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International Journal of Differential Equations
Volume 2012 (2012), Article ID 570283, 30 pages
Qualitative Analysis of Coating Flows on a Rotating Horizontal Cylinder
1School of Mathematical Sciences, Claremont Graduate University, 710 N. College Avenue, Claremont, CA 91711, USA
2School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Received 11 May 2012; Accepted 1 August 2012
Academic Editor: Sining Zheng
Copyright © 2012 Marina Chugunova and Roman M. Taranets. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. We obtain sufficient conditions for finite speed of support propagation and for waiting time phenomena by application of a new extension of Stampacchia's lemma for a system of functional equations.
The time evolution of thickness of a viscous liquid film spreading over a solid surface under the action of the surface tension and gravity can be described by lubrication models [1–5]. These models approximate the full Navier-Stokes system that describes the motion of the liquid flow. Thin films play an increasingly important role in a wide range of applications, for example, packaging, barriers, membranes, sensors, semiconductor devices, and medical implants [6–8].
In this paper we consider the dynamics of a viscous incompressible thin fluid film on the outer surface of a horizontal circular cylinder that is rotating around its axis in the presence of a gravitational field. The motion of the liquid film is governed by four physical effects: viscosity, gravity, surface tension, and centrifugal forces. These are reflected in the parameters: : the radius of the cylinder, : its rate of rotation (assumed constant), : the acceleration due to gravity, : the kinematic viscosity, : the fluid's density, and : the surface tension. These parameters yield three independent dimensionless numbers: the Reynolds number , , and the Weber number . The understanding of coating flow dynamics is important for industrial printing process where rotating cylinder transports the coating material in the form of liquid paint. The rotating thin fluid film can exhibit variety of different behaviour including: interesting pattern formations (“shark teeth” and “duck bill” patterns), fluid curtains, hydroplaning drops, and frontal avalanches [8–10]. As a result, the coating flow has been the subject of continuous study since the pioneering model was derived in 1977 by Moffatt (see ): The surface tension and inertial effects were neglected in (1.1). Here is the thickness of the fluid film, is a rotation angle, and is a time variable. The linear stability of rigidly rotating films on a rotating circular cylinder under three-dimensional disturbances was examined in [12, 13]. It was shown that the most unstable mode for thin film flows on the surface of a cylinder is the purely axial one that leads to so-called “ring instabilities”. During the past decade, coating and rimming problems attracted many researchers who analyzed different types of flow regime asymptotically [14–17] and numerically [18–20]. For a detailed review of a growing literature on different types of thin film flows please see  and references there in.
The coating flow is generated by viscous forces due to cylinder's surface motion relative to the fluid. There is no temperature gradient, hence the interface does not experience a shear stress. If the cylinder is fully coated there is only one free boundary where the liquid meets the surrounding air. Otherwise, there is also a free boundary (or contact line) where the air and liquid meet the cylinder's surface.
The asymptotic evolution equation for the thickness of the fluid film with the surface tension effect: was derived by Pukhnachev  in 1977. It is valid under the assumptions that the fluid film is thin and its slope is small . Later in 2009, taking into account inertial effects, Kelmanson  presented a more general model: He analyzed, asymptotically and numerically, diverse effects of inertia in both small- and large-surface-tension limits.
We should mention that all three lubrication approximation models described above were based on the assumption of the no-slip boundary condition. It is well known  that the combination of constant viscosity and no-slip boundary conditions at the liquid-solid interface yields a logarithmic divergence in the rate of dissipation at moving contact line, that is, an infinite energy is needed to make the droplet expand. The most common way to overcome this difficulty is to introduce effective slip conditions (see (2.1)) that indeed removes the force singularity at advancing contact lines (see ).
The main goal of our paper is to study waiting time phenomenon for the coating flows under an assumption of effective slip conditions, that is, we analyze (2.1) that is a modified version of (1.3). Our approach is based on now well-established nonlinear PDE analysis for degenerate higher order parabolic equations.
The sufficient conditions: for , for , (where and are some positive constants) on nonnegative initial data, for the occurrence of waiting time phenomena were derived by Dal Passo et al.  for the classic thin film equation: These results were based on an energy method developed in  for quasilinear parabolic equations. To the best of our knowledge, there is only one publication , where the waiting time phenomenon in the classic thin film equation (1.4) was discovered for for . The result was obtained by means of matching asymptotic methods and was supported by numerous numerical simulations. For more general nonlinear degenerate parabolic equations with nonlinear lower order terms the waiting time phenomenon was analyzed in [29–31].
It is well known  that the similarity solutions of the second order nonlinear parabolic equation: subject to prescribing appropriate initial data, demonstrate the existence of a waiting-time phenomena before the free boundary moves. The comparison theorem, that is not applicable in our case, then enabled a number of results to be obtained about the existence and length of waiting times for general initial data. Our approach is completely different and based on local entropy/energy functional estimates.
