**2.5. MRF models**

352 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

**Figure 2.** Classification of time-independent non-Newtonian fluid

MRF responds to the external field, where the particles are held together to form chains parallel to the applied field. The interaction between the particles impedes to a certain level of the shear stress without breaking and simultaneously increases the viscosity of the fluids [32]. In many cases, the effect of MRF is described by Bingham Plastic model [33]. A modified or extended Bingham model, or a combination of Bingham model with other models such as viscous and coulomb friction have also been used to describe the behaviour of MRF [34]. In the absence of an external field, MRF behave like a normal fluid which is known as Newtonian fluid. There are many factors that influence the rheological properties of controllable MRF such as concentration and density of particles, particle size and shape distribution, properties of the carrier fluid [35], additional additives, applied field and temperature. The relationships of all these factors are very complex and are important in establishing methodology to improve efficiency of these fluids for suitable applications. Excellent MRF must have low viscosity and coercivity of particles without the influence of an external magnetic field and can achieve maximum yield stress in the presence of the external magnetic field. Gross [8] in his invention related to the valve for magnetic fluids, found that the advantage of large particle sizes or heavy suspensions can increase the size of the gap which also increases the flow of the fluid. Conversely, the large particles of the magnetically active phase of MRF lead to a strong tendency for particles to settle out of the

Some of the techniques are typically necessary in order to increase the yield stress; either by increasing the volume fraction of MR particles or by increasing the strength of the applied magnetic field. However, neither of these techniques is desirable since a higher volume fraction of the MR particles can add significant weight to the MR devices as well as increases the overall off-state viscosity of the material. In that connection, restricting the size and

**2.4. Rheology of MRF** 

liquid phase [19].

MRF models play an important role in the development of MRF based devices. Moreover, accurate models that can predict the performance of these MRF devices are an important part of implementation of such devices. MRF demonstrates nonlinear behavior when subjected to external magnetic fields. The rheological behavior of these materials can be separated into distinct preyield and post-yield regimes. A wide variety of nonlinear models have been used to characterize MRF, including the Bingham plastic model [37, 38], the biviscous model [39], the Herschel–Bulkley model [40, 41], and Eyring plastic model [42]. Although there have been several models have been developed and applied for MRF the two most popular models have been widely used with reasonable accuracy and computational cost are the Bingham plastic model and the Herschel–Bulkley plastic model. Therefore, in this chapter these two constitutive models are used.

i. Bingham plastic model

The so-called Bingham plastic model includes a variable rigid perfectly plastic element connected in parallel to a Newtonian viscosity element. This model assumes that the fluid exhibits shear stress proportional to shear rate in the post-yield region and can be expressed as [37, 38]

$$
\tau = \tau\_y(H) \text{sgn}(\dot{\boldsymbol{\chi}}) + \eta \dot{\boldsymbol{\chi}} \tag{10}
$$

where *τ* is the shear stress in the fluid, *τy* is the yielding shear stress controlled by the applied field *H*, *η* is the post-yield viscosity independent of the applied magnetic field, γ is the shear strain rate and sgn(·) is the signum function. That is, the fluid is in a state of rest and behaves viscoelastically until the shear stress is greater than the critical value *τy*, whereas it moves like a Newtonian fluid when such a critical value is exceeded. The Bingham plastic model is shown in Figure 3 to represent the field-dependent behaviour of the yield stress. The simplicity of this two-parameter model has led to its wide use for representation of field-controllable fluids, especially ER and MRF.

ii. Herschel-Bulkley plastic model

In cases where the fluid experiences post-yield shear thickening or shear thinning, especially when the MRF experiences high shear rate, this choice of constitutive equation can result in an overestimation. In this case, the Herschel-Bulkley plastic model is more suitable [43]. The Herschel-Bulkley model can be expressed by

$$\tau = (\tau\_y(H)\text{sgn}(j) + K\left|j\right|^{1/m})\text{sgn}(j)\tag{11}$$

where *K* is the consistency parameter and *m* is fluid behavior index of the MRF. For *m* > 1, equation (4) represents a shear thinning fluid, while shear thickening fluids are described by *m* < 1. Note that for *m* = 1 the Herschel–Bulkley model reduces to the Bingham model.

