Spontaneous Core Rotation in Ferrofluid Pipe Flow

Ferrofluid flow along a tube of radius $R$ in a constant axial magnetic field is revisited. Our analytical solution and numerical simulations predict a transition from an initially axial flow to a steady swirling one. The swirl dynamo arises above some critical pressure drop and magnetic field strength. The new flow pattern consists of two phases of different symmetry: The flow in the core resembles Poiseuille flow in a rotating tube of the radius $r_{*}<R$, where each fluid element moves along a screw path, and the annular layer of the thickness $R-r_{*}$, where the flow remains purely axial. These phases are separated by a thin domain wall. The swirl appearance is accompanied with a sharp increase in the flow rate that might serve for the detection of the swirling instability.

Figure 1

Stability diagram in the $P-h$ plane (a). In the region of swirling flow (shaded) coordinates of any point $K$ determine the radius $r_{*}=b/a$ of a rotating liquid cylinder. Sketch of the swirling flow (b).

Figure 2

Volume flow rate $Q$ as a function of the pressure gradient $P$ for ferrofluid with $\mu =9$ at different values of magnetic field $h$. Solid lines—purely axial flow, dashed lines—swirling flow.

Figure 3

Axial, $v_z(r)$, and azimuthal, $v_{\phi }(r)$, components of the velocity for ferrofluid with $\mu =9$ at $P=11$ and two values of magnetic field $h$ are solid lines. Profiles $v_z(r)$ obtained when the azimuthal component of the velocity is suppressed ($v_{\phi }=0$) are dashed lines. Inset: Enlarged view of $v_{\phi }(r)$ and function $\mathrm{\Lambda }(r)$ for $h=8.8$ in the vicinity of $r_{*}=0.8931$. The width of $\mathrm{\Lambda }(r)$ is $\mathrm{\Delta }r=0.0138$; its maximum magnitude ${\mathrm{\Lambda }}_{\max }=2.940$.

Figure 4

Magnetization components ${\tilde{M}}_i(r)$ for ferrofluid with $\mu =9$ at $P=11$ and $h=8.8$ shown by solid lines. Dashed lines of ${\tilde{M}}_r(r)$ and ${\tilde{M}}_z(r)$ continue when the azimuthal component of the velocity is suppressed ($v_{\phi }=0$); in this case ${\tilde{M}}_{\phi }=0$. Inset: Enlarged view in the vicinity of $r_{*}=0.8931$.