applied sciences

Article Modeling the Response of Magnetorheological Fluid Dampers under Seismic Conditions

Darson Dezheng Li 1 , Declan Finn Keogh 1, Kevin Huang 1, Qing Nian Chan 1, Anthony Chun Yin Yuen 1 , Chris Menictas 1, Victoria Timchenko 1 and Guan Heng Yeoh 1,2,*

1 School of Mechanical and Manufacturing Engineering, the University of New South Wales, UNSW, Sydney, NSW 2052, Australia; [email protected] (D.D.L.); [email protected] (D.F.K.); [email protected] (K.H.); [email protected] (Q.N.C.); [email protected] (A.C.Y.Y.); [email protected] (C.M.); [email protected] (V.T.) 2 Australian Nuclear Science and Technology Organization (ANSTO), Locked Bag 2001, Kirrawee, NSW 2232, Australia * Correspondence: [email protected]; Tel.: +61-2-9385-4763



Received: 1 August 2019; Accepted: 24 September 2019; Published: 8 October 2019


Abstract: Magnetorheological (MR) fluid is a smart material fabricated by mixing magnetic-responsive particles with non-magnetic-responsive carrier fluids. MR fluid dampers are able to provide rapid and reversible changes to their damping coefficient. To optimize the efficiency and effectiveness of such devices, a computational model is developed and presented where the flow field is simulated using the computational fluid dynamics approach, coupled with the magnetohydrodynamics model. Three different inlet pressure profiles were designed to replicate real loading conditions are examined, namely a constant pressure, a sinusoidal pressure profile, and a pressure profile mimicking the 1994 Northbridge earthquake. When the MR fluid damper was in its off-state, a linear pressure drop between the inlet and the outlet was observed. When a uniform perpendicular external magnetic field was applied to the annular orifice of the MR damper, a significantly larger pressure drop was observed across the annular orifice for all three inlet pressure profiles. It was shown that the fluid velocity within the magnetized annular orifice decreased proportionally with respect to the strength of the applied magnetic field until saturation was reached. Therefore, it was clearly demonstrated that the present model was capable of accurately capturing the damping characteristics of MR fluid dampers.

Keywords: computational fluid dynamics; magneto-hydrodynamic model; magnetorheological fluid damper; seismic control

1. Introduction Advancements in nanotechnology have allowed for the fabrication of micrometer- and nanometer-sized magnetic-responsive particles, and when mixed with non-magnetic-responsive carrier fluids and at least one type of thiocarbamate and/or thiophosphrous additives, a type of smart fluid material is developed, known as the magnetorheological (MR) fluid [1]. Under the application of external magnetic fields, magnetic-responsive particles interact with each other through dipole moments to form chain-like structures along the direction of the applied magnetic field [2]. These chain structures contribute to resisting shear stress in the perpendicular direction to the magnetic field. Therefore, by varying the magnitude of external magnetic fields, MR fluids can exhibit a rapid, reversible, and tuneable transition between a liquid state (off-state) and a more viscous semisolid state (on-state) [3]. The initial development of MR fluid devices can be traced back to the late 1940s by Rabinow [4]. Carbonyl iron particles obtained from the thermal decomposition of iron pentacarbonyl are the most

Appl. Sci. 2019, 9, 4189; doi:10.3390/app9194189 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 4189 2 of 16 Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 17 Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 17 commonlycommonly usedused magnetic-responsive magnetic-responsive particles particles in MR in fluids MR duefluids to their due large to saturationtheir large magnetization. saturation commonly used magnetic-responsive particles in MR fluids due to their large saturation magnetization.Typical base fluids Typical used base in MRfluids fluids used include in MR mineralfluids include and silicone mineral oils, and synthetic silicone hydrocarbons,oils, synthetic magnetization. Typical base fluids used in MR fluids include mineral and silicone oils, synthetic hydrocarbons,polyethers, polyesters, polyethers, and polyesters, water [5]. and water [5]. hydrocarbons, polyethers, polyesters, and water [5]. Bell et et al. al. [6] [6] studied studied the the influences influences of of particle particle sh shapeape on on the the properties properties of of MR MR fluids. fluids. Experimental Experimental Bell et al. [6] studied the influences of particle shape on the properties of MR fluids. Experimental resultsresults showed showed that that the the off-state off-state viscosity viscosity of of MR MR fl fluidsuids containing containing only only wire wiress was was substantially substantially higher higher results showed that the off-state viscosity of MR fluids containing only wires was substantially higher thanthan MR MR fluids fluids containing containing only spherical particles. The The effects effects of particle particle size size on on the the properties properties of of MR MR than MR fluids containing only spherical particles. The effects of particle size on the properties of MR fluidsfluids were examined inin ReferencesReferences [[7,8].7,8]. ItIt waswas shownshown thatthat largerlarger magnetic-responsive magnetic-responsive particles particles had had a fluids were examined in References [7,8]. It was shown that larger magnetic-responsive particles had alarger larger yield yield stress stress compared compared to particles to particles with with smaller smaller sizes. sizes. However, However, MR fluids MR withfluids smaller with smaller particle a larger yield stress compared to particles with smaller sizes. However, MR fluids with smaller particlesizes showed sizes showed higher stabilityhigher stability against against sedimentation. sedimentation. Lemaire Lemaire et al. [ 7et] explainedal. [7] explained these etheseffects effects as the particle sizes showed higher stability against sedimentation. Lemaire et al. [7] explained these effects asresults the results of fluctuations of fluctuations of particle of particle positions positions in the particlein the particle chain. chain. as the results of fluctuations of particle positions in the particle chain. Figure 11 showsshows thethe threethree basicbasic modesmodes ofof operationsoperations ofof MRMR fluidfluid devicesdevices [[9].9]. The The valve valve mode, mode, or or Figure 1 shows the three basic modes of operations of MR fluid devices [9]. The valve mode, or thethe pressure-driven pressure-driven flow flow mode, mode, is is often often observed observed in in dampers, dampers, shock shock absorbers, absorbers, and and servo-valves servo-valves [5]. [5]. the pressure-driven flow mode, is often observed in dampers, shock absorbers, and servo-valves [5]. The shear mode is often often utilized utilized in clutches clutches and and br brakes.akes. The The squeeze squeeze mode mode has has not not been been as as thoroughly thoroughly The shear mode is often utilized in clutches and brakes. The squeeze mode has not been as thoroughly studiedstudied as the previousprevious twotwo operating operating modes modes but but can can be be used used as as low low amplitude amplitude vibration vibration dampers dampers [9]. [9].studied as the previous two operating modes but can be used as low amplitude vibration dampers [9].

