 20 25 35 2A Calculated (inches) 0.0015 0.04 0.09 0 19 2A Measured (inches) 0.0010 0.04 0,10 0.22 TABLE 2 Max. acceleration = l0.7g X = 3.4" t2 - 16 milliseco h = 0.25" Max. (2A) = 3.48" /? = 3549 rad/seco t (millisec.) 5 25 35 2A Calculated (inches) 0.0057 0.12 0,24 2A Measured (inches) 0.0010 0.15 0.25 TABLE 3 Max. acceleration = 12.81g X = 3.6 t2 = 14 millisec. h = 0.25" Max. (2A) = 3.09" = 3.24 rad/seco t (millisec.) 5 25 35 2A Calculated (inches) 0.0015 0.12 0.22 2A Measured (inches) 0.0010 0.10 0.20

-22 TABLE 4 Max. acceleration = 12.8g = 3.4" t2 = 14 milliseco h = 0.25" Max. (2A) = 3.27"?6 - 3.68 rad/seco t (millisec.) 5 25 30 35 2A Calculated (inches) 0.006 0.14 0.20 0.26 2A Measured (inches) 0.001 0.12 0o16 0.22 TABLE 5 Max. acceleration = 18.65g X = 306" t2 11 millisec. h = 0.25" Max. (2A) - 3.16" \a 3.16 rad/seco t (millisec.) 5 25 40 2A Calculated (inches) 0.012 0.19 0.34 2A Measured (inches) 0.010 0.19 0.30 TABLE 6 Max. acceleration = 18.65g X = 3.3 t2 = 11 millisec. h = 0.25" Max. (2A) = 2.72" = 3.68 rad/sec. t (millisec.) 5 25 40 2A Calculated (inches) 0.004 0.20 0.45 2A Measured (inches) 0.010 0.18 0.35,,

-23 TABLE 7 Max. acceleration = 11.65g X = 3.4" t2 14 milliseco h = 0.5" Max. (2A) = 1.12" \a = 4.52 rad/seco t (millisec.) 5 40 50 2A Calculated (inches) 0.003 0.14 0.18 2A Measured (inches) 0.001 0,12 0.20 TABLE 8 Max. acceleration = 11.65g X = 3.5" t2 = 14 milliseco h = 0.5" Max. (2A)= 1.41" /a = 4.27 rad/seco t (millisec.) 5 25 35 45 2A Calculated (inches) 0.006 0.10 0,18 0.22 2A Measured (inches) 0.008 0.10 0.15 0.20 TABLE 9 Max. acceleration = 13.42g X = 3.2" t2 = 12 milliseco h = 0,5" Max. (2A) = 0.61" a = 5,0 rad/seco t (millisec.) 5 40 70 2A Calculated (inches) 0.004 0.09 0.18 2A Measured (inches) 0.001 0.10 0.18

-24 TABLE 10 Max. acceleration = 13.42g X = 3.4" t2 12 milliseco h = 0.5" Max. (2A) = 1.11" a - 4.51 rad/sec. t (millisec.) 5 20 30 40 2A Calculated (inches) 0.0014 0.05 0o10 0,15 2A Measured (inches) 0.0020 0o04 0.09 0o15 TABLE 11 Max. acceleration = 19.2g X = 3.1" t2 12 millisec, h = 005" Max. (2A) = 0.57" t = 5.3 rad/sec. t (millisec.) 5 30 40 2A Calculated (inches) 0.0017 0.06 0.09 2A Measured (inches) 0.0020 0.08 0.12 TABLE 12 Max, acceleration = 19.2g X = 3.3" t2 = 12 millisec. h = 0.5" Max. (2A) = 2.34" \a = 3.43 rad/sec. t (millisec.) 5 15 25 35 2A Calculated (inches) 0.012 0.05 0,13 0o21 2A Measured (inches) 0.010 0.05 0.11 0,18.....,.... ~~~~~~~~~~~~~~~~~~~~~f ~

