On the drag-out problem in liquid film theory
|
|
- Kevin Reynolds
- 6 years ago
- Views:
Transcription
1 J. Fluid Mec. (28), vol. 617, pp c 28 Cambridge University Press doi:1.117/s x Printed in te United Kingdom 28 On te drag-out problem in liquid film teory E. S. BENILOV AND V. S. ZUBKOV Department of Matematics, University of Limerick, Ireland (Received 15 February 28 and in revised form 4 September 28) We consider an infinite plate being witdrawn (at an angle α to te orizontal, wit a constant velocity U) from an infinite pool of viscous liquid. Assuming tat te effects of inertia and surface tension are weak, Derjaguin (C. R. Dokl. Acad. Sci. URSS, vol. 9, 194, p. 1.) conjectured tat te load l, i.e. te tickness of te liquid film clinging to te plate, is l =(μu/ρg sin α) 1/2,wereρ and μ are te liquid s density and viscosity, and g is te acceleration due to gravity. In te present work, te above formula is derived from te Stokes equations in te limit of small slopes of te plate (witout tis assumption, te formula is invalid). It is sown tat te problem as infinitely many steady solutions, all of wic are stable but only one of tese corresponds to Derjaguin s formula. Tis particular steady solution can only be singled out by matcing it to a self-similar solution describing te non-steady part of te film between te pool and te film s tip. Even toug te near-pool region were te steady state as been establised expands wit time, te upper, non-steady part of te film (wit its tickness decreasing towards te tip) expands faster and, tus, occupies a larger portion of te plate. As a result, te mean tickness of te film is 1.5 times smaller tan te load. 1. Introduction Many industrial coating processes can be modelled by a simple setting were an infinite sloping plate is witdrawn from an infinite pool of viscous liquid (see figure 1a). Given tat te plate s speed is constant and, eventually, a steady state is establised, one usually needs to know te so-called load, i.e. te tickness of te liquid film tat te plate carries away from te pool. Tis classical problem was originally formulated by Derjaguin in 194, wo conjectured tat, if surface tension is negligible and te plate is being witdrawn slowly (ence, te effect of inertia is weak), te load l is ( ) 1/2 μu l =, (1.1) ρg sin α were ρ and μ are te liquid density and dynamic viscosity, U is te plate speed, α is te angle between te plate and te orizontal, and g is te acceleration due to gravity. In 1945, Derjaguin supported is conjecture by a quantitative argument; formula (1.1) as also been verified experimentally (Derjaguin & Titievskaya 1945) and numerically (Jin, Acrivos & Münc 25). Note, owever, tat Derjaguin s (1945) argument supporting formula (1.1) was more intuitive tan rigorous (see a discussion in Appendix A), and it left several address for correspondence: eugene.benilov@ul.ie
2 284 E. S. Benilov and V. S. Zubkov y * Liquid film U (a) α Pool x * x (b) x = t x Figure 1. Formulation of te problem: (a) te setting, (b) te matematical model. important issues unresolved. First, one can sow tat te problem admits infinitely many steady solutions, all corresponding to different loads and it is unclear wy te observed value, (1.1), is selected among all te oters. Secondly, since formula (1.1) does not ave a rigorous matematical foundation, it is difficult to establis its pysical limitations. In tis work, we sall remedy te above sortcomings. We sall demonstrate tat te observed steady state (and, ence, te load) is, paradoxically, determined by te unsteady part of te film adjacent to its tip (no matter ow far tat as moved away). Furtermore, it will be sown tat te film s tickness is close to te load only near te pool s edge, wile most of te region between te pool and film s tip is unsteady. As a result, te load (being a steady-state caracteristic) is not representative of te mean tickness of te film drawn from te pool but differs from it by an order-one factor. Finally, formula (1.1) will be sown to old only for moderate values of te plate s slope.
3 On te drag-out problem in liquid film teory 285 Te paper as te following structure: in 2, te problem is formulated matematically; in, we examine te case were te plate s slope is small; and, in 4, te general case is tackled. 2. Formulation of te problem Let te x -axis of te (x,y ) coordinate system (asterisks indicate tat te corresponding variables are dimensional) be directed along te plate and downwards see figure 1(a). We sall also introduce te (similarly oriented) velocities (u,v ), te pressure p, te film s tickness, and te time t Te limit of zero surface tension Following Derjaguin (1945), we sall assume tat inertia and surface tension are negligible, in wic case te flow is governed by te Stokes equations, ( ( 2 u 2 v p x = ρg sin α + μ x u y 2 ), p y = ρg cos α + μ x v y 2 ), (2.1) u + v =. (2.2) x y Te no-slip and no-flow conditions at te plate are u = U, v = at y =. (2.) Te kinematic condition at te liquid s surface can be written in a form reflecting mass conservation, + t x wereas te dynamic condition is were u dy =, (2.4) S n = at y =, (2.5) 2μ u ( u p μ + v ) [ S = ( x y x ] u μ + v ) 2μ v, n = x (2.6) p 1 y x y are te stress tensor and a (not necessarily unit) normal to te liquid s surface. Note tat our setting involves a contact point, separating te dry and wet parts of te plate wic, generally speaking, requires a certain boundary condition (see Dussan V. 1979; Sikmurzaev 26, and references terein). As a result, a boundary layer occurs near te contact point. Te widt of tis layer, owever, is extremely small, and all of te previous researcers dealing wit te drag-out problem neglected its effect. Accordingly, we sall assume tat te effect of dewetting is negligible and te contact point moves togeter wit te plate wic corresponds to te following boundary condition: = at x = Ut. (2.7)
4 286 E. S. Benilov and V. S. Zubkov At large distances from te edge of te pool, te pool s surface is unperturbed, i.e. (tan α)x as x +. (2.8) Te boundary-value problem (2.1) (2.8) determines te flow in te domain Ut <x < +, <y <. Next we introduce te following non-dimensional variables: x = εx, y = y l l, t = εut, = l l, (2.9) u = u U, v = v εu, p = (1 + ε2 ) 1/2 p, (2.1) ρgl were l is defined by (1.1) and ε =tanα (te use of symbol ε does not necessarily imply tat tan α 1). Substituting (2.9) (2.1) into te Stokes set (2.1) (2.8), we obtain p 2 u x =1+ε2 x + 2 u 2 y, 2 ( 2ε 2 u x p ( u ε 2 v + ε2 y x u x + v y p y = 1+ε4 2 v x 2 + ε2 2 v y 2, (2.11) =, (2.12) u = 1, v = at y =, (2.1) ) ( ) u x v + ε2 = y x at y =, (2.14) ) ( x 2ε 2 v ) y p = at y =, (2.15) t + u dy =, (2.16) x = at x = t, (2.17) x as x +. (2.18) In wat follows, equations (2.11) (2.18) will be used wen te plate s slope is orderone. If, owever, it is small, ε 1, one can take advantage of te lubrication approximation. Following te usual routine of lubrication analysis, one can treat (2.11) (2.15) as a boundary-value problem for u, v, andp and express tem as expansions in ε 2,e.g. )( u = 1+ (y y2 1 ) + O(ε 2 ). (2.19) 2 x Substituting tis equality into (2.16) and omitting te O(ε 2 ) terms, we obtain t + ( ) + x =. (2.2) x