(a) (b) (c) (a) (b) (c) Figure 1. BasicBasic operational operational modes modes for for MR MR fluid fluid devices: devices: ( (aa)) valve valve mode, mode, ( (bb)) shear shear mode, mode, and and ( (cc)) squeezesqueezeFigure 1. mode. mode. Basic operational modes for MR fluid devices: (a) valve mode, (b) shear mode, and (c) squeeze mode. Among these these applications, applications, MR MR fluid fluid dampers dampers have have received received the the most most extensive extensive research research interest. interest. MR fluid fluidAmong dampers, these applications, first first introduc introduced MRed fluid in 1992 dampers by Carlson have received and Chrzan the most [10], [10 extensive], utilize researchthe controllable interest. propertiesMR fluid dampers,of MR fluids, fluids, first including introduc viscosity viscosityed in 1992 and and by shear shear Carlson stress, stress, and while while Chrzan only only requiring[10], requiring utilize a a low lowthe power powercontrollable input. input. Theproperties peak power of MR requirement fluids, including in some viscosity commercially and shear av available stress,ailable while dampers dampers only requiringis is less less than than a low10 10 W, W,power where input. the powerThe peak fromfrom power aa small small requirement camera camera battery battery in some would would commercially allow allow for morefor av moreailable than than one dampers hourone ofhour is continuous less of continuousthan operation10 W, whereoperation [11 ,12the]. [11,12].Thepower schematic fromThe schematic a diagramsmall camera diagram of a typical battery of a MRtypical would fluid MR allow damper fluid for damper is more shown than is inshown Figureone hourin2 .Figure of continuous 2. operation [11,12]. The schematic diagram of a typical MR fluid damper is shown in Figure 2.

Figure 2. Schematic diagram of an MR fluid damper [13]. Figure 2. Schematic diagram of an MR fluidfluid damper [[13].13].

Compared to to conventional dampers, dampers, the the dampin dampingg coefficient coefficient of of MR MR fluid fluid dampers can be continuouslyCompared and to quickly conventional adjusted dampers, by modifying the dampin the strengthg coefficient of the applied of MR external fluid dampers magnetic can field. be continuously and quickly adjusted by modifying the strength of the applied external external magnetic magnetic field. field. Hence, they they can be effectively effectively used as shock abso absorbersrbers to increase passenger comfort and and safety in automobileHence, they ground can be vehicleseffectively [14–18], used asrailway shock vehiclesabsorbers [19–23], to increase and aircraft passenger [24–27]. comfort They and can safety also bein builtautomobile into various ground civil vehicles structures, [14–18], such railway as building vehicless [28–30], [19–23], bridges and aircraft [31,32], [24–27].and piping They systems can also [33], be built into various civil structures, such as buildings [28–30], bridges [31,32], and piping systems [33], Appl. Sci. 2019, 9, 4189 3 of 16 automobile ground vehicles [14–18], railway vehicles [19–23], and aircraft [24–27]. They can also be built into various civil structures, such as buildings [28–30], bridges [31,32], and piping systems [33], in order to minimize their responses to vibrations caused by strong winds and earthquakes. Studies on these systems were reviewed by Housner et al. [34], Spencer Jr. and Nagarajaiah [35], and Jung et al. [36]. In order to optimize the performance of MR fluid dampers in these applications, it is of fundamental importance to quantitatively examine the response (output) of MR fluid dampers under various input pressure profiles. In some modern studies [2,37–39], MR fluids and other particulate mixtures are modelled using coupled computational fluid dynamics and discrete phase modelling approaches. However, this method is very computationally intensive, and can only be applied in modelling MR fluids in microscale configurations. The macroscale behavior of MR fluid dampers has been experimentally investigated in various studies [40–42]. However, effective approaches in numerically modelling such systems at the device level remain underdeveloped. In this study, a macroscopic computational model was developed to simulate the response of MR fluid dampers under various input pressure profiles, including constant pressure, a sinusoidal pressure profile, and a seismic pressure based on the Northridge earthquake in 1994, utilizing the computational fluid dynamics and magnetohydrodynamics models. The effectiveness of damping was investigated by examining the static pressure at different locations along the annular orifice of the MR fluid damper under a range of applied external magnetic fields. This novel computational approach enabled the macroscale simulation of MR fluids by significantly reducing the computational resources required compared to other computational approaches, where only microscale simulations are possible. It can also significantly reduce the human and economic resources required in the optimization of MR fluid dampers when compared to experimental approaches.

2. Mathematical Model

2.1. Computational Fluid Dynamics In computational fluid dynamics, the continuity and Navier–Stokes governing equations are used to simulate the flow behavior. Given the small size of the geometry, and hence the small hydrodynamic length scale, the flow can be assumed to be laminar. Also, assuming MR fluids are incompressible, the continuity equation of the fluid can be written as:

u = 0 (1) ∇· where u is the vector field of the mixture velocity. The momentum of the fluid is governed by the Navier–Stokes equation:

∂u 2 1 + (u )u = νe f f u + p + S (2) ∂t ·∇ − ∇ ρe f f ∇ where t is the flow time; νe f f and ρe f f are the respective effective kinematic viscosity and density of the MR fluid; p is the flow pressure gradient, which contributes to the internal source term; and S is the ∇ external source term due to the magnetic response of the MR fluid. The effective kinematic viscosity and density of the MR fluid is considered to be the weighted average between the carrier fluid (subscript f ) and the particle suspensions (subscript p). The effective density of the MR fluid is given as:

