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					                                                                                                   UNIFIED ANALYSIS OF CHANNEL FLOW OF MAGNETORHEOLOGICAL FLUIDS
                                                                                                                                                               Authors: Barkan Kavlicoglu, Faramarz Gordaninejad, and
                                                                                                                                                                                     Xiaojie Wang
                                        Abstract                                                                                                                                                                                          University of Nevada, Reno
A new universal approach for the flow analysis of magneto-rheological (MR) fluids through                                                                                                                                     Department of Mechanical Engineering
channels with various surface topologies is presented in this study. It is attempted to define the

pressure loss of an MR fluid in channel flow in terms of applied magnetic field, channel surface
                                                                                                                                                                                                                                Composite and Intelligent Materials Laboratory
properties and flow rate without using the concept of yield stress, therefore eliminating the

confusion in the surface dependency and definition of shear yield stress. A channel flow

rheometer with interchangeable channel walls was built for this purpose. To examine different                                             Table 1. Dimensions of the Test Channel.
surface topographies first the channel surface roughness is varied and in the second part of the                                                                                                                                                                                                                                        Non-Dimensional Modeling
experimental study regular shaped grooves are machined to the channel surface, and models for

these two topographies are developed. A relation for non-dimensional friction factor is                                                                                                                                                                                                                                       The experimental data is processed for all surface topologies and all magnetic fields to model the

developed to predict the pressure loss of an MR fluid. This non-dimensional equation for friction                                                                                                                                                                                                                             friction factor associated with MR fluid channel flow as described in the previous section.                                                      The
                                                                                                           Two different sets of surface topologies are examined in this study. In the first part the effects of
factor is developed in terms of Mason number (ratio of viscous forces to magnetic forces) and                                                                                                                                                                                                                                 Mason number at the channel wall is defined as:
                                                                                                           channel surface roughness is examined. The surface roughness of the channel wall is varied as
dimensionless surface topology parameters. It is demonstrated that the pressure loss across the            0.4, 1.6, 3.2, 6.4, and 12.7   . For comparison purposes Figure 3 and Figure 4 provide the SEM                                                                                                                                                                                                           •
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Figure. 10 Comparison of the friction factor from the experimental results and Eq. (18) for rough
                                                                                                                                                                                                                                                                                                                                                                                                         8η f γ w
MR fluid flow channel is significantly affected by the surface properties.                                                                                                                                                                                                                                                                                                          Mnw =                                                                              (3)            surfaces.
                                                                                                           images of the flow surfaces of 1.6     and 12.7 rough surfaces.                                                                                                                                                                                                                        µ 0µ β
                                                                                                                                                                                                                                                                                                                                                                                                                    H MR 2
Using the non-dimensional model, the pressure loss for various magnetic fields and volumetric

flow rates can be represented by a single master curve for a given channel surface topology                                                                                                                                                                                                                                   and the friction factor is considered as:

without an assumption of a constitutive model for MR fluids.
                                       Methods                                                                                                                                                                                                                                                                                                                                        cf =
                                                                                                                                                                                                                                                                                                                                                                                                  1 ρ V2
                                                                                                                                                                                                                                                                                                                                                                                                   2 f m

In order to relate the pressure loss across an MR fluid flow channel to the applied magnetic field,
the flow rate and channel surface topology without using a constitutive model that utilizes the                                                                                                                                                                                                                               where,    is the MR fluid density and                  is the mean fluid velocity given as Vm =
                                                                                                                                                                                                                                                                                                                                                                                                             , in which    is
concept of the shear yield stress it is needed to relate the total shear stress, τ , and shear strain                                                                                                               Figure. 6 Pressure drop and displacement profile for 3.2      surface roughness for various magnetic      the channel cross-sectional area. The geometric parameters and the material properties for the
rate,   γ to the measured quantities: pressure drop, ∆P , and volumetric flow rate, Q . The value of                                                                                                                                                                                                                         MR fluid and that are used in Eqs. (3) and (4) are listed in Table 3 and Table 4, respectively.

the shear stress at the wall,   τw is determined by pressure difference across the channel as:
                                                   g ∆P                                      (1)                                                                                                                                                                                                                                                  Table 3. Material Properties and Geometric Parameters.
                                               τw = c
                                                    2l c

where        is the channel gap and       is the channel length. For a non-Newtonian fluid flow, the                              Figure. 3 SEM image of 1.6    rough sample.
shear stress at the wall can be defined in terms of Q as (details can be found in[1]):                                                                                                                                                                                                                                                                                                                                                                                                Figure. 11 Comparison of the friction factor from the experimental results and Eq. (19) for grooved