7. APPLICATION TO POST-ROLL INSTABILITY OF STOCK ON FOURDRINIER WIRES The striking effect of acceleration variation on the amplitude of surface disturbances has been demonstrated both theoretically and experimentally. The pertinence of this effect on post-roll instability of stock on a Fourdrinier wire can be seen from the acceleration schedule of the stock as it passes over the table rolls. On top of a table roll the acceleration is downward. Then it increases to a large positive value at a region of reverse curvature which must exist as the wire leaves the roll. As the wire has to become horizontal eventually, the reverse curvature will have to pass over to a region of concave-downward curvature again. This is a region of downward acceleration. Thus the acceleration schedule-is much like that given in Figure 5. The latter part of the schedule (Figure 6) is chiefly responsible for the amplitude increase. One point of great interest is the phase shift of the surface disturbances as they leave the table rolls. A qualitative explanation can be supplied here, For simplicity consider the acceleration schedule described by Figure 3. (The same general conclusion can be reached by considering more complicated and realistic acceleration schedules.) In the region of upward acceleration the solution is given by (28) instead of (27). In (28) s2 can be very large if a is large, and a large S2 means a greater frequency and a shorter period of time required for a phase shift of 1800, i.e., for the ridges to become troughs and vice versa. -25

-26 The growth of surface instability as the stock passes over the table rolls has been adequately explained in Reference 1. In Reference 4 the velocity distribution in the post-roll region was considered a possible cause of an instability of the Taylor-Goirtler type, with the presence of growing longitudinal vortex tubes. Although this remains a possibility, the violence of amplitude growth in the post-roll region seems to favor the variability of acceleration as the main cause of instability, particularly since this explanation is entirely independent of the velocity distribution in the post-roll region. The authors believe that the variability of acceleration is indeed the main cause not only of post-roll instability, but also of the instability of the free surface as the stock leaves the slice. This paper serves as a final report on the research sponsored by TAPPI on free-surface instability for the past several yearso The writer of this paper (C.-S. Yih), who has served as supervisor of this research, wishes to express his sincere thanks to TAPPI for this sponsorship, and to the members of the several fluid-mechanics committees of TAPPI, for their interest and many stimulating discussions. The authors also wish to thank Dr. W. R. Debler and Messrs, Milo Kaufman and Ao Engerer for assistance in the experimental program,

REFERENCES 1. Yih, C.-S.. "Instability of a Surface, " Proc. Roy. Soc. A, Rotating Liquid Film with a Free Vol. 258, pp. 63-86, 1960. 2. Yih, C.-S., "Stability of a Rotating Liquid Film, " No. 6, pp. 524-527, 1962. TAPPI, Vol. 45, 3. Debler, W. R. and Yih, C.-S., "On the Instability of Stock on a Fourdrinier Wire," TAPPI, Vol. 45, No. 4, ppo 272-279, 1962. 4. Yih, C.-S. and Spengos, A. C., "Free-Surface Instability," Vol. 42, No. 5, pp. 398-4035 1959. TAPPI, 5. Benjamin, T. B. and Ursell, F., "Instability of the Plane Free Surface of a Liquid in Vertical Periodic Motion," Proc. Roy. Soc. A, Vol. 225, pp. 505-515, 1954. -27

—-I I a b c a = a1. Wave Attains Maximum Height. a Changes Less Than Flat. to a2, Which is a1. Surface is a = a2. Wave Attains Next Maximum Height. Figure 1 I ro!o l a b c a = al. Wave Attains Maximum Height. a Changes to a2, Which is Greater Than al. Surface is Flat. a = a2. Wave Attains Next Maximum Height. Figure 2

-29 a(t) A t to t, Figure 3. An Idealized Acceleration Schedule. O(t) t + Figure 4. A More Realistic Acceleration Schedule.

-30 a(t) t Figure 5. An Acceleration Schedule Representing the Actual Situation on a Fourdrinier Wire. a(t) t Figure 6. That Portion of Figure 5 Used for Experimental Verification of the Theory.

PItlate i:. Apparatus. Pl.]ate +II (. Acce1lerat ion Graph.

Pl.ate AIX. Wave Form Soon After a2 (Nearly F qua:l to -g) is Reached by a, After Deereasing From a tanrge Posibtive Val.ue. Plate IV. A Few Mli A.:,-Seconds After the Condition in Pl]ate II:I: Was Reached.