2.1 General problem

The velocity in the liquid film is constrained by

(2.1) $$\begin{eqnarray}\frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}(rv_{r})+\frac{\unicode[STIX]{x2202}v_{z}}{\unicode[STIX]{x2202}z}=0.\end{eqnarray}$$

Invoking the lubrication approximation, i.e. neglecting terms ${\sim}(\unicode[STIX]{x1D6FF}/R)^{2}$ and smaller, the $r$ -component of the Navier–Stokes equation takes the form

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}v_{r}}{\unicode[STIX]{x2202}z^{2}}=\frac{\text{d}p}{\text{d}r},\end{eqnarray}$$

where we have $p=p(r)$ . Boundary conditions at the impermeable disk are given by

(2.3a,b ) $$\begin{eqnarray}v_{r}(r,0)=0,\quad v_{z}(r,0)=0,\end{eqnarray}$$

and at the permeable disk by

(2.4a,b ) $$\begin{eqnarray}v_{r}(r,\unicode[STIX]{x1D6FF})=\langle v_{r}\rangle _{s}(r,\unicode[STIX]{x1D6FF})-\frac{\sqrt{\unicode[STIX]{x1D705}}}{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}v_{r}}{\unicode[STIX]{x2202}z}(r,\unicode[STIX]{x1D6FF}),\quad v_{z}(r,\unicode[STIX]{x1D6FF})=\langle v_{z}\rangle _{s}(r,\unicode[STIX]{x1D6FF})+\dot{\unicode[STIX]{x1D6FF}},\end{eqnarray}$$

where $\langle \boldsymbol{v}\rangle _{s}$ is the superficial average velocity in the porous disk (Whitaker Reference Whitaker1999). The first boundary condition in (2.4), for finite values of the dimensionless parameter $\unicode[STIX]{x1D6FC}$ , allows for ‘slip’ between the liquid and porous disk (Beavers & Joseph Reference Beavers and Joseph1967), and in the second boundary condition $\dot{\unicode[STIX]{x1D6FF}}=\text{d}\unicode[STIX]{x1D6FF}/\text{d}t$ .

Integration of (2.2) with respect to $z$ and applying the boundary conditions in (2.3a ) and (2.4a ) gives

(2.5) $$\begin{eqnarray}v_{r}=-\frac{\unicode[STIX]{x1D6FF}^{2}}{2\unicode[STIX]{x1D707}}\frac{\text{d}p}{\text{d}r}\left[\frac{1+2\unicode[STIX]{x1D70E}}{1+\unicode[STIX]{x1D70E}}\frac{z}{\unicode[STIX]{x1D6FF}}-\left(\frac{z}{\unicode[STIX]{x1D6FF}}\right)^{2}\right]+\frac{\langle v_{r}\rangle _{s}(r,\unicode[STIX]{x1D6FF})}{1+\unicode[STIX]{x1D70E}}\frac{z}{\unicode[STIX]{x1D6FF}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}=\sqrt{\unicode[STIX]{x1D705}}/\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}$ . Substitution of (2.5) in (2.1) gives

(2.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}v_{z}}{\unicode[STIX]{x2202}z}=\frac{\unicode[STIX]{x1D6FF}^{2}}{2\unicode[STIX]{x1D707}}\frac{1}{r}\frac{\text{d}}{\text{d}r}\left(r\frac{\text{d}p}{\text{d}r}\right)\left[\frac{1+2\unicode[STIX]{x1D70E}}{1+\unicode[STIX]{x1D70E}}\frac{z}{\unicode[STIX]{x1D6FF}}-\left(\frac{z}{\unicode[STIX]{x1D6FF}}\right)^{2}\right]-\frac{1}{1+\unicode[STIX]{x1D70E}}\frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(r\langle v_{r}\rangle _{s}(r,\unicode[STIX]{x1D6FF})\right)\frac{z}{\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

Integration of (2.6) over $z$ from $0$ to $\unicode[STIX]{x1D6FF}$ with the boundary conditions in (2.3b ) and (2.4b ) gives

(2.7) $$\begin{eqnarray}\frac{1}{r}\frac{\text{d}}{\text{d}r}\left(r\frac{\text{d}p}{\text{d}r}\right)=12\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D6FF}^{3}}[\dot{\unicode[STIX]{x1D6FF}}+\langle v_{z}\rangle _{s}(r,\unicode[STIX]{x1D6FF})]+\frac{6}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D6FF}^{2}}\frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}(r\langle v_{r}\rangle _{s}(r,\unicode[STIX]{x1D6FF})),\end{eqnarray}$$

which is the Reynolds equation for the liquid film. We note that the stress from (2.6) is $(\unicode[STIX]{x1D6FF}/R)^{2}$ times smaller than the stress from (2.7).

The superficial average velocity $\langle \boldsymbol{v}\rangle _{s}$ in the porous disk is constrained by

(2.8) $$\begin{eqnarray}\frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}(r\langle v_{r}\rangle _{s})+\frac{\unicode[STIX]{x2202}\langle v_{z}\rangle _{s}}{\unicode[STIX]{x2202}z}=0.\end{eqnarray}$$

The superficial average streamfunction $\langle \unicode[STIX]{x1D713}\rangle _{s}$ is defined by the relations

(2.9a,b ) $$\begin{eqnarray}\langle v_{r}\rangle _{s}=-\frac{1}{r}\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D713}\rangle _{s}}{\unicode[STIX]{x2202}z},\quad \langle v_{z}\rangle _{s}=\frac{1}{r}\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D713}\rangle _{s}}{\unicode[STIX]{x2202}r}.\end{eqnarray}$$

We assume flow in the porous disk is governed by Darcy’s law, which takes the form

(2.10a,b ) $$\begin{eqnarray}\langle v_{r}\rangle _{s}=-\frac{\unicode[STIX]{x1D705}}{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}\langle p\rangle _{i}}{\unicode[STIX]{x2202}r},\quad \langle v_{z}\rangle _{s}=-\frac{\unicode[STIX]{x1D705}}{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}\langle p\rangle _{i}}{\unicode[STIX]{x2202}z},\end{eqnarray}$$

where $\langle p\rangle _{i}$ is the intrinsic average pressure within the porous disk (Whitaker Reference Whitaker1999). Combining (2.8) and (2.10), we have

(2.11) $$\begin{eqnarray}\frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(r\frac{\unicode[STIX]{x2202}\langle p\rangle _{i}}{\unicode[STIX]{x2202}r}\right)+\frac{\unicode[STIX]{x2202}^{2}\langle p\rangle _{i}}{\unicode[STIX]{x2202}z^{2}}=0,\end{eqnarray}$$

which is solved subject to $\unicode[STIX]{x2202}\langle p\rangle _{i}/\unicode[STIX]{x2202}r(0,z)=0$ and the boundary condition

(2.12) $$\begin{eqnarray}\langle p\rangle _{i}(r,\unicode[STIX]{x1D6FF})=p(r).\end{eqnarray}$$

Using (2.10) and (2.11), we can write (2.7) as

(2.13) $$\begin{eqnarray}\frac{1}{r}\frac{\text{d}}{\text{d}r}\left(r\frac{\text{d}p}{\text{d}r}\right)=12\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x1D6FF}^{3}}-12\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D705}}{\unicode[STIX]{x1D6FF}^{3}}\frac{\unicode[STIX]{x2202}\langle p\rangle _{i}}{\unicode[STIX]{x2202}z}(r,\unicode[STIX]{x1D6FF})+\frac{6}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D705}}{\unicode[STIX]{x1D6FF}^{2}}\frac{\unicode[STIX]{x2202}^{2}\langle p\rangle _{i}}{\unicode[STIX]{x2202}z^{2}}(r,\unicode[STIX]{x1D6FF}),\end{eqnarray}$$

which is solved subject to $\text{d}p/\text{d}r(0)=0$ and $p(R)=0$ . The coupled system of (2.11)–(2.13) applies to both configurations (i) and (ii). To solve (2.11) for $\langle p\rangle _{i}$ , additional boundary conditions must be prescribed.