                                                      6Q  2n + 1                           (2)                                                                                                                                                                                                                                                                                                                                                                                      surfaces.
                                             γw =               
                                                     gc 2 wc  3n 

The term in paranthesis in Eq. (2) can be treated as the Rabinowitch correction factor for flow in                                                                                                                                                                                                                                                                                                                                                                                    It can also be concluded that the pressure drop across a flow channel is directly proportional to

uniform rectangular channels. For a Newtonian fluid, n=1; for an MR fluid, n can be determined as                                                                                                                                                                                                                                                                    Table 4. MR Fluid Properties.                                                                                    for a given surface topology, as follows:

a function of magnetic field, B, from the flow curve plots of ln τ (or ln ∆P ) against ln Q for various
magnetic fields.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      ∆P ∝ H MR (2λ − m )Q 2−λ                                             (11)

                   Experimental study and
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      It can be stated that, the change in the magnetic field along the channel has a more profound
                 Non-Dimensional Modeling                                                                                                                                                                                                                                                                                                                                                                                                                                             effect when compared to the effects of varying the volumetric flow rate.

The main goal of the experimental study was to perform rheological measurements on MR fluids                                                                                                                        Figure. 7 Comparison of pressure drop for 1.6, 3.2 and 12.7      rough surfaces for various magnetic                                                                                                                                                              To predict the pressure loss for different surface topologies, Eqs. (9) and (10) can be written in
                                                                                                                                  Figure. 4 SEM image of 12.7    rough sample.
and to develop a non-dimensional friction factor model for MR fluids for different surface                                                                                                                          fields with plexi-glass smooth surface.                                                                   After examining all experimental data and plotting the friction factor as a function of Mason                                                           dimensional form as:
                                                                                                           Nine different groove configurations made of aluminum were tested, additionally an ungrooved
topologies. A piston driven flow type rheometer with a rectangular channel was built. In                                                                                                                                                                                                                                      number, it is concluded that the friction factor can be modeled as:                                                                                                                                                                                                 −1.937
                                                                                                           configuration was also tested to compare the groove width and depth effects. Figure 5 presents                                                                                                                                                                                                                                                                                                                                           0.719                      •
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                lc ρ f Q 2  ε                           8 ηf γw            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              ∆Prough   = 341.1        2      
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             β 2 µµ
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           ) H MR Bsat 0.881 
addition, with this flow rheometer it would be possible to examine the surface roughness and               the groove cross-section and geometric dimensions. The groove depth, d and width, w are varied                                                                                                                                                                                                                                                                                                                        g c Ach  g c                                  1.119                    (12)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      f
grooved channel wall effects on the MR fluid channel flow. The MR fluid was pressurized to flow            to determine the effects of groove depth and width on the MR fluid pressure drop across the                                                                                                                                                      c f ,MR = A (channel and surface topography, H MR ) Mnwλ
                                                                                                                                                                                                                                                                                                                                                                                                                  −                                                    (5)                                                                                                                        −1.937
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       2           0.756                        •
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                lc ρ f Q  d                      8 ηf µ µ f γ w
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          0                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              ∆Pgroove = 28.7          2                                                                 (13)
through the channel between two parallel-arranged magnetic poles by means of a hydraulic                   channel. The groove configurations are given in Table 2.                                                                                                                                                                                                                                                                                                                                                              g c Ach  g c                 ( 0 f ) MR sat 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            β 2 µ µ 0.011 H 1.011 B 0.989 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          
actuator. An electromagnetic coil was built and installed to the middle of the channel to permit                                                                                                                                                                                                                              where,    is called the power index for the Mason number. The index parameter, A , is a function of

the application of magnetic flux density normal to the flow direction. The pressure drop, is                                                                                                                                                                                                                                  both surface topography and magnetic field intensity and can be expressed as:                                                                           Using Eqs. (12) and (13) one can predict the pressure loss associated with MR fluid flow through

measured using two pressure transducers across the channel. The flow rheometer is connected                                                                                                                                                                                                                                                                                                                                                                                           channels with rough and grooved walls by just having the knowledge of surface properties,
to an Instron Model 8821S servo hydraulic actuator system. The input profile was a double ramp                                                                                                                                                                                                                                                                                                     µ0µ f H MR                                                        (6)            magnetic field acting on the channel and fluid physical properties such as the viscosity, density
                                                                                                                                                                                                                                                                                                                                                                       A = f (surface topography)             
                                                                                                                                                                                                                                                                                                                                                                                                   Bsat                                                                             and saturation magnetization. Eqs. (12) and (13) are valid for a channel gap of 1 mm and
displacement profile, which generates constant velocity.              The electromagnet activation input                          Figure. 5 Grooved channel cross-section and dimensions.
current for each velocity profile was varied as 0 A, 0.5 A, 1.0 A, 2.0 A and 3.0 A. The experimental                                                                                                                                                                                                                          where, m is a parameter to be determined from experimental results and                                                             is the saturation    magnetic fields less than 0.405 T.
                                                                                                           Table 2. Dimensions of Different Groove Configurations Tested in the Flow Type Rheometer.
setup and schematic of the experiments are shown in Figure 1 and Figure 2, respectively. The                                                                                                                                                                                                                                  magnetization of MR fluid. Consequently, Eq. (5) can be written as:

