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Fig. 2 Plane buckled shape of a water column impinging on the flat end of a screwdriver

Buckling of Inviscid Streams

There is an intriguing analogy between the buckling of elastic solid columns and the meandering behaviour of inviscid streams, both of which can be analysed through the static equilibrium of the system. In this context, a finite-size control volume is drawn around the stream. As depicted in Fig. 1, if the stream and its control volume have a thickness on the order of D, and the stream's cross-section area is A, then the control volume (or a thin-walled boundary surrounding the stream) satisfies two conditions necessary for infinitesimal sinusoidal buckling, similar to those observed in elastic systems:

Fig 1. Translational and rotational equilibrium of a finite-size stream

1. The control volume undergoes axial compression due to impulse and reaction forces. This provides insight into the underlying buckling mechanisms of inviscid streams, akin to the behavior observed in elastic structures. The force balance analysis, reveals that the control volume of an inviscid stream experiences axial compression, with the force C expressed as: C = ρV²A Eq.(1)
2. In addition to axial compression, when subjected to bending, the control volume develops a resistive bending moment proportional to the induced curvature: M=-ρV²I${\partial ^{2} \over \partial x^{2}}y$ Eq. (2)

In Eq.(2), I represents the area moment of inertia of the stream's cross-section and (-$Y''$) denotes the local curvature of the infinitesimally deformed control volume. This equation is analogous to the elastic beam bending equation M=−EI$Y''$, where EI represents the bending stiffness. In the case of inviscid streams, the product ρV² plays the role of the modulus of elasticity, indicating that the stream exhibits mechanical properties akin to elasticity. This is evident when attempting to bend a high-Reynolds-number stream confined in a thin-walled hose.

The static equilibrium of the control volume requires both translational and rotational equilibrium. The translational equilibrium is ensured by the balance of axial forces C. However, as in Euler's buckling theory for solid columns, rotational equilibrium must also be maintained. When the forces C are not perfectly aligned, the rotational equilibrium equation is given by:

−M(x)+CY+$M_{0}$=0 Eq.(3)

or, substituting expressions Eq.(1) and Eq.(2),

(ρV²I)$Y''$ + (ρV²A)Y + $M_{0}$=0 Eq.(4)

The condition for static rotational equilibrium suggests that the equilibrium shape of a nearly straight stream column follows a sinusoidal pattern with a small amplitude and a characteristic wavelength:

$\lambda _{B}=2\pi {\biggl (}{\frac {I}{A}}{\biggr )}^{1/2}={\begin{cases}{\frac {\pi }{2}}D,&{\text{circular cross section }}\\{\frac {\pi }{3^{1/2}}}D,&{\text{rectangular cross section }}\end{cases}}$ Eq.(5)

which means that in an order of magnitude sense,

$\lambda _{B}\sim 2D$ Eq.(6)

The buckling wavelength $\lambda _{B}$ serves as a geometric property of the control volume, typically corresponding to a length that is roughly twice the transverse dimension D. The scaling relationship $\lambda _{B}\sim D$ predicted by the buckling theory of inviscid streams aligns with the first empirical scaling law observed during transition.

Before demonstrating how the buckling property also accounts for the second transition scaling law, the following observations are pertinent:

1. The buckling wavelength of an inviscid stream is unique (approximately of order DDD) because the compressive load ρV²A is consistently proportional to the elasticity modulus ρV² This characteristic sharply distinguishes the buckling behaviour of inviscid streams from that of elastic solid columns, where the parameters C and E are independent. As a result, in solid columns, there exists an infinite range of $\lambda _{B}$​ values (an additional degree of freedom), from which we must identify a discrete sequence of special $\lambda _{B}$ values that satisfy end-clamping conditions. In contrast, for inviscid streams, the buckling wavelength is unique, and end-boundary conditions are not significant.
2. The buckling theory of inviscid streams involves the equilibrium of a finite-sized region within the flow field, marking a significant departure from conventional fluid mechanics methodologies, which typically begin with the Navier–Stokes equations and consider small fluid packets
3. Although the proportionality $\lambda _{B}\sim D$ is universally applicable, the control volume with a transverse dimension D is selected arbitrarily. Any fluid fibre, or any control volume of thickness D' not equal to D, meets the conditions for infinitesimal buckling. However, among this infinite set of fibres, only a specific class exists in a state of unstable equilibrium. The instability of inviscid flow and the identification of certain fluid fibres as unstable represent a distinct flow property and arise from a separate theoretical framework known as hydrodynamic stability.
4. The buckling property, or the scaling law $\lambda _{B}\sim D$, is commonly observed in natural flows and can also be demonstrated in laboratory settings. A substantial photographic record documenting these phenomena is available in Refs, highlighting examples such as river meandering, the waving of flags and the undulating descent of paper ribbons, the buckling of fast liquid jets moving through the air, the wrinkling of two-dimensional fluid layers pushed from one end, and the sinuous structure of turbulent plumes. One can easily visualise the buckling scaling described in Eq. (6) by placing an obstacle beneath a capillary water column falling from a faucet. Fig. 2 illustrates the plane buckled shape of a water column impacting the flat end of a screwdriver, with the left side depicting a direct view and the right side showing a view through a side mirror. The observed sinuous deformation primarily occurs in one plane, resembling the cigarette smoke plume and the locally measured $\lambda _{B}$/D ratio consistently approximates to the order of 2, as indicated in Eq. (6) .

References

Article prepared by:

Asati Buland Rakesh (Roll.No. 21135031), IIT BHU (Varanasi)

Ashish Kumar Yadav (Roll.No. 21135032), IIT BHU (Varanasi)

Kapil Soni (Roll.No. 21135069), IIT BHU (Varanasi)

Mudit Pathak (Roll.No. 21135084), IIT BHU (Varanasi)

Rahul Kumar (Roll.No. 21135108), IIT BHU (Varanasi)