ρ = (1 α)ρ + αρp (3) e f f − f where α is the volume fraction of the magnetic-responsive particles. The effective dynamic viscosity µe f f of a two-phase mixture containing rigid spherical particles was first predicted by Einstein [43] as: Appl. Sci. 2019, 9, 4189 4 of 16

µe f f = µ f (1 + k1α) (4) whereAppl.µ f Sci.is the2019, dynamic9, x FOR PEER viscosity REVIEW of the continuous phase (carrier fluid), and k1 is the Einstein4 of 17 shape factor. For rigid spheres, the Einstein shape factor is given by k1 = 2.5. However, this approximation is onlywhere ideal for𝜇 low is the particle dynamic volume viscosity fractions of the continuous (α < 0.05). phase (carrier fluid), and k1 is the Einstein shape Forfactor. higher For rigid particle spheres, volume the fractions,Einstein shape Vand factor [44] proposedis given by the k1 = following 2.5. However, expression, this approximation which accounts is only ideal for low particle volume fractions (α < 0.05). for both mutual hydrodynamic interactions and particle collisions: For higher particle volume fractions, Vand [44] proposed the following expression, which accounts for both mutual hydrodynamic interactions" and particle collisions:# k α + θ(k k )α2 = 1 2 1 µe f f µ f exp αθ+−()− α2 (5) kkk1211 BHα μμ= exp eff f −− α (5) 1 BH where k1 is the Einstein shape factor, k2 is the shape factor for collision doublets, θ is the collision timewhere constant, k1 is and the BEinsteinH is the shape hydrodynamic factor, 𝑘 is interaction the shape factor constant. for collision For spherical doublets, particles θ is the [collision44], k1 = 2.5, k2 = 3.175,time constant,θ = 4, and andB BHH =is 0.609.the hydrodynamic interaction constant. For spherical particles [44], k1 = 2.5, k2 = 3.175, θ = 4, and BH = 0.609. Once the effective dynamic viscosity µe f f is determined, the effective kinematic viscosity νe f f can be calculatedOnce via: the effective dynamic viscosity 𝜇 is determined, the effective kinematic viscosity 𝜈 can be calculated via: µe f f νe f f = (6) μρe f f ν = eff eff ρ (6) 2.2. Magneto-Hydrodynamic Model eff The magneto-hydrodynamic (MHD) model simulates the interaction between an electrically conducting2.2. Magneto-Hydrodynamic fluid flow and the appliedModel electromagnetic field. There are two main aspects of the coupling between theThe fluidmagneto-hydrodynamic flow field and the (MHD) magnetic model field: simulates electromagnetic the interaction induction between andan electrically Lorentz force. Electromagneticconducting fluid induction flow and is thethe eappliedffect of electromagne electric currenttic field. induction There are as atwo result main of aspects the movement of the of conductivecoupling material between in the a magneticfluid flow field.field and Lorentz the magnet forceic is field: generated electromagnetic as a result induction between and the Lorentz interaction of theforce. electric Electromagnetic current and induction the magnetic is the field.effect of electric current induction as a result of the movement of conductive material in a magnetic field. Lorentz force is generated as a result between the In MR fluid dampers, the macroscopic physics can be understood as occurring due to a Lorentz interaction of the electric current and the magnetic field. force. AsIn illustrated MR fluid dampers, in Figure the3, whenmacros thecopic electromagnetic physics can be understood coil is in theas occurring on-state, due a magnetic to a Lorentz field B is generatedforce. As inillustrated the horizontal in Figure direction. 3, when the When electromagnetic the damper coil is is actuated,in the on-state, the MRa magnetic fluid experiences field B is a changinggenerated magnetic in the field. horizontal According direction. to Faraday’sWhen the lawdamper of induction, is actuated, an the electromotive MR fluid experiences force (EMF) a is inducedchanging and amagnetic current field. will flowAccording around to Faraday’s the annular law orificeof induction, with currentan electromotive density forcej. The (EMF) interaction is betweeninduced the currentand a current density will and flow magnetic around fieldthe annular creates orifice a force with that current opposes density the motion j. The ofinteraction the fluid [45]. This forcebetween is called the current a Lorentz density Force and and magnetic is equal field to thecrea crosstes a force product that of opposes the current the motion density of andthe fluid magnetic field [45].j B This. It is force this is interaction called a Lorent of thez Force fluid and and is magneticequal to the field cross that product creates of the a damping current density force and will magnetic× field 𝒋𝑩. It is this interaction of the fluid and magnetic field that creates a damping force result in a higher pressure differential across the annular orifice. and will result in a higher pressure differential across the annular orifice.

Figure 3. Schematic of the magnetohydrodynamics model for a magnetorheological fluid damper. Figure 3. Schematic of the magnetohydrodynamics model for a magnetorheological fluid damper. Appl. Sci. 2019, 9, 4189 5 of 16

In the MHD model, Maxwell’s equations in the following form are used to describe the electromagnetic fields: B = 0 (7) ∇ · ∂B E = (8) ∇ × − ∂t where B is the magnetic field in T, and E is the electric field in V/m. In addition:

∂j H = j+ (9) ∇ × ∂t D = q (10) ∇ · where H and D are the induction fields for the magnetic and electric fields, respectively. q is the electric charge density in C/m3, and j is the electric current density vector in A/m2. For media with ∂D sufficiently high conductivity, the electric charge density q and the displacement current ∂t are considered negligible. The induction fields H and D can be determined by:

