Formation of the Capillary Ridge on the Free Surface Dynamics of Second-Grade Fluid Over an Inclined Locally Heated Plate

1 Introduction

The understanding of the capillary ridge formation on a gravity-driven thin liquid film flowing over a substrate is of interest in a wide range of applications including painting and coating process [1], [2], contact lens manufacture [3], microchip fabrication process [4], [5], and microfluidic devices [6]. Geometrical structure of the flow, surface tension, and the temperature of the fluid at the substrate are the prime reason for the evolution of the capillary ridge in such fluid flow problems. The fluid flowing over a topography exhibits capillary ridge caused by the topographic features and the surface tension [7], [8]. In most of the industrial applications of microfluidic devices, the motion of the thin layer of fluid flows is optimised by the imposed temperature gradient on the substrate. In these processes, the uneven distribution of temperature at the substrate causes the surface tension gradient at the free surface of the flows and results in a Marangoni stress [9]. These Marangoni stresses at the heated substrate oppose the gravitational flow, which results in the formation of a capillary ridge. The capillary ridge depicting the bump appearing at the front of the spreading liquid leads to the nonuniformity in the thin film coating. Hence, it is important to comprehend this phenomenon when the properties of the coating film rely upon the film thickness.

The analysis of the formation of the capillary ridge and its stability is studied by many authors, for example, Kalliadasis et al. [10] used the lubrication approximation for deriving the evolution equation of the film thickness of a viscous Newtonian fluid flowing over a particular trench or mound-like surface to investigate the position and reason for the ridge development in the flow. It is found that the ridge develops before and after the topography, and the ridge height increases, particularly on a precipitous depth substrate. Tiwari et al. [11] have studied the stability and dynamics of thin liquid films flowing over locally heated surfaces under lubrication framework. They have reported that the thermocapillary stress at the heated surface caused the formation of the capillary ridge, and it is found to be unstable when the Marangoni number exceeds a critical value. Later, the stability analysis has been extended including the topographical features in the noninertial coating flows over a locally heated surface [12]. The structure formation in a film falling down a vertical plate with a built-in rectangular heater was experimentally and numerically investigated by Frank and Kabov [13]. Authors have found the effects of the Reynolds and Weber number on the critical Marangoni number at the onset of instability.

There are many interfacial liquid films flows in industrial applications such as polymeric fluids, food products, and geological materials such as lava and glacier flow, which demonstrate flow characteristics that cannot be adequately described by the linear viscous fluid model. There are several models available to explain the rheology of a non-Newtonian fluid. The flow of a non-Newtonian gravity–driven liquid film on an inclined angle is discussed in depth [14], [15], [16], [17], [18], [19], [20], [21] in a different context of fluid configurations and rheology. One class of viscoelastic fluid is of the differential type [22], namely, second order. These materials lose their memory sharply, and the range of application of this model is restricted to materials that are slightly viscoelastic. Markovitz and Coleman [23] have reported that a 5.4 % arrangement of polyisobutylene in cetane at 30 °C pursues the second-order liquid properties. Many authors [24], [25] have investigated the flow of second-grade fluid on a nonhorizontal surface and reported the influence of second-grade parameters on the flow fields. Many of the research as mentioned above is devoted to the development of second-grade models for the study of thin-film flow over a stretching sheet in the boundary layer approximation framework. Recently, Panda et al. [26] have derived the thin-film equation systematically for a second-grade non-Newtonian fluid over an unsteady stretching sheet. In this work, the authors have discussed the effect of the second-grade viscoelastic parameter and the stretching rate on the free surface profile. Thus, a natural question is how the viscoelastic setting influences the capillary ridge height in the flow of second-grade fluid on an inclined, locally heated plate. This could have an interesting practical outcome to control the undesired capillary ridge on a free surface.

