Based on the postulates \vec{\rm v} = [0; v = v(r, z, t); w = \rm \dot h/2] and ∇p = dp / dr , we reduce the continuity equation to:
\rm \frac{1}{r} \frac{\partial(rv)}{\partial r} +\frac{\partial w}{\partial z} =0\qquad\qquad\qquad(E.4.9.1)
and the equation of motion to:
\rm\frac{\partial}{\partial r} p(r,t)=\frac{\partial}{\partial z} \tau_{rz}\qquad\qquad\qquad(E.4.9.2)
Integration of Eq. (E.4.9.1) across the gap \rm-\frac{h}{z} \leq z\leq \frac{h}{2} requires:
\rm\int dw=-\int\frac{1}{r}\frac{\partial(r\,v)}{\partial r} dz\qquad\qquad\qquad(E.4.9.3)
Specifically,
\int\limits_{-h/2}^{h/2}dw =w\left\lgroup \frac{h}2\right\rgroup -w\left\lgroup -\frac{h}2\right\rgroup =\frac{\dot{h}}2-\left\lgroup-\frac{\dot{h}}2\right\rgroup =\dot{h}\qquad\qquad\qquad\rm(E.4.9.4a)
and
\rm-\int\limits_{-h/2}^{h/2}\frac{1}r\frac{\partial(rv)}{\partial r} dz=-\frac{1}{r} \frac{d(r\overline{v} )}{dr} \int\limits_{-h/2}^{h/2}dz=-\frac{1}{r} \frac{d}{dr} (r\overline{v} )h\qquad\qquad\qquad(E.4.9.4b)
so that
\rm\mathrm{\dot h\equiv\!\!\frac{{d\mathrm{h}}}{{d}t}}=-\frac{1}{\mathrm{r}}\,\frac{\mathrm{d}}{{d}\mathrm{r}}\,(\mathrm{rh}\bar{v})\qquad\qquad\qquad(E.4.9.5a)
where
\rm{\overline{{v}}}={\frac{2}{\mathop{h}}}\int\limits_0^{h/2}v({r},{z};{t}){d}z\qquad\qquad\qquad(E.4.9.5b)
Before integrating Eq. (E.4.9.2) across the gap, \rm τ_{rz} has to be defined.
(i) Newtonian fluid: \rm\tau_{rz}\approx \mu\frac{\partial v}{\partial z} \qquad\qquad\qquad(E.4.9.6)
Thus, Eq. (E.4.9.2) reads:
\rm\mu\frac{\partial^2v}{\partial z^2} =\frac{dp}{dr} \qquad\qquad\qquad(E.4.9.7)
subject to v(z = ± h/2) = 0. Integration yields:
\rm\mathrm{v}={\frac{\mathrm{h}^{2}}{2\mu{}}}{}\left\lgroup \frac{dp}{dr}\right\rgroup \left[\left\lgroup{\frac{\mathrm{z}}{\mathrm{h}}}\right\rgroup ^{2}-{\frac{1}{4}}\right]\qquad\qquad\qquad(E.4.9.8)
Evaluating the average radial velocity as needed in Eq. (E.4.9.3), we have:
\rm\mathrm{h}{\overline{{\mathrm{v}}}}={\frac{1}{\mu}}\left\lgroup\frac{dp}{dr}\right\rgroup \int\limits_{0}^{\mathrm{h}/2}\left\lgroup\mathrm{z}^{2}-{\frac{\mathrm{h}^{2}}{4}}\right\rgroup \mathrm{d}\mathrm{z}:={\frac{-\mathrm{h}^{3}}{12\mu}}{\frac{\mathrm{d}\mathrm{p}}{\mathrm{d}\mathrm{r}}}\qquad\qquad\qquad(E.4.9.9)
so that
\rm{\frac{1}{r}}{\frac{\mathrm{d}}{\mathrm{d}\mathrm{r}}}\left\lgroup\mathrm{r}\,{\frac{\mathrm{d}\mathrm{p}}{\mathrm{d}\mathrm{r}}}\right\rgroup ={\frac{12\mu}{\mathrm{h}^{3}}}{\frac{\mathrm{d}\mathrm{h}}{\mathrm{dt}}}\qquad\qquad\qquad(E.4.9.10)
subject to p(r = R) = 0 and dp/dr = 0. Thus, double integration yields:
\rm\mathrm{p}={\frac{3\mu R^{2}}{\mathrm{h}^{3}}}{\dot{\mathrm{h}}}\left[\left\lgroup{\frac{\mathrm{r}}{\mathrm{R}}}\right\rgroup ^{2}\,-1\right]\qquad\qquad\qquad(E.4.9.11)
Now the load can be evaluated as:
\rm{ L}={\int\limits_0^\mathrm{R}}{pd}{ A}=2\pi\int\limits_{0}^{\mathrm{R}}\mathrm{pr}{ d}{ r}:=\frac{3\pi}{2}\,\mathrm{\mu}{ R}^{4}\dot h\Big/h^3\qquad\qquad\qquad(E.4.9.12)
(ii) Power-law fluid:
\rm\tau_{rz}=K\left\lgroup\frac{\partial u}{\partial z} \right\rgroup ^{n-1}\qquad\qquad\qquad(E.4.9.13)
Starting over with Eq. (E.4.9.2) for this basic non-Newtonian fluid, the load is:
\rm\mathrm{L}={\frac{2\pi(2+1/\mathrm{n})^{\mathrm{n}}}{\mathrm{n}+3}}\,\mathrm{K}\,\mathrm{sgn}(\mathrm{\dot{h}})\,|\,{\dot{\mathrm{h}}}\,|^{\mathrm{n}}\,\mathrm{h}^{-(2\mathrm{n}+1)}\,\mathrm{R}^{\mathrm{n}+3}\qquad\qquad\qquad(E.4.9.14)
(iii) Linear viscoelastic fluid: \rm\tau_{rz}=\int\limits_{-\infty }^tG(t-t^\prime)\frac{\partial u(t^\prime)}{\partial z} dt^\prime\qquad\qquad\qquad (E.4.9.15)
leads to:
\rm\mathrm{L}={\frac{3\pi\mathrm{R}^{4}}{2}}\int\limits_{0}^{\mathrm{t}}\mathrm{Q}(\mathrm{t-t^{\prime}})\,(\mathrm{\dot h}\,/\,\mathrm{h}^{3})\,\mathrm{d}t^{\prime}\qquad\qquad\qquad(E.4.9.16)
Graphs (for Case(i)):
Comments:
As expected, the radial squeeze-film velocity v(z) is parabolic due to the approaching disks creating at time t with \rm\dot h_{disk} a pressure gradient dp/dr which is driving the fluid outwards. The pressure distribution (see Eq. (E.4.9.11)) clearly depends on the disk-approach speed \rm\dot h_{disk} =\dot h(L) and the disk radius. In turn, the load determines the disk speed (see Eq. (E.4.9.12)).