1 H = B (11) µ

D = εE (12) where µ is the magnetic permeability and ε is the electric permittivity. The electric current density vector j may be determined using two different approaches, namely the magnetic induction method and the electric potential method. The magnetic induction method is derived from Ohm’s law and the Maxwell’s equation, whereas the electric potential method solves the electric potential equation to determine the current density using Ohm’s law. In this study, the electric potential method is used. The current density is defined by Ohm’s law using:

j = σE (13) where σ is the electrical conductivity of the medium. In a flow under an external magnetic field B, Ohm’s law can be expressed as: j = σ(E + U B) (14) × where U is the velocity of the fluid flow. In the electric potential approach, the electric field E is given as: ∂A E = ϕ (15) −∇ − ∂t where ϕ is the scalar potential and A is the vector potential. When the magnetic field is static and much less than B0, the Ohm’s law shown in Equation (14) can be expressed as:

j = σ( ϕ + (U B )) (16) −∇ × 0 Due to the principle of conservation of electric charges in media with sufficiently high conductivity:

j = 0 (17) ∇· Therefore, the electric potential equation can be written as:

2ϕ = (U B ) (18) ∇ ∇ · × 0 Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 17

Appl. Sci. 2019, 9, 4189 ∇=∇⋅×2ϕ () 6 of 16 UB0 (18)

Finally, the following equation is derived based on the Ohm’s law and Maxwell’s equations: Finally, the following equation is derived based on the Ohm’s law and Maxwell’s equations: ∂B 1 ∂B+⋅∇=()UB1 ∇+⋅∇2 BBU() + (U )B = 2B + (B )U (19)(19) ∂∂tt · ∇ μσµσ∇ · ∇

The coupling between between the the MHD model and the Navier–Stokes equation is achieved via the additional source term S𝑺 in in Equation Equation (2) (2) due due to to the the Lorentz Lorentz force, force, given given by: by:

SF==× jB (20) S = FLorentzLorentz = j B (20) × 3. Numerical Model 3. Numerical Model 3.1. Material Properties 3.1. Material Properties The MR fluid used in this computational model is assumed to be a mixture of water at 20 ◦C The MR fluid3 used in this computational model is assumed to be a mixture3 of water at 20 °C (𝜌 (ρ f = 999 kg/m , µ f = 1.003 cP) and spherical iron particles (ρ f = 7870 kg/m ) with a particle volume 3 3 = 999 kg/m , 𝜇 = 1.003 cP) and spherical iron particles (𝜌 = 7870 kg/m ) with a particle volume fraction α = 0.20. The effective density ρe f f and viscosity µe f f were determined using Equations (3) and fraction(5), respectively. 𝛼 = 0.20. The The magnetic effective permeability density 𝜌 of and the MRviscosity fluid was𝜇 takenwere todetermined be 4 H/m, andusing the Equations electrical (3)conductivity and (5), respectively. of the mixture The was magneticσ = 15,000 permeability S/m. of the MR fluid was taken to be 4 H/m, and the electrical conductivity of the mixture was 𝜎 = 15,000 S/m. 3.2. Computational Domain 3.2. Computational Domain In order to simplify the underlying problem and reduce the computational power required, only MR fluidIn around order to the simplify annular the orifice underlying region (seeproblem Figure and2) wasreduce modeled the computational in this study. power A two-dimensional required, only MRaxisymmetric fluid around configuration the annular was orifice implemented region to(see reduce Figure the 2) computational was modeled eff ort.in this The study. computational A two- dimensionaldomain is shown axisymmetric in Figure configuration4. Numerical was results implemented were extracted to reduce at sixthe equallycomputational spaced effort. locations The computationalthroughout the domain domain is marked shown as inO 1Figure–O6 The 4. shaded Numeri regioncal results between were locations extractedO3 atand sixO 4equallyrepresents spaced the locationsannular orifice throughout where the the domain MR fluid marked is affected as ① by– the⑥The external shaded magnetic region field.between locations ③ and ④ represents the annular orifice where the MR fluid is affected by the external magnetic field.

Figure 4. Computational domain with results extraction locations. Figure 4. Computational domain with results extraction locations. 3.3. Mesh 3.3. Mesh To numerically obtain the solutions for the aforementioned governing equations, a structured meshTo consisting numerically of rectangular obtain the elementssolutions was for createdthe aforementioned for the two-dimensional governing equations, computational a structured domain meshsince suchconsisting elements of rectangular are considered elements to be was the mostcreated suitable for the due two-dimensional to its high conformity computational to the geometry domain sinceand low such aspect elements ratio. are considered to be the most suitable due to its high conformity to the geometry and lowA mesh aspect sensitivity ratio. analysis was performed in order to arrive at a mesh-independent solution. To thisA mesh end, based sensitivity on the analysis unmagnetized was performed case, a mesh in or convergenceder to arrive study at a mesh-independent was conducted by graduallysolution. Todecreasing this end, thebased element on the sizeunmagnetized and observing case, the a mesh changes convergence in the maximum study was velocity conducted in the by fluidgradually flow, decreasingas shown in the Figure element5. It size can beand seen observing that the the maximum changes velocityin the maximum reached itsvelocity asymptotic in the value fluid (toflow, four as shownsignificant in Figure figures) 5. onceIt can the be mesh seen reachedthat the amaximum 0.25 mm element velocity size. reached For thisits asymptotic element size, value there (to were four a significanttotal number figures) of 48,000 once mesh the mesh elements, reached and a the 0.25 maximum mm element velocity size. within For this the element domain size, was there 0.3737 were m/s. a total number of 48,000 mesh elements, and the maximum velocity within the domain was 0.3737 m/s. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 17 Appl. Sci. 2019, 9, 4189 7 of 16 Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 17 0.3738

0.3736 0.3738 0.3736 0.3734 0.3734 0.3732 0.3732 0.373 0.373

Maximum Velocity (m/s) Velocity Maximum 0.3728

Maximum Velocity (m/s) Velocity Maximum 0.3728 0.3726 0.37260 40,000 80,000 120,000 160,000 0 40,000 80,000 120,000 160,000 NumberNumber of Elements of Elements

