⇦Chinese Physics Letters, 2019, Vol. 36, No. 1, Article code 014701 Unsteady Liquid Film Flow with a Prescribed Free-Surface Velocity Tiegang Fang (方铁钢)**, Fujun Wang (王甫军) Affiliations Mechanical and Aerospace Engineering Department, North Carolina State University, Raleigh NC 27695, USA Received 11 September 2018, online 25 December 2018 **Corresponding author. Email: tfang2@ncsu.edu Citation Text: Fang T G and Wang F J 2019 Chin. Phys. Lett. 36 014701    Abstract A liquid film flow over a flat plate is investigated by prescribing the unsteady interface velocity. With this prescribed surface velocity, the governing Navier–Stokes (NS) equations are transformed into a similarity ordinary differential equation, which is solved numerically. The flow characteristics is controlled by an unsteadiness parameter $S$ and the flow direction parameter ${\it \Lambda}$. The results show that solutions only exist for a certain range of the unsteadiness parameter, i.e., $S\leqslant 1$ for ${\it \Lambda} =-1$ and $S\leqslant -2.815877$ for ${\it \Lambda} =1$. In the solution domain, the dimensionless liquid film thickness $\beta $ decreases with $S$ for both the cases. The wall shear stress increases with the decrease of $S$ for ${\it \Lambda} =-1$. However, for ${\it \Lambda} =-1$ the shear stress magnitude first decreases and then increases with the decrease of $S$. There are no zero crossing points for the velocity profiles for both the cases. The profiles of velocity stay either positive or negative all the time, except for the wall zero velocity. Consequently, the vertical velocity becomes a monotonic function. To maintain the prescribed velocity, mass transpiration is generally needed, but for the shrinking film case it is possible to have an impermeable wall. The results are also an exact solution to the full NS equations. DOI:10.1088/0256-307X/36/1/014701 PACS:47.10.ad, 47.15.Cb © 2019 Chinese Physics Society Article Text Liquid film flow has wide practical applications in industry and scientific research. Wang[1] first investigated the flow of a liquid film over an unsteady stretching sheet. The governing Navier–Stokes (NS) equations were transformed into a similarity equation and solved numerically. The problem was further extended to the axisymmetric configuration for a disk by Usha and Sridharan.[2] Anderson et al.[3] generalized Wang's problem from a Newtonian fluid to a power-law non-Newtonian fluid. The heat transfer problem of Wang's work was first analyzed by Anderson et al.[4] and Chen investigated the heat transfer problem for the power-law fluid[5] and by including viscous dissipation effects.[6] Dandapat et al.[7] studied the liquid film flow over an unsteady stretching sheet by considering the dependence of surface tension on fluid temperature. Xu et al.[8] investigated the liquid film flow and heat transfer for nanofluids. In many applications, a liquid film is often formed on a surface by liquid jet impingement or droplet impact on to a solid plate. For these situations, the plate is often stationary and the liquid film interface moves in a certain velocity. Therefore, liquid film flows with a prescribed interface velocity may have important applications not only for theoretical fluid mechanics but also for real engineering practices. The objective of this work is to analyze the liquid film flow over a stationary plate with a prescribed interface velocity. Consider a two-dimensional laminar incompressible fluid film flow over a stationary flat plate. The fluid film has a finite thickness $h(t)$. The film is assumed to be stable and stays flat all the time. It is also treated as a free surface at the interface with the ambient gas, assuming that the ambient gas pressure is $p_{0}$. It is assumed that the film free surface has a velocity $U_{h}(x,t)={\it \Lambda} \frac{bx}{1-\alpha t}$ with $b$ being a positive real number, $\alpha$ a real number, and ${\it \Lambda}$ the direction parameter. Without loss of generality, we assume ${\it \Lambda} =\pm 1$. When ${\it \Lambda} =1$, the film moves along the positive $x$-direction, and for ${\it \Lambda} =-1$ the film moves towards the origin. The $x$-axis runs along the plate and the $y$-axis is perpendicular to it. The governing NS equations of this problem read $$\begin{alignat}{1} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=\,&0,~~ \tag {1} \end{alignat} $$ $$\begin{alignat}{1} \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\,&-\frac{1}{\rho}\frac{\partial p}{\partial x}+\upsilon \Big(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\Big),~~ \tag {2} \end{alignat} $$ $$\begin{alignat}{1} \frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=\,&-\frac{1}{\rho}\frac{\partial p}{\partial y}+\upsilon \Big(\frac{\partial^{2}v}{\partial x^{2}}+\frac{\partial^{2}v}{\partial y^{2}}\Big),~~ \tag {3} \end{alignat} $$ where $u$ and $v$ are the velocity components in the $x$ and $y$ directions, respectively, $\upsilon$ is the kinematic viscosity, $\rho$ is the fluid density, and $p$ is the fluid pressure. The boundary conditions (BCs) include $$\begin{align} u(x,0,t)=\,&0,~~ \tag {4a}\\ u(x,h,t)=\,&U_{h},~~ \tag {4b}\\ \frac{\partial u}{\partial y}(x,h,t)=\,&0.~~ \tag {4c} \end{align} $$ The governing equations can be transformed into a similarity equation by employing a stream function and a similarity variable in the form of $$\begin{align} {\it \Psi} (x,y,t)=\,&\sqrt \frac{b\upsilon}{1-\alpha t} xf(\eta),~~ \tag {5a}\\ \eta =\,&\frac{y\sqrt b}{\sqrt {\upsilon (1-\alpha t)}}.~~ \tag {5b} \end{align} $$ Then, the velocity components are given by $u=\frac{\partial {\it \Psi}}{\partial y}=\frac{bxf'(\eta)}{1-\alpha t}$ and $v=-\frac{\partial {\it \Psi}}{\partial x}=-\sqrt \frac{b\upsilon}{1-\alpha t} f(\eta)$. The pressure field can be obtained from Eq. (3) as follows: $$\begin{alignat}{1} -p(y,t)=\rho \Big(\int {\frac{\partial v}{\partial t}dy} +\frac{v^{2}}{2}-\upsilon \frac{\partial v}{\partial y}\Big)+C(t).~~ \tag {6} \end{alignat} $$ Substituting these variables into the governing equations yields $$\begin{align} f'''+ff''-f^{'2}-S\Big(\frac{\eta}{2}f''+f'\Big)=0~~ \tag {7} \end{align} $$ subjected to the associated BCs $$\begin{align} f'(0)=0,~~f'(\beta)={\it \Lambda} =\pm 1,~~f''(\beta)=0,~~ \tag {8} \end{align} $$ where $S=\frac{\alpha}{b}$ is the unsteadiness parameter ($S>0$ is an accelerating flow and $S < 0$ is a decelerating flow), $\beta$ is the dimensionless film thickness to be determined, and the prime denotes differentiation with respect to $\eta$. As defined above, ${\it \Lambda}$ is a parameter showing the film flow direction. When ${\it \Lambda} =1$, the liquid film moves from the origin, representing a spreading process during liquid jet impinging or droplet impact processes. For ${\it \Lambda} =-1$, the liquid film moves towards the origin, representing a receding process during a droplet impact. The thickness of the film $h(t)$ is given by $h(t)=\beta \sqrt \frac{\upsilon (1-\alpha t)}{b}$. Matching the surface vertical velocity $\dot{h}(t)=-\frac{\alpha \beta \sqrt \upsilon}{2\sqrt {b(1-\alpha t)}}=-\sqrt \frac{b\upsilon}{1-\alpha t} f(\beta)$ yields another boundary condition $f(\beta)=\frac{S\beta}{2}$. To simplify the calculation, a further transformation as $f(\eta)=\beta F(\frac{\eta}{\beta})=\beta F(\xi)$ yields[9] $$\begin{align} \dddot{F}+\beta^{2}\Big[F\ddot{F}-\dot{F}^{2} -S\Big(\frac{\xi}{2}\ddot{F}+\dot{F}\Big)\Big]=0~~ \tag {9} \end{align} $$ subjected to the BCs $$\begin{alignat}{1} \dot{F}(0)=0,~~\dot{F}(1)={\it \Lambda},~\ddot{F}(1)=0, ~~F(1)=\frac{S}{2},~~ \tag {10} \end{alignat} $$ where the dot denotes the derivative with respect to $\xi$. Based on this transformation, $f''(0)=\ddot{F}(0)/\beta$. Considering the conditions at the film interface, the pressure can be rewritten as $$\begin{alignat}{1} \!\!\!\!\!\!\!\!p(y,t)=\frac{\rho \nu b}{1\!-\!\alpha t}\Big[\frac{Sf(\eta)\eta}{2}\!-\!\frac{f^{2}(\eta)}{2}\!-\!f'(\eta)\Big]\!+\!C(t),~~ \tag {11} \end{alignat} $$ where $C(t)=p_{0}-\frac{\rho \nu b}{1-\alpha t}[\frac{S^{2}\beta^{2}}{8}-f'(\beta)]$ by applying the pressure balance at the interface. Because the analytical solutions to Eqs. (9) and (10) are unavailable, the similarity equations were solved for different parameters using the embedded function NDSolve in Mathematica 10. Default calculation precision of Mathematica was used and the precision is less than 10$^{-7}$. Before solving the current problem, the Mathematica code was validated using the results in Ref. [9] for unsteady stretching film cases. Further, the numerical solutions were also compared with the analytical approximations for extreme conditions. In the following, numerical solutions will be discussed and analyzed first.