We also analyze speed of support propagation and obtain an upper bound on it for the modified version of (1.3) (see (2.1)). The first finite speed results for nonnegative generalized solutions of the classic thin film equation (1.4) were obtained in [33, 34] for the case and , respectively. For more general types of thin film equations the finite speed of support propagation phenomenon was studied in [35–39] (see also references there in).
The outline of our paper is as follows. We first prove for the long-time existence of a generalized weak solution and then prove that it can have an additional regularity in Section 2. In Sections 3 and 4 we show finite speed support propagation in the “slow” convection case (): for and waiting time phenomena for , accordingly. The general strategy is to use an extension of Stampacchia's lemma for a system of functional equations (see Lemma 3.1 , where this extension is proved for a single equation and Lemma A.2 in , where this extension is proved for systems in the homogeneous case). This result to our knowledge is new and might be of independent interest. We leave as an open problem the “fast” convection case (): finite speed of support propagation and sufficient conditions for waiting time phenomenon.
2. Existence and Regularity of Solutions
We are interested in the existence of nonnegative generalized weak solutions to the following initial-boundary value problem: where , , , , , , and such that ote that (1.3) is a particular case of (2.1) that corresponds to and .
Definition 2.1. A generalized weak solution of problem is a nonnegative function satisfying where . The solution satisfies (2.1) in the following sense: for all with ; for at all points of the lateral boundary where
Because the second term of (2.4) has an integral over rather than over , the generalized weak solution is “weaker” than a standard weak solution. Here, is short hand for . This short hand is used throughout: the time interval included in is to be inferred from the context it appears in.
A key object for proving additional properties of a generalized weak solution is an integral quantity introduced by Bernis and Friedman : the “entropy” . The function is defined by where By construction, is a nonnegative convex function on . For , the linear part of is chosen to ensure that has a positive lower bound on . Also in the statement of Theorem 2.2 we use an “-entropy”, , where is a nonnegative convex function on . The linear part of is chosen to ensure that has a positive lower bound on if . If , the -entropy is the same as the entropy (2.7).
Theorem 2.2. (a) (Existence). Let and the nonnegative initial data , satisfy Then for any time there exists a nonnegative generalized weak solution, , on in the sense of the Definition 2.1. Furthermore, Let then the weak solution satisfies (b) (Regularity). If the initial data also satisfies for some , then the nonnegative generalized weak solution has the extra regularity and .
The theorem above was proved earlier in  for the case only. We note that the analogue of Theorem 4.2 in  also holds: there exists a nonnegative weak solution with the integral formulation If initial data satisfy finite -entropy condition, that is, then one can prove existence of nonnegative solutions with some additional regularity properties and use an integral formulation  to define them that is similar to that of (2.16) except that the second integral is replaced by the results of one more integration by parts (there are no terms). It is worth to mention that for the case the finite entropy assumption in Theorem 2.2 can be omitted because it does not impose any restriction on nonnegative initial data. One needs to impose finite entropy and finite -entropy conditions on initial data if only.
2.1. Regularized Problem
Given , , a regularized parabolic problem, similar to one that was studied by Bernis and Friedman  can be written as:
where The in (2.20) makes the problem (2.17) regular (i.e., uniformly parabolic). The parameter is an approximating parameter which has the effect of increasing the degeneracy from to . The nonnegative initial data, , is approximated via The term in (2.21) “lifts” the initial data so that they are smoothing from to . By Eĭdel’man [44, Theorem 6.3, p.302], the regularized problem has a unique classical solution for some time . For any fixed value of and , by Eĭdel’man [44, Theorem 9.3, p.316] if one can prove a uniform in time a priori bound for some longer time interval ) and for all then Schauder-type interior estimates [44, Corollary 2, p.213] imply that the solution can be continued in time to be in .
Although the solution is initially positive, there is no guarantee that it will remain nonnegative. The goal is to take , in such a way that , the solutions converge to a (nonnegative) limit, , which is a generalized weak solution, and inherits certain a priori bounds. This is done by proving various a priori estimates for that are uniform in and and hold on a time interval that is independent of and . As a result, will be a uniformly bounded and equicontinuous (in the norm) family of functions in . Taking will result in a family of functions that are classical, positive, unique solutions to the regularized problem with . Taking will then result in the desired generalized weak solution . This last step is where the possibility of nonunique weak solutions arise; see  for simple examples of how such constructions applied to can result in two different solutions arising from the same initial data.
2.2. A Priori Estimates
We use an integral quantity based on a function chosen such that This is analogous to the “entropy” function first introduced by Bernis and Friedman .
Lemma 2.3. Let satisfy (2.21) and be built from a nonnegative function that satisfies the hypotheses of Theorem 2.2. Then there exist , and time such that if , , and is a solution of the problem (2.17)–(2.21) with initial data , then for any the following inequalities: hold. The energy (see (2.13)) satisfies The time and the constants are independent of and .
The proof of existence of , , , , and is constructive; how to find them and what quantities determine them are shown with details in Section 4.
Lemma 2.3 yields uniform-in--and- bounds for , , , and . However, these bounds are found in a different manner than in earlier work for the equation , for example. Although the inequality (2.24) is unchanged, the inequality (2.23) has an extra term involving . In the proof, this term was introduced to control additional, lower-order terms. This idea of a “blended” -entropy bound was first introduced by Shishkov and Taranets for long-wave stable thin film equations with convection .
The final a priori bounds for positive, classical solutions use the following functions, parameterized by for , where is given by (2.9). In the following lemma, we restrict ourselves to the case ; note that for such .
Lemma 2.4. Assume and are from Lemma 2.3, , and . Assume and that is a positive, classical solution of the problem (2.17)–(2.21) with initial data satisfying Lemma 2.3. If the initial data is built from which also satisfies then there exists such that holds for all and is independent of and is determined by , , , , , and . Here Furthermore, if then are uniformly bounded.
The proof of existence starts from a construction of a classical solution on that satisfies the hypotheses of Lemma 2.3 if and . Taking the regularizing parameter, , to zero, one proves that there is a limit and that is a generalized weak solution. After that additional nonlinear estimates are required to analyze properties of the limit ; specifically to show that it is strictly positive, classical, and unique. Hence, the a priori bounds given by Lemmas 2.3 and 2.4 are applicable to . This allows us to take the approximating parameter, , to zero and to construct the desired nonnegative generalized weak solution of Theorems 2.2 (see, e.g., ).
2.3. Long-Time Existence of Solutions
Proof of Lemma 2.5. By (2.13), The linear-in-time bound (2.14) on then implies Using the estimate (see [41, Lemma 4.1, page 1837]) and Young's inequality: Using this in (2.34), the desired bound (2.31) follows immediately.
3. Finite Speed of Support Propagation
Theorem 3.1. Let . Assume is nonnegative, and . Then the solution of Theorem 2.2 has finite speed of support propagation, that is, there exists a continuous nondecreasing function , such that for all .
In the following theorem, we find the explicit upper bounds of the for a solution of the corresponding Cauchy problem with a compactly supported nonnegative initial data . Note that the definition of generalized weak solution of the Cauchy problem is as Definition 2.1 except that is replaced by and the relation (2.6) is dropped. Using Lemma 2.5, we can show that the upper estimate of from Theorem 3.1 is independent on therefore the solution from Theorem 2.2 can be extended to be identically zero for and thus is a solution on the line for all . Performing a similar procedure in , we obtain a compactly supported nonnegative solution of the Cauchy problem for all and Theorem 2.2 holds with .
Theorem 3.2. Let . Assume is nonnegative, , and is a solution of the Cauchy problem. Then the following estimates: for all if , for small enough time if , are valid. Here the constants depend on the parameters problem and initial data only.
3.1. Proof of Theorem 3.1 for the Case
Lemma 3.3. Let such that , on , and . Assume that , and . Then there exist constants dependent on , , , , and , independent of , such that for all
Let , and let . For an arbitrary and we consider the families of sets We introduce a nonnegative cutoff function from the space with the following properties: Next we introduce our main cut-off functions such that for all and possess the following properties: for all , . Choosing , from (3.1) we arrive at for all , where and is enough small. We apply the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in the region to a function with , , , , , and under the conditions: Integrating the resulted inequalities with respect to time and taking into account (3.5), we arrive at the following relations: where . These inequalities are true provided that Simple calculations show that inequalities (3.6) and (3.8) hold with some if and only if . The finite speed of propagations follows from (3.7) by applying Lemma B.3 with . Hence, where .
3.2. Proof of Theorem 3.2 for the Case
We can repeat the previous procedure from Section 3.1 for and we obtain instead of (3.7), and where Now we need to estimate . With that end in view, we obtain the following estimates: where , and depends on initial data only. Really, applying the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in to a function with , , , , , and under the condition , we deduce that Integrating (3.14) with respect to time and taking into account the Hölder inequality (), we arrive at the following relations: From (3.15), due to (A.16) (as ) and (2.31), we find (3.13).
3.3. Proof of Theorem 3.1 for the Case
Lemma 3.4. Let and . Let such that in and on , and . Then there exist constants dependent on , , , and , independent of and , such that for any
Let be denoted by (3.4). Setting into (3.17), after simple transformations, we obtain for all for all . We apply the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in the region to a function with , , , , , and under the conditions: Integrating the resulted inequalities with respect to time and taking into account (3.19), we arrive at the following relations: where . These inequalities are true provided that Simple calculations show that inequalities (3.20) and (3.22) hold with some if and only if . Since all integrals on the right-hand sides of (3.21) vanish as , the finite speed of propagations follows from (3.21) by applying Lemma B.3 with and sufficiently small . Hence,
3.4. Proof of Theorem 3.2 for the Case
Suppose that , for all , , and . Since the time interval is small, we can assume that . Hence, for all , we can take (up to regularization) in (3.18). As a result, we obtain for all , . Using the Hardy type inequality where and , we deduce that whence for all , . Substituting (3.27) in (3.24), we get for all , . By the Nirenberg-Gagliardo, Hölder and Young inequalities, after simple transformations, for , we have Substituting the estimates (3.29) in (3.28) and making the standard iterative procedure for small enough , we arrive at the inequality where , , , . Thus, (3.30) yields for all and , where , , , . By Lemma B.3, from (3.31) we find that , where . As for any , we have .
4. Waiting Time Phenomenon
Let for all , and where . Let us assume that the function satisfies the flatness conditions. Namely, for every the following estimate: is valid.
Theorem 4.1. Let . Assume is nonnegative, and for all , that is, the condition (4.1) is valid, and the flatness condition (4.2) also holds.
Then for the solution of Theorem 2.2 (with ) there exists the time depending on the known parameters only such that where is the constant from the flatness condition. Note, that as .
Remark 4.2. Let the initial data satisfy the following properties:(1) if then we suppose that
(2) if then we suppose that .
These assumptions on the initial data are sufficient for the validity of flatness condition (4.2) and guarantee the appearance of the WTP, that is, the validation of property (4.3).
Remark 4.3. Note that due to Lemma 2.5 we have the estimate . Therefore, using this inequality in (3.1) with , we could also obtain the waiting time phenomenon by the application of Theorem 2.1 from  with , , , and .
Proof of Theorem 4.1. Similar to (3.10) for and we obtain Let us check that all conditions of Lemma B.4 are satisfied. We denote by Taking in (4.5) and passing to the limit , due to the boundedness of functions and , we deduce This implies that the condition (i) of Lemma B.4 is fulfilled. Because of the assumption (4.2) on the function , we can find such that the condition (ii) of Lemma B.4 is valid for all . Here goes to infinity as . Hence, the application of Lemma B.4 ends the proof.
A. Proofs of a Priori Estimates
The first observation is that the periodic boundary conditions imply that classical solutions of (2.17) conserve mass: Further, (2.21) implies as . Also, we will relate the norm of to the norm of its zero-mean part as follows: where . We will use the Poincaré inequality which holds for any zero-mean function in Also used will be an interpolation inequality [47, Theorem 2.2, page 62] for functions of zero mean in : where , , , . It follows that for any zero-mean function in where To see that (A.5) holds, consider two cases. If , then by (A.3), is controlled by . By the Hölder inequality, is then controlled by . If then by (A.4), is controlled by where . By the Poincaré inequality, is controlled by .
Proof of Lemma 2.3. In the following, we denote the solution by whenever there is no chance of confusion.
To prove the bound (2.23) one starts by multiplying (2.17) by , integrating over , and using the periodic boundary conditions (2.18) yields By Cauchy and Young inequalities, due to (A.3)–(A.5), it follows from (A.7) that where , , + . Multiplying (2.17) by , integrating over , and using the periodic boundary conditions (2.18), we obtain where . Further, from (A.8) and (A.9) we find where . Applying the nonlinear Grönwall lemma  to with yields for all , where Using the , convergence of the initial data and the choice of (see (2.21)) as well as the assumption that the initial data has finite entropy (2.11), the times converge to a positive limit and the upper bound in (A.11) can be taken finite and independent of and for and sufficiently small. Therefore there exists and and such that the bound (A.11) holds for all and with replacing and for all
Using the uniform bound on that (A.11) provides, one can find a uniform-in--and- bound for the right-hand-side of (A.10) yielding the desired a priori bound (2.23). Similarly, one can find a uniform-in--and- bound for the right-hand-side of (A.9) yielding the desired a priori bound (2.24). The time and the constant are determined by , , , , , , , , and .
To prove the bound (2.25), multiply (2.17) by , integrate over , integrate by parts, use the periodic boundary conditions (2.18) to find (2.25).
Proof of Lemma 2.4. In the following, we denote the positive, classical solution by whenever there is no chance of confusion.
Multiplying (2.17) by , integrating over , taking , and using the periodic boundary conditions (2.18) yield