FigureFigureFigure 5. Mesh 5. 5. Mesh Meshconvergence convergence study. study. study. 3.4. Numerical Scheme 3.4. Numerical3.4. Numerical Scheme Scheme The finite volume method was used to discretize the fluid governing equations. Semi-Implicit The finiteThe volume finite volume method method was used was toused discretize to discretize the fluidthe fluid governing governing equations. equations. Semi-Implicit Semi-Implicit Method for Pressure Linked Equations (SIMPLE) and Pressure-Implicit with Splitting of Operators Method forMethod Pressure for Pressure Linked Linked Equations Equations (SIMPLE) (SIMPLE) and andPressure-Implicit Pressure-Implicit with with Splitting Splitting of of Operators Operators (PISO)(PISO) schemes schemes were were tested tested for for the the coupling coupling betweenbetween pressure pressure and and velocity, velocity, whilst whilst first-order first-order and and (PISO) schemes were tested for the coupling between pressure and velocity, whilst first-order and second-ordersecond-order upwind upwind methods methods were were tested tested forfor spatialspatial discretization. discretization. The The underrelaxation underrelaxation factor factor for for second-order upwind methods were tested for spatial discretization. The underrelaxation factor for the pressurethe pressure correction correction using using the the SIMPLESIMPLE scheme scheme was was set set to a to value a value of 0.3. of For 0.3. the For PISO the scheme, PISO scheme,the the pressure correction using the SIMPLE scheme was set to a value of 0.3. For the PISO scheme, the the underrelaxationunderrelaxation factor was was set set to to unity, unity, taking taking adva advantagentage of ofthe the solutions solutions of the of thepressure pressure correction correction underrelaxation factor was set to unity, taking advantage of the solutions of the pressure correction in twoin two stages. stages. Owing Owing to to the the simplicity simplicity inin thethe geometry setup setup and and mesh mesh grids, grids, the thedifferences differences in in in two stages. Owing to the simplicity in the geometry setup and mesh grids, the differences in numericalnumerical results results between between these these two two schemes schemes werewere observed to to be be minimal minimal (≈0.1%). ( 0.1%). For Forsimplicity, simplicity, numericalthe results SIMPLE between and second-order these two upwindschemes schemes were observed were adopted. to be Transient minimal solutions (≈0.1%).≈ wereFor simplicity, attained in the SIMPLE and second-order upwind schemes were adopted. Transient solutions were attained in −4 the SIMPLEtime and by marching second-order the computations upwind schemes with a fixedwere time adopted. step of Transient 10 4s. At each solutions time step, were the attained governing in time by marching the computations with a fixed time step of 10− s. At each time step, the governing time by marchingequations thewere computations iteratively solved with and a fixed deemed time to step be converged of 10−4 s. Atuntil each the timeRoot-Mean-Square step, the governing (RMS) equationsresiduals were fell iterativelybelow 10−6. solved and deemed to be converged until the Root-Mean-Square (RMS) equations were iteratively solved6 and deemed to be converged until the Root-Mean-Square (RMS) residuals fell below 10− . residuals fell below 10−6. 3.5. Boundary Conditions 3.5. Boundary Conditions 3.5. Boundary TheConditions boundary conditions applied in this study are shown in Figure 6. The top and bottom surfacesThe boundary of the conditionscomputational applied domain in this were study defined are shown as non-slip in Figure stationary6. The topwalls and to bottomrealistically surfaces ofThe the representboundary computational the conditions surfaces domain of theapplied werepiston definedin and this housing stud as non-slipy of are the shownMR stationary fluid in damper. Figure walls 6. to The realistically top and represent bottom the surfacessurfaces of the of thecomputational piston and housing domain of were the MR defined fluid damper.as non-slip stationary walls to realistically represent the surfaces of the piston and housing of the MR fluid damper.

Figure 6. Boundary conditions applied to the computational domain.

The inlet and outlet of the domain were defined by specifying the pressure at these locations. Figure 6. Boundary conditions applied to the computational domain. The outlet pressureFigure 6. wasBoundary set to beconditions the gauge applied pressure to the for computationalall the simulations domain. conducted in this study, whileThe inlet three and different outlet cases of the of inlet domain pressure were were defined investigated, by specifying as shown the in pressure Figure 7. at The these first locations. case The inlet and outlet of the domain were defined by specifying the pressure at these locations. The outletwas a constant pressure pressure was set of to 𝑃 be the = 500 gauge Pa. The pressure second for case all involved the simulations the application conducted of an oscillating in this study, The outlet pressure was set to be the gauge pressure for all the simulations conducted in this study, whilesinusoidal three di ffpressureerent cases with ofrespect inlet to pressure time: were investigated, as shown in Figure7. The first case while three different cases of inlet pressure were investigated, as shown in Figure 7. The first case was a constant pressure of Pinlet = 500 Pa.() The=+ second case ( involvedω ) the application of an oscillating was a constant pressure of 𝑃 = 500 Pa. PtTheinlet second500 case 250sin involved t the application of an oscillating(21) sinusoidal pressure with respect to time: sinusoidal pressure with respect to time: where 𝜔2𝜋𝑓 and f = 10 Hz. P()inlet=+(t) = 500 + 250 (ω sin )(ωt) (21) Ptinlet 500 250sin t (21) = wherewhere 𝜔2𝜋𝑓ω 2andπ f and f = 10f = Hz.10 Hz. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 17

 0 Pa 0≤≤t 3.4375  ω ()− ≤≤  936.414sin tt 3.4375 Pa 3.4375 4.375  − ω ()− ≤≤  1560.69sintt 4.375 Pa 4.375 6.25 Pt()= inlet  − ω ()− ≤≤ (22)  1248.552sin tt 6.25 Pa 6.25 7.5  936.414sinω ()tt− 7.5 Pa 7.5≤≤ 9.84   0.2907 ω ()− ≤ ≤ 936.414et sin 9.84 Pa 9.84 t25 The third case is based on the Northridge earthquake in 1994 [46]. According to the displacement

Appl.measured Sci. 2019, during9, 4189 the earthquake, the inlet pressure was scaled and approximated using a piecewise8 of 16 function, as shown in Equation (22).