It is noteworthy that, in the above, the post-yield parameters of the MRF such as the postyield viscosity, the consistency parameter and the fluid behavior index are assumed to be independent on the applied magnetic field. In practice, these parameters are slightly affected by the applied magnetic field. Zubieta et al. [44] have proposed field-dependent plastic models for MRF based on the original Bingham plastic and Herschel-Bulkley plastic models. The models were then applied in several researches [45, 46] with experimental agreement. In the field dependent Bingham and Herschel-Bulkley model, the rheological properties of MRF depend applied magnetic field and can be estimated by the following equation

**Figure 3.** Viscoplastic models often used to describe MR fluids

$$Y = Y\_{\alpha} + (Y\_0 - Y\_{\alpha}) \langle \mathcal{Z}e^{-B\alpha\_{\mathcal{S}\mathcal{Y}}} - e^{-2B\alpha\_{\mathcal{S}\mathcal{Y}}} \rangle \tag{12}$$

where *Y* stands for a rheological parameters of MRF such as yield stress, post yield viscosity, fluid consistency and flow behavior index. The value of parameter *Y* tends from the zero applied field value *Y0* to the saturation value *Y*∞*.* α*YS* is the saturation moment index of the *Y*  parameter. *B* is the applied magnetic density. The values of *Y0, Y*∞*,* α*YS* are determined from experimental results using curve fitting method.

### **3. MRF mode of operation and its application**

#### **3.1. Valve mode**

Figure 4a schematically show the valve mode which have been used in many MR devices where the flow of the MRF between motionless plates or a duct is created by a pressure drop. The magnetic field, which is applied perpendicular to the direction of the flow, is used to change the rheological properties of the MRF in order to control the flow. Therefore, the increase in yield stress or viscosity alters the velocity profile of the fluid in the gap between two plates. A typical velocity profile for Bingham-plastic of the valve mode is illustrated in Figure 4b. The velocity profile contains a pre-yield region, where the velocity gradient is zero across the plug region. The velocity profile of MRF between two parallel plates can be represented by the following relation [47]

$$\begin{aligned} u\_1(y) &= \frac{n}{n+1} (\frac{\Delta P}{KL})^{1/n} \left[ (\frac{d-\delta}{2})^{1+1/n} + (\frac{2y+\delta}{2})^{1+1/n}; \ u\_2(y) \right] = \frac{n}{n+1} (\frac{\Delta P}{KL})^{1/n} (\frac{d-\delta}{2})^{1+1/n} \\ u\_3(y) &= \frac{n}{n+1} (\frac{\Delta P}{KL})^{1/n} \left[ (\frac{d-\delta}{2})^{1+1/n} - (\frac{2y-\delta}{2})^{1+1/n} \right] \end{aligned} \tag{13}$$

Here, *n*=1/*m*, *u1* and *u3* are velocity profiles of the post-yield flow regions adjacent to the walls of the rectangular duct, and *u2* is the velocity profile across the central pre-yield or plug region. δ is the plug region thickness, which is a key parameter of the flow. As the field increases, so does the pre-yield thickness, thereby, constricting the flow through the duct, increasing the pressure drop. A high resistance produced by the valve mode can be used in many applications such as dampers, valves and actuators [48-51].

**Figure 4.** Valve mode in the MR application

#### **3.2. Shear mode**

354 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

τ τ

**Figure 3.** Viscoplastic models often used to describe MR fluids

parameter. *B* is the applied magnetic density. The values of *Y0, Y*

applied field value *Y0* to the saturation value *Y*

experimental results using curve fitting method.

**3.1. Valve mode** 

**3. MRF mode of operation and its application** 

1/ ( ( )sgn( ) )sgn( ) *<sup>m</sup>*

γγ

where *K* is the consistency parameter and *m* is fluid behavior index of the MRF. For *m* > 1, equation (4) represents a shear thinning fluid, while shear thickening fluids are described by *m* < 1. Note that for *m* = 1 the Herschel–Bulkley model reduces to the Bingham model.