cpl-36-1-014701-fig1.png
Fig. 1. Solutions of $\ddot{F}(0)$, $\beta$, $F(0)$ and $f''(0)$ as a function of $S$.
cpl-36-1-014701-fig2.png
Fig. 2. Profiles of $F(\xi)$, $\dot{F}(\xi)$ and $\ddot{F}(\xi)$ under different values of $S$ for ${\it \Lambda} =-1$.
The numerical results of $\beta$, $\ddot{F}(0)$, $F(0)$ and $f''(0)$ are shown in Fig. 1. Interesting variation behavior can be seen in the results, showing different trends for ${\it \Lambda} =-1$ and ${\it \Lambda} =1$. The solution existence domains for the two different cases are different. For ${\it \Lambda} =-1$, there is an upper limit of $S\cong 1$, while for ${\it \Lambda} =1$ the existence limit is about $S\cong -2.815877$. The limiting value of $\ddot{F}(0)$ is zero for ${\it \Lambda} =1$ but it is approaching to infinity for ${\it \Lambda} =-1$. The value of $\ddot{F}(0)$ increases with the decrease of $S$ for both the values of ${\it \Lambda}$ and approaches to ${\it \Lambda}$ with $S\to -\infty$. For $\beta$ both the cases show similar trends and the film thickness decreases with $S$ and it approaches to infinity when $S$ approaches to the upper domain limit. The value of $f''(0)$ also follows a similar trend to $\ddot{F}(0)$ but due to the variation of $\beta$ with $S$, the trend for ${\it \Lambda} =-1$ becomes different and it first increases and then decreases with the decrease of $S$. The value of $F(0)$ follows the same trend for both the cases, which decreases with $S$. The results also imply that to maintain a similarity solution there should be a mass transpiration at the plate surface in general. For ${\it \Lambda} =-1$, both mass suction and mass injection are possible, while for ${\it \Lambda} =1$ a mass injection is required. It is interesting that for ${\it \Lambda} =-1$ solution exists for an impermeable surface, which is an important case for a non-porous plate. This can be used to analyze the unsteady receding film flow during the droplet impact process on a solid surface. Profiles of $F(\xi)$, $\dot{F}(\xi)$, and $\ddot{F}(\xi)$ for ${\it \Lambda} =-1$ are shown in Fig. 2 for a few typical values of $S$. It is seen that for $S=1$, the profiles show boundary layer behavior and the velocity variation stays close to the wall in a very small distance from the wall. For all the cases, $F(\xi)$ and $\dot{F}(\xi)$ both decrease with the distance from the wall, and $\dot{F}(\xi)$ always stays negative while $\ddot{F}(\xi)$ shows non-monotonic behavior. For ${\it \Lambda} =1$, the three profiles show quite different behaviors as shown in Fig. 3. Here $\dot{F}(\xi)$ always stays positive and $\ddot{F}(\xi)$ first increases with $\xi$, then reaches a peak and finally decreases to zero. At the limiting value of $S=-2.8158773$, the profiles shift toward the liquid interface. As demonstrated by the numerical results, when the negative unsteadiness parameter $S$ becomes larger, there is a definite trend observed for both $\beta$ and $\ddot{F}(0)$. To obtain more insights for the solution behavior for large negative $S$, a further analysis is carried out. Assuming $S=-A$ with $A$ being a positive real number, by defining $F(\xi)=-\frac{A}{2}+\phi (\xi)$ and substituting it into Eq. (9), we obtain $$\begin{alignat}{1} \!\!\!\!\!\!\dddot \phi +\beta^{2}\Big[\Big(-\frac{A}{2}+\phi\Big)\ddot{\phi}-\dot{\phi}^{2} +A(\frac{\xi}{2}\ddot{\phi}+\dot{\phi})\Big]=0~~ \tag {12} \end{alignat} $$ subjected to BCs $$\begin{alignat}{1} \dot{\phi}(0)=0,~~\dot{\phi}(1)={\it \Lambda},~\ddot{\phi}(1)=0, ~~\phi (1)=0.~~ \tag {13} \end{alignat} $$ From the results in Fig. 1, we have known that $\beta$ decreases with the increase of $A$. Assuming for large $A$, $\beta$ is approximated as $\beta^{2}=\sigma^{2}A^{\gamma}$ with $\gamma$ less than zero. Substituting this approximation in Eq. (12) at the large $A$ limit yields $$\begin{align} \dddot \phi +\sigma^{2}A^{\gamma +1}\Big(-\frac{1}{2}\ddot{\phi} +\frac{\xi}{2}\ddot{\phi}+\dot{\phi}\Big)=0.~~ \tag {14} \end{align} $$ Depending on the value of $\alpha$, Eq. (14) can be simplified to different equations. If $-1 < \gamma < 0$, for large $A$, Eq. (14) becomes $$\begin{align} -\frac{1}{2}\ddot{\phi}+\frac{\xi}{2}\ddot{\phi}+\dot{\phi}=0.~~ \tag {15} \end{align} $$ A general solution to Eq. (15) is obtained as follows: $$\begin{align} \phi (\xi)=\frac{C_{1}}{1-\xi}+C_{2}.~~ \tag {16} \end{align} $$ This solution cannot satisfy any two of the BCs (13). Therefore, $-1 < \gamma < 0$ is not a proper assumption. If $\gamma < -1$, Eq. (14) becomes $$\begin{align} \dddot \phi =0.~~ \tag {17} \end{align} $$ The general solution is a parabolic function $$\begin{align} \phi (\xi)=C_{1}\xi^{2}+\xi C_{2}+C_{3},~~ \tag {18} \end{align} $$ which cannot satisfy the BCs. A feasible option is assumed to be $\gamma =-1$, with Eq. (14) becoming $$\begin{align} \dddot \phi +\sigma^{2}\Big(-\frac{1}{2}\ddot{\phi}+\frac{\xi}{2}\ddot{\phi}+\dot{\phi}\Big)=0.~~ \tag {19} \end{align} $$ Fortunately, Eq. (19) has a closed form solution as $$\begin{alignat}{1} \!\!\!\!\!\phi (\xi)=-\frac{{\it \Lambda}}{\sigma}e^{-\frac{1}{4}\sigma^{2}(\xi^{2}+1)}e^{\frac{\sigma^{2}\xi}{2}}\sqrt \pi \mathrm{Erfi}\Big[\frac{\sigma}{2}(1-\xi)\Big],~~ \tag {20} \end{alignat} $$ where $\mathrm{Erfi}(x)$ is the imaginary error function, and $\sigma$ can be obtained by solving the nonlinear algebraic equation $$\begin{align} e^{\frac{\sigma^{2}}{4}}-\frac{1}{2}\sigma \sqrt \pi \mathrm{Erfi}\Big[\frac{\sigma}{2}\Big]=0,~~ \tag {21} \end{align} $$ which gives $\sigma \cong 1.848278$. From Eq. (20), we can derive $$\begin{align} \phi'(\xi)=\,&{\it \Lambda} \Big\{1-\frac{\sigma}{2}\sqrt \pi (1-\xi)e^{-\frac{1}{4}\sigma^{2}(-1+\xi)^{2}}\\ &\cdot\mathrm{Erfi}\Big[\frac{\sigma}{2}(1-\xi)\Big]\Big\},~~ \tag {22} \end{align} $$ and $$\begin{align} \phi''(\xi)=\,&\frac{\it \Lambda}{4}\Big\{2\sigma^{2}(1-\xi)-\sqrt \pi [-2+\sigma^{2}(-1+\xi)^{2}]\sigma\\ &\cdot e^{-\frac{1}{4}\sigma^{2}(-1+\xi)^{2}}\mathrm{Erfi} \Big[\frac{\sigma}{2}(1-\xi)\Big]\Big\}.~~ \tag {23} \end{align} $$ Based on the analytical solutions, the following approximated equations can be obtained $$\begin{align} \beta (S\to -\infty)=\,&\frac{\sigma}{\sqrt {-S}}=\frac{1.848278}{\sqrt A},~~ \tag {24} \end{align} $$ $$\begin{align} \ddot{F}(0)(S\to -\infty)=\,&{\it \Lambda},~~ \tag {25} \end{align} $$ $$\begin{align} f''(0)(S\to -\infty)=\,&{\it \Lambda} 0.541044A^{\frac{1}{2}},~~ \tag {26} \end{align} $$ $$\begin{align} F(0)(S\to -\infty)=\,&\frac{S}{2}-\frac{2{\it \Lambda}}{\sigma^{2}}.~~ \tag {27} \end{align} $$ The dashed lines in Fig. 1 show the results of the approximations and it is seen that the estimates closely follow the numerical results for all four parameters.
cpl-36-1-014701-fig3.png
Fig. 3. Profiles of $F(\xi)$, $\dot{F}(\xi)$ and $\ddot{F}(\xi)$ under different values of $S$ for ${\it \Lambda} =1$.
The physical implications of the current work are of interest and will be briefly highlighted. The liquid film flow over a plate has many applications in engineering systems and industries, such as spray coating,[10] spray and film cooling,[11] and power generation.[12] In addition, liquid film flow is also observed in liquid droplet impact on a solid surface during the spreading and receding processes.[13,14] In a recent work, a new unsteady liquid film flow was studied and the results were related to the droplet receding process after impact on a solid surface.[15] The analytical analysis can provide important insights about the dynamics of the liquid film flow. The format of the unsteady velocity employed in this work is of practical significance and was first proposed in the analysis of an unsteady stagnation point flow by Yang,[16] and it was used to investigate the unsteady stagnation flow on a cylinder. The current analysis and results serve the same purposes for applications in liquid film flow during the liquid jet impinging or droplet impact process on a solid surface. It should be noted that, due to the specially prescribed surface velocity format, it can be quite difficult to implement and can observe such kinds of flow in most of the common applications for liquid film flows over a flat surface unless the 2D impinging liquid jet is well controlled in a specified time-dependent manner to achieve the surface motion with the special spatial and temporal variation. The main contribution of the current work is to provide an interesting exact solution to the governing NS equations. This solution not only enriches the exact solution library to the famous equations but can also be used as benchmark problems to validate numerical codes. In conclusion, the liquid film flow over a stationary flat plate with a prescribed free surface velocity has been investigated. The governing NS equations are transformed into a similarity ordinary differential equation under the specified velocity functions. The similarity equation is solved numerically and the limiting conditions are analyzed theoretically. Two flow directions are analyzed. It is found that the solutions only exist for a certain range of the unsteadiness parameter for both flow directions. Different velocity and shear stress behaviors are observed for the two different flow directions while the velocity shows monotonic variation without zero-crossing points. However, the shear stress shows non-monotonic behavior for most of the conditions. The current results also provide an exact solution to the full 2D NS equations. References Liquid film on an unsteady stretching surfaceThe Axisymmetric Motion of a Liquid Film on an Unsteady Stretching SurfaceFlow of a power-law fluid film on an unsteady stretching surfaceHeat transfer in a liquid film on an unsteady stretching surfaceHeat transfer in a power-law fluid film over a unsteady stretching sheetEffect of viscous dissipation on heat transfer in a non-Newtonian liquid film over an unsteady stretching sheetThermocapillarity in a liquid film on an unsteady stretching surfaceFlow and heat transfer in a nano-liquid film over an unsteady stretching surfaceAnalytic solutions for a liquid film on an unsteady stretching surfaceReview on film cooling of liquid rocket enginesProgress in miniature liquid film combustors: Double chamber and central porous fuel inlet designsInertia dominated drop collisions. II. An analytical solution of the Navier–Stokes equations for a spreading viscous filmDrop dynamics after impact on a solid wall: Theory and simulationsLiquid film flow over an unsteady moving surface with a new stretching velocity
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