Qualitative Analysis of Coating Flows on a Rotating Horizontal Cylinder
Marina Chugunova1and Roman M. Taranets2
Academic Editor: Sining Zheng
Received11 May 2012
Accepted01 Aug 2012
Published09 Sept 2012
Abstract
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.
1. Introduction
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 [1β5]. 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 [6β8].
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 (βshark teethβ and βduck billβ patterns), fluid curtains, hydroplaning drops, and frontal avalanches [8β10]. As a result, the coating flow has been the subject of continuous study since the pioneering model was derived in 1977 by Moffatt (see [11]):
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 βring instabilitiesβ. During the past decade, coating and rimming problems attracted many researchers who analyzed different types of flow regime asymptotically [14β17] and numerically [18β20]. For a detailed review of a growing literature on different types of thin film flows please see [21] 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 [22] 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 [23] 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 [24] 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 [25]).
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. [26] for the classic thin film equation:
These results were based on an energy method developed in [27] for quasilinear parabolic equations. To the best of our knowledge, there is only one publication [28], 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 [29β31].
It is well known [32] 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 [35β39] (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 βslowβ 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 Lemmaββ3.1 [26], where this extension is proved for a single equation and LemmaββA.2 in [37], 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 βfastβ 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 ,ββ,ββ,ββ,ββ,ββββ, and such that
ote that (1.3) is a particular case of (2.1) that corresponds to and .
We consider a generalized weak solution in the following sense [40, 41].
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 βweakerβ 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 [42]: the βentropyβ . 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 β-entropyβ, , 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 ,ββ then the nonnegative generalized weak solution has the extra regularity and .
The theorem above was proved earlier in [41] for the case only. We note that the analogue of Theorem 4.2 in [42] 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 [43] 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 ,ββ, a regularized parabolic problem, similar to one that was studied by Bernis and Friedman [42] 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) βliftsβ the initial data so that they are smoothing from to . By EΔdelβman [44, Theorem 6.3, p.302], the regularized problem has a unique classical solution for some time . For any fixed value of and , by EΔdelβman [44, Theorem 9.3, p.316] if one can prove a uniform in time a priori bound for some longer time interval ββ) 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 ,ββ in such a way that ββ, the solutions converge to a (nonnegative) limit, , which is a generalized weak solution, and ββ 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 [40] 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
Our first task is to derive a priori estimates for classical solutions of (2.17)β(2.21). The lemmas given in this section are proved in the Section 4.
We use an integral quantity based on a function chosen such that
This is analogous to the βentropyβ function first introduced by Bernis and Friedman [42].
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)β(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 ,ββ,ββ,ββ, 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 βblendedβ -entropy bound was first introduced by Shishkov and Taranets for long-wave stable thin film equations with convection [30].
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)β(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 ,ββ,ββ,ββ,ββ,ββ and . Here
Furthermore, if then
are uniformly bounded.
The -entropy, , was first introduced for in [45] and an a priori bound like that of Lemma 2.4 and regularity results like those of Theorem 2.2 were found simultaneously and independently in [40, 43].
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., [41]).
2.3. Long-Time Existence of Solutions
Lemma 2.5. Let be a generalized solution of Theorem 2.2. Then
where is defined in (2.13), , and
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.
This -estimate will be used to extend the short-time existence of a solution to the long-time existence result of Theorem 2.2 (see [41, Proof of Theorem 3, page 1838]).
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 ,ββ 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:ββ for all if ,ββ for small enough time if , are valid. Here the constants depend on the parameters problem and initial data only.
The following lemma contains the local entropy estimate. The proof of Lemma 3.3 is similar to (A.16), (A.29), therefore it is omitted.
Lemma 3.3. Let such that , on , and . Assume that , and . Then there exist constants dependent on ,ββ,ββ,ββ, 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 ,ββ. 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 ,ββ,ββ,ββ,ββ, 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 .
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 ,ββ,ββ,ββ,ββ, and under the condition , we deduce that
Integrating (3.14) with respect to time and taking into account the HΓΆlder inequality (), we arrive at the following relations:
From (3.15), due to (A.16) (as ) and (2.31), we find (3.13).
Inserting (3.13) into (3.12), we obtain after straightforward computations that
The following lemma contains the local energy estimate. The proof of Lemma 3.4 is Appendix A.