It is noteworthy that, in the above, the post-yield parameters of the MRF such as the postyield viscosity, the consistency parameter and the fluid behavior index are assumed to be independent on the applied magnetic field. In practice, these parameters are slightly affected by the applied magnetic field. Zubieta et al. [44] have proposed field-dependent plastic models for MRF based on the original Bingham plastic and Herschel-Bulkley plastic models. The models were then applied in several researches [45, 46] with experimental agreement. In the field dependent Bingham and Herschel-Bulkley model, the rheological properties of

MRF depend applied magnetic field and can be estimated by the following equation

 γ

2

 α (12)

*YS* are determined from

*YS* is the saturation moment index of the *Y* 

∞*,* α

<sup>0</sup> ( )(2 ) *SY SY B B YY Y Y e e* − − ∞ ∞ =+ − −

where *Y* stands for a rheological parameters of MRF such as yield stress, post yield viscosity, fluid consistency and flow behavior index. The value of parameter *Y* tends from the zero

> ∞*.* α

Figure 4a schematically show the valve mode which have been used in many MR devices where the flow of the MRF between motionless plates or a duct is created by a pressure drop. The magnetic field, which is applied perpendicular to the direction of the flow, is used

α

(11)

= + *<sup>y</sup> H K*

The second working mode for controllable fluid devices is the direct shear mode. An MRF is situated between two surfaces, whereby one surface slides or rotates in relation to the other, with a magnetic field applied perpendicularly to the direction of motion of these shear surfaces. Figure 5 shows the concept of shear mode in MRF application

**Figure 5.** The concept of shear mode

The direct shear mode has been studied thoroughly especially in the MR damper technology. Masri et al. [52] proposed a curve fitting technique for representing the nonlinear restoring force of an ER device in order to characterize the ER material behaviour under static and dynamic loading over a wide range of electric fields. Spencer et al. [53] developed a phenomenological model which is based on the improved Bouc-Wen hysteresis model to represent MR dampers. Moreover, Wereley et al. have proposed a nondimensional approach to model different types of shear damper (linear shear mode, rotary drum and rotary disc damper) [54]. In the research, the Bingham–plastic, biviscous, and Herschel–Bulkley models are considered. In terms of the behaviour of the damper under conditions of high-velocity and high field input, Lee et al. [55] recommended the Herschel-Bulkley shear model to analyze the performance of impact damper systems. Furthermore, Neelakantan et al. [56] incorporated a volume fraction profile of particles with an analytical technique for calculating the torque transmitted in clutches experiencing particle centrifuging. The effect of centrifuging at high rotational speeds and the subsequent sealing problems associated with it can be mitigated by the proposed model. Extraordinary features of the direct shear mode like simplicity, fast response, simple interface between electrical power input and mechanical power output using magnetic fields, and controllability are features that make MRF technology suitable for many applications such as dampers, brakes, clutches and polishing devices [56-59].

#### **3.3. Squeeze mode**

The third working mode of MRF is the squeeze mode shown in Figure 6. This mode has not been widely investigated. Squeeze mode operates when a force is applied to the plates in the same direction of a magnetic field to reduce or expand the distance between the parallel plates causing a squeeze flow. In squeeze mode, the MRF is subjected to dynamic (alternate between tension and compression) or static (individual tension or compression) loadings. As the magnetic field charges the particles, the particle chains formed between the walls become rigid with rapid changes in viscosity. The displacements engaged in squeeze mode are relatively very small (few millimeters) but require large forces. The squeeze mode was disclosed by Stanway et al. [60] in 1992. They studied the usage of ER fluids in squeeze mode and found that the yield stress produced under DC excitation could be several times greater than the shear mode. The same outcome was later confirmed by Monkman [61] for fluids under compressive stress. Consequently, systematic investigations have been carried out by many researchers to evaluate the mechanical and electrical properties of ER and MRF in squeeze flow. Despite the fact that the Bingham plastic model was used to describe the behaviour of ER fluid in shear mode, Nilsson and Ohlson [62] have not recommended to utilize that model in squeeze mode. Bingham parameters tested from shear mode are not well-founded for the calculation of the squeeze mode behaviour. Sproston et al. [63] characterized the performance of ER fluids in dynamic squeeze mode using a bi-viscous model under a constant potential difference or by a constant field. Later on, Sproston and El Wahed [64] utilized the model to assess the fluid's response to a step-change in the applied field and the influence of the size of solid phase. Even though the model was useful to predict the peak values of the input and transmitted forces [65], a more refined model is needed, according to authors, to predict the detailed temporal variations. Therefore, a new approach of modified bi-viscous model was developed by El Wahed et al. in order to model the behaviour of an ER squeeze mode cell under dynamic conditions [66, 67]. Furthermore, Yang and Zhu [68] extended this latter model by incorporating Navier slip condition to obtain the radial velocity, pressure gradient, pressure and squeeze force.