This work focuses on the systematic derivation of the dynamic film thickness equation of the second-grade fluid flowing over a locally heated inclined surface, analogous to that of Panda et al. [26]. The motivation of the study is to investigate the effects of the second-grade parameter on the capillary ridge, which is applicable to analyse the free surface of polymeric fluid flowing down an inclined locally heated plate. The following section formulates the mathematical model for the description of the flow of second-grade fluid on an inclined locally heated plate. This section is followed by the long-wave approximation for the derivation of thin film dynamic equation. A brief description of the numerical procedure to solve the derived thin film equation is presented in the following section. The effects of different flow parameters on the height of the capillary ridge are discussed in the penultimate section, and concluding remarks are drawn in the last section.

Figure 1:

Geometry of the problem.

2 Mathematical Formulation

Consider a two-dimensional, laminar, incompressible flow of second-grade non-Newtonian fluid [26] over a locally heated planar plate inclined at an angle Ω (0<Ω⩽π/2) with the fixed horizontal axis. The coordinate system (Fig. 1) is chosen such that x measures the distance along the plate, and z is the transverse coordinate such that z = 0 corresponds to the surface of the plate. The fluid domain of characteristic length 2L in the x direction is very large as compared to the z direction such that the aspect ratio is very small, i.e. ϵ=(h0/L)≪1, where h₀ represents the average depth of the fluid. The fluid with initial thickness δ(x), at the free surface boundary, is in contact with the ambient air. Due to inclination of the sheet, the fluid starts flowing downhill with velocity field V(x,z,t)=(u,w)(x,z,t), where u(x,z,t) and w(x,z,t) are the velocity components in x and z directions of the fluid flow, respectively. Here, the variable t stands for time.

The flow surface is assumed to be a good conductor of heat. A heater is embedded in a plate that produces temperature TS(x) at the plate surface z = 0 and creates a temperature difference ΔT. For example [27], at a reference point x = 0, the temperature difference between the sheet surface temperature at origin (TS0=TS(x=0)) and the ambient air temperature (T_a) gives ΔT=TS0−Ta. The thermal difference ΔT is responsible for temperature distribution T(x,z,t) across the fluid. This circulation of heat imposes that the film thickness h(x,t) of the fluid varies across the flow surface. In this problem, the fluid properties such as density ρ, dynamic viscosity μ, thermal conductivity κ, and the specific heat c_p are assumed constants. However, the surface tension σ is considered a linear function of temperature. The linear variation of this fluid property reads

where σ₀ is the surface tension at the reference ambient air temperature, i.e. σ0=σ(Ta). Also, Kσ=ΔTσ0(−dσdT)T=Ta is a parameter measuring the rate of change with respect to temperature. The governing equations of flow in vector form under the aforementioned physical conditions are as follows:

1. Continuity equation:

   (2)∇⋅V=0,

2. Momentum equation:

   (3)ρDVDt=∇⋅𝝉+ρg,

3. Temperature equation:

   (4)ρcpDTDt=κ∇2T+trace(𝝉⋅(∇V)).

The operator used in the equations is the nabla operator ∇=(∂/∂x,∂/∂z), and D/Dt is the substantial derivative, i.e. D/Dt=∂/∂t+V⋅∇. The second term of (3) is the gravitational force with gravity g=(gsinΩ,−gcosΩ), which is acting at an angle Ω with the inclined slope, and trace stands for the trace operator. The fluid of interest is a non-Newtonian differential-type second-grade fluid whose constitutive relation between the stress 𝝉 and the strain rate is given by Rivlin and Ericksen [28],

where p is the fluid pressure, I is the identity tensor, the material constants α1,α2 are the first and second normal stress coefficients, and the Rivlin–Ericksen tensors A₁ and A₂ are given as

Here, the superscript (^∗) stands for the transpose operator.