Finally, the force exerted by the liquid film on the porous disk is obtained from

(2.14) $$\begin{eqnarray}{\mathcal{F}}=2\unicode[STIX]{x03C0}\int _{0}^{R}p(r)r\,\text{d}r,\end{eqnarray}$$

where, as noted previously, we are neglecting terms from (2.6) that are $(\unicode[STIX]{x1D6FF}/R)^{2}$ times smaller than those being retained.

2.2 Solution for the sealed-face configuration (i)

First, we consider the sealed-face configuration (i) shown in figure 1 for which the boundary conditions for (2.11) are

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\langle p\rangle _{i}}{\unicode[STIX]{x2202}z}(r,\unicode[STIX]{x1D6FF}+b)=0, & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle \langle p\rangle _{i}(R,z)=0. & \displaystyle\end{eqnarray}$$

The solution to (2.11) subject to (2.12), (2.15) and (2.16) can be written as

(2.17) $$\begin{eqnarray}\langle p\rangle _{i}=\mathop{\sum }_{n=1}^{\infty }B_{n}\frac{\text{J}_{0}(\unicode[STIX]{x1D6FD}_{n}r)\cosh [\unicode[STIX]{x1D6FD}_{n}(\unicode[STIX]{x1D6FF}+b-z)]}{\cosh (\unicode[STIX]{x1D6FD}_{n}b)},\end{eqnarray}$$

where $\text{J}_{0}(x)$ is the Bessel function of the first kind. The $\unicode[STIX]{x1D6FD}_{n}$ are found from the roots of $\text{J}_{0}(\unicode[STIX]{x1D6FD}_{n}R)=0,n=1,2,3,\ldots \,$ , and

(2.18) $$\begin{eqnarray}B_{n}=\frac{2}{R^{2}\text{J}_{1}^{2}(\unicode[STIX]{x1D6FD}_{n}R)}\int _{0}^{R}p(r)\text{J}_{0}(\unicode[STIX]{x1D6FD}_{n}r)r\,\text{d}r,\quad n=1,2,3,\ldots .\end{eqnarray}$$

Substituting (2.17) in (2.13), we obtain

(2.19) $$\begin{eqnarray}\frac{1}{r}\frac{\text{d}}{\text{d}r}\left(r\frac{\text{d}p}{\text{d}r}\right)=12\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x1D6FF}^{3}}+\frac{12}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D705}}{\unicode[STIX]{x1D6FF}^{3}}\mathop{\sum }_{n=1}^{\infty }B_{n}\left[(1+\unicode[STIX]{x1D70E})\tanh (\unicode[STIX]{x1D6FD}_{n}b)+\frac{1}{2}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FD}_{n}\right]\unicode[STIX]{x1D6FD}_{n}\text{J}_{0}(\unicode[STIX]{x1D6FD}_{n}r).\end{eqnarray}$$

Integrating (2.19) twice with respect to $r$ gives

(2.20) $$\begin{eqnarray}p=-3\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D707}\frac{R^{2}}{\unicode[STIX]{x1D6FF}^{3}}\dot{\unicode[STIX]{x1D6FF}}\!\left[1-\left(\frac{r}{R}\right)^{2}\right]-\frac{12}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D705}R}{\unicode[STIX]{x1D6FF}^{3}}\mathop{\sum }_{n=1}^{\infty }B_{n}\!\left[(1+\unicode[STIX]{x1D70E})\frac{\tanh (\unicode[STIX]{x1D6FD}_{n}b)}{\unicode[STIX]{x1D6FD}_{n}R}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\!\text{J}_{0}(\unicode[STIX]{x1D6FD}_{n}r).\end{eqnarray}$$

Substitution of (2.20) in (2.18) gives

(2.21) $$\begin{eqnarray}B_{n}=-24\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FF}}\frac{R^{2}}{\unicode[STIX]{x1D6FF}^{3}}\frac{1}{(\unicode[STIX]{x1D6FD}_{n}R)^{3}\text{J}_{1}(\unicode[STIX]{x1D6FD}_{n}R)}\left\{1+\frac{12}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D705}R}{\unicode[STIX]{x1D6FF}^{3}}\left[(1+\unicode[STIX]{x1D70E})\frac{\tanh (\unicode[STIX]{x1D6FD}_{n}b)}{\unicode[STIX]{x1D6FD}_{n}R}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}.\end{eqnarray}$$

Substituting (2.21) in (2.20) we obtain

(2.22) $$\begin{eqnarray}\frac{p\unicode[STIX]{x1D6FF}^{3}}{\unicode[STIX]{x1D707}R^{2}}=-3\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\dot{\unicode[STIX]{x1D6FF}}\mathop{\sum }_{n=1}^{\infty }\frac{8}{\bar{\unicode[STIX]{x1D6FD}}_{n}^{3}}\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\tanh (\bar{\unicode[STIX]{x1D6FD}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FD}}_{n}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}\frac{\text{J}_{0}(\bar{\unicode[STIX]{x1D6FD}}_{n}r/R)}{\text{J}_{1}(\bar{\unicode[STIX]{x1D6FD}}_{n})},\end{eqnarray}$$

where $\bar{\unicode[STIX]{x1D6FD}}_{n}=\unicode[STIX]{x1D6FD}_{n}R$ , $\unicode[STIX]{x1D700}=b/R$ and $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}$ (note that $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D6F9}/\unicode[STIX]{x1D700}$ ). Substituting (2.21) in (2.17) we obtain

(2.23) $$\begin{eqnarray}\displaystyle \frac{\langle p\rangle _{i}\unicode[STIX]{x1D6FF}^{3}}{\unicode[STIX]{x1D707}R^{2}} & = & \displaystyle -3\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\dot{\unicode[STIX]{x1D6FF}}\mathop{\sum }_{n=1}^{\infty }\frac{8}{\bar{\unicode[STIX]{x1D6FD}}_{n}^{3}}\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\tanh (\bar{\unicode[STIX]{x1D6FD}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FD}}_{n}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{\text{J}_{0}(\bar{\unicode[STIX]{x1D6FD}}_{n}r/R)\cosh \left(\bar{\unicode[STIX]{x1D6FD}}_{n}\unicode[STIX]{x1D700}[1-(z-\unicode[STIX]{x1D6FF})/b]\right)}{\text{J}_{1}(\bar{\unicode[STIX]{x1D6FD}}_{n})\cosh (\bar{\unicode[STIX]{x1D6FD}}_{n}\unicode[STIX]{x1D700})},\end{eqnarray}$$

and from (2.5) using (2.22) and (2.23), we find

(2.24) $$\begin{eqnarray}\displaystyle \frac{v_{r}\unicode[STIX]{x1D6FF}}{R} & = & \displaystyle -\frac{12}{1+4\unicode[STIX]{x1D70E}}\dot{\unicode[STIX]{x1D6FF}}\left[\left(1+2\unicode[STIX]{x1D70E}+2\frac{\unicode[STIX]{x1D6FF}}{R}\unicode[STIX]{x1D6F6}\right)\frac{z}{\unicode[STIX]{x1D6FF}}-(1+\unicode[STIX]{x1D70E})\left(\frac{z}{\unicode[STIX]{x1D6FF}}\right)^{2}\right]\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{n=1}^{\infty }\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\tanh (\bar{\unicode[STIX]{x1D6FD}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FD}}_{n}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}\frac{\text{J}_{1}(\bar{\unicode[STIX]{x1D6FD}}_{n}r/R)}{\bar{\unicode[STIX]{x1D6FD}}_{n}^{2}\text{J}_{1}(\bar{\unicode[STIX]{x1D6FD}}_{n})}.\end{eqnarray}$$