600

500

400

300

200

Inlet Pressure (Pa) Inlet Pressure 100

0 0 5 10 15 20 25 Time (s)

(a) 800 700 600 500 400 300 200 Inlet Pressure (Pa) Inlet Pressure 100 0 00.511.522.5 Time (s)

Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 17 (b)

2000 1500 1000 500 0 -500 -1000 Inlet Pressure (Pa) Inlet Pressure Time (s) -1500 -2000 0 5 10 15 20 25

(c)

FigureFigure 7. Velocity7. Velocity profile profile applied applied atat thethe inlet of of the the annular annular orifice: orifice: (a) (constanta) constant pressure pressure of 500 of Pa, 500 (b Pa,) (b)sinusoidal pressure pressure profile, profile, and and (c ()c seismic) seismic pressure pressure profile profile based based on onthe the Northridge Northridge Earthquake Earthquake in in 1994.1994.

4. Results and Discussion Simulation results for the constant, sinusoidal, and seismic response cases are presented. In the constant pressure case, the results for different magnetic field strengths are presented, while in the sinusoidal pressure and seismic response cases, the results for a single magnetic field strength of B = 1.0 T are presented.

4.1. Constant Pressure Figure 8 shows the static pressure measured across the annular orifice when a constant inlet pressure of 500 Pa and magnetic field strengths from 0 T to 1.5 T were applied. In the absence of a magnetic field, the resistance to shear flow experienced by the damper was due to the viscosity and the yield strength of the composite fluid. In this case, a linear pressure drop was observed between the inlet and the outlet of the computational domain, and the pressure drop across the annular orifice, between locations ③ and ④, was 53.82 Pa.

① ② ③ ④⑤⑥ 500

B = 0T 400 B = 0.5T B = 1.0T B = 1.5T 300

200 Pressure (Pa) Pressure

100

0 0 0.1 0.2 0.3 x (m)

Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 17

2000 1500 1000 500 Appl. Sci. 2019, 9, 4189 9 of 16 0 -500 -1000 Inlet Pressure (Pa) Inlet Pressure  Time (s)  0 Pa 0 t 3.4375 -1500  ≤ ≤  -2000 936.414 sin[ω(t 3.4375)] Pa 3.4375 t 4.375  − ≤ ≤ 0 5 10 15 20 25  1560.69 sin[ω(t 4.375)] Pa 4.375 t 6.25 Pinlet(t) = − − ≤ ≤ (22)  [ (t )] t  1248.552 sin ω 6.25 Pa 6.25 7.5 − − ≤ ≤ 936.414 sin[ω(t 7.5)] Pa 7.5 t 9.84  (c)  − ≤ ≤ 936.414e0.2907 sin[ω(t 9.84)] Pa 9.84 t 25 Figure 7. Velocity profile applied at the inlet of the annular− orifice: (a) constant≤ ≤ pressure of 500 Pa, (b) Thesinusoidal third casepressure is based profile, on and the Northridge(c) seismic pressure earthquake profile in based 1994 [on46 ].the According Northridge to Earthquake the displacement in measured1994. during the earthquake, the inlet pressure was scaled and approximated using a piecewise function, as shown in Equation (22). 4. Results and Discussion 4. Results and Discussion Simulation results for the constant, sinusoidal, and seismic response cases are presented. In the constantSimulation pressure results case, forthethe results constant, for different sinusoidal, magnetic and seismic field strengths response are cases presented, are presented. while Inin thethe constantsinusoidal pressure pressure case, and the seismic results response for diff cases,erent magneticthe results field for a strengths single magnetic are presented, field strength while inof theB = sinusoidal1.0 T are presented. pressure and seismic response cases, the results for a single magnetic field strength of B = 1.0 T are presented. 4.1. Constant Pressure 4.1. Constant Pressure Figure 8 shows the static pressure measured across the annular orifice when a constant inlet pressureFigure of8 500 shows Pa and the magnetic static pressure field strengths measured from across 0 T the to 1.5 annular T were orifice applied. when In athe constant absence inlet of a pressuremagnetic offield, 500 the Pa andresistance magnetic to shear field flow strengths experienced from 0 Tby to the 1.5 damper T were was applied. due to In the the viscosity absence ofand a magneticthe yield field,strength the resistanceof the composite to shear fluid. flow experiencedIn this case, bya linear the damper pressure was drop due was to the observed viscosity between and the yieldthe inlet strength and the of outlet the composite of the computational fluid. In this domain, case, a linearand the pressure pressure drop drop was across observed the annular between orifice, the inletbetween and locations the outlet ③ of and the ④ computational, was 53.82 Pa. domain, and the pressure drop across the annular orifice, between locations O3 and O4 , was 53.82 Pa.

① ② ③ ④⑤⑥ 500

B = 0T 400 B = 0.5T B = 1.0T B = 1.5T 300

200 Pressure (Pa) Pressure

100

0 0 0.1 0.2 0.3 x (m)

Figure 8. Static pressure distribution along the length of the channel with various magnitudes of externally applied magnetic fields.

In the cases where the annular orifice was magnetized, as the piston moved, a current was induced in the MR fluid. While in microscopic models, it has been shown that chain-like aggregates form and resist shear flow [2], in this macroscopic model, it was the resulting Lorentz force acting in opposition to the fluid flow that resulted in damping. This was demonstrated by the greater pressure differential Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 17

Figure 8. Static pressure distribution along the length of the channel with various magnitudes of externally applied magnetic fields.