The constitutive relation (5) is derived based on the second-order corrections for viscoelastic effects. As this relation is invariant under the transformation (Dunn and Fosdick [29]) and compatible with hydrodynamics, therefore, the material constants obey the following restriction:

The third relation (7) is the consequence of satisfying the Clausius–Duhem inequality by fluid motion, and a second relation arises due to the assumptions that specific Helmholtz free energy of the fluid takes its minimum value in equilibrium. A fluid satisfying model (5) with αi<0 (i=1,2) is termed a second-order fluid and with αi>0 is termed as second-grade fluid. Although a second-order fluid obeys model (5) with α1<α2, α1<0, it exhibits undesirable instability characteristic (Fosdick and Rajagopal [30]). The second-order approximation is valid at a low shear rate (Dunn and Rajagopal [31]).

A certain amount of energy is stored up in the material due to the flow of viscoelastic fluid as strain energy in addition to the dissipation of heat due to viscosity. Hence, the last term in the energy (4) is retained due to dissipation.

The boundary conditions and initial conditions are described under which the proposed fluid flow problem is desirably well posed. The proposed boundary conditions at the inclined surface z = 0 are no-slip, no-penetration, and imposed surface temperature distributions. Whereas at the free surface, i.e. at z=h(x,t), the normal and tangential stress balance conditions, Newton’s law of cooling, and the kinematic condition act. Mathematically, it can be written as

and the free surface boundary conditions at z=h(x,t) are

and

where n=(−hx/1+hx2,1/1+hx2), t=(1/1+hx2,hx/1+hx2) are the unit normal and unit tangent vectors at the free surface from the fluid towards the air, respectively. The rate of heat transport from the liquid to the ambient gas phase is represented by K_g. Assuming the velocity of the ambient air is zero, the air pressure is denoted by p_a.

The initial conditions are as follows:

3 Long-Wave Approximation

The long-wave theory is used to derive the one-dimensional thin film equation. We follow the analysis of Mukhopadhyay and Chattopadhyay [32] but extended to include the complex rheology of second-grade non-Newtonian fluid.

The flow parameters such as velocity, pressure, and temperature are expanded using power series expansion in terms of the aspect ratio ϵ up to a first-order approximation. This gives two sets of partial differential equations of orders O(1) and O(ϵ). The expansion of the velocities, pressure, and temperature is given as follows:

The leading-order equations are obtained after substituting (25) in the dimensionless model (15–23). The zero-order problem reads

The corresponding boundary conditions at leading order are as follows:

and at the free surface, z=h(x,t),

The solution of the zeroth-order problem (26–31) is

The first-order problem is the collection of coefficients of 0(ϵ) terms.

and

The boundary conditions are as follows:

and at the free surface z=h(x,t):,

and

We solve the first-order problem (33–40) using the solutions of the zeroth-order problem, i.e. (32). The horizontal velocity component u₁ and the temperature field T₁ of the first-order problem are as follows:

where

The temperature distribution can be now obtained from (32d) and (41b) as

Following [34], the shear stress acting on the fluid can be written in nondimensional form as

where τ₀ is the shear stress of the leading-order problem given by

and the shear stress for the first-order problem reads

Using solutions of zero-order and the first-order problems for the velocity fields u₀, w₀, and u₁ given in (32) and (41), the expression for the shear stress can be written in terms of the free surface height at the plate, i.e. at z = 0 as

4 Numerical Procedure

The thin film (50) is a time-dependent partial differential equation of first order in time and fourth order in space. To solve numerically, we discretised the model equation implicitly on a uniform grid in a finite volume framework as described in [26], [27].

The discrete form of the equation reads

where Fi+1/2n+1=F(xi+Δx/2,tn+1), and Δt and Δx are the time step and grid length, respectively.

The face value and the gradient at the internal nodes are evaluated as

Figure 2:

Evolution of free surface with time. Corresponding temperature profiles. Film thickness profile with parameters ϵ = 0.01, Ω=π/2, Kσ=0.1, Mw = 2, Re = 1, Pr = 1, We = 10, and Ec = 0.1. (a) Film thickness profiles at different times, (b) corresponding temperature profile (54).

Similarly, for the higher-order derivatives terms are approximated, e.g.

and,

The nonlinear implicit discretised system is solved using MATLAB solver fsolve routine. This routine solves the nonlinear equations using Trust–Region–Dogleg (Levenberg–Marquardt) method. The convergence is achieved usually in fewer than 10 iterations, and the convergence criterion is that the norm of the residuals should be less than 10⁻⁷ on the grid size Δx=40/201.

5 Results and Discussion

In this section, we first begin our discussion on the dynamics of the capillary ridge and then discuss how the flow parameters influence the capillary ridge. In the left panel of Figure 2, the free surface profiles are given at different times for the Newtonian fluid (K = 0) and for the non-Newtonian second-grade fluid (K = 1). The imposed temperature on the solid plate is considered as

Figure 3:

Influence of time and non-Newtonian parameter K on the free surface profile with ϵ = 0.01, Ω=π/2, Kσ=0.1, Mw = 2, Re = 1, Pr = 1, We = 10, and Ec = 0.1.

The temperature profile is shown in the right panel of Figure 2. The other flow parameters used in the simulation are reported in the figure caption. It illustrates that the Marangoni stress at the surface of the plate opposes the gravitational flow, which leads to the formation of a capillary ridge. The downstream capillary ridge moves towards the direction of the flow with the advancement of time. It can be noted that the downstream surge in height is higher for the second-grade fluid (K = 1) in comparison to the Newtonian fluid (K = 0). Physically, for the second-grade fluid, the tensile stress between the fluid layer is higher, and consequently, it resists the motion of the fluid, which increases the amplitude of the downstream ridge. For the higher value of K, the amplitude of the ridge increases both in the upward and downward direction as shown in Figure 3. In other words, as the fluid is more non-Newtonian, the capillary ridge gets bigger. Thus, the capillary ridge height is sensitive to the parameters.

Figure 4 demonstrates the effects of increasing the Marangoni number, Mw, of the free surface profile. The free surface profile is plotted at fixed time (nondimensional) t = 1 varying the Marangoni number and viscoelastic parameter. The gravity-driven stream-wise flow is opposed by the thermocapillary forces at the upstream edge (x = 0) of the plate, where the temperature gradient is steep, and the velocity decreases near the free surface. The surface tension dominates at the front of the flow and therefore causes the capillary ridge. It can be seen that the capillary ridge height is increasing with the increasing value of the Marangoni number. Physically, the higher the Marangoni number, the higher the surface tension force in comparison to viscous force, and consequently, it influences the levelling of the fluid. This observation is consistent with that discussed by Tiwari et al. [11] for the flow of Newtonian fluid over locally heated surfaces. It is also seen that the height is greater for the second-grade fluid due to the viscoelastic effects.

Figure 4:

Influence of the Marangoni number, Mw, and the viscoelastic parameter, K, on the free surface with ϵ = 0.01, Ω=π/2, Kσ=0.1, Re = 1, Pr = 1, We = 10, and Ec = 0.1 at time t = 1.

Figure 5:

Influence of the Reynolds number, Re, on the capillary ridge with ϵ = 0.01, Ω=π/2, Kσ=0.1, Mw = 2, K = 1, Pr = 1, We = 10, and Ec = 0.1, at a fixed time (nondimensional) t = 1.

Figure 6:

Influence of the Weber number, We, on the capillary ridge with ϵ = 0.01, Ω=π/2, Kσ=0.1, Mw = 2, K = 1, Re = 1, Pr = 1, and Ec = 0.1, at a fixed time (nondimensional) t = 1.

Figure 7:

Effect of the Eckert number, Ec, on the thin film height with ϵ = 0.01, Ω=π/2, Kσ=0.1, Mw = 2, K = 1, Pr = 1, Re = 1, and We = 10, at a fixed time (nondimensional) t = 1.

As shown in Figure 5, we found that the viscosity of the fluid influences the capillary ridge. For larger Reynolds number, the capillary ridge height increases. The capillary ridge with higher Reynolds number flow (Re = 100) moves forward, and its height decreases due to less resistance to the flow. Similar behaviour is also observed for the effect of Weber number on the capillary ridge (Fig. 6). The Weber number characterises the ratio of inertial force to surface tension force. The smaller the Weber number, the higher the surface tension force, which significantly influences the capillary ridge. The higher surface tension holds the fluid to move, and as a result, the capillary height increases, and the fluid takes more time to travel.