Using (2.23) with (2.10) and (2.9) we obtain

(2.25) $$\begin{eqnarray}\displaystyle \frac{\langle \unicode[STIX]{x1D713}\rangle _{s}}{R^{2}} & = & \displaystyle -3\dot{\unicode[STIX]{x1D6FF}}\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D6F6}\mathop{\sum }_{n=1}^{\infty }\frac{8}{\bar{\unicode[STIX]{x1D6FD}}_{n}^{3}}\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\tanh (\bar{\unicode[STIX]{x1D6FD}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FD}}_{n}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{(r/R)\text{J}_{1}(\bar{\unicode[STIX]{x1D6FD}}_{n}r/R)\sinh \left(\bar{\unicode[STIX]{x1D6FD}}_{n}\unicode[STIX]{x1D700}[1-(z-\unicode[STIX]{x1D6FF})/b]\right)}{\text{J}_{1}(\bar{\unicode[STIX]{x1D6FD}}_{n})\cosh (\bar{\unicode[STIX]{x1D6FD}}_{n}\unicode[STIX]{x1D700})}.\end{eqnarray}$$

Finally, substituting (2.22) in (2.14) we obtain

(2.26) $$\begin{eqnarray}\frac{2{\mathcal{F}}}{3\unicode[STIX]{x1D707}\unicode[STIX]{x03C0}R^{4}}=-\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\frac{\dot{\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x1D6FF}^{3}}\mathop{\sum }_{n=1}^{\infty }\frac{32}{\bar{\unicode[STIX]{x1D6FD}}_{n}^{4}}\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\tanh (\bar{\unicode[STIX]{x1D6FD}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FD}}_{n}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1},\end{eqnarray}$$

(note that $\sum _{n=1}^{\infty }32/\bar{\unicode[STIX]{x1D6FD}}_{n}^{4}=1$ ). For the no-slip case ( $\unicode[STIX]{x1D70E}=0$ ), the expressions in (2.22) and (2.26) simplify to those given by Murti (Reference Murti1974b ).

2.3 Solution for the sealed-edge configuration (ii)

For the sealed-edge configuration (ii) shown in figure 1, we have the boundary conditions

(2.27) $$\begin{eqnarray}\displaystyle & \displaystyle \langle p\rangle _{i}(r,\unicode[STIX]{x1D6FF}+b)=p_{0}, & \displaystyle\end{eqnarray}$$

(2.28) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\langle p\rangle _{i}}{\unicode[STIX]{x2202}r}(R,z)=0. & \displaystyle\end{eqnarray}$$

The solution of (2.11) subject to (2.12), (2.27) and (2.28) can be written as

(2.29) $$\begin{eqnarray}\langle p\rangle _{i}=p_{0}+(p_{0}-\bar{p})\left(\frac{z-\unicode[STIX]{x1D6FF}}{b}-1\right)+\mathop{\sum }_{n=1}^{\infty }A_{n}\frac{\text{J}_{0}(\unicode[STIX]{x1D6FC}_{n}r)\sinh [\unicode[STIX]{x1D6FC}_{n}(\unicode[STIX]{x1D6FF}+b-z)]}{\sinh [\unicode[STIX]{x1D6FC}_{n}b]},\end{eqnarray}$$

where

(2.30) $$\begin{eqnarray}\bar{p}=\frac{2}{R^{2}}\int _{0}^{R}p(r)r\,\text{d}r\end{eqnarray}$$

is the average pressure in the liquid film. The $\unicode[STIX]{x1D6FC}_{n}$ are found from the roots of $\text{J}_{1}(\unicode[STIX]{x1D6FC}_{n}R)=0,n=0,1,2,\ldots \,$ , which includes $\unicode[STIX]{x1D6FC}_{0}=0$ , and the $A_{n}$ are given by

(2.31) $$\begin{eqnarray}A_{n}=\frac{2}{R^{2}\text{J}_{0}^{2}(\unicode[STIX]{x1D6FC}_{n}R)}\int _{0}^{R}p(r)\text{J}_{0}(\unicode[STIX]{x1D6FC}_{n}r)r\,\text{d}r,\quad n=1,2,3,\ldots .\end{eqnarray}$$

The total mass flow rate ${\mathcal{W}}$ from the reservoir to the porous disk is given by

(2.32) $$\begin{eqnarray}\frac{{\mathcal{W}}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}}=-2\int _{0}^{R}\langle v_{z}\rangle _{s}(r,b+\unicode[STIX]{x1D6FF})r\,\text{d}r=\frac{\unicode[STIX]{x1D705}R^{2}}{\unicode[STIX]{x1D707}b}(p_{0}-\bar{p}),\end{eqnarray}$$

where the second equality is obtained using (2.10b ) and (2.29).

Substitution of (2.29) in (2.13) leads to

(2.33) $$\begin{eqnarray}\displaystyle \frac{1}{r}\frac{\text{d}}{\text{d}r}\left(r\frac{\text{d}p}{\text{d}r}\right) & = & \displaystyle 12\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D6FF}^{3}}\left[\dot{\unicode[STIX]{x1D6FF}}-\frac{\unicode[STIX]{x1D705}}{\unicode[STIX]{x1D707}b}(p_{0}-\bar{p})\right]\nonumber\\ \displaystyle & & \displaystyle -\,\frac{12}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D705}}{\unicode[STIX]{x1D6FF}^{3}}\mathop{\sum }_{n=1}^{\infty }A_{n}\left[(1+\unicode[STIX]{x1D70E})\coth (\unicode[STIX]{x1D6FC}_{n}b)+\frac{1}{2}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}_{n}\right]\unicode[STIX]{x1D6FC}_{n}\text{J}_{0}(\unicode[STIX]{x1D6FC}_{n}r).\qquad\end{eqnarray}$$

Integrating (2.33) twice with respect to $r$ gives

(2.34) $$\begin{eqnarray}\displaystyle p & = & \displaystyle -3\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D707}\frac{R^{2}}{\unicode[STIX]{x1D6FF}^{3}}\left[\dot{\unicode[STIX]{x1D6FF}}-\frac{\unicode[STIX]{x1D705}}{\unicode[STIX]{x1D707}b}(p_{0}-\bar{p})\right]\left[1-\left(\frac{r}{R}\right)^{2}\right]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{12}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D705}R}{\unicode[STIX]{x1D6FF}^{3}}\mathop{\sum }_{n=1}^{\infty }A_{n}\left[(1+\unicode[STIX]{x1D70E})\frac{\coth (\unicode[STIX]{x1D6FC}_{n}b)}{\unicode[STIX]{x1D6FC}_{n}R}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right][\text{J}_{0}(\unicode[STIX]{x1D6FC}_{n}R)-\text{J}_{0}(\unicode[STIX]{x1D6FC}_{n}r)].\qquad\end{eqnarray}$$

Substitution of (2.34) in (2.31) gives

(2.35) $$\begin{eqnarray}\displaystyle A_{n} & = & \displaystyle 12\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D707}\frac{R^{2}}{\unicode[STIX]{x1D6FF}^{3}}\left(\dot{\unicode[STIX]{x1D6FF}}-\frac{{\mathcal{W}}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}}\right)\frac{1}{(\unicode[STIX]{x1D6FC}_{n}R)^{2}\text{J}_{0}(\unicode[STIX]{x1D6FC}_{n}R)}\nonumber\\ \displaystyle & & \displaystyle \times \,\left\{1+\frac{12}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D705}R}{\unicode[STIX]{x1D6FF}^{3}}\left[(1+\unicode[STIX]{x1D70E})\frac{\coth (\unicode[STIX]{x1D6FC}_{n}b)}{\unicode[STIX]{x1D6FC}_{n}R}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1},\end{eqnarray}$$

where (2.32) has been used. Using (2.34) and (2.35), the pressure in the liquid film is given by

(2.36) $$\begin{eqnarray}\displaystyle \frac{p\unicode[STIX]{x1D6FF}^{3}}{\unicode[STIX]{x1D707}R^{2}} & = & \displaystyle -3\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\left(\dot{\unicode[STIX]{x1D6FF}}-\frac{{\mathcal{W}}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}}\right)\mathop{\sum }_{n=1}^{\infty }\frac{4}{\bar{\unicode[STIX]{x1D6FC}}_{n}^{2}}\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\coth (\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FC}}_{n}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}\nonumber\\ \displaystyle & & \displaystyle \times \,\left[1-\frac{\text{J}_{0}(\bar{\unicode[STIX]{x1D6FC}}_{n}r/R)}{\text{J}_{0}(\bar{\unicode[STIX]{x1D6FC}}_{n})}\right],\end{eqnarray}$$

where $\bar{\unicode[STIX]{x1D6FC}}_{n}=\unicode[STIX]{x1D6FC}_{n}R$ . Combining (2.30), (2.32) and (2.36), $p_{0}$ is given by

(2.37) $$\begin{eqnarray}\displaystyle \frac{p_{0}\unicode[STIX]{x1D6FF}^{3}}{\unicode[STIX]{x1D707}R^{2}} & = & \displaystyle \frac{{\mathcal{W}}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}}\frac{\unicode[STIX]{x1D700}}{\unicode[STIX]{x1D6F6}}-\frac{3}{2}\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\left(\dot{\unicode[STIX]{x1D6FF}}-\frac{{\mathcal{W}}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}}\right)\mathop{\sum }_{n=1}^{\infty }\frac{8}{\bar{\unicode[STIX]{x1D6FC}}_{n}^{2}}\nonumber\\ \displaystyle & & \displaystyle \times \,\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\coth (\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FC}}_{n}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}.\end{eqnarray}$$

To obtain $\langle p\rangle _{i}$ , (2.32), (2.35) and (2.37) are substituted in (2.29), which gives

(2.38) $$\begin{eqnarray}\displaystyle \frac{\langle p\rangle _{i}\unicode[STIX]{x1D6FF}^{3}}{\unicode[STIX]{x1D707}R^{2}} & = & \displaystyle \frac{{\mathcal{W}}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}}\frac{\unicode[STIX]{x1D700}}{\unicode[STIX]{x1D6F6}}\frac{z-\unicode[STIX]{x1D6FF}}{b}-\frac{3}{2}\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\left(\dot{\unicode[STIX]{x1D6FF}}-\frac{{\mathcal{W}}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{n=1}^{\infty }\frac{8}{\bar{\unicode[STIX]{x1D6FC}}_{n}^{2}}\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\coth (\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FC}}_{n}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}\nonumber\\ \displaystyle & & \displaystyle \times \,\left[1-\frac{\text{J}_{0}(\bar{\unicode[STIX]{x1D6FC}}_{n}r/R)\sinh \left(\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700}[1-(z-\unicode[STIX]{x1D6FF})/b]\right)}{\text{J}_{0}(\bar{\unicode[STIX]{x1D6FC}}_{n})\sinh (\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700})}\right].\end{eqnarray}$$

From (2.5) using (2.36) and (2.38), we find

(2.39) $$\begin{eqnarray}\displaystyle \frac{v_{r}\unicode[STIX]{x1D6FF}}{R} & = & \displaystyle \frac{6}{1+4\unicode[STIX]{x1D70E}}\left(\dot{\unicode[STIX]{x1D6FF}}-\frac{{\mathcal{W}}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}}\right)\left[\left(1+2\unicode[STIX]{x1D70E}+2\frac{\unicode[STIX]{x1D6FF}}{R}\unicode[STIX]{x1D6F6}\right)\frac{z}{\unicode[STIX]{x1D6FF}}-(1+\unicode[STIX]{x1D70E})\left(\frac{z}{\unicode[STIX]{x1D6FF}}\right)^{2}\right]\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{n=1}^{\infty }\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\coth (\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FC}}_{n}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}\frac{\text{J}_{1}(\bar{\unicode[STIX]{x1D6FC}}_{n}r/R)}{\bar{\unicode[STIX]{x1D6FC}}_{n}\text{J}_{0}(\bar{\unicode[STIX]{x1D6FC}}_{n})}.\end{eqnarray}$$

Using (2.38) with (2.10) and (2.9) we obtain

(2.40) $$\begin{eqnarray}\displaystyle \frac{\langle \unicode[STIX]{x1D713}\rangle _{s}}{R^{2}} & = & \displaystyle -\frac{1}{2}\frac{{\mathcal{W}}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}}\left(\frac{r}{R}\right)^{2}+\frac{3}{2}\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D6F6}\left(\dot{\unicode[STIX]{x1D6FF}}-\frac{{\mathcal{W}}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{n=1}^{\infty }\frac{8}{\bar{\unicode[STIX]{x1D6FC}}_{n}^{2}}\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\tanh (\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FC}}_{n}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{(r/R)\text{J}_{1}(\bar{\unicode[STIX]{x1D6FC}}_{n}r/R)\cosh \left(\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700}[1-(z-\unicode[STIX]{x1D6FF})/b]\right)}{\text{J}_{0}(\bar{\unicode[STIX]{x1D6FC}}_{n})\sinh (\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700})}.\end{eqnarray}$$

Substitution of (2.36) in (2.14) gives

(2.41) $$\begin{eqnarray}\displaystyle \frac{2{\mathcal{F}}}{3\unicode[STIX]{x1D707}\unicode[STIX]{x03C0}R^{4}} & = & \displaystyle -\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\frac{1}{\unicode[STIX]{x1D6FF}^{3}}\left(\dot{\unicode[STIX]{x1D6FF}}-\frac{{\mathcal{W}}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}}\right)\mathop{\sum }_{n=1}^{\infty }\frac{8}{\bar{\unicode[STIX]{x1D6FC}}_{n}^{2}}\nonumber\\ \displaystyle & & \displaystyle \times \,\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\coth (\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FC}}_{n}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}\end{eqnarray}$$

(note that $\sum _{n=1}^{\infty }8/\bar{\unicode[STIX]{x1D6FC}}_{n}^{2}=1$ ).

We consider two cases for configuration (ii). One is the externally pressurized bearing case where the mass flow rate from the reservoir to the porous disk ${\mathcal{W}}$ is prescribed. The other case is when the reservoir pressure is equal to the ambient pressure ( $p_{0}=0$ ) so that, similar to configuration (i), liquid flows from the porous disk at a permeable external surface. In this case, from (2.37), ${\mathcal{W}}$ is found to be

(2.42) $$\begin{eqnarray}\displaystyle \frac{{\mathcal{W}}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}} & = & \displaystyle \dot{\unicode[STIX]{x1D6FF}}\left[1-\left(1+\frac{3}{2}\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D6F6}}{\unicode[STIX]{x1D700}}\mathop{\sum }_{n=1}^{\infty }\frac{8}{\bar{\unicode[STIX]{x1D6FC}}_{n}^{2}}\right.\right.\nonumber\\ \displaystyle & & \displaystyle \times \left.\left.\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\coth (\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FC}_{n}}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}\right)^{-1}\right].\end{eqnarray}$$

Note that for squeezing flow $\dot{\unicode[STIX]{x1D6FF}}<0$ we have ${\mathcal{W}}<0$ , i.e. liquid flows from the porous disk to the reservoir [ $(..)^{-1}=1-{\mathcal{W}}/\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}\dot{\unicode[STIX]{x1D6FF}}$ ]. Substitution of (2.42) in (2.41) gives

(2.43) $$\begin{eqnarray}\displaystyle \frac{2{\mathcal{F}}}{3\unicode[STIX]{x1D707}\unicode[STIX]{x03C0}R^{4}} & = & \displaystyle -\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\frac{\dot{\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x1D6FF}^{3}}\nonumber\\ \displaystyle & & \displaystyle \times \left(1+\frac{3}{2}\frac{1+\unicode[STIX]{x1D70E}}{1+4\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x1D6F6}}{\unicode[STIX]{x1D700}}\mathop{\sum }_{n=1}^{\infty }\frac{8}{\bar{\unicode[STIX]{x1D6FC}}_{n}^{2}}\!\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\!\left[(1+\unicode[STIX]{x1D70E})\frac{\coth (\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FC}_{n}}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\!\right\}^{-1}\right)^{-1}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{n=1}^{\infty }\frac{8}{\bar{\unicode[STIX]{x1D6FC}}_{n}^{2}}\left\{1+\frac{12\unicode[STIX]{x1D6F6}}{1+4\unicode[STIX]{x1D70E}}\left[(1+\unicode[STIX]{x1D70E})\frac{\coth (\bar{\unicode[STIX]{x1D6FC}}_{n}\unicode[STIX]{x1D700})}{\bar{\unicode[STIX]{x1D6FC}}_{n}}+\frac{1}{2}\frac{\unicode[STIX]{x1D6FF}}{R}\right]\right\}^{-1}.\end{eqnarray}$$

3.1 Velocity and pressure fields

First, we consider the sealed-face configuration (i). Figure 2 shows the pressure and radial velocity fields in the liquid film for constant squeeze velocity $-\dot{\unicode[STIX]{x1D6FF}}$ . For the impermeable disk case ( $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D6F6}=0$ ), the expression in (2.22) gives $p\propto [1-(r/R)^{2}]$ and the expression in (2.24) gives $v_{r}\propto (r/R)[(z/\unicode[STIX]{x1D6FF})-(z/\unicode[STIX]{x1D6FF})^{2}]$ . From this figure we see that the pressure profile flattens, which, in turn, leads to a reduction in the radial velocity, as the permeability of the disk $\unicode[STIX]{x1D6F6}$ increases. Pressure and streamfunction contours within the porous disk are presented in figure 3, which shows that liquid flows into the porous disk axially and then, because of the impermeable upper surface, flows in the radial direction.

Figure 2.

Pressure (a) from (2.22) and radial velocity at $r/R=0.5$ (b) from (2.24) in liquid film for configuration (i) with constant squeezing velocity $-\dot{\unicode[STIX]{x1D6FF}}$ for different values of permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}=0.01,0.1,1.0$ (solid lines top to bottom) with $\unicode[STIX]{x1D700}=0.1$ , $\unicode[STIX]{x1D6FF}/R=0.01$ and $\unicode[STIX]{x1D70E}=0$ . The dashed line is for the impermeable disk.

Figure 3.

Pressure (a) $-\langle p\rangle _{i}\unicode[STIX]{x1D6FF}^{3}/\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FF}}R^{2}$ from (2.23) and streamfunction (b) $-\langle \unicode[STIX]{x1D713}\rangle _{s}/\dot{\unicode[STIX]{x1D6FF}}R^{2}$ from (2.25) contours in a porous disk for configuration (i) with constant squeezing velocity $-\dot{\unicode[STIX]{x1D6FF}}$ for permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}=0.1$ with $\unicode[STIX]{x1D700}=0.1$ , $\unicode[STIX]{x1D6FF}/R=0.01$ and $\unicode[STIX]{x1D70E}=0$ .

We now examine the sealed-edge configuration (ii) for the case where $p_{0}=0$ so that liquid flows from the porous disk to the reservoir. Again, for the impermeable disk case ( $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D6F6}=0$ ), the expression in (2.36) simplifies to $p\propto [1-(r/R)^{2}]$ and the expression in (2.39) simplifies to $v_{r}\propto (r/R)[(z/\unicode[STIX]{x1D6FF})-(z/\unicode[STIX]{x1D6FF})^{2}]$ . Figure 4 shows the pressure and radial velocity fields in the liquid film for constant squeeze velocity $-\dot{\unicode[STIX]{x1D6FF}}$ . As with configuration (i), the pressure profile flattens and the radial velocity decreases as $\unicode[STIX]{x1D6F6}$ increases; however, the dependence on $\unicode[STIX]{x1D6F6}$ is stronger for configuration (ii) compared to configuration (i). For large permeability ( $\unicode[STIX]{x1D6F6}=1.0$ ), the liquid-film pressure is nearly uniform and the radial velocity approaches zero.

Figure 4.

Pressure (a) from (2.36) and radial velocity at $r/R=0.5$ (b) from (2.39) for configuration (ii) with constant squeezing velocity $-\dot{\unicode[STIX]{x1D6FF}}$ for different values of permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}=0.01$ , 0.1, 1.0 (solid lines top to bottom) with $\unicode[STIX]{x1D700}=0.1$ , $\unicode[STIX]{x1D6FF}/R=0.01$ and $\unicode[STIX]{x1D70E}=0$ . The dashed line is for the impermeable disk.

As noted earlier, approximate solutions for configuration (ii) have been reported (Murti Reference Murti1974a ; Verma Reference Verma1980) by assuming $\langle p\rangle _{i}$ is a linear function of $z$ . In this case, the pressure within the porous disk is found using the boundary conditions in (2.12) and (2.27) and is given by $\langle p\rangle _{i}=p_{0}+(p_{0}-p)[(z-\unicode[STIX]{x1D6FF})/b-1]$ . We note, however, that this solution is not consistent with (2.8), (2.10a ) and (2.10b ). Nevertheless, using this assumption (and setting $p_{0}=0$ , $\unicode[STIX]{x1D70E}=0$ ) the solution of (2.13) can be written as

(3.1) $$\begin{eqnarray}\frac{p\unicode[STIX]{x1D6FF}^{3}}{\unicode[STIX]{x1D707}R^{2}}=-\dot{\unicode[STIX]{x1D6FF}}\frac{\unicode[STIX]{x1D700}}{\unicode[STIX]{x1D6F6}}\left[1-\frac{\text{I}_{0}(\sqrt{12\unicode[STIX]{x1D6F6}/\unicode[STIX]{x1D700}}r/R)}{\text{I}_{0}(\sqrt{12\unicode[STIX]{x1D6F6}/\unicode[STIX]{x1D700}})}\right],\end{eqnarray}$$

where $\text{I}_{0}(x)$ is the modified Bessel function of the first kind. This expression is in good agreement with (2.36) for $\unicode[STIX]{x1D700}\ll 1$ , but over-predicts the liquid-film pressure for larger $\unicode[STIX]{x1D700}$ .

Pressure and streamfunction contours within the porous disk for configuration (ii) are presented in figure 5. From this figure we see that flow in the porous disk is mainly in the axial direction except near the outer edge of the disk. The detailed view given in figure 6 shows the flow splits where a portion of the liquid (for $r/R\gtrsim 0.93$ ) entering the porous disk from the film flows back into the film, that is $\langle v_{z}\rangle _{s}(r,\unicode[STIX]{x1D6FF})<0$ . This occurs because liquid flow in the radial direction encounters the impermeable surface at $r=R$ , which results in a stagnation point. Hence, the flow resistance along a path back to liquid film is smaller than along a path through the porous disk to the reservoir. Pressure and streamfunction contours for a thick porous disk ( $\unicode[STIX]{x1D700}=1$ ) are presented in figure 5. Here, we see a rather pronounced flow splitting where fluid entering the disk for $r/R\gtrsim 0.55$ returns to the liquid film. The approximate solution where $\langle p\rangle _{i}$ is assumed to be a linear function of $z$ does not capture the more complicated pressure contours given by the exact solution, nor will it predict the flow splitting that is shown in figures 6 and 7. From (2.10b ) and (2.23), it is easy to show that $\langle v_{z}\rangle _{s}(r,\unicode[STIX]{x1D6FF})>0$ for $0\leqslant r\leqslant R$ , which means that for configuration (i), fluid within the porous disk never flows back into the liquid film. For configuration (ii), however, flow splitting becomes more pronounced with decreasing $\unicode[STIX]{x1D6F6}$ and vanishes only in the limit $\unicode[STIX]{x1D700}\rightarrow 0$ .

Figure 5.

Pressure (a) $-\langle p\rangle _{i}\unicode[STIX]{x1D6FF}^{3}/\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FF}}R^{2}$ from (2.38) and streamfunction (b) $-\langle \unicode[STIX]{x1D713}\rangle _{s}/\dot{\unicode[STIX]{x1D6FF}}R^{2}$ from (2.40) contours in a porous disk for configuration (ii) with constant squeezing velocity $-\dot{\unicode[STIX]{x1D6FF}}$ for permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}=0.1$ with $\unicode[STIX]{x1D700}=0.1$ , $\unicode[STIX]{x1D6FF}/R=0.01$ and $\unicode[STIX]{x1D70E}=0$ .

Figure 6.

Detailed view of pressure (a) $-\langle p\rangle _{i}\unicode[STIX]{x1D6FF}^{3}/\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FF}}R^{2}$ from (2.38) and streamfunction (b) $-\langle \unicode[STIX]{x1D713}\rangle _{s}/\dot{\unicode[STIX]{x1D6FF}}R^{2}$ from (2.40) contours in a porous disk for configuration (ii) with constant squeezing velocity $-\dot{\unicode[STIX]{x1D6FF}}$ for permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}=0.1$ with $\unicode[STIX]{x1D700}=0.1$ , $\unicode[STIX]{x1D6FF}/R=0.01$ and $\unicode[STIX]{x1D70E}=0$ .

Figure 7.

Pressure (a) $-\langle p\rangle _{i}\unicode[STIX]{x1D6FF}^{3}/\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FF}}R^{2}$ from (2.38) and streamfunction (b) $-\langle \unicode[STIX]{x1D713}\rangle _{s}/\dot{\unicode[STIX]{x1D6FF}}R^{2}$ from (2.40) contours in a porous disk for configuration (ii) with constant squeezing velocity $-\dot{\unicode[STIX]{x1D6FF}}$ for permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}=0.1$ with $\unicode[STIX]{x1D700}=1$ , $\unicode[STIX]{x1D6FF}/R=0.01$ and $\unicode[STIX]{x1D70E}=0$ .

It is of interest to examine the rate at which the liquid flows into the porous disk relative to the rate at which it flows from the liquid film at $r=R$ . To examine this quantitatively, we consider the fraction of liquid from the film that flows into the porous disk for constant squeeze velocity $-\dot{\unicode[STIX]{x1D6FF}}$ , which is defined as

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}=-\frac{1}{\dot{\unicode[STIX]{x1D6FF}}}\frac{2}{R^{2}}\int _{0}^{R}\langle v_{z}\rangle _{s}(r,\unicode[STIX]{x1D6FF})r\,\text{d}r=\frac{\unicode[STIX]{x1D705}}{\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FF}}}\frac{2}{R^{2}}\int _{0}^{R}\frac{\unicode[STIX]{x2202}\langle p\rangle _{i}}{\unicode[STIX]{x2202}z}(r,\unicode[STIX]{x1D6FF})r\,\text{d}r,\end{eqnarray}$$

where the second equality follows using (2.10b ). The fraction of liquid that flows, or is squeezed, out of the film at $r=R$ is $1-\unicode[STIX]{x1D6F7}$ ; for the case of an impermeable disk, $\unicode[STIX]{x1D6F7}=0$ . The dependence of $\unicode[STIX]{x1D6F7}$ on $\unicode[STIX]{x1D6F6}$ for configurations (i) and (ii) is shown in figure 8. For constant squeeze velocity $-\dot{\unicode[STIX]{x1D6FF}}$ , since the film thickness $\unicode[STIX]{x1D6FF}$ decreases ( $\unicode[STIX]{x1D6F6}$ increases), $\unicode[STIX]{x1D6F7}$ increases during squeezing. In figure 8 we see for a thin porous disk ( $\unicode[STIX]{x1D700}=0.1$ ) that $\unicode[STIX]{x1D6F7}$ is larger for configuration (ii) compared with configuration (i), and the opposite is true for a thick porous disk ( $\unicode[STIX]{x1D700}=1.0$ ). This is explained by the fact that increasing the disk thickness for configuration (ii) leads to a larger resistance for flow to the reservoir, while for configuration (i) an increase in $\unicode[STIX]{x1D700}$ lowers the resistance for flow out of the disk at $r=R$ . Figure 8 also shows that the dependence of $\unicode[STIX]{x1D6F7}$ on disk thickness $\unicode[STIX]{x1D700}$ is stronger for configuration (ii) compared to (i). As we have just seen (see figures 6 and 7), the degree of flow separation and flow from the porous disk back into the film ( $\langle v_{z}\rangle _{s}(r,\unicode[STIX]{x1D6FF})<0$ ) for configuration (ii) increases with increasing $\unicode[STIX]{x1D700}$ .

Figure 8.

Fraction of liquid from the film that flows into the porous disk as defined in (3.2) for configurations (i) and (ii) with constant squeezing velocity $-\dot{\unicode[STIX]{x1D6FF}}$ as a function of permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}$ for $\unicode[STIX]{x1D70E}=0$ and $\unicode[STIX]{x1D6FF}/R=0.01$ .