In the cases where the annular orifice was magnetized, as the piston moved, a current was Appl.induced Sci. 2019 in the, 9, 4189 MR fluid. While in microscopic models, it has been shown that chain-like aggregates10 of 16 form and resist shear flow [2], in this macroscopic model, it was the resulting Lorentz force acting in opposition to the fluid flow that resulted in damping. This was demonstrated by the greater pressure differentialbetween locations betweenO3 locationsand O4 . The ③ pressure and ④. dropThe pressure increased drop to 177.55 increased Pa, 333.63 to 177.55 Pa, and Pa, 409.7333.63 Pa Pa, for and the B409.7= 0.5 Pa T, for 1.0 the T, andB = 0.5 1.5 T, T cases,1.0 T, and respectively. 1.5 T cases, respectively. When the magnetic magnetic field field was was increased increased from from BB = =0 0T Ttoto B B= 0.5= 0.5 T, and T, and again again from from B =B 0.5= T0.5 to T B to = B1 T,= 1there T, there was wasa linear a linear increase increase in the in pressure the pressure differential differential across across locations locations ③ andO3 and ④. OHowever,4 . However, as theas the magnetic magnetic field field was was increased increased to B to = 1.5B = T,1.5 the T, pressure the pressure difference difference asymptotically asymptotically increased. increased. This Thisresulted resulted from from the MR the MRfluid fluid approaching approaching its itsmagnetic magnetic saturation saturation point. point. Increases Increases in in magnetic magnetic field field strength above this point will only serve to to margin marginallyally increase the damping coefficient coefficient of the damper. In Figure 99,, thethe unmagnetizedunmagnetized casecase hadhad aa fluidfluid velocityvelocity of 0.374 mm/s/s at at location location ⑥O6 downstream from thethe inletinlet condition. condition. In In the the magnetized magnetized cases, cases, the the fluid fluid velocity velocity was was much much lower lower downstream downstream from fromthe inlet the with inlet a with deacceleration a deacceleration upon entering upon entering the annular the orificeannular after orifice location afterO3 locationand an acceleration③ and an accelerationupon exiting upon after locationexiting afterO4 . This location deceleration ④. This and deceleration acceleration and of the acceleration fluid in the of magnetized the fluid in cases the weremagnetized noticeably cases absent were fromnoticeably the unmagnetized absent from the case. unmagnetized The deceleration case. of The the deceleration fluid was caused of the by fluid the wasLorentz caused force by generated the Lorentz after force location generatedO3 and after the location re-acceleration ③ andwas the causedre-acceleration by the fluidwas enteringcaused by a thelow-pressure fluid entering zone a afterlow-pressure location Ozone4 . The after downstream location ④ fluid. The velocitiesdownstream at location fluid velocitiesO6 were at 0.284 location m/s, ⑥0.163 were m/ s,0.284 and m/s, 0.0946 0.163 m/ sm/s, for and the B0.0946= 0.5 m/s T, 1.0 for T, the and B 1.5= 0.5 T T, cases, 1.0 T, respectively. and 1.5 T cases, The lowerrespectively. velocity The in lowerthe magnetized velocity in cases the wasmagnetized characteristic cases ofwas MR characteristic fluids and was of causedMR fluids by theand increases was caused in apparent by the viscosityincreases andin apparent viscoelasticity viscosity of theand fluid viscoelasticity in the presence of the of fluid magnetic in the fieldspresence [5]. of magnetic fields [5].

① ② ③ ④⑤⑥

0.50 B = 0T B = 0.5T B = 1.0T 0.40 B = 1.5T

0.30

0.20 Velocity (m/s)

0.10

0.00 0 0.1 0.2 0.3 x (m)

Figure 9. Velocity distribution along length of the channel with various magnitudes of externally Figureapplied 9. magnetic Velocity fields. distribution along length of the channel with various magnitudes of externally applied magnetic fields. 4.2. Sinusoidal Pressure Profile 4.2. Sinusoidal Pressure Profile The model was shown to simulate the expected characteristics of MR fluid dampers in the constant pressureThe case.model The was response shown to of thesimulate system the to expected a variable characteristics sinusoidal pressure of MR inletfluid conditiondampers within the a peak-to-peakconstant pressure amplitude case. The of 500 response Pa was of tested the system with magnetic to a variable field strengthssinusoidal of pressureB = 0 T, 0.5inlet T, condition 1.0 T, and 1.5with T a applied peak-to-peak to the annularamplitude orifice. of 500 This Pa wa wass tested to simulate with magnetic the response field of strengths the MR fluidof B = damper 0 T, 0.5 whenT, 1.0 T,used and in 1.5 civil T applied structures to the to dampen annular theorifice. vibration This was induced to simulate from strong the response wind conditions. of the MR fluid damper whenIn used Figure in civil 10a–d, structures between to locations dampen theO1 to vibrationO3 , there induced was a large from pressure strong wind variation, conditions. and between locationsIn FigureO4 to 10a–d,O6 , there between was a locations much smaller ① to pressure③, there variation was a large across pressure all cases, variation, indicating and between that the locationsfluid velocity ④ to was ⑥, dampedthere was after a much the annularsmaller orifice.pressure The variation largest across pressure all cases, drops indicating once the flow that hasthe developed across locations O3 and O4 at t = 1.32 s were 88.46 Pa, 131.99 Pa, 228.45 Pa, and 320.01 Pa for the B = 0 T, 0.5 T, 1.0 T, and 1.5 T cases, respectively. As expected, the magnetized cases had a much higher pressure drop across the annular orifice than the unmagnetized case. When the magnetic field strength was increased across the cases, the pressure drop across the annular orifice could be Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 17 fluid velocity was damped after the annular orifice. The largest pressure drops once the flow has developed across locations ③ and ④ at t = 1.32 s were 88.46 Pa, 131.99 Pa, 228.45 Pa, and 320.01 Pa for the B = 0 T, 0.5 T, 1.0 T, and 1.5 T cases, respectively. As expected, the magnetized cases had a muchAppl. Sci. higher2019, 9 , 4189pressure drop across the annular orifice than the unmagnetized case. When11 ofthe 16 magnetic field strength was increased across the cases, the pressure drop across the annular orifice could be seen to almost linearly increase with the peak-to-peak pressure differences across locations ③seen and to ④, almost increasing linearly as increase the field with strength the peak-to-peak was increased, pressure as illustrated differences in Figure across 11. locations O3 and O4 , increasingMeanwhile, as the in field the strength constant was pressure increased, case, asthe illustrated pressure drop in Figure tapered 11. off as the field strength was increasedMeanwhile, toward inthe the saturation constant point pressure of the case, particles; the pressure this could drop not tapered be seen off inas the the sinusoidal field strength case. was In theincreased sinusoidal toward case, the there saturation was a constant point of increase the particles; as the this magnetic could field not be strength seen in was the sinusoidalincreased from case. BIn = the 1.0sinusoidal T to 1.5 T. case, While there this was seems a constant to suggest increase that asthe the saturation magnetic field field strength strength was was not increased being approached,from B = 1.0 at T toB 1.5= 1.5 T. T, While there this should seems be to asymptotic suggest that beha thevior. saturation However, field this strength did not was occur not in being the sinusoidalapproached, case at asB the= 1.5 velocity T, there of shouldthe fluid be throug asymptotich the annular behavior. orifice However, was lo thiswer didthan not in the occur constant in the pressuresinusoidal case. case This as the was velocity evident of thein the fluid significan throughtly the lower annular pressure orifice wasdrops lower for the than sinusoidal in the constant case shownpressure in case.Figure This 11, despite was evident the theoretical in the significantly mean inlet pressure lower pressure of the two drops conditions for the being sinusoidal the same, case andshown in the in Figurevertical 11 shift, despite of the the inlet theoretical pressures mean prof inletile. Although pressure of the the inlet two pressure conditions profile being was the set same, to 500and + in 250sin the vertical(𝜔𝑡) , which shift of should the inlet create pressures a mean profile. pressure Although identical the inletto the pressure constant profile pressure was setcase, to because500 + 250 ofsin the( ωresistancet), which due should to the create magnetic a mean field, pressure the actual identical mean to inlet the constant pressure pressure achieved case, was because 434.54 Paof thefor resistancethe sinusoidal due tocase. the magneticThis lower field, mean the pressu actualre mean meant inlet that pressure the fluid achieved velocity was in 434.54the annular Pa for orificethe sinusoidal did not case.approach This the lower requisite mean pressurevelocity meantrequired that for the the fluid current velocity channeled in the annularthrough orifice the fluid did tonot result approach in saturation. the requisite This velocityindicates required that if th fore frequency the current of channeled the wave throughwas decreased, the fluid the to pressure result in dropsaturation. of the Thisdamper indicates under thatsinusoidal if the frequency loading would of the waveapproach was the decreased, values in the the pressure constant drop pressure of the case.damper Actual under wind sinusoidal loading loadingconditions would would approach have a themuch values lower in frequency the constant than pressure the inlet case. condition Actual usedwind in loading the model conditions and this would accounted have a for much the lower poorer frequency performance than theof the inlet damper condition when used loaded in the with model a sinusoidaland this accounted profile. In for general, the poorer it appears performance that ofthe the damper damper was when more loaded effective with aat sinusoidal damping profile.lower frequencyIn general, profiles it appears as thatthese the allowed damper the was flow more in etheffective annular at damping orifice to lower fullyfrequency develop. Regardless, profiles as these the damperallowed thestill flow achieved in the annularsignificant orifice damping to fully develop.and the Regardless,model was the shown damper to still capture achieved the significant relevant characteristicsdamping and theof MR model fluid was dampers. shown to capture the relevant characteristics of MR fluid dampers.

