Measurement of Near-Bed Velocity and Bottom Shear Stress by Ferrofluids
In physical modelling of hydraulic processes, the knowledge of wall shear stress is fundamental to evaluate the energy dissipation of the flow and to assess the flow interaction with the bottom. Furthermore, the interaction between flow and solid walls influences to a great extent the characteristic of the flow itself. For such reasons measurements of wall shear stresses are extremely important.
In the past, several instruments have been developed and used to measure the shear stresses generated at the bottom in the presence of waves and currents.
Simple instruments, such as Preston tubes and Stanton tubes, have been used to measure bed shear stresses . Advantages of such instruments are the robustness. However, because their application is restricted to flows where the normal law of the wall is valid, they are affected by several limits as, for example, the tube dimensions must be smaller than wall layer and the difference between total (dynamic+static) and static pressure must be large.
Nowadays, velocity measurements are often carried out using system based on the acoustic and optic methodologies, such as ADV (Acoustic Doppler Velocimetry), LDA (Laser Doppler Anemometry), PIV (Particle image velocimetry) and PTV (Particle tracking Velocimetry). Some application of such instruments can be found in    end . Bottom shear stresses are derived from such instruments adopting theoretical approaches (e.g. log-fit, momentum integral method, etc). The advantage of such instruments is their reliability. However, such instruments have the following disadvantages: both acoustic and optical techniques cannot measure very close to the bottom, either due to the size of the sampling volume of the acoustic probe or to undesired reflection and disturbances from the bottom itself; the use of optical instruments is limited in large flumes; and measurements with such instruments are extremely difficult in the presence of sediments. Furthermore, for example at the bottom of sea waves, the velocity profiles are not logarithmic which is an underlying assumption of some instrument.
Direct measurements of bed shear stresses can be performed also by means of flush-mounted shear plates, which integrate the force over a relatively large area    . The system allows to resolve shear stress of O(1 mNm−2). However, one of the main problems while using shear plates is the tradeoff between sensor spatial resolution and the sensor capability to measure small forces.
Another class of instruments is the one of thermal sensors, to which hot film anemometers belong  . The system allows to measure turbulent fluctuations. Unfortunately, besides the well-known fragility of the sensors, hot film techniques are traditionally limited by difficulties in obtaining a unique calibration relationship between heat transfer and wall shear-stresses, from the reduction in sensitivity and complications in the dynamic response due to frequency-dependent conductive heat transfer into the substrate, and by measurement errors associated with mean temperature drift.
To overcome some of the above mentioned limits, recent studies     demonstrated the possibility to use ferrofluids sensors to measure near-bed velocity. Another promising method has been presented recently by  using sensor films with arrays of flexible micro-pillars sensing the wall shear stress by their bending in the flow. Ferrofluids sensors can be applied in the presence of sandy bottoms, over which state-of-the-art-instruments usually fail, and are characterized by a high robustness and a low cost.
Ferrofluids are two-state systems made up by small ferromagnetic particles (with size 1 nm-15 nm) dispersed in an organic non-magnetic solvent . A surfactant covers the nano-particles in order to prevent their agglomeration, which would be otherwise induced by the Van Der Waals forces and the magnetic forces. Such fluids behave like Newtonian fluid when the applied magnetic field is null or very small. Whereas if they are exposed to a magnetic field, their viscosity is strongly modified. At the microscopic scale, long chains of particles are formed in the direction of the magnetic field, whereas at the macro-scale a series of spikes aligned along the magnetic field appears. Such a phenomenon is called “the Rosensweig effect” . An example of Rosensweig effect is shown in Figure 1I and in Figure 1II. The use of a relatively large amount of ferrofluid O(10 ml) causes a formation of hexagonal conical patterns which are aligned along the magnetic field lines (see Figure 1b). A single fluidic pattern with a conical shape is generated using a volume of ferrofluid on the order of 2 ml, with a spike height approximately of 1 mm (see Figure 1III and Figure 1IV).
Considering the scheme of Figure 1d, the forces acting in the ferrofluid spike are: i) its weight; ii) the hydrostatic force; iii) the external magnetic field. In addition to the aforementioned forces, when the ferrofluid is subject to water flow, the effects of the dynamics force must be considered. In particular, the ferrofluid is subject to drag force, lift force, the friction generated on the surface of ferrofluid spike and the drag force generated at the bottom of the ferrofluid spike.
The order of magnitude of such forces are as follows : i)ferrofluid weight O(10-5 N); ii) drag force due the pressure in flow direction O(10-7-10-4 N); iii) lift force O(10-7-10-4 N); iv) friction generated on the surface of ferrofluid spike O(10-9-10-6 N); v) drag force generated at the bottom of the ferrofluid spike O(104 N);
As it can be seen from the above list, drag force generated at the bottom of the ferrofluid spike is the largest force in the balance (greater of several order of magnitude). Therefore, the other forces can be neglected. The magnetic force exerted by permanent magnets, although not directly measured, should be large enough to maintain the ferrofluid shape. Thus, the magnetic force must compensate the drag force generated at the bottom of the ferrofluid spike. Consequently, it can be concluded that, under flow action, the dynamic equilibrium lead to deformations and displacements of the ferrofluid spike which are both proportional to the magnitude of the bottom shear stress. Figure 2II and Figure 2III qualitatively show how the displacement of the ferrofluid spike increases as the bed shear stress increases. It follows that through an appropriate calibration of the system, shear stress measurements can be obtained by recovering the deformation of the spike.
State of art on the ferrofluid sensor calibration
With the aim to study the ferrofluid sensor under different flow conditions, in the last years, several tests were carried out at the Hydraulic Laboratory of the Department of Civil Engineering and Architecture, University of Catania. A preliminary analysis on the ferrofluid sensor was carried out in . In such study the tests were performed using a circular channel, named “annular cell” (Taylor Couette System), placed on a rotating platform that is driven to an alternate rotation motion with given amplitude and period to obtain a controlled fluid motion. The ferrofluid was placed on the bottom surface where a magnetic force was applied through the permanent magnet. The ferrofluid shape was monitored through a web-cam. The images of web-cam were used to estimate the displacement of the top apex of ferrofluid spike due to water flow. In the experimental campaign of <ref name = "Musumeci2015a"\> the ferrofluid shape was monitored through an inductive readout strategy which makes use of two planar coils located at the bottom of the flume (see Figure 3). The ferrofluid spike was placed initially at the center of the two coils. The movement of the ferrofluid spike produced a variation in the magnetic permeability and consequently a variation of the inductance in the coils. In this way, the perturbation of the magnetic field generated by the displacement of the spike is transduced into a voltage variation by a suitable conditioning circuit.
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