As noted in the previous section, the sealed-edge configuration (ii) includes the case of an externally pressurized bearing where the mass flow rate ${\mathcal{W}}$ from the reservoir to the porous disk is prescribed. Figure 9 shows the pressure and radial velocity fields in the liquid film for the case of a constant film thickness ( $\dot{\unicode[STIX]{x1D6FF}}=0$ ). From this figure we again see that the pressure profile flattens and radial velocity decreases as $\unicode[STIX]{x1D6F6}$ increases (see figure 4). Pressure and streamfunction contours within the porous disk for this case are presented in figure 10, which shows that the pressure field varies primarily in the axial direction leading to a flow in the axial direction. The modification of the lubricated squeezing flow technique for generating equibiaxial deformations in polymer melts (Venerus et al. Reference Venerus, Shiu, Kashyap and Hosttetler2010) is based on this type of flow where the liquid (lubricant) film thickness is constant.

Figure 9.

Pressure (a) from (2.36) and radial velocity at $r/R=0.5$ (b) from (2.39) for configuration (ii) with constant ${\mathcal{W}}$ and $\dot{\unicode[STIX]{x1D6FF}}=0$ for permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}=0.01$ , 0.1, 1.0 (solid lines top to bottom) with $\unicode[STIX]{x1D700}=0.1$ , $\unicode[STIX]{x1D6FF}/R=0.01$ and $\unicode[STIX]{x1D70E}=0$ .

Figure 10.

Pressure (a) $\langle p\rangle _{i}\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}^{3}/\unicode[STIX]{x1D707}{\mathcal{W}}$ from (2.38) and streamfunction (b) $\langle \unicode[STIX]{x1D713}\rangle _{s}\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}/{\mathcal{W}}$ from (2.40) contours in a porous disk for configuration (ii) with constant ${\mathcal{W}}$ and $\dot{\unicode[STIX]{x1D6FF}}=0$ for permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}=0.1$ with $\unicode[STIX]{x1D700}=0.1$ , $\unicode[STIX]{x1D6FF}/R=0.01$ and $\unicode[STIX]{x1D70E}=0$ .

3.2 Load-bearing behaviour

It is of great interest to understand the load-bearing behaviour of porous-walled thrust bearings and its dependence on the properties of the porous disk. The force exerted on the permeable disk ${\mathcal{F}}$ is due to the pressure field that develops in the liquid film, which is required to drive flow in the radial direction. As a reference, we consider the impermeable disk case ( $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D6F6}=0$ ). For this case, expressions for ${\mathcal{F}}$ for configurations (i) and (ii) given in (2.26) and (2.43), respectively, simplify to

(3.3) $$\begin{eqnarray}\frac{2{\mathcal{F}}}{3\unicode[STIX]{x1D707}\unicode[STIX]{x03C0}R^{4}}=-\frac{\dot{\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x1D6FF}^{3}}.\end{eqnarray}$$

For constant squeezing velocity $-\dot{\unicode[STIX]{x1D6FF}}$ , we see that ${\mathcal{F}}\rightarrow \infty$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ .

Figure 11 shows the dependence of force on the porous disk ${\mathcal{F}}$ on the permeability parameter $\unicode[STIX]{x1D6F6}$ for the sealed-face configuration (i). According to (3.3), we see that ${\mathcal{F}}$ is reduced relative to the impermeable disk case. The load-bearing capacity is further reduced by slip as shown by the dependence of ${\mathcal{F}}$ on $\unicode[STIX]{x1D70E}$ . Results for the sealed-edge configuration (ii) with $p_{0}=0$ are presented in figure 12, where it is evident that ${\mathcal{F}}$ is also reduced relative to the impermeable disk case and is further reduced by slip. Comparison of figures 11 and 12 shows that, for a thin porous disk $\unicode[STIX]{x1D700}=0.1$ , the load-bearing capacity of configuration (ii) is smaller than configuration (i). For example, for $\unicode[STIX]{x1D6F6}=1$ , the load-bearing capacity of configuration (ii) is roughly a factor of 10 smaller than configuration (i). The dependence of ${\mathcal{F}}$ on $\unicode[STIX]{x1D6F6}$ in these figures can be understood when one considers the results in figure 8, which show that increasing $\unicode[STIX]{x1D6F6}$ leads to an increase in $\unicode[STIX]{x1D6F7}$ , thereby reducing the radial velocity (and pressure) in the liquid film.

Figure 11.

Force on porous disk from (2.26) for configuration (i) with constant squeezing velocity $-\dot{\unicode[STIX]{x1D6FF}}$ as a function of permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}$ for different values of slip parameter $\unicode[STIX]{x1D70E}=\sqrt{\unicode[STIX]{x1D705}}/\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}=0$ , 0.01, 0.1, 1.0 (solid lines top to bottom) with $\unicode[STIX]{x1D700}=0.1$ and $\unicode[STIX]{x1D6FF}/R=0.01$ .

Figure 12.

Force on porous disk from (2.43) for configuration (ii) with constant squeezing velocity $-\dot{\unicode[STIX]{x1D6FF}}$ as a function of permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}$ for different values of slip parameter $\unicode[STIX]{x1D70E}=\sqrt{\unicode[STIX]{x1D705}}/\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}=0$ , 0.01, 0.1, 1.0 (solid lines top to bottom) with $\unicode[STIX]{x1D700}=0.1$ and $\unicode[STIX]{x1D6FF}/R=0.01$ .

The load-bearing behaviours of configurations (i) and (ii) for a thick porous disk ( $\unicode[STIX]{x1D700}=1$ ) are presented in figure 13. A comparison of these results with those in figures 11 and 12 indicates that increasing $\unicode[STIX]{x1D700}$ leads to a reduction in the load-bearing capacity for configuration (i) and an increase in the load-bearing capacity for configuration (ii). The latter is explained by the fact that for configuration (ii) the flow is primarily in the axial direction and an increase in porous disk thickness requires a larger pressure in the liquid film to drive the flow. For configuration (i), an increase in disk thickness moves the impermeable upper surface away from the liquid film, thereby decreasing the pressure required to drive flow through the porous disk at permeable external surfaces. The load-bearing capacities of configurations (i) and (ii) coincide for $\unicode[STIX]{x1D700}\approx 0.6$ .

An expression for ${\mathcal{F}}$ using the approximate solution described above is obtained by substitution of (3.1) in (2.14) leading to

(3.4) $$\begin{eqnarray}\frac{2{\mathcal{F}}}{3\unicode[STIX]{x1D707}\unicode[STIX]{x03C0}R^{4}}=-\frac{\dot{\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x1D6FF}^{3}}\frac{2\unicode[STIX]{x1D700}}{3\unicode[STIX]{x1D6F6}}\left[1-\frac{2\text{I}_{1}(\sqrt{12\unicode[STIX]{x1D6F6}/\unicode[STIX]{x1D700}})}{\sqrt{12\unicode[STIX]{x1D6F6}/\unicode[STIX]{x1D700}}\text{I}_{0}(\sqrt{12\unicode[STIX]{x1D6F6}/\unicode[STIX]{x1D700}})}\right],\end{eqnarray}$$

which is equivalent to the expression given by Verma (Reference Verma1980). For thin porous disks $\unicode[STIX]{x1D700}\ll 1$ , predictions for ${\mathcal{F}}$ from (3.4) are consistent with those from the exact solution in (2.43). However, as shown in figure 13 for $\unicode[STIX]{x1D700}=1$ , the expression in (3.4) from the approximate model significantly over-predicts  ${\mathcal{F}}$ .

Figure 13.

Force on porous disk from (2.26) for configuration (i) and from (2.43) for configuration (ii) with constant squeezing velocity $-\dot{\unicode[STIX]{x1D6FF}}$ as a function of permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}$ for $\unicode[STIX]{x1D70E}=0$ with $\unicode[STIX]{x1D700}=1.0$ and $\unicode[STIX]{x1D6FF}/R=0.01$ . The dashed curve is the approximate solution for configuration (ii) given in (3.4).

We now examine the dynamics of liquid-film thinning for constant applied force ${\mathcal{F}}$ . Again, we use the impermeable case as a reference. Integration of (3.3) with $\unicode[STIX]{x1D6FF}(0)=\unicode[STIX]{x1D6FF}_{0}$ gives

(3.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D6FF}_{0}}=\frac{1}{\sqrt{1+4{\mathcal{F}}\unicode[STIX]{x1D6FF}_{0}^{2}t/3\unicode[STIX]{x1D707}\unicode[STIX]{x03C0}R^{4}}},\end{eqnarray}$$

which shows that $\unicode[STIX]{x1D6FF}\rightarrow 0$ only for $t\rightarrow \infty$ . Figure 14 shows the liquid-film thickness as a function of time for configurations (i) and (ii) without a slip velocity ( $\unicode[STIX]{x1D70E}=0$ ) for different values of the permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}_{0}^{3}$ . These results were obtained by a straightforward numerical integration of (2.26) and (2.43) with $\unicode[STIX]{x1D6FF}(0)=\unicode[STIX]{x1D6FF}_{0}$ . As expected the rate of liquid-film drainage is increased relative to the impermeable case, and, for $\unicode[STIX]{x1D700}=0.1$ , the rate of drainage is larger for configuration (ii) compared to configuration (i). Including a slip velocity ( $\unicode[STIX]{x1D70E}\neq 0$ ) decreases ${\mathcal{F}}$ (see figures 11 and 12) and would further increase the rate of liquid film drainage. From figure 14 we see that complete drainage of the liquid film occurs at a finite time, which we denote as $t_{f}$ : $\unicode[STIX]{x1D6FF}(t_{f})=0$ . The dependence of $t_{f}$ on the permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}_{0}^{3}$ for configurations (i) and (ii) for $\unicode[STIX]{x1D700}=0.1,1$ is shown in figure 15. As expected, $t_{f}$ decreases with increasing permeability $\unicode[STIX]{x1D6F6}$ , which was previously reported for configuration (i) (Murti Reference Murti1974b ; Knox et al. Reference Knox, Wilson, Duffy and McKee2015). For configuration (i) we see that increasing the thickness of the porous disk $\unicode[STIX]{x1D700}$ decreases the time for complete film drainage, while for configuration (ii) increasing $\unicode[STIX]{x1D700}$ leads to an increase in $t_{f}$ . Also shown in figure 15 is $t_{f}$ from the approximate solution for configuration (ii) determined by numerical integration of (3.4), which over-predicts $t_{f}$ for the thicker porous disk $\unicode[STIX]{x1D700}=1$ .

Figure 14.

Liquid-film thickness as a function of time from (2.26) for configuration (i) and from (2.43) for configuration (ii) with constant squeezing force ${\mathcal{F}}$ for different values of permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}_{0}^{3}=0.01$ , 0.1, 1.0 (solid lines top to bottom) with $\unicode[STIX]{x1D700}=0.1$ , $\unicode[STIX]{x1D6FF}_{0}/R=0.01$ and $\unicode[STIX]{x1D70E}=0$ . The dashed line is for the impermeable disk given by (3.5).

Figure 15.

Time for complete drainage of liquid-film thickness $\unicode[STIX]{x1D6FF}(t_{f})=0$ for configurations (i) and (ii) with constant squeezing force ${\mathcal{F}}$ as a function of permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}_{0}^{3}$ with $\unicode[STIX]{x1D700}=0.1,1.0$ , $\unicode[STIX]{x1D6FF}_{0}/R=0.01$ and $\unicode[STIX]{x1D70E}=0$ . The dot-dashed line is for the approximate solution from (3.4).

Finally, we consider the load-bearing characteristics of configuration (ii) when the liquid mass flow rate ${\mathcal{W}}$ from the reservoir to the porous disk is prescribed, or an externally pressurized bearing. This allows for a thrust bearing where the liquid film thickness can be controlled. Figure 16 shows the dependence of the load-bearing capacity on the permeability parameter $\unicode[STIX]{x1D6F6}$ for constant ${\mathcal{W}}$ with constant liquid-film thickness ( $\dot{\unicode[STIX]{x1D6FF}}=0$ ). From this figure we see that ${\mathcal{F}}$ decreases with increasing disk permeability $\unicode[STIX]{x1D6F6}$ and that a slip velocity further reduces load-bearing capacity. While ${\mathcal{F}}$ increases with decreasing $\unicode[STIX]{x1D6F6}$ , the reservoir pressure $p_{0}$ , which can be computed from (2.37), increases.

Figure 16.

Force on porous disk from (2.41) for configuration (ii) with constant ${\mathcal{W}}$ and $\dot{\unicode[STIX]{x1D6FF}}=0$ as a function of permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}^{3}$ for different values of slip parameter $\unicode[STIX]{x1D70E}=\sqrt{\unicode[STIX]{x1D705}}/\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}=0$ , 0.01, 0.1, 1.0 (solid lines top to bottom) with $\unicode[STIX]{x1D700}=0.1$ and $\unicode[STIX]{x1D6FF}/R=0.01$ .

When operated with a prescribed liquid mass flow rate, the thrust bearing can also be used to lift an applied load. To demonstrate this, (2.41) is numerically integrated with the initial condition $\unicode[STIX]{x1D6FF}(0)=0$ ; the constant applied force ${\mathcal{F}}$ is set by the expression in (2.41) at steady state ( $\dot{\unicode[STIX]{x1D6FF}}=0$ ) and depends on the final liquid-film thickness $\unicode[STIX]{x1D6FF}(t\rightarrow \infty )=\unicode[STIX]{x1D6FF}_{\infty }$ . As shown in figure 17, prescribing a constant ${\mathcal{W}}$ causes the liquid-film thickness to increase, lifting the load, to the final liquid-film thickness $\unicode[STIX]{x1D6FF}_{\infty }$ . Alternatively, one could examine the time evolution of the film thickness if the reservoir pressure $p_{0}$ were prescribed, and ${\mathcal{W}}$ would be determined by (2.37).

Figure 17.

Liquid-film thickness as a function of time by integration of (2.41) for configuration (ii) with constant ${\mathcal{W}}$ for different values of permeability parameter $\unicode[STIX]{x1D6F6}=\unicode[STIX]{x1D705}R/\unicode[STIX]{x1D6FF}_{\infty }^{3}=0.001$ , 0.01, 0.1 (solid lines top to bottom) with $\unicode[STIX]{x1D700}=0.1$ , $\unicode[STIX]{x1D6FF}_{\infty }/R=0.01$ and $\unicode[STIX]{x1D70E}=0$ . The squeezing force ${\mathcal{F}}$ is determined from figure 17 with $\unicode[STIX]{x1D6FF}\rightarrow \unicode[STIX]{x1D6FF}_{\infty }$ .