Pressure distribution in unsteady sink and source flows

Basic flow generated in a viscous incompressible fluid by a “point” sink (source) of mass is revised. In practice, such flow can be modeled by sucking (pushing) fluid from a thin tube with a small porous sphere at one end. Intuitively, by sucking (pushing) fluid, one creates low (high) pressure near the origin and a positive (negative) radial pressure gradient drives the fluid to (from) the origin. A simple analysis, however, shows that the pressure distribution for both steady flows is the same. Then a question arises: How does the fluid “know” in what direction to flow? To explain this “paradox” an unsteady flow is considered and the pressure terms responsible for the flow direction are derived.

Figure 1

The three terms of the pressure distribution (11) generated by a point impulsively started source with $Q\left(t\right)=Q_{0}H\left(t\right)$. The case of the source, $Q_0=-5\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$, is shown in (a), and the case of the sink, $Q_0=-5\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, is shown in (b). The pressure singularities at the origin are shown schematically by the vertical lines.

Figure 2

The three terms of the pressure distribution (12) generated by a point source whose intensity increases with time as $Q\left(t\right)=q_{0}t$. The case of the source, $q_0=5\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, is shown in (a), and the case of the sink, $q_0=-5\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, is shown in (b). The signs of the first two terms (shown by solid lines) depend on the sign of $q_0$ and determine the flow direction. The third term does not depend on the flow direction; it grows with time as $\sim t^2$ and is shown by the dotted line for $t=0.1\mathrm{s}$ and by the dashed line for $t=1\mathrm{s}$.

Figure 3

The estimated time evolution of the mean pressure on the small, radius $R$, porous sphere of the experimental apparatus mentioned in the Introduction. In this example $R=0.1\mathrm{cm}, Q\left(t\right)=q_{0}t$, and $q_0=\pm 5\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-2}$. Solid line – source, dashed line – sink.