**Figure 6.** The concept of squeeze mode

356 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

clutches and polishing devices [56-59].

**3.3. Squeeze mode** 

The direct shear mode has been studied thoroughly especially in the MR damper technology. Masri et al. [52] proposed a curve fitting technique for representing the nonlinear restoring force of an ER device in order to characterize the ER material behaviour under static and dynamic loading over a wide range of electric fields. Spencer et al. [53] developed a phenomenological model which is based on the improved Bouc-Wen hysteresis model to represent MR dampers. Moreover, Wereley et al. have proposed a nondimensional approach to model different types of shear damper (linear shear mode, rotary drum and rotary disc damper) [54]. In the research, the Bingham–plastic, biviscous, and Herschel–Bulkley models are considered. In terms of the behaviour of the damper under conditions of high-velocity and high field input, Lee et al. [55] recommended the Herschel-Bulkley shear model to analyze the performance of impact damper systems. Furthermore, Neelakantan et al. [56] incorporated a volume fraction profile of particles with an analytical technique for calculating the torque transmitted in clutches experiencing particle centrifuging. The effect of centrifuging at high rotational speeds and the subsequent sealing problems associated with it can be mitigated by the proposed model. Extraordinary features of the direct shear mode like simplicity, fast response, simple interface between electrical power input and mechanical power output using magnetic fields, and controllability are features that make MRF technology suitable for many applications such as dampers, brakes,

The third working mode of MRF is the squeeze mode shown in Figure 6. This mode has not been widely investigated. Squeeze mode operates when a force is applied to the plates in the same direction of a magnetic field to reduce or expand the distance between the parallel plates causing a squeeze flow. In squeeze mode, the MRF is subjected to dynamic (alternate between tension and compression) or static (individual tension or compression) loadings. As the magnetic field charges the particles, the particle chains formed between the walls become rigid with rapid changes in viscosity. The displacements engaged in squeeze mode are relatively very small (few millimeters) but require large forces. The squeeze mode was disclosed by Stanway et al. [60] in 1992. They studied the usage of ER fluids in squeeze mode and found that the yield stress produced under DC excitation could be several times greater than the shear mode. The same outcome was later confirmed by Monkman [61] for fluids under compressive stress. Consequently, systematic investigations have been carried out by many researchers to evaluate the mechanical and electrical properties of ER and MRF in squeeze flow. Despite the fact that the Bingham plastic model was used to describe the behaviour of ER fluid in shear mode, Nilsson and Ohlson [62] have not recommended to utilize that model in squeeze mode. Bingham parameters tested from shear mode are not well-founded for the calculation of the squeeze mode behaviour. Sproston et al. [63] characterized the performance of ER fluids in dynamic squeeze mode using a bi-viscous model under a constant potential difference or by a constant field. Later on, Sproston and El Wahed [64] utilized the model to assess the fluid's response to a step-change in the applied field and the influence of the size of solid phase. Even though the model was useful to The stress produced by the squeeze mode is the highest stress among other modes and can be used in damping vibrations with low amplitudes and high dynamic forces [69, 70]. In vibration isolation of structural system, the unwanted vibration in a relatively high frequency range can be attenuated by activating the MR mount. Examples for vibration control are isolation engine mount [71], turbo-machinery [72] and squeeze film damper [73]. Another interesting application on the squeeze mode is related to haptic devices where the user can feel the resistance forces by touching and moving a tool [74].
