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Experimental study of the dynamics of magneto-rheological fluid droplet impact

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Abstract

The kinematics of a magneto-rheological fluid droplet impact on a smooth surface, subjected to external magnetic field, was studied theoretically and experimentally. A time-dependent one-dimensional model of the impact, as typified by the droplet top center point height kinematics is developed. A series of experiments were conducted in order to validate the theoretical model. The shape changes generated during the impact process were recorded using a digital high-speed camera. Our novel kinematical model based on a variable damping function shows very good agreement with the experimental data for a wide range of Mason and Reynolds numbers.

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Abbreviations

a :

Acceleration

B :

Magnetic field

Ca:

Capillary number

\( D_{0} \) :

Droplet initial diameter

\( D_{\text{C}} \) :

Circle diameter of a contact line

\( D_{\text{M}}^{*} \) :

Dimensionless maximum spread factor

\( D_{\text{P}} \) :

Particles mean size diameter

\( F_{\sigma } \) :

Surface tension force vertical component

FPS:

Frame per second

Fr:

Froude Number (Eq. 3)

G :

Gravity constant

\( \hat{H} \) :

Normalized top center point \( \left( {\hat{H}(t) = H(t)/D_{0} } \right) \)

\( \hat{H}_{\infty } \) :

Normalized steady state minimum height

i :

Index

k :

Coefficients (Eq. 21)

Ma:

Mason number (Eq. 18)

MR:

Magneto-rheological

R 0 :

Impact radius

Re:

Reynolds number (Eq. 17)

\( \text{Re}^{0} \) :

Reynolds number at B = 0

RMS:

Root mean square

RSQ:

Square of the correlation coefficient

T :

Time

\( \hat{t} \) :

Non-dimensional time (Eq. 4)

V :

Droplet volume

V d :

Droplet velocity

V 0 :

Impact velocity

V S :

Settling velocity

W :

Droplet weight

We:

Weber number (Eq. 16)

X, Y, Z :

Cartesian coordinate system

\( \hat{Z} \) :

Non-dimensional longitudinal axis (Eq. 5)

\( \hat{Z}_{\text{CM}} \) :

Non-dimensional center-of-mass position

α:

Parameter

β:

Quadratic function (Eq. 14)

\( \dot{\gamma } \) :

Shear rate

\( \Upgamma \) :

Function (Eq. 21)

δ:

Parameter

ε:

Coefficient (Eq. 22)

η:

Shear apparent viscosity

ηF :

Magneto-rheological carrier fluid viscosity

κ i :

Damping function coefficients (Eqs. 8, 9)

λ:

Parameter

ΛC :

Variable damping function

Λ1, Λ2 :

Variable damping function components (Eq. 10)

ξ:

Geometrical shape factor

ρ:

Magneto-rheological fluid density

υ1, υ2 :

Coefficients (Eq. 19)

ρF :

Fluid carrier density

\( \rho_{S} \) :

Solid particle density

σ :

Surface tension

τ s :

Settling time constant \( \left( {\tau_{\text{s}} = {{D_{0} } \mathord{\left/ {\vphantom {{D_{0} } {V_{\text{S}} }}} \right. \kern-\nulldelimiterspace} {V_{\text{S}} }}} \right) \)

μ 0 :

Vacuum permeability

Φ:

Function correlating the normalized steady state minimum height with Reynolds number at B = 0 (Eq. 19)

Ψ:

Dumping function exponential parameter

ΨI :

Function (defined in Eq. 22)

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  1. Faculty of Aerospace Engineering, Technion Israel Institute of Technology, 32000, Haifa, Israel

    Shai Rahimi & Daniel Weihs

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  1. Shai Rahimi
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  2. Daniel Weihs
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Correspondence to Daniel Weihs.

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Rahimi, S., Weihs, D. Experimental study of the dynamics of magneto-rheological fluid droplet impact. Exp Fluids 53, 1577–1589 (2012). https://doi.org/10.1007/s00348-012-1376-3

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