800

600

400

200

0 0T Inlet

Pressure (Pa) Pressure -200 0T Location (1) 0T Location (2) -400 0T Location (3) 0T Location (4) 0T Location (5) -600 0T Location (6) 0T Outlet -800 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time (s)

(a)

Figure 10. Cont. Appl. Sci. 2019, 9, 4189 12 of 16

Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 17

800

600

400

200

0 0.5T Inlet

Pressure (Pa) Pressure -200 0.5T Location (1) 0.5T Location (2) -400 0.5T Location (3) 0.5T Location (4) 0.5T Location (5) -600 0.5T Location (6) 0.5T Outlet -800 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time (s) (b) 800

600

400

200

0 1.0T Inlet 1.0T Location (1) Pressure (Pa) Pressure -200 1.0T Location (2) 1.0T Location (3) -400 1.0T Location (4) 1.0T Location (5) -600 1.0T Location (6) 1.0T Outlet -800 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time (s)

(c) 800

600

400

200

0 1.5T Inlet

Pressure (Pa) Pressure -200 1.5T Location (1) 1.5T Location (2) -400 1.5T Location (3) 1.5T Location (4) -600 1.5T Location (5) 1.5T Location (6) 1.5T Outlet -800 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time (s) (d)

Figure 10. Change in static pressure at various measurement locations with respect to time under a sinusoidal inlet pressure profile and an externally applied magnetic field of: (a) B = 0 T, (b) B = 0.5 T, (c) B = 1 T, and (d) B = 1.5 T. Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 17

Figure 10. Change in static pressure at various measurement locations with respect to time under a Appl. Sci. 2019sinusoidal, 9, 4189 inlet pressure profile and an externally applied magnetic field of: (a) B = 0 T, (b) B = 0.5 T, 13 of 16 (c) B = 1 T, and (d) B = 1.5 T.

450

400

350

300

250

200

150 Pressure drop (Pa) drop Pressure 100 Sinusoidal Inlet Pressure Profile 50 Constant Inlet Pressure Profile 0 00.511.5 Applied Magnetic Field (T)

Figure 11. Pressure drop between locations O3 and O4 at various magnitudes of external magnetic fields ③ ④ for theFigure sinusoidal 11. Pressure inlet pressuredrop between profile locations and the constant and at inlet various pressure magnitudes cases. of external magnetic fields for the sinusoidal inlet pressure profile and the constant inlet pressure cases. 4.3. Seismic Response 4.3. Seismic Response Another possible application of MR fluid dampers is in the control of the vibrational characteristics Another possible application of MR fluid dampers is in the control of the vibrational of civil structures under seismic conditions. The suitability of the model in simulating the response characteristics of civil structures under seismic conditions. The suitability of the model in simulating of MRthe fluid response dampers of MR to fluid real dampers earthquake to real conditions earthquake was conditions examined was using examined a piecewise using a piecewise pressure inlet conditionpressure that inlet approximated condition that the approximated 1994 Northridge the 1994 earthquake, Northridge and earthquake, the results and are shownthe results in Figure are 12. As expected,shown in theFigure system 12. As experienced expected, the the system largest experi pressureenced the drