A point force makes a jet only when it is strong

What a jet keeps, and what it collects and exactly similar, and one number short solved the round jet the way it is always solved: as a boundary layer. Far from its source a laminar jet is slender, its flow nearly parallel to its axis, and dropping the terms that slenderness makes small gives Schlichting’s solution — a shape function with no parameter, a momentum flux that never changes, and a volume flux that grows as $8\pi\nu x$ whatever the jet’s strength. The second essay checked those properties to parts in ten million and named what the approximation cannot say: what happens near the source, sideways, behind it, and when the jet is too weak to be slender at all.

A force without the flow that makes it used the opposite limit, a point force in a fluid with no inertia, Stokes’s point force, which is not slender at all. Between those two limits there is an exact solution of the full equations, and it is the subject here.

A conical flow #

A point force $P$ applied at the origin of a fluid at rest has no length in it; the only length the flow could use is $\nu/U$ for some speed, and there is no given speed either. So the velocity must fall off as $1/r$ and depend on direction only — a conical flow — and for such a flow the Navier–Stokes equations reduce to ordinary differential equations in the angle. Landau found their solution in 1944, and Squire independently in 1951:

$$u_r = \frac{2\nu}{r}\left[\frac{A^2 - 1}{(A - \cos\theta)^2} - 1\right], \qquad u_\theta = -\frac{2\nu}{r}\,\frac{\sin\theta}{A - \cos\theta},$$

with $\theta$ measured from the direction of the force and one constant, $A > 1$, set by the force’s strength. A force small against $\rho\nu^2$ has $A$ large; a force large against it has $A$ close to one. The natural measure of strength is the jet Reynolds number $\sqrt{P/\rho}/\nu$, the square root of the force in units of $\rho\nu^2$.

The solution is exact at every strength — no slenderness, no small parameter. The only thing it assumes is the point: a real jet issues from a nozzle of some size, and the conical flow is what that jet looks like from distances much larger than the nozzle.

One number, because there is no length #

The conical form is forced by dimensions, and seeing why makes the single parameter unsurprising. The force divided by the density, $P/\rho$, has the units of a velocity squared times a length squared, and so does $\nu^2$. Their ratio is a pure number, and there is no other: no length, no velocity, no time is given by a point force in an unbounded fluid at rest. So every property of the flow that is itself dimensionless — an angle, a ratio of speeds, a share of the momentum — can depend only on $P/\rho\nu^2$, and every dimensional one must be built from $\nu$ and the distance $r$ at which it is measured. That is counting what matters at its starkest: one group, and the whole family of flows from a bacterium’s push to a fire hose’s is a curve in it.

The same counting explains why a two-dimensional version would be different. A line force per unit length divided by the density has the units of a velocity squared times a length — one power of length fewer — so it does fix a length, $\nu^2\rho/P$, and a line force has no conical solution. Its creeping limit is worse than different: the flow with no solution is Stokes’s paradox, the absence of any steady creeping flow past a cylinder that dies away far from it, which is the two-dimensional point force failing to exist as a flow with a quiet far field. In three dimensions the point force exists at every strength.

Weak and strong #

At a jet Reynolds number of one the streamlines are Stokes’s: the fluid is pushed outwards across almost the whole forward hemisphere — the cone of outflow is 89.4° from the axis — and drawn in across the rear one, the familiar pattern of a body being dragged through a viscous fluid seen from the body. At a jet Reynolds number of a hundred the picture is a jet. The outflow is confined to a cone 22.8° from the axis, and outside it the fluid moves inwards from every direction, forward and back, turning into the jet near the axis. A strong push does not push the fluid in front of it; it draws fluid in from all round and throws it forward in a narrow stream.

That is entrainment, seen whole. The boundary-layer jet entrains too, but it has no “outside”: its solution ends where the jet’s velocity has fallen to nothing, and the inflow that feeds it is a weak sideways velocity at the edge of its domain. The exact solution shows where that inflow comes from — from behind the source as much as from beside the jet.

Where the cone is, and why it closes #

The cone that divides outflow from inflow has a closed form. The radial velocity changes sign where $(A^2-1) = (A-\cos\theta)^2$, that is at $\cos\theta_0 = A - \sqrt{A^2-1}$. For a weak force $A$ is large, the right-hand side is about $1/2A$, and $\theta_0$ is a hair short of ninety degrees: the Stokes limit’s forward hemisphere. For a strong force $A$ is close to one, $A - \sqrt{A^2 - 1} \approx 1 - \sqrt{2(A-1)}$, and the cone’s half-angle is about $(8(A-1))^{1/4}$ — which, with $A - 1$ falling as one over the force, closes as the inverse fourth root of the force, or the inverse square root of the jet Reynolds number. The core’s half-width closes as the inverse first power. So a strong jet is a narrow core inside a much wider cone of slow outflow, inside a sphere of slow inflow, and each layer narrows at its own rate. The outermost layer is the one no boundary-layer solution has, because it is not slender at any strength.

From a cosine to a pencil #

The radial velocity round a circle shows the transition. For the weak force it is close to Stokes’s $\cos\theta$: as large backwards as forwards, positive ahead and negative behind. As the force grows, the outflow narrows — at a jet Reynolds number of a hundred it has fallen to half its axis value three degrees from the axis — and the inflow becomes weak and nearly uniform over the rest of the sphere. A narrow fast outflow and a broad slow inflow carry the same volume. The inflow’s slowness is also why it is so easily missed in a photograph of dye: a streak in the outflow moves hundreds of times faster than one at the side, and a short exposure records the jet and not the fluid feeding it.

Between the two the profile has no simple shape, and that middle range — jet Reynolds numbers of a few to a few tens — is where small laminar jets in the laboratory often sit: a hypodermic needle’s jet of water, an ink-jet’s stream before break-up, the jet a swimming organism makes with a cilium. None of them is a Stokeslet and none is Schlichting’s jet.

When the boundary-layer jet becomes right #

Schlichting’s jet predicts a half-width angle that falls as one over the jet Reynolds number. The exact one follows it from above at strong forces and departs from it at weak ones: at a jet Reynolds number of 10 the half-widths are 26.3° and 27.8°, at 30 they are 9.93° and 10.2°, and at 100, 3.00° and 3.02°. The boundary-layer prediction is good to a per cent once the jet is a few degrees wide — which is to say once it is slender, the approximation’s own condition — and it errs on the wide side before that. The cone of outflow closes much more slowly, from 90° at weak forces to 22.8° at 100 and 7.3° at 1,000: the jet’s core narrows as one over the Reynolds number, its outer region, where the profile has fallen to a small fraction of the axis value, much more slowly.

The axis first, the edges last #

Two quantities converge on Schlichting’s at different rates. The axis speed starts at two-thirds of his — Stokes’s value — overshoots it by 7.3 per cent near a jet Reynolds number of 14, and is within a per cent by a hundred. The outflow volume, the fluid crossing a sphere outwards, starts near nothing and reaches 0.922 of Schlichting’s $8\pi\nu r$ at a jet Reynolds number of a hundred and 0.992 at a thousand.

The difference is where each quantity lives. The axis speed is set near the axis, where the jet is slender soonest; the volume flux is an integral over the whole outflow cone, most of which is the jet’s slow outer region, and that region is where the boundary-layer approximation is weakest — the flow there is not nearly parallel to the axis. So a measurement of a small jet’s centreline speed can agree with the boundary-layer jet at a Reynolds number where a measurement of its entrainment does not, and the second is the one that matters for mixing.

The momentum is the force, exactly #

The relation between the constant $A$ and the force is

$$\frac{P}{2\pi\rho\nu^2} = \frac{32A}{3(A^2-1)} + 8A - 4A^2\ln\frac{A+1}{A-1},$$

and it is checked here by not using it. The force must equal the momentum carried out through any sphere round the origin — convected momentum, pressure and viscous stress together — and integrating those three over a sphere from the velocity field and Landau’s pressure, at a jet Reynolds number of about seven, returns the applied force to three parts in a thousand million. At that strength the three share the work: in the Stokes limit the pressure carries exactly a third and the viscous stress two-thirds, and as the force grows the convected momentum takes over, which is why the boundary-layer jet keeps only the convective flux. Continuity holds at a sample point to $10^{-8}$; at a vanishing force the radial velocity is Stokes’s $P\cos\theta/4\pi\mu r$ to a quarter of a per cent; and at a jet Reynolds number of three thousand the axis speed is Schlichting’s $3P/8\pi\rho\nu r$ to three parts in a hundred thousand.

From a bacterium to a needle #

The range of strengths in the figures is the range of real small jets. A swimming bacterium pushes on the water with about a piconewton; in water that is a jet Reynolds number of 0.03, deep in the Stokes limit, which is why its wake is not a jet at all but the hemispherical push and pull of the first figure — a swimmer that cannot go backwards lives entirely in that regime. A jet of water a metre a second from a fine hypodermic needle a third of a millimetre across carries about seventy micronewtons of momentum: a jet Reynolds number of about 260, a slender jet whose axis Schlichting’s solution describes to a fraction of a per cent and whose entrainment it describes to a few per cent. Between them lie the jets of microfluidic mixers and of an animal’s cilia, at Reynolds numbers of one to thirty, where neither limit holds and the exact solution is the only description.

How the work is shared #

The momentum flux through a sphere has three parts — momentum carried by the moving fluid, pressure, and viscous stress — and the exact solution says how they share the force as it grows. At a jet Reynolds number of a tenth the convected part is zero to four figures, the pressure carries a third and the viscous stress two-thirds, Stokes’s split exactly. At seven the three are 0.386, 0.177 and 0.438. At thirty-two the convected momentum carries 0.969 of the force and the viscous stress 0.045, while the pressure has turned slightly negative — the low pressure in the jet’s core, pulling backwards on the sphere’s forward cap. At three hundred the convected momentum is the whole of it to a part in a thousand. A boundary-layer jet keeps only the convected part, and the figures’ convergence is that part taking over.

What a nozzle adds #

A real jet’s source has a size and a profile, and the conical solution describes it from far enough away that both are forgotten. How far is a matter of the jet’s own momentum: a nozzle’s jet carries the same momentum flux as a point force, so far downstream it is the same flow, displaced by a virtual origin whose distance from the nozzle grows with the jet Reynolds number. That displacement is the one place the nozzle’s details survive in the far field. In the creeping limit the nozzle’s flow joins the Stokeslet within a few diameters, because there is no inertia to carry its profile; in the strong limit the slender jet keeps a memory of the nozzle for hundreds of diameters, and it is in that long transition that measured laminar jets usually lie.

The virtual origin is also how a measured jet is compared with the solution. On the axis the conical flow’s speed falls exactly as the inverse of the distance from the point force, so the reciprocal of the measured axis speed, plotted against distance from the nozzle, is a straight line once the nozzle is forgotten. Its slope gives the momentum flux, the force the equivalent point source must supply, and the place where it crosses zero is where that source sits. A measured jet whose reciprocal axis speed is not yet straight is still remembering its nozzle.

The convention: a point force and a jet Reynolds number #

The force $P$ is the total momentum the source supplies per unit time, the same quantity as a jet’s momentum flux. The jet Reynolds number is $\sqrt{P/\rho}/\nu$; the figures’ strengths run from 0.03 to 1,000. Angles are from the force’s direction. The fluid is incompressible and Newtonian, the flow steady and axisymmetric, and the source a point.

What the picture cannot show #

A real jet comes from a nozzle, and near the nozzle — within a few nozzle diameters — its flow is set by the nozzle’s profile, not by the conical solution; the conical flow is the far field. Laminar jets also become unstable: Schlichting’s profile goes unstable to helical disturbances at a jet Reynolds number of a few tens, and a free laminar jet at a hundred is rarely laminar for long, so the strong end of the figures describes a flow that is difficult to produce. The solution is steady, and the start-up of a jet — the vortex ring at its head, which a ring’s own motion describes — is outside it. And the inflow far from the axis, while exact, is weak, and in a real tank any slight convection or stratification overwhelms it. The solution is also silent about heat and buoyancy: a warm jet in cool water carries a buoyancy flux as well as a momentum flux, and far enough downstream every such jet becomes a plume, whose spreading is set by the buoyancy and not by the push — a second length, the one the point force lacked, entering through gravity.

Who found it, and when #

Landau published the solution in 1944, in a short note, and it appears in Landau and Lifshitz’s Fluid Mechanics as a worked problem; Squire found it independently in 1951 and studied its jet-like limit. Schlichting’s boundary-layer round jet is from 1933. Batchelor’s textbook draws the streamlines at several strengths. The quantitative statements here about when the boundary-layer jet becomes accurate, for the axis and for the entrainment separately, are computed from the exact solution.

Still open: a jet issuing from a wall #

The point force acts in open fluid. A jet issuing through a hole in a wall cannot draw fluid from behind its source, because the wall is there, and its entrainment must come from beside it alone. Squire’s solution has a version for that: the same conical form with a no-slip condition on a plane through the origin, solved with the constant and the wall condition together. The next calculation solves it and asks how much the wall reduces the entrainment at a given jet Reynolds number, how close to the wall the inflow’s boundary layer lies, and whether the wall’s jet becomes Schlichting’s at a higher Reynolds number than the free one because it has lost the supply from behind.