Viscoelastic film flows over an inclined substrate with sinusoidal topography. I. Steady state

We consider the steady flow of a viscoelastic liquid film over an inclined wall with sinusoidal corrugations of arbitrary wavelength and depth. We develop a computational model and carry out detailed numerical simulations based on the finite-element method to investigate this flow. To this end, we solve the two-dimensional momentum and mass conservation equations while employing the Phan-Thien-Tanner (PTT) constitutive model to account for the rheology of the viscoelastic material. An elliptic grid generation scheme is used to follow the large deformations of the liquid film. We perform a thorough parametric analysis to investigate the combined effects of elasticity, inertia, and capillary and viscous forces on the characteristics of the steady flow. The results of our simulations indicate that fluid elasticity suppresses interfacial deformation at low flow rates, whereas at moderate values of $\mathrm{Re}$ it enhances the deformation considerably. In the latter case, elastic forces may even give rise to the formation of a static hump and a cusp at the free surface, the size of which increases with the relaxation time of the liquid. The resonance of the liquid film with the substrate undulations is also enhanced by shear thinning. Interestingly, it is predicted that under certain conditions the transition to the inertia regime is not smooth and a hysteresis loop arises, which is the signature of an abrupt change of the film shape, since its high deformations cannot be sustained. Additionally, we have performed calculations for a wide range of different geometrical characteristics of the substrate. We find that viscoelastic effects become more pronounced in the case of long-wavelength wall undulations, while for substrates with short wavelengths the effect of shear thinning is less significant due to the presence of vortices inside the corrugations.

Figure 1

Cross section of the film flowing over a sinusoidal substrate inclined with respect to the horizontal by an angle $\alpha$. $L^{*}$ and $A^{*}$ are the length and the depth of the unit cell of the substrate, respectively. $H^{*}$ is the film height at the inlet of the periodic unit cell.

Figure 2

Comparison of our predictions for an inclined substrate at $\alpha ={45}^{\mathrm{o}}$ with previous studies by Nguyen and Bontozoglou [19] for $\mathrm{Re}=25$, $\mathrm{Ka}=200$, $A=5.97,$ and $L=29.87$. (b), (c) Experimental (red lines) [31] and theoretical (blue lines) streamline patterns for Elbesil 145 oil ($\mathrm{Ka}=1.27$) for $\mathrm{Re}=6.28$ and $8.84$, respectively. The inclination angle is $\alpha ={10}^{\mathrm{o}},$ and geometric characteristics are $L=13.765$ and $A=5.506$.

Figure 3

Streamlines of the steady states of the film flow over the substrate: (a) $\mathrm{Re}=2$, (b) $\mathrm{Re}=8$, (c) $\mathrm{Re}=15$, and (d) $\mathrm{Re}=20$ for $\mathrm{Wi}=1$ and $\varepsilon =0$.

Figure 4

(a) Relative amplitude of the free surface $A_{\mathrm{rel}}$ and (b) the phase shift of the free surface to the wall corrugations as a function of $\mathrm{Re}$ for various values of $\mathrm{Wi}$ using the Oldroyd-B model ($\varepsilon =0$). For low Re dual minima appear in the free surface as seen in the insets (i, ii) for $\mathrm{Re}=2.7$ and $2.65$, respectively.

Figure 5

Spatial variation of the normal stress component (${\tau }_{p,xx}$) using the Oldroyd-B model for (a) $\mathrm{Wi}=0.5$ and (b) $\mathrm{Wi}=1.5$ calculated at $\mathrm{Re}=15$.

Figure 6

(a) Relative amplitude of the free surface $A_{\mathrm{rel}}$ as a function of the Reynolds number for various values of $\varepsilon$ at $\mathrm{Wi}=1$. (b), (c) Film shape and spatial variation of shear stress component (${\tau }_{p,yx}$) before and after the hysteresis loop, respectively, for $\mathrm{Re}=14.38$, $\mathrm{Wi}=1$, and $\varepsilon =0.25$.

Figure 7

Relative amplitude of the free surface $A_{\mathrm{rel}}$ as a function of (a) Reynolds number and (b) for various values of $\mathrm{Ka}$ at $\mathrm{Wi}=1$, $\varepsilon =0.05$, $\mathrm{and}\beta =0.1$.

Figure 8

Film shape and spatial variation of (a), (b) shear stress component (${\tau }_{p,yx}$) and (c), (d) normal stress component (${\tau }_{p,xx}$). The flow parameters are (a), (c) $\mathrm{Ka}=1.0$ and (b), (d) $\mathrm{Ka}=10$ calculated at $\mathrm{Re}=7.2$ and $\mathrm{Re}=28.7$, respectively, for $\mathrm{Wi}=1$, $\varepsilon =0.05,$ and $\beta =0.1$.

Figure 9

Relative amplitude of the free surface $A_{\mathrm{rel}}$ as a function of Reynolds number for various values of solvent viscosity ratio $\beta$ for $\mathrm{Wi}=1.0$ and $\varepsilon =0.05$.

Figure 10

(a) Free surface deformation as a function of the Reynolds number for various values of $A$. (b)–(d) Film shape and spatial variation of the shear stress component (${\tau }_{p,yx}$) for $A=2.7,5.50,10.30$ for $\mathrm{Re}=12.9,14.6,15.4$, respectively. The remaining parameters are $\mathrm{Wi}=1$, $\varepsilon =0.25$, $\beta =0.1,$ and the inclination angle is set at $\alpha ={10}^o$.

Figure 11

(a) Relative amplitude of the free surface $A_{\mathrm{rel}}$ as a function of the Reynolds number for various values of $L$. (b)–(d) Film shape and spatial variation of the shear stress component (${\tau }_{p,yx}$) for $L=7.5,25,50$ for $\mathrm{Re}=8.5,23.25,24$, respectively. The remaining parameters are $\mathrm{Wi}=1$, $\varepsilon =0.25,$ and $\beta =0.1$. The inclination angle is set at $\alpha ={10}^{\mathrm{o}}$.

Figure 12

Relative amplitude of the free surface $A_{\mathrm{rel}}$ as a function of (a) the Reynolds number and (b) Weber number for various values of inclination angles for $\mathrm{Wi}=1$, $\varepsilon =0.25,$ and $\beta =0.1$.