dimensions of the test channel are given in Table 1.
                                                                                                                                                                                                                                                                                                                                                                                                                                           •
                                                                                                                                                                                                                                                                                                                                                                                                                                      8η f γ w
                                                                                                                                                                                                                                                                                                                                                                                                                                                            −λ                                      Conclusions and Future Work
                                                                                                                                                                                                                                                                                                                                                       c f ,MR = f (surface topography )                                                                             (7)
                                                                                                                                                                                                                                                                                                                                                                                                                                 m                  m 
                                                                                                                                                                                                                                                                                                                                                                                         (µ0µ                              f ) β
                                                                                                                                                                                                                                                                                                                                                                                                                               1−              2− m

                                                                                                                                                                                                                                                                                                                                                                                                                                    λ   2
                                                                                                                                                                                                                                                                                                                                                                                                                                          H MR λ Bsat λ 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     In this study it is attempted to develop a new approach to model for magnetorheological (MR)
                                                                                                                                                                                                                    Figure. 8 Total pressure loss for various magnetic fields for a piston velocity of 2.54 mm/s.                                                                                                                                                                     fluid flow in channel with different surface topologies. The surface roughness of the channel is
                                                                                                                                                                                                                                                                                                                              Here, the term in the last parenthesis of Eq. (7) is referred as the modified Mason Number, and                                                         varied in the first part of the experiments; secondly the channel surface is modified by machining
                                                                                                                                                                                                                    Figure 9 presents the pressure loss for configurations G0, G1, G2, G3 and G6 for         of 0 T, 0.19 T   defined as:                                                                                                                                             grooves to the surface. A non-dimensional model is developed for each surface that relates the
                                                                                                                                                                                                                    and 0.40 T corresponding to 0 A, 0.5 A and 3.0 A of input current, respectively. Similarly, for the                                                                                                 •                                                             friction factor of MR fluid to the surface properties, volumetric flow rate and magnetic field. The
                                                                                                                                                                                                                                                                                                                                                                                                             8η f γ w
                                                                                                                                                                                                                    off-state case (0 A) the pressure losses are almost identical in the tested range of the volumetric                                            Mnw,mod =                                                                                           (8)            experimental studies demonstrated that the pressure loss for an MR fluid across a channel with
                                                                                                                                                                                                                                                                                                                                                                                                  1− m         (2−m ) m
                                                                                                                                  Experimental Results                                                              flow rate for all samples. This would suggest that the grooved surface configurations does not
                                                                                                                                                                                                                                                                                                                                                                                    (µ µ )
                                                                                                                                                                                                                                                                                                                                                                                       0     f
                                                                                                                                                                                                                                                                                                                                                                                                         β H MR λ Bsat λ