Lemma 3.4. Let and . Let such that in and on , and . Then there exist constants dependent on ,ββ,ββ, 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 ,ββ,ββ,ββ,ββ, 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,
Suppose that ,ββ 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 ,ββ. Using the Hardy type inequality
where and , we deduce that
whence
for all , . Substituting (3.27) in (3.24), we get
for all ,ββ. By the Nirenberg-Gagliardo, HΓΆlder 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
whereββββ,ββ,ββββ,ββββ. Thus, (3.30) yields
for all and , where ,ββ,ββ,ββ. 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 [46] with ,ββ,ββ, 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.
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)β(A.5), it follows from (A.7) that
where ,ββ,ββ + . 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 GrΓΆnwall lemma [48] to with yields
for all , where
Using the ,ββ 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 ,ββ,ββ,ββ,ββ, , , , 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
Caseββ1ββ. The coefficient multiplying in (A.14) is positive and can therefore be used to control the term on the right-hand side of (A.14). Specifically, using the Cauchy-Schwartz inequality and the Cauchy inequality,
Using the bound (A.15) in (A.14), due to (A.5), yields
where + . Using the Cauchy inequality in (A.7) and taking , after applying the Cauchy-Schwartz inequality and (A.5), yields
where + . Using (A.16) yields
where . Applying the nonlinear GrΓΆnwall lemma [48] to with yields
for all :
The bound (A.19) holds for all where is from Lemma 2.3 and for all where is from Lemma 2.3. 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 (2.27), the times converge to a positive limit and the upper bound in (A.19) can be taken finite and independent of . Therefore there exists and such that the bound (A.19) holds for all with replacing and for all
where is the time from Lemma 2.3. Using the uniform bound on that (A.19) provides, one can find a uniform-in- bound for the right-hand-side of (A.16) yielding the desired bound
which holds for all and all . Note, (A.22) implies that for all that and are contained in balls in and respectively, that is,
From these estimates follows immediately (2.30). Caseββ2 . For the coefficient multiplying in (A.14) is negative. However, we will show that if then one can replace this coefficient with a positive coefficient while also controlling the term on the right-hand side of (A.14). Using the Cauchy-Schwartz inequality, it is easy to show that
Using (A.24) in (A.14) yields
Note that if then all the terms on the left-hand side of (A.25) are positive. We now control the term on the right-hand side of (A.25). By integration by parts and the periodic boundary conditions
Applying the Cauchy inequality to (A.26) yields
Using inequality (A.27) in (A.25) yields
Adding to both sides of (A.28) and using the inequality (A.24) yields
where +. Recall the bound (A.17). As before, by the Cauchy inequality,
Using (A.30) in (A.17) yields
where + . Using (A.29) yields
where . The rest of the proof now continues as in the case. Specifically, one finds a bound
for all :
The time is defined as in (A.21) and the uniform bound (A.33) used to bound the right-hand side of (A.29) yields the desired bound
Proof of Lemma 3.4. Let . Multiplying (2.17) by , and integrating on , yields
We now bound the terms ,ββ,ββ, and . First,
Now, multiplying (2.17) by , and integrating on , using the Young's inequality, letting , we obtain the following estimate:
where . If we now add inequalities (A.36) and (A.41), in view of (A.37)β(A.39), then, applying Lemma B.1, choosing , and letting , we obtain (3.17).
B. Auxiliary Lemmas
Lemma B.1 (see [34, 38]). Let , be a bounded convex domain with smooth boundary, and let for , and for . Then the following estimates hold for any strictly positive functions such that on and :
where is an arbitrary nonnegative function such that the tangential component of is equal to zero on , and the constant is independent of .
Lemma B.2 (see [49]). If is a bounded domain with piecewise-smooth boundary, , , , and , then there exist positive constants and ββ is unbounded depending only on , , , , and such that the following inequality is valid for every :
Lemma B.3 (see [37]). Let and let . Assume that are nonnegative nonincreasing functions satisfying the conditions:
with real constants , and for , and for . Let , and let the function be such that at a some . Then there exists a positive constant depending on , and such that for all , where . Note, if then .
Lemma B.4. Let , and let , . Assume that , are nonnegative nonincreasing functions satisfying the conditions:
with real constants , and . Let the functions
and be such that(i) for some the inequality holds for all ,(ii) for all ,ββwhere and . Then for all .
Proof. Let us denote by . Raising both side of (B.4) to the power and summing with respect to , we deduce
Choosing , we arrive at
whence we find that
where , . Applying [27, Lemmaββ4] to , taking into account the conditions (i) and (ii), we obtain for all .
Acknowledgment
The research of Dr. R. Taranets leading to these results has received funding from the European Communityβs Seventh Framework Programme (FP7/2007-2013) under Grant Agreement no PIIF-GA-2009-254521β.
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