Figure 7 shows the changing capillary ridge height with a variation of the Eckert number. The Eckert number (Ec) (dissipation of energy) [35] is used to describe the heating up of the fluid as a consequence of dissipation effects. The decrease in Ec implies that the temperature difference dominates the kinetic energy of the fluid. As a result, the surface tension gradient develops, which caused the formation of a ridge with a higher amplitude. The rise in the dissipation of energy enhances the temperature in the fluid, and it works against the viscous fluid stresses. Thus, as Ec increases, the capillary ridge height decreases.

It can be concluded that the capillary ridge height is higher for the small Weber number due to the high surface tension effect, but for the high Eckert number flow, a smaller ridge with a spreading of shape is observed due to the transformation of kinetic energy into heat by the viscous dissipation.

Further, the tabulated values of the shear stress and the rate of heat transfer at the plate are given in Table 1 to understand the combined effects of the Weber number and Eckert number. The absolute values of the shear stress and the temperature gradient at the plate z = 0 are calculated from (47) and (43), respectively, at the fixed place on the plate x = 0 and at a fixed time t = 1 (nondimensional). The value of the other flow parameters is kept constant and given in the table caption. The results show that the value of the shear stress increases with the increasing Weber number, while decreasing effect is observed for the rate of heat transfer at the plate. It is also evident from the tabulated data that the trends of the absolute value of temperature gradient at the plate are increasing with an increase in Eckert number. As the rate of heat transfer at the plate is higher for the higher value of the Eckert number, the frictional force decreases.

Figure 8:

Effect of inclination angle Ω on the thin film height with ϵ = 0.01, Kσ=0.1, Mw = 2, K = 1, Pr = 1, Re = 1, We = 400, and Ec = 0.1 at a fixed time (nondimensional) t = 1.

In the next, we explore the effect of Prandtl number on the capillary ridge height in the free surface of the thin liquid film. To analyse this effect, the simulations were run for different values of Prandtl number, and the maximum height of the free surface (h_max) is then calculated at a fixed time t = 1 (nondimensional). The computed results are presented in Table 2. The nondimensional Prandtl number is the ratio of viscous diffusion rate to thermal diffusion rate. The fluid with small Prandtl number has high thermal conductivity and tends to flow freely. We can see from the tabulated results that when the Prandtl number increases the viscosity dominates the fluid; thereby, it experiences the resistance, which increases the amplitude of the ridges.

Figure 8 shows that as the inclination angle increases, the capillary ridge height gets bigger. There is no significant change in the ridge height for the higher inclination angle due to the shear thickening nature of the fluid.

6 Conclusion

The flow of a thin second-grade fluid flowing over an inclined locally heated surface was considered. The mathematical model for the evolution of the free surface of the thin film height is derived using the long-wave approximation. It is observed that the Marangoni stress influences the dynamics of the fluid. These results are consistent with the work of Tiwari et al. [11], who have analysed the free surface profile and its stability of the flow of a Newtonian fluid on an inclined locally heated surface. The effects of Reynolds number, Weber number, and Eckert number on the capillary ridge height are significant. The analysis explained the dependency of the Weber number and Eckert number on the capillary ridge and the rate of heat transfer and the shear stress at the surface of the plate. It was found that the height of the capillary ridge is more pronounced for the second-grade fluid in comparison to the Newtonian fluid. Now, the obvious question is how the fluid parameters such as elasticity influence the stability of the flow. The present investigation revealed the important influence of the viscoelastic properties of the flow on the free surface of the film height and inspired to investigate in the future the stability of the second-grade liquid film flowing down an inclined heated surface. Thus, the derived free surface equation will serve as a starting point in this direction. This work will also provide a scope to explore the free surface stability of the coating polymer liquid, which is slightly viscoelastic.