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      different surface properties is directly proportional to H MR (2 λ − m )Q 2−λ , where m is a parameter for
                                                                                                                                                                                                                    affect the pressure loss of MR fluid without the applied magnetic field. However, when the                The final step in developing the non-dimensional model for friction factor of MR fluids is to                                                           channel surface topography and                  is the power index that is determined from experimental
                                                                                                           Figure 6 presents a typical experimental result for channel with 3.2      surface roughness and for
                                                                                                                                                                                                                    magnetic field is applied, there is an apparent increase in the pressure loss when compared with          determine the function f in terms of channel and surface properties. After all experimental data                                                        results. It is proposed that the power index,                    , is only a function of MR fluid properties, since it
                                                                                                           various magnetic fields,        . As can be seen, the pressure drop increases as the magnetic field
                                                                                                                                                                                                                    the ungrooved surface.                                                                                    are analyzed, it is concluded that the friction factor for channel with rough surfaces and grooved                                                      is observed that      does not change with surface properties. Therefore, the surface topology
                                                                                                           increases for a given profile. Similar results were obtained for all configurations and different
                                                                                                                                                                                                                                                                                                                              channel surfaces can be given as follows:                                                                                                               affects the        term and consequently affects the MR fluid pressure loss. Using the relation
                                                                                                           velocities. The pressure loss for different surface roughnesses as a function of the volumetric flow
                                                                                                           rate are given in Figure 7 for magnetic field strengths of 0 T, 0.191 T and 0.405 T corresponding to                                                                                                                                                                                  0.719                  •
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     developed for the friction factor of MR fluid channel flow, the pressure drop of an MR fluid can be
                                                                                                                                                                                                                                                                                                                                                                                    ε                           8η f γ w              
                                                                                                                                                                                                                                                                                                                                                         c f , MR ,rough     = 341.1 
                                                                                                           0, 0.5 and 3.0 amp input currents, respectively. The pressure loss of ungrooved configuration is                                                                                                                                                                          gc                 ( 0 f)
                                                                                                                                                                                                                                                                                                                                                                                                          β 2 µ µ 0.119 H 1.119 B 0.881 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      (9)            estimated without referring to a constitutive model or using the concept of shear yield stress.
                                                                                                                                                                                                                                                                                                                                                                                                                           MR    sat
                                                                                                           also presented for comparison. It can be concluded that for the off-state case (0 T) the viscous                                                                                                                                                                                                                                                                           The experiments in this study are performed for a channel gap of 1 mm. Research is ongoing on
                                                                                                                                                                                                                                                                                                                                                                                             0.756                                    •
                                                                                                                                                                                                                                                                                                                                                                                    d                                         8η f γ w                             (10)           the gap effects on the non-dimensional friction factor of MR fluids. Experiments are in progress
                                   Figure. 1 Experimental setup.                                           pressure losses are almost identical in the tested range of the volumetric flow rate for all the                                                                                                                                              c f , MR , groove   = 28.7  
                                                                                                                                                                                                                                                                                                                                                                                                      ( 0
                                                                                                                                                                                                                                                                                                                                                                                                      β 2 µµ
                                                                                                                                                                                                                                                                                                                                                                                                                                ) H MR Bsat 0.989 
                                                                                                                                                                                                                                                                                                                                                                                     gc                                              1.011
                                                                                                                                                                                                                                                                                                                                                                                                                           f
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     for channel gaps of 200     . In addition, the effects of iron particle volume percentage of MR fluids
                                                                                                           samples, which suggests that the surface roughness does not affect the viscous pressure loss.
                                                                                                                                                                                                                                                                                                                              It is observed from experimental data that the power index,                                                       , in Eqs. (9) and (10) only depends   is also under investigation. With these studies completed, it would be possible to define the MR
                                                                                                           However, when the magnetic field is applied, the effect of surface roughness is apparent. For a
                                                                                                                                                                                                                                                                                                                              on the material properties of the MR fluid.                            is neither a function of the surface topology nor the                            fluid pressure drop for different gap sizes, different particle volume percentages and surface
                                                                                                           given magnetic field as the surface roughness increases, an increase in the pressure loss is
                                                                                                                                                                                                                                                                                                                              applied magnetic field. For Lord MRF-132AD fluid, the power index is determined to be 1.937.                                                            properties.
                                                                                                           observed. The increase in the pressure loss in more effective at high magnitude magnetic fields,

                                                                                                           such as for 0.4 T. Figures 8 presents the MR fluid pressure loss for all surface roughnesses and
                                                                                                                                                                                                                                                                                                                              The friction factor for the given MR fluid in channel flow with different surface topologies is                                                                                 Acknowledgements
                                                                                                                                                                                                                                                                                                                              determined as:
                                                                                                           magnetic fields for actuator piston velocity of 2.54 mm/s. The viscous pressure drop is not
                                                                                                                                                                                                                                                                                                                              Figures 10 and 11 present the friction factor as a function of modified Mason Number for rough                                                          The authors would like to thank partial support of the U. S. Army Research Office for this project
                                                                                                           affected by the surface roughness for all velocities. It can also be concluded that, by increasing the
                                                                                                                                                                                                                                                                                                                              surfaces and grooved surfaces, respectively. It can be observed that Eqs. (9) and (10) represent all                                                    under the direction of Dr. Gary Anderson, program director. The assistance of MR. Gregory
                                                                                                           surface roughness it would be possible to obtain the same pressure drop by decreasing the
                                                                                                                                                                                                                    Figure. 9 Comparison of pressure drop for grooved channel wall configurations G0, G1, G2, G3 and          experimental data for a given surface topology with a single non-dimensional curve.                                                                     Hitchcock from Advanced Materials and Devices Inc., Reno, Nevada is acknowledged.
                                                                                                           magnetic filed. This conclusion is essential in the point of power consumption of an MR fluid
                         Figure. 2 Schematic of the experimental setup.                                                                                                                                             G6 for various magnetic fields.