5 On te drag-out problem in liquid film teory 287 Note tat te first term in te brackets corresponds to entrainment of te liquid by te plate s motion, te second term describes te effect of gravity, and te last one te ydrostatic pressure gradient due to variations of te film s tickness. Similar equations ave been derived, for similar problems, by Moriarty, Scwartz & Tuck (1991), Bertozzi & Brenner (1997), and Kalliadasis, Bielarz & Homsy (2). Finally, te asymptotic equivalents of te boundary conditions (2.17) (2.18) are = at x = t, (2.21) x as x +. (2.22) Te boundary-value problem (2.2) (2.22) is illustrated in figure 1(b) Te effect of surface tension As mentioned before, tis paper is concerned wit te limit of negligible surface tension, but it is still instructive to briefly discuss te capillary-modified version of equation (2.2). Its derivation is very similar to tat of (2.2), resulting in t + x ( + x + γ x ) =, were σ (tan α) γ = (2.2) μu is te non-dimensional parameter caracterizing te importance of capillary effects (σ is te surface tension, recall also tat μ is te viscosity and U, te plate s velocity). To illustrate expression (2.2), let te liquid under consideration be glycerine at 2 C, for wic μ 1.41kg m 1 s 1, σ.6 N m 1. Ten, for a moderately small angle α = 1π, and a velocity U =.1m 9 s 1, formula (2.2) yields γ.4 wic can be safely regarded as a small parameter. Furtermore, wit te exception of mercury, surface tensions of common liquids range witin.2.8 N m 1, wereas te corresponding viscosities may differ by up to two orders of magnitude. Te viscosity of industrial polymers, for example, often reaces 4 kg s 1 m 1, wile tat of liquid foods suc as ketcup, mustard, or peanut butter ranges from 5 to 25 kg s 1 m 1. Tus, in many applications, γ is very small and capillary effects are negligible.. Te small-slope limit It appears obvious (and will be verified later) tat, even toug a steady state emerges near te edge of te pool, te film s tip always remains non-steady. At large times (t 1), te two regions become well separated and can be examined separately see.1. Ten, in.2, te asymptotic results will be verified and complemented by direct simulation of te small-slope equation (2.2)..1. Asymptotic results for t 1 For a steady solution, = (x), equation (2.2) yields + d dx = q (.1)
6 288 E. S. Benilov and V. S. Zubkov 2 1 c b a x Figure 2. Te film s tickness (x) of a steady drag-out flow (determined by te boundary-value problem (.1) (.4)), for various values of te non-dimensional load: (a) =1, (b) = 2, (c) =1.7. Solutions (a) and(b) correspond to te endpoints of range (.5). were te constant of integration q is, pysically, te non-dimensional flux of liquid carried by te plate. We sall also require as x, (.2) were is te non-dimensional load, i.e. te tickness of te film carried by te plate infinitely far from te pool. Taking te limit x in (.1) and taking into account (.2), we obtain q = + 1. (.) Finally, condition (2.22) (wic matces te film to te pool) yields x as x +. (.4) Equation (.1) is separable and, tus, boundary-value problem (.1) (.4) can be readily solved (te actual solution is not presented ere, as it is bulky but several examples are sown in figure 2). Most importantly, solutions exist for all 1, i.e. te steady-state problem leaves te load undetermined. Te allowable range of, owever, can be reduced by analysing expression (2.19) for te liquid s velocity. Estimating its sign at y =, one can see tat, for > 2, te film s top layers slide back towards te pool wic corresponds to a source of liquid located at minus-infinity and makes te solution irrelevant pysically. Still, all values in te range 1 2 (.5) appear to give rise to pysically meaningful steady solutions. Only one of tese, owever, corresponds to te correct (experimentally verified) formula (1.1) for te dimensional load. We note tat paradoxes resulting from multiple solutions can often be resolved by a stability analysis (wic would usually sow tat all solutions, except one, are unstable). In te present case, owever, all solutions corresponding to range (.5) are stable (see Appendix B). As it turns out, te correct value of te load can only be identified by studying te non-steady part of te film. Fortunately, tat appens to be described by a
7 On te drag-out problem in liquid film teory 289 relatively simple self-similar solution wic, owever, does not necessarily matc te steady solution near te pool. Te only value of for wic te two solutions do matc is te correct one. To find te solution describing te non-steady part of te film, let (x,t)=t 1 f (ξ), (.6) were ξ =(x + t)t (.7) (tis substitution corresponds to a self-similar beaviour in te reference frame associated wit te film s tip). Substitution of (.6) (.7) into equation (2.2) yields [ ] f f + f ξ (f 1) =. (.8) Te condition at te tip of te film, (2.21), implies f () =. We ave found tree distinct asymptotics of equation (.8) compatible wit te above boundary condition (and are reasonably sure tat none as been missed): f aξ 1/4 + 4ξ + O( ξ 5/4) as ξ + (.9) 7 or f = ξ + a exp[ ξ 1 2lnξ + O(ξ)] as ξ + (.1) (in bot cases, a is arbitrary) or f = aξ + a (a 1)ξ 2 + O(ξ ) as ξ + (.11) (a 1). To distinguis between te tree possible asymptotics, note tat te evolutionary equation (2.2) preserves te slope at te film s tip (see Appendix C). Ten, assuming tat our solution originates from te initial condition were te pool s surface is unperturbed (ence, / x = 1), we conclude tat te desired asymptotic is (.1). Asymptotics (.9) implies an infinite slope at ξ =, wereas (.11) implies a finite one, but it still does not matc te derivative of te initial condition. At large ξ, in turn, equation (.8) yields f = ξ 1/2 + 1 ln ξ + b + O( ξ 1/2 ln 2 ξ ) as ξ + (.12) 12 or f = ξ + b(1 + ξ 1 +9ξ 2 ln ξ)+cξ 2 + O(ξ ln ξ) as ξ +, were b and c are arbitrary. It can be verified, owever, tat te latter asymptotic cannot be matced to te steady solution (x) near te pool and, tus, is irrelevant to te problem at and. To matc te remaining asymptotic, (.12), to (x), assume tat x is far from te pool, but not too close to te film s tip, 1 x t (.1) (wic, obviously, implies tat t 1). Ten, in solution (.6) (.7), we can replace f wit asymptotics (.12) and obtain t 1 (x + t)t 1. (.14)
8 29 E. S. Benilov and V. S. Zubkov f ξ Figure. Te sape of te self-similar part of te film (i.e. te numerical solution of te boundary-value problem (.8), (.1), (.12)). Te dotted lines correspond to te asymptotics given by te first term of (.1) and te first two terms of (.12). Next, recall tat, in region (.1), te steady solution (x) is close to its limiting value ence, it can be matced to te limiting value (.14) of te self-similar solution only if =1. (.15) Recalling ow (and, ence, ) were non-dimensionalized (see (2.9) (2.1)), one can see tat (.15) agrees wit formula (1.1) for te dimensional load l. To complete our asymptotic results, we present te (implicit) solution of te steady problem (.1) (.4) corresponding to te correct value =1, ( ) ln =9x. (.16) +2 1 We sall also need te function f (ξ) wic describes te sape of te non-steady part of te film. To find it, we solved te boundary-value problem (.8), (.1), (.12) numerically. Instead of zero and plus-infinity, te boundary conditions were set at a small and large values of ξ, wit te corresponding values of f determined by te first term of asymptotic (.1) and te first two terms of asymptotic (.12), respectively. Te resulting problem was solved troug an iterative procedure based on Newton s metod, and te solution is sown in figure. Summarizing tis subsection, we present te composite asymptotic solution, (x,t) (x)+t 1 f [(x + t)t] 1, (.17) were (x) is te steady solution emerging near te pool (given by (.16)); f (ξ) describes te sape of te film s non-steady upper part (determined by (.8), (.1), (.12)); and te unity represents te common part of te two solutions. In te limit t, (.17) olds for all values of x..2. Simulation of te evolutionary equation (2.2) Since our asymptotic solution (.17) is applicable only for large t, it cannot be traced back to t =. As a result, we cannot verify tat it corresponds to te initial condition tat we are interested in te one corresponding to te pool s surface being unperturbed, i.e. = x at t =. (.18) Tis issue can only be resolved by direct simulation of te small-slope equation (2.2). Te initial/boundary-value problem (2.2) (2.22), (.18) was integrated using te finite-element solver available in te COMSOL Multipysics package. Te solution
9 On te drag-out problem in liquid film teory 291 (a) (b) x Figure 4. Te evolution of te film for te small-slope case (determined by te initial/ boundary-value problem (2.2) (2.22), (.18)). Te curves are labelled wit te corresponding values of te time t. Te solid curves sow te numerical solution, te dotted curves sow: (a) te composite asymptotic solution (.17), (b) te steady solution (.16) and te self-similar solution (.6) (.7). computed, as well as te composite asymptotic solution (.17), are sown in figures 4 and 5. One can see tat, at t = 1, te two solutions agree reasonably well; at t = 2, tey are difficult to distinguis; and, for t, virtually indistinguisable. In addition to te composite asymptotic solution, we sow its components (te steady and self-similar solutions) in figure 4(b). Observe tat, even for te long-term evolution sown in figure 6(a), tere seems to be no visible sectoin were te solution is orizontal, i.e. equal to te load. Moreover, te emerging steady state becomes visible only if a small area near te pool s edge is enlarged (as in figure 6(b)). Te reason for tis is tat te steady-state region (wit approximately equal to te load) expands muc slower tan te non-steady region (wit variable ) ence, te portion of te plate occupied by te former is muc smaller tan tat occupied by te latter. Tis circumstance as crucial implications for te mean tickness of te film (wic is te most important caracteristic of te drag-out process). If te solution outside te pool were steady, te mean tickness would be equal to te load l, but, since te film is mainly unsteady and its tickness varies from l to, te mean is less tan l.
10 292 E. S. Benilov and V. S. Zubkov x Figure 5. Te evolution of te film for te small-slope case (determined by (2.2) (2.22), (.18)). Te curves are labelled wit te corresponding values of te time t. Te solid curves sow te numerical solution, te dotted curves sow te composite asymptotic solution (.17) (for t, te numerical and asymptotic curves are virtually indistinguisable). (a) x (b) x Figure 6. Te evolution of te film for te small-slope case (determined by (2.2) (2.22), (.18)). Te curves are labelled wit te corresponding values of te time t. Te solid curves sow te numerical solution, te dotted curve sows te steady solution (.16). Te area saded in panel (a) corresponds (b) (te latter illustrating te emergence of te steady state near te edge of te pool).
11 On te drag-out problem in liquid film teory x Figure 7. Te same as in figure 5, but for te initial condition (.19) (.2) instead of (.18). To calculate te mean tickness for t 1, one can approximate te profile of te self-similar solution, f (ξ), by its large-distance asymptotics (.12) (wic is not valid only in a relatively small region adjacent to te film s tip). Tus, taking into account tat te lengt of te film grows linearly wit time, we obtain 1 mean non-dimensional tickness = lim t 1 (x + t) t dx = 2 t t. t Dimensionally, tis implies tat te film s mean tickness is 1.5 times smaller tan te load given by formula (1.1). Finally, we sall discuss wat appens if te pool s surface is initially perturbed. In particular, let te initial condition ave non-unit derivative at te pools s boundary, i.e. ( ) 1 at t =. x x= Ten, since te slope at te film s tip is conserved (see Appendix C), suc initial conditions seem to be incompatible wit our self-similar solution (.6) (.7), (.1) for wic / x at te film s tip equals unity. Te simplest option in tis case appears to be replacing te boundary condition (.1) wit (.11) wic allows non-unit derivative at ξ =, i.e. attefilm stip. It turns out, owever, tat te resulting boundary-value problem, (.8), (.11) (.12), as no solution. (Tis is not a matematical teorem but rater a conclusion based on numerical evidence.) To resolve tis issue, te small-slope equation (2.2) was simulated for various initial conditions wit non-unit derivative at te film s tip. Typical results, computed for = x + kxe x at t =, (.19) wit k = 1, (.2) are sown in figure 7. One can see tat te asymptotic solution (.17) agrees wit te numerical one everywere except te vicinity of te film s tip. Most importantly, te disagreement region rapidly contracts wit time and as no influence watsoever on te load or oter global caracteristics.
12 294 E. S. Benilov and V. S. Zubkov α Figure 8. Te non-dimensional load (as determined by te Stokes equations (2.11) (2.18)) vs. te angle α of te plate s slope. Te dotted line corresponds to te small-α asymptotic result, (.15). Te same pattern was observed for oter values of k, as well as oter initial conditions. 4. Te case of finite slope If te slope of te film s surface is not small, te Stokes equations (2.11) (2.18) do not involve small parameters and can only be solved numerically. Tis task as been carried out originally by Jin et al. (25) for a more general model including surface tension. In te present work, in turn, surface tension is neglected, as we need to verify our asymptotic results presented above. Te Stokes equations (2.11) (2.18) were integrated using te Moving Mes (ALE) and Incompressible Navier Stokes modules of te COMSOL Multipysics. Te parameters were cosen in suc a way tat, in all runs, te Reynolds number was 1 4, so inertia was negligible. Starting from an appropriate initial condition, te problem was simulated until a steady state ad been establised everywere in te computational region, after wic te load could be measured. Te results are sown in figure 8. One can see tat te small-slope approximation works for fairly large angles, α 5. Furtermore, its error in tis range is 1, wic is a great deal better tan expected. Indeed, te small-slope equation (2.2) was derived by omitting O[(tan α) 2 ] and smaller terms from te Stokes set ence, for α 5, te truncation error sould be about (tan 5 ) 2.5. Tis discrepancy suggests tat some (probably more tan one) of te next-order corrections to te asymptotic load cancel out, improving te effective accuracy of te leading-order formula. Observe also te abrupt turn wit wic te numerical curve splits from its asymptotic counterpart at α 5. It appears tat a pitcfork bifurcation occurs at tis point: te old solution loses stability and te system switces to a lower branc. Suc an explanation implies tat tere exists anoter stable solution, corresponding to te upper branc (i.e. wit a load iger tan te observed one) owever, it as never appeared in our simulations. To resolve te discrepancy, recall tat te evolution begins from te initial condition wit a zero load; as a result, te system cannot reac te ig-load solution witout
13 On te drag-out problem in liquid film teory 295 passing troug te low-load one. Te latter, owever, is a steady state, and, once it is reaced, te system stops evolving and remains tere indefinitely. In oter words, te ig-load solution, if it exists, cannot be computed if te liquid is initially at rest and te pool s surface is unperturbed nor can it be observed (for te same reason) in te real world. 5. Summary and concluding remarks We ave examined te flow of a viscous liquid induced by an infinite sloping plate being witdrawn from an infinite pool. It was conjectured (Derjaguin 194) tat te load l, i.e. te tickness of te liquid film clinging to te plate, is given by formula (1.1). Tis conjecture, owever, was not supported by a consistent calculation, and none as followed since 194. In te present paper, formula (1.1) for l is derived from te Stokes set under an additional assumption tat te plate s slope is small, in wic case a relatively simple asymptotic equation, (2.2), can be derived. It is sown tat (2.2) as infinitely many steady solutions, all of wic are stable, but only one of tese corresponds to formula (1.1). Tis particular steady solution, (.16), is ten identified by matcing it to a self-similar solution for te film s non-steady upper part (described by (.6) (.7), (.8), (.1), (.12)). Te two asymptotic solutions are eventually combined as a composite solution, (.17), wic is applicable everywere. It is demonstrated (by direct simulation of equation (2.2), see.2) tat te composite solution is an attractor and, tus, provides large-time asymptotics for a wide class of initial conditions. It is also sown tat, even toug te region were te steady state as been establised expands wit time, te upper part of te film (wit its tickness decreasing towards te tip) expands faster. As a result, te mean value of te film s tickness is 1.5 times smaller tan te load. We ave also carried out direct simulations of te Stokes set and, tus, sowed tat te small-slope approximation is valid wen te angle of te plate s slope is less tan, approximately, 5. Remarkably, its accuracy witin tis range is about 1, wic is exceptionally ig. Finally, it would be interesting to extend te present approac to capillary effects and compare te results wit te extensive body of literature on steady drag-out flows wit surface tension (Landau & Levic 1942; Wilson 1982, etc.). Tis work was supported by te Science Foundation Ireland troug te Matematics Applications Consortium for Science and Industry (MACSI). Appendix A. Te paper by Derjaguin (1945) Unlike our work, Derjaguin (1945) (wose paper will be referred to as D45) assumes te plate to be motionless and te level of liquid in te pool to be dropping at a constant speed. Tis setting can be obtained from ours by canging te latter to te reference frame co-moving wit te plate. Te two approaces differ also in a more substantial way: unlike our work, D45 neglects te pressure gradient associated wit variations of te film s tickness. In terms of our non-dimensional variables, tis amounts to x 1. (A 1)
14 296 E. S. Benilov and V. S. Zubkov Accordingly, te last term in te small-slope equation (2.2) can be omitted, wic yields t + ( ) + = (A2) x (observe tat te cange of variables x co moving = x + t transforms (A 2) into D45 s equation (8)). Te general solution of (A 2) can be readily found (in an implicit form) by te metod of caracteristics, x = F ()+( 1+ 2 )t. (A ) Te function F () is determined by te initial condition, wic, somewat unexpectedly, D45 assumes to be = for t =, x > (A4) (see D45 s equation (1)). Eventually, D45 concludes tat F () = and (A ) yields x = +1. (A 5) t Note tat D45 s condition (A 4) implies tat te angle between te plate and te pool s surface is 9, wereas a more sensible assumption would be tat it is α. In terms of our non-dimensional variables (see (2.9) (2.1)), te latter amounts to = x for t =, x >. Accordingly, te undetermined function in (A ) becomes F ()=, and (A ) yields x = t t 1. (A 6) 2 2t Tis solution rectifies te inconsistency resulting from te incorrect initial condition (A 4). Note also tat, as t, te correct and incorrect solutions, (A 6) and (A 5), forget teir respective initial conditions and merge. Unfortunately, D45 contains anoter inconsistency, one tat cannot be easily rectified. It as several manifestations: first, it follows from (A 6) tat x t as x +, wic is inconsistent wit te boundary condition (2.22) and, tus, does not matc te pool. Secondly, even if it did (i.e. if x for large x), it would imply / x 1, wic violates assumption (A 1) on wic te original equation (A 2) is based. Tirdly, assumption (A 1) is also violated near te film s tip (x t) and te liquid level (x ). In oter words, D45 sould ave used te full equation (2.2) instead of te truncated one, (A 2). (Note tat (A 2) is still valid were te film s slope is small, e.g. far from bot te film s tip and te pool (provided tese regions are far apart, i.e. for large t). In tis case, te corresponding limit of solution (A 6) coincides wit te correct solution (.6) (.7) were f (ξ) is replaced wit its large-ξ asymptotics, i.e. te first term of (.12).).
15 On te drag-out problem in liquid film teory 297 Appendix B. Stability of steady solutions of equation (2.2) Let = (x)+ (x,t), (B 1) were is a steady solution of te small-slope equation (2.2) and is a disturbance. Substituting (B 1) into (2.2) and linearizing, we obtain t + ( + 2 d 2 x dx ) =. (B 2) x We sall confine ourselves to solutions wit exponential dependence on t, (x,t)=φ(x) e λt, for wic (B 2) yields λφ + d dx ( φ + 2 φ 2 φ d dx ) dφ =. (B ) dx We sall assume tat te disturbance is bounded at infinity, i.e. φ < as x ±. (B 4) (B ) (B 4) form an eigenvalue problem, were φ is te eigenfunction and λ is te eigenvalue. If tere exists an eigenvalue suc tat Reλ >, te steady state (x) is unstable. Observe tat, since as x, (B ) yields ( ) ( ) λ λ φ C 1 exp x + C 2 exp x as x, (B 5) were C 1,2 are constants of integration. Ten, it follows tat φ does not tend to zero (but oscillates) only if Reλ <, Imλ =; suc solutions are stable and, tus, do not need furter examination. In all oter cases, condition (B 4) requires one of te constants C 1,2 be zero, and (B 5) results in φ as x. (B 6) To analyse te beaviour of φ as x +, we sall need te corresponding asymptotics of (x) (wic can be extracted from (.1) (.4)): = x +x 1 + O(x 2 ) as x +. Ten, as follows from (B ), φ = C 1 [1 + O(x 1 )] + C 2 [x 2 + O(x )] as x +. Note tat solutions wit C 1 involve canges of te liquid s level in te pool far away from its edge and, tus, are irrelevant to te stability of te drag-out flow (moreover, it can be sown tat tey imply λ =, i.e. are stable anyway). As a result, we can assume φ as x +. (B 7) In wat follows, it is convenient to move te boundary conditions (B 6) (B 7) from infinity to finite points, φ = at x = x ±, (B 8)
16 298 E. S. Benilov and V. S. Zubkov ten take te limit x, x + + (te advantages of tis approac will be explained later). Next, we introduce a new variable ψ suc tat φ = ψ ) ( 2 exp 1 dx. In terms of ψ, te eigenvalue problem (B ), (B 8) is λ ψ ) [ ( ( 2 )] exp 1 dx = 1 d dψ 2 dx dx exp 1 dx, (B 9) ψ = at x = x ±. (B 1) Now, multiply (B 9) by ψ (were te asterisk denotes complex conjugate), integrate from x to x +, integrate te rigt-and side by parts, and take into account boundary conditions (B 1), wic yields x+ ( ψ 2 ) 2 1 x+ 2 ( ) λ exp dx dx = x 1 dψ 2 1 x dx exp dx dx. (B 11) Tis equality proves tat all eigenvalues λ are real and negative ence, te steady state under consideration is asymptotically stable. Ten, in te limit x ± ±,all λ are real and non-positive, i.e. te steady state is eiter asymptotically or neutrally stable. (Note tat wen te region were an eigenvalue problem is defined tends to infinity, some of te eigenvalues may accumulate near zero.) Observe tat, if te boundary conditions were set at infinity from te start, te integrals in (B 11) would ave infinite limits and we would need to deal wit teir convergence (wic is not difficult in principle, but involves cumbersome algebra involving large-distance asymptotics of ψ and te coefficients of equation (B 9)). Appendix C. Conservation of te film s slope at its tip Denote te film s slope at its tip by s, i.e. ( ) s = x, x= t and consider ( ) ds 2 dt = t x 2. (C 1) x 2 x= t Recalling tat (x,t) is governed by te small-slope equation (2.2), one can replace te t-derivative in (C 1) troug te x-derivatives and obtain { ( ds dt = x x x + ) [( ) 2 ( ) 2 ]} 2 x x x 2 x x x= t Since, by definition, te film s tickness vanises at te tip (i.e. () x = t = ), te rigt-and side of te above equality is zero. REFERENCES Bertozzi, A. L. & Brenner, M. P Linear stability and transient growt in driven contact lines. Pys. Fluids 9, 5 59.
17 On te drag-out problem in liquid film teory 299 Derjaguin, B. 194 Tickness of liquid layer adering to walls of vessels on teir emptying and te teory of poto- and motion-picture film coating. C. R. (Dokl.) Acad. Sci. URSS 9, Derjaguin, B On te tickness of te liquid film adering to te walls of a vessel after emptying. Acta Pysicocim. URSS 2, Derjaguin, B. V. & Titievskaya, A. S An experimental study of te tickness of a liquid layer left on a rigid wall by a retreating meniscus. Dokl. Acad. Nauk SSSR 5, 4. Dussan V., E. B On te spreading of liquids on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mec. 11, Jin, B., Acrivos, A. & Münc, A. 25 Te drag-out problem in film coating. Pys. Fluids 17, 16. Kalliadasis, S., Bielarz, C. & Homsy, G. M. 2 Steady free-surface tin film flows over topograpy. Pys. Fluids 12, Landau, L. & Levic, B Dragging of liquid by a plate. Acta Pysiocim. USSR 17, Moriarty, J. A., Scwartz, L. W. & Tuck E. O Unsteady spreading of tin liquid films wit small surface tension. Pys. Fluids A, Sikmurzaev,Y.D.26 Singularities at te moving contact line. Matematical, pysical and computational aspects. Pysica D 217, Wilson,S.D.R.1982 Te drag-out problem in film coating teory. J. Engng Mats 16,
Steady rimming flows with surface tension
J. Fluid Mec. (8), vol. 597, pp. 9 8. c 8 Cambridge University Press doi:.7/s79585 Printed in te United Kingdom 9 Steady rimming flows wit surface tension E. S. BENILOV,M.S.BENILOV AND N. KOPTEVA Department
More informationA = h w (1) Error Analysis Physics 141
Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.
More informationThe derivative function
Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative
More informationNumerical analysis of a free piston problem
MATHEMATICAL COMMUNICATIONS 573 Mat. Commun., Vol. 15, No. 2, pp. 573-585 (2010) Numerical analysis of a free piston problem Boris Mua 1 and Zvonimir Tutek 1, 1 Department of Matematics, University of
More informationNUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,
NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing
More information3.4 Worksheet: Proof of the Chain Rule NAME
Mat 1170 3.4 Workseet: Proof of te Cain Rule NAME Te Cain Rule So far we are able to differentiate all types of functions. For example: polynomials, rational, root, and trigonometric functions. We are
More informationLecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator.
Lecture XVII Abstract We introduce te concept of directional derivative of a scalar function and discuss its relation wit te gradient operator. Directional derivative and gradient Te directional derivative
More informationRecall from our discussion of continuity in lecture a function is continuous at a point x = a if and only if
Computational Aspects of its. Keeping te simple simple. Recall by elementary functions we mean :Polynomials (including linear and quadratic equations) Eponentials Logaritms Trig Functions Rational Functions
More informationf a h f a h h lim lim
Te Derivative Te derivative of a function f at a (denoted f a) is f a if tis it exists. An alternative way of defining f a is f a x a fa fa fx fa x a Note tat te tangent line to te grap of f at te point
More informationCopyright c 2008 Kevin Long
Lecture 4 Numerical solution of initial value problems Te metods you ve learned so far ave obtained closed-form solutions to initial value problems. A closedform solution is an explicit algebriac formula
More informationConsider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.
Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions
More information1 Calculus. 1.1 Gradients and the Derivative. Q f(x+h) f(x)
Calculus. Gradients and te Derivative Q f(x+) δy P T δx R f(x) 0 x x+ Let P (x, f(x)) and Q(x+, f(x+)) denote two points on te curve of te function y = f(x) and let R denote te point of intersection of
More informationHow to Find the Derivative of a Function: Calculus 1
Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te
More informationQuasiperiodic phenomena in the Van der Pol - Mathieu equation
Quasiperiodic penomena in te Van der Pol - Matieu equation F. Veerman and F. Verulst Department of Matematics, Utrect University P.O. Box 80.010, 3508 TA Utrect Te Neterlands April 8, 009 Abstract Te Van
More informationNotes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.
Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But
More informationSECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY
(Section 3.2: Derivative Functions and Differentiability) 3.2.1 SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY LEARNING OBJECTIVES Know, understand, and apply te Limit Definition of te Derivative
More informationHOMEWORK HELP 2 FOR MATH 151
HOMEWORK HELP 2 FOR MATH 151 Here we go; te second round of omework elp. If tere are oters you would like to see, let me know! 2.4, 43 and 44 At wat points are te functions f(x) and g(x) = xf(x)continuous,
More informationPreface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
Preface Here are my online notes for my course tat I teac ere at Lamar University. Despite te fact tat tese are my class notes, tey sould be accessible to anyone wanting to learn or needing a refreser
More informationFinding and Using Derivative The shortcuts
Calculus 1 Lia Vas Finding and Using Derivative Te sortcuts We ave seen tat te formula f f(x+) f(x) (x) = lim 0 is manageable for relatively simple functions like a linear or quadratic. For more complex
More information1watt=1W=1kg m 2 /s 3
Appendix A Matematics Appendix A.1 Units To measure a pysical quantity, you need a standard. Eac pysical quantity as certain units. A unit is just a standard we use to compare, e.g. a ruler. In tis laboratory
More informationMaterial for Difference Quotient
Material for Difference Quotient Prepared by Stepanie Quintal, graduate student and Marvin Stick, professor Dept. of Matematical Sciences, UMass Lowell Summer 05 Preface Te following difference quotient
More informationDifferential Calculus (The basics) Prepared by Mr. C. Hull
Differential Calculus Te basics) A : Limits In tis work on limits, we will deal only wit functions i.e. tose relationsips in wic an input variable ) defines a unique output variable y). Wen we work wit
More informationEffects of Radiation on Unsteady Couette Flow between Two Vertical Parallel Plates with Ramped Wall Temperature
Volume 39 No. February 01 Effects of Radiation on Unsteady Couette Flow between Two Vertical Parallel Plates wit Ramped Wall Temperature S. Das Department of Matematics University of Gour Banga Malda 73
More informationChapter 5 FINITE DIFFERENCE METHOD (FDM)
MEE7 Computer Modeling Tecniques in Engineering Capter 5 FINITE DIFFERENCE METHOD (FDM) 5. Introduction to FDM Te finite difference tecniques are based upon approximations wic permit replacing differential
More information2.8 The Derivative as a Function
.8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open
More informationQuantum Numbers and Rules
OpenStax-CNX module: m42614 1 Quantum Numbers and Rules OpenStax College Tis work is produced by OpenStax-CNX and licensed under te Creative Commons Attribution License 3.0 Abstract Dene quantum number.
More informationAverage Rate of Change
Te Derivative Tis can be tougt of as an attempt to draw a parallel (pysically and metaporically) between a line and a curve, applying te concept of slope to someting tat isn't actually straigt. Te slope
More informationPrediction of Coating Thickness
Prediction of Coating Tickness Jon D. Wind Surface Penomena CE 385M 4 May 1 Introduction Tis project involves te modeling of te coating of metal plates wit a viscous liquid by pulling te plate vertically
More informationChapter 2 Limits and Continuity
4 Section. Capter Limits and Continuity Section. Rates of Cange and Limits (pp. 6) Quick Review.. f () ( ) () 4 0. f () 4( ) 4. f () sin sin 0 4. f (). 4 4 4 6. c c c 7. 8. c d d c d d c d c 9. 8 ( )(
More informationMath 31A Discussion Notes Week 4 October 20 and October 22, 2015
Mat 3A Discussion Notes Week 4 October 20 and October 22, 205 To prepare for te first midterm, we ll spend tis week working eamples resembling te various problems you ve seen so far tis term. In tese notes
More informationPre-Calculus Review Preemptive Strike
Pre-Calculus Review Preemptive Strike Attaced are some notes and one assignment wit tree parts. Tese are due on te day tat we start te pre-calculus review. I strongly suggest reading troug te notes torougly
More informationTHE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225
THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:
More informationOn the absence of marginal pinching in thin free films
On te absence of marginal pincing in tin free films P. D. Howell and H. A. Stone 6 August 00 Abstract Tis paper concerns te drainage of a tin liquid lamella into a Plateau border. Many models for draining
More informationExam 1 Review Solutions
Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),
More informationContinuity and Differentiability Worksheet
Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;
More informationSome Review Problems for First Midterm Mathematics 1300, Calculus 1
Some Review Problems for First Midterm Matematics 00, Calculus. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd,
More informationlecture 26: Richardson extrapolation
43 lecture 26: Ricardson extrapolation 35 Ricardson extrapolation, Romberg integration Trougout numerical analysis, one encounters procedures tat apply some simple approximation (eg, linear interpolation)
More informationPolynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximatinga function fx, wose values at a set of distinct points x, x, x,, x n are known, by a polynomial P x suc
More information1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t).
. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd, periodic function tat as been sifted upwards, so we will use
More informationQuaternion Dynamics, Part 1 Functions, Derivatives, and Integrals. Gary D. Simpson. rev 01 Aug 08, 2016.
Quaternion Dynamics, Part 1 Functions, Derivatives, and Integrals Gary D. Simpson gsim1887@aol.com rev 1 Aug 8, 216 Summary Definitions are presented for "quaternion functions" of a quaternion. Polynomial
More informationSFU UBC UNBC Uvic Calculus Challenge Examination June 5, 2008, 12:00 15:00
SFU UBC UNBC Uvic Calculus Callenge Eamination June 5, 008, :00 5:00 Host: SIMON FRASER UNIVERSITY First Name: Last Name: Scool: Student signature INSTRUCTIONS Sow all your work Full marks are given only
More informationAnalytic Functions. Differentiable Functions of a Complex Variable
Analytic Functions Differentiable Functions of a Complex Variable In tis capter, we sall generalize te ideas for polynomials power series of a complex variable we developed in te previous capter to general
More informationPhysically Based Modeling: Principles and Practice Implicit Methods for Differential Equations
Pysically Based Modeling: Principles and Practice Implicit Metods for Differential Equations David Baraff Robotics Institute Carnegie Mellon University Please note: Tis document is 997 by David Baraff
More informationMathematics 105 Calculus I. Exam 1. February 13, Solution Guide
Matematics 05 Calculus I Exam February, 009 Your Name: Solution Guide Tere are 6 total problems in tis exam. On eac problem, you must sow all your work, or oterwise torougly explain your conclusions. Tere
More informationMath 2921, spring, 2004 Notes, Part 3. April 2 version, changes from March 31 version starting on page 27.. Maps and di erential equations
Mat 9, spring, 4 Notes, Part 3. April version, canges from Marc 3 version starting on page 7.. Maps and di erential equations Horsesoe maps and di erential equations Tere are two main tecniques for detecting
More information2.1 THE DEFINITION OF DERIVATIVE
2.1 Te Derivative Contemporary Calculus 2.1 THE DEFINITION OF DERIVATIVE 1 Te grapical idea of a slope of a tangent line is very useful, but for some uses we need a more algebraic definition of te derivative
More information2.11 That s So Derivative
2.11 Tat s So Derivative Introduction to Differential Calculus Just as one defines instantaneous velocity in terms of average velocity, we now define te instantaneous rate of cange of a function at a point
More informationTime (hours) Morphine sulfate (mg)
Mat Xa Fall 2002 Review Notes Limits and Definition of Derivative Important Information: 1 According to te most recent information from te Registrar, te Xa final exam will be eld from 9:15 am to 12:15
More informationNotes on wavefunctions II: momentum wavefunctions
Notes on wavefunctions II: momentum wavefunctions and uncertainty Te state of a particle at any time is described by a wavefunction ψ(x). Tese wavefunction must cange wit time, since we know tat particles
More information1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point
MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note
More informationSection 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is
Mat 180 www.timetodare.com Section.7 Derivatives and Rates of Cange Part II Section.8 Te Derivative as a Function Derivatives ( ) In te previous section we defined te slope of te tangent to a curve wit
More informationDerivatives. By: OpenStaxCollege
By: OpenStaxCollege Te average teen in te United States opens a refrigerator door an estimated 25 times per day. Supposedly, tis average is up from 10 years ago wen te average teenager opened a refrigerator
More informationName: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ).
Mat - Final Exam August 3 rd, Name: Answer Key No calculators. Sow your work!. points) All answers sould eiter be,, a finite) real number, or DNE does not exist ). a) Use te grap of te function to evaluate
More informationSolution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4.
December 09, 20 Calculus PracticeTest s Name: (4 points) Find te absolute extrema of f(x) = x 3 0 on te interval [0, 4] Te derivative of f(x) is f (x) = 3x 2, wic is zero only at x = 0 Tus we only need
More informationSECTION 1.10: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES
(Section.0: Difference Quotients).0. SECTION.0: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES Define average rate of cange (and average velocity) algebraically and grapically. Be able to identify, construct,
More informationLIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT
LIMITS AND DERIVATIVES Te limit of a function is defined as te value of y tat te curve approaces, as x approaces a particular value. Te limit of f (x) as x approaces a is written as f (x) approaces, as
More information3.1 Extreme Values of a Function
.1 Etreme Values of a Function Section.1 Notes Page 1 One application of te derivative is finding minimum and maimum values off a grap. In precalculus we were only able to do tis wit quadratics by find
More informationch (for some fixed positive number c) reaching c
GSTF Journal of Matematics Statistics and Operations Researc (JMSOR) Vol. No. September 05 DOI 0.60/s4086-05-000-z Nonlinear Piecewise-defined Difference Equations wit Reciprocal and Cubic Terms Ramadan
More information. If lim. x 2 x 1. f(x+h) f(x)
Review of Differential Calculus Wen te value of one variable y is uniquely determined by te value of anoter variable x, ten te relationsip between x and y is described by a function f tat assigns a value
More informationCubic Functions: Local Analysis
Cubic function cubing coefficient Capter 13 Cubic Functions: Local Analysis Input-Output Pairs, 378 Normalized Input-Output Rule, 380 Local I-O Rule Near, 382 Local Grap Near, 384 Types of Local Graps
More informationComment on Experimental observations of saltwater up-coning
1 Comment on Experimental observations of saltwater up-coning H. Zang 1,, D.A. Barry 2 and G.C. Hocking 3 1 Griffit Scool of Engineering, Griffit University, Gold Coast Campus, QLD 4222, Australia. Tel.:
More informationCombining functions: algebraic methods
Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)
More information3. Using your answers to the two previous questions, evaluate the Mratio
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 0219 2.002 MECHANICS AND MATERIALS II HOMEWORK NO. 4 Distributed: Friday, April 2, 2004 Due: Friday,
More informationExam in Fluid Mechanics SG2214
Exam in Fluid Mecanics G2214 Final exam for te course G2214 23/10 2008 Examiner: Anders Dalkild Te point value of eac question is given in parentesis and you need more tan 20 points to pass te course including
More information, meant to remind us of the definition of f (x) as the limit of difference quotients: = lim
Mat 132 Differentiation Formulas Stewart 2.3 So far, we ave seen ow various real-world problems rate of cange and geometric problems tangent lines lead to derivatives. In tis section, we will see ow to
More informationNUMERICAL DIFFERENTIATION
NUMERICAL IFFERENTIATION FIRST ERIVATIVES Te simplest difference formulas are based on using a straigt line to interpolate te given data; tey use two data pints to estimate te derivative. We assume tat
More information0.1 Differentiation Rules
0.1 Differentiation Rules From our previous work we ve seen tat it can be quite a task to calculate te erivative of an arbitrary function. Just working wit a secon-orer polynomial tings get pretty complicate
More information1 1. Rationalize the denominator and fully simplify the radical expression 3 3. Solution: = 1 = 3 3 = 2
MTH - Spring 04 Exam Review (Solutions) Exam : February 5t 6:00-7:0 Tis exam review contains questions similar to tose you sould expect to see on Exam. Te questions included in tis review, owever, are
More informationClick here to see an animation of the derivative
Differentiation Massoud Malek Derivative Te concept of derivative is at te core of Calculus; It is a very powerful tool for understanding te beavior of matematical functions. It allows us to optimize functions,
More informationPolynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximating a function f(x, wose values at a set of distinct points x, x, x 2,,x n are known, by a polynomial P (x
More informationMAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points
MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to 8. 1. lim f (x) 0 2. lim
More informationThe Dynamic Range of Bursting in a Model Respiratory Pacemaker Network
SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 4, No. 4, pp. 117 1139 c 25 Society for Industrial and Applied Matematics Te Dynamic Range of Bursting in a Model Respiratory Pacemaker Network Janet Best, Alla Borisyuk,
More informationLab 6 Derivatives and Mutant Bacteria
Lab 6 Derivatives and Mutant Bacteria Date: September 27, 20 Assignment Due Date: October 4, 20 Goal: In tis lab you will furter explore te concept of a derivative using R. You will use your knowledge
More informationSection 3.1: Derivatives of Polynomials and Exponential Functions
Section 3.1: Derivatives of Polynomials and Exponential Functions In previous sections we developed te concept of te derivative and derivative function. Te only issue wit our definition owever is tat it
More information6. Non-uniform bending
. Non-uniform bending Introduction Definition A non-uniform bending is te case were te cross-section is not only bent but also seared. It is known from te statics tat in suc a case, te bending moment in
More informationOrder of Accuracy. ũ h u Ch p, (1)
Order of Accuracy 1 Terminology We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, wic can be for instance te grid size or time step in a numerical
More information232 Calculus and Structures
3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE
More informationNumerical Differentiation
Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function
More information1. State whether the function is an exponential growth or exponential decay, and describe its end behaviour using limits.
Questions 1. State weter te function is an exponential growt or exponential decay, and describe its end beaviour using its. (a) f(x) = 3 2x (b) f(x) = 0.5 x (c) f(x) = e (d) f(x) = ( ) x 1 4 2. Matc te
More informationINTRODUCTION AND MATHEMATICAL CONCEPTS
INTODUCTION ND MTHEMTICL CONCEPTS PEVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips of sine,
More information1 + t5 dt with respect to x. du = 2. dg du = f(u). du dx. dg dx = dg. du du. dg du. dx = 4x3. - page 1 -
Eercise. Find te derivative of g( 3 + t5 dt wit respect to. Solution: Te integrand is f(t + t 5. By FTC, f( + 5. Eercise. Find te derivative of e t2 dt wit respect to. Solution: Te integrand is f(t e t2.
More informationNew Streamfunction Approach for Magnetohydrodynamics
New Streamfunction Approac for Magnetoydrodynamics Kab Seo Kang Brooaven National Laboratory, Computational Science Center, Building 63, Room, Upton NY 973, USA. sang@bnl.gov Summary. We apply te finite
More informationTHE STURM-LIOUVILLE-TRANSFORMATION FOR THE SOLUTION OF VECTOR PARTIAL DIFFERENTIAL EQUATIONS. L. Trautmann, R. Rabenstein
Worksop on Transforms and Filter Banks (WTFB),Brandenburg, Germany, Marc 999 THE STURM-LIOUVILLE-TRANSFORMATION FOR THE SOLUTION OF VECTOR PARTIAL DIFFERENTIAL EQUATIONS L. Trautmann, R. Rabenstein Lerstul
More informationPractice Problem Solutions: Exam 1
Practice Problem Solutions: Exam 1 1. (a) Algebraic Solution: Te largest term in te numerator is 3x 2, wile te largest term in te denominator is 5x 2 3x 2 + 5. Tus lim x 5x 2 2x 3x 2 x 5x 2 = 3 5 Numerical
More informationMath 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0
3.4: Partial Derivatives Definition Mat 22-Lecture 9 For a single-variable function z = f(x), te derivative is f (x) = lim 0 f(x+) f(x). For a function z = f(x, y) of two variables, to define te derivatives,
More informationSin, Cos and All That
Sin, Cos and All Tat James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 9, 2017 Outline Sin, Cos and all tat! A New Power Rule Derivatives
More informationINTRODUCTION AND MATHEMATICAL CONCEPTS
Capter 1 INTRODUCTION ND MTHEMTICL CONCEPTS PREVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips
More informationHigher Derivatives. Differentiable Functions
Calculus 1 Lia Vas Higer Derivatives. Differentiable Functions Te second derivative. Te derivative itself can be considered as a function. Te instantaneous rate of cange of tis function is te second derivative.
More informationWhy gravity is not an entropic force
Wy gravity is not an entropic force San Gao Unit for History and Pilosopy of Science & Centre for Time, SOPHI, University of Sydney Email: sgao7319@uni.sydney.edu.au Te remarkable connections between gravity
More informationCHAPTER (A) When x = 2, y = 6, so f( 2) = 6. (B) When y = 4, x can equal 6, 2, or 4.
SECTION 3-1 101 CHAPTER 3 Section 3-1 1. No. A correspondence between two sets is a function only if eactly one element of te second set corresponds to eac element of te first set. 3. Te domain of a function
More informationThe Krewe of Caesar Problem. David Gurney. Southeastern Louisiana University. SLU 10541, 500 Western Avenue. Hammond, LA
Te Krewe of Caesar Problem David Gurney Souteastern Louisiana University SLU 10541, 500 Western Avenue Hammond, LA 7040 June 19, 00 Krewe of Caesar 1 ABSTRACT Tis paper provides an alternative to te usual
More informationMAT244 - Ordinary Di erential Equations - Summer 2016 Assignment 2 Due: July 20, 2016
MAT244 - Ordinary Di erential Equations - Summer 206 Assignment 2 Due: July 20, 206 Full Name: Student #: Last First Indicate wic Tutorial Section you attend by filling in te appropriate circle: Tut 0
More informationDedicated to the 70th birthday of Professor Lin Qun
Journal of Computational Matematics, Vol.4, No.3, 6, 4 44. ACCELERATION METHODS OF NONLINEAR ITERATION FOR NONLINEAR PARABOLIC EQUATIONS Guang-wei Yuan Xu-deng Hang Laboratory of Computational Pysics,
More informationQuantum Theory of the Atomic Nucleus
G. Gamow, ZP, 51, 204 1928 Quantum Teory of te tomic Nucleus G. Gamow (Received 1928) It as often been suggested tat non Coulomb attractive forces play a very important role inside atomic nuclei. We can
More information1 2 x Solution. The function f x is only defined when x 0, so we will assume that x 0 for the remainder of the solution. f x. f x h f x.
Problem. Let f x x. Using te definition of te derivative prove tat f x x Solution. Te function f x is only defined wen x 0, so we will assume tat x 0 for te remainder of te solution. By te definition of
More informationSymmetry Labeling of Molecular Energies
Capter 7. Symmetry Labeling of Molecular Energies Notes: Most of te material presented in tis capter is taken from Bunker and Jensen 1998, Cap. 6, and Bunker and Jensen 2005, Cap. 7. 7.1 Hamiltonian Symmetry
More informationNumerical Analysis MTH603. dy dt = = (0) , y n+1. We obtain yn. Therefore. and. Copyright Virtual University of Pakistan 1
Numerical Analysis MTH60 PREDICTOR CORRECTOR METHOD Te metods presented so far are called single-step metods, were we ave seen tat te computation of y at t n+ tat is y n+ requires te knowledge of y n only.
More informationDifferentiation. Area of study Unit 2 Calculus
Differentiation 8VCE VCEco Area of stud Unit Calculus coverage In tis ca 8A 8B 8C 8D 8E 8F capter Introduction to limits Limits of discontinuous, rational and brid functions Differentiation using first
More informationTheoretical Analysis of Flow Characteristics and Bearing Load for Mass-produced External Gear Pump
TECHNICAL PAPE Teoretical Analysis of Flow Caracteristics and Bearing Load for Mass-produced External Gear Pump N. YOSHIDA Tis paper presents teoretical equations for calculating pump flow rate and bearing
More informationA Modified Distributed Lagrange Multiplier/Fictitious Domain Method for Particulate Flows with Collisions
A Modified Distributed Lagrange Multiplier/Fictitious Domain Metod for Particulate Flows wit Collisions P. Sing Department of Mecanical Engineering New Jersey Institute of Tecnology University Heigts Newark,
More information