Boundary-layer flows over deforming surfaces

In this paper a formulation of the incompressible Navier-Stokes equations is introduced which allows one to model boundary-layer flows induced by the motion of a deforming surface. Such a formulation may be used to model flows relevant in a wide variety of industries from polymer processing to glass manufacturing. We show that for particular sheet geometries and velocities, similarity solutions may be obtained that account for sheet thinning (or thickening) and roughness patterns observed in extrusion-type processes.

Figure 1

Schematic diagrams of boundary-layer flows developing over (a) a stretching thinning sheet and (b) a periodic rough surface. In both cases the profile of the surface is denoted by $s\left(x\right)$, the surface has a wall velocity denoted by $U_w\left(x\right)$, and it is this nonconstant wall velocity that results in the development of a boundary-layer profile denoted by $u(x,y)$.

Figure 2

Plots of the streamwise and wall-normal velocity components for the case when $\gamma =2$ and $U_w=\sqrt{{}_0{U}_w^2+2\xi /3}=\sigma$. In this case both $u_0$ and $v_0$ are independent of $\xi$. Given that no exact analytical solution for $\widehat{f}$ exists in the case when $\gamma =2$, a shooting method that makes use of a fourth-order Runge-Kutta integrator, twinned with a secant root finding scheme, was employed to solve (3). As part of this solution process we determine that ${\widehat{f}}_{ZZ}^{″}(Z=0)\approx 0.8300$ and that $\widehat{f}(Z\to \infty )={\widehat{f}}_{\infty }\approx 1.0625$. In the limit as $Z\to \infty$, then $v_0\to -\sqrt{2/3}{\widehat{f}}_{\infty }$.

Figure 3

In (a) the wall velocity $U_w$, the approximate wall velocity, $U_w^{\text{approx}}=\xi +0.2260$, and the exponentially thinning sheet profile, $s\left(\xi \right)=e^{-\xi }$, are plotted against $\xi$. In (b) the wall velocity, given an identical thinning sheet profile, is plotted for a range of $\gamma$ values. In both plots the dashed black line corresponds to the wall velocity result for a sheet undergoing linear stretching.

Figure 4

Plots of the streamwise (a) and wall-normal (b) velocity components for a range of $\xi$ values. In this case the sheet is thinning exponentially: $s\left(\xi \right)=e^{-\xi }$. Crane's flat plate solutions are given by the dashed black curves.

Figure 5

In (a) the wall velocity $U_w$, the approximate wall velocity, $U_w^{\text{approx}}={(\xi +1.1819)}^{-\frac{1}{3}}$, and the logarithmically thickening sheet profile, $s\left(\xi \right)=ln(e+\xi )$, are plotted against $\xi$. In (b) and (c), respectively, solutions for the streamwise velocity $u_0$ and wall-normal velocity $v_0+s_{\xi }^{\prime }{u}_0$ are mapped back to the unscaled boundary-layer coordinate system ($\xi ,\eta +s$). The solid black line indicates the surface of the thickening decelerating sheet.

Figure 6

In (a) the wall velocity $U_w$, the approximate wall velocity, $U_w^{\text{approx}}={({}_0{U}_w^{\gamma }+M\xi )}^{\frac{1}{1+\gamma }}$, and the small-amplitude rough profile, $s\left(\xi \right)=\varepsilon [1-cos(2\pi \xi \left)\right]$, are plotted against $\xi$. In (b) the cyclical velocities $U_0$ and $V_0$ are plotted against the boundary-layer coordinate $Z$. The solid curves correspond to the solutions at $2\xi =(n-1)$, the dotted curves are the solutions at $2\xi =(n-\frac{1}{2})$, for $n=1,2,3,\cdots$. In both instances $\varepsilon =1/5, {}_0{U}_w=2$, and $\gamma =2$.

Figure 7

In (a) a 3D-plot of the streamwise velocity $u_0$ against $\xi$ and $Z$. In (b) the solution for $u_0$ has been mapped back to the unscaled boundary-layer coordinate system ($\xi ,\eta +s$). In both plots the roughness parameter, $\varepsilon$, has been set equal to one-fifth; the initial wall velocity, ${}_0{U}_w$, is set equal to 1; and the surface is moderately accelerating with $\gamma =2$. In (b) the solid black line indicates the surface of the rough sheet.

Figure 8

In (a) we plot the absolute value of the difference between the numerical solutions for the streamwise velocity component at $\xi =2.5$, with the corresponding boundary-layer solution at the same point, for a range of values of the small parameter ${Re}^{-1/2}$. In (b) we plot an identical comparison for the wall-normal velocity component.

Figure 9

Comparison between the finite-element and boundary-layer solutions across the computational domain for (a) ${Re}^{-1/2}=0.05$ and (b) ${Re}^{-1/2}=0.04$. The same colorbar scale is used in both instances so that the reduction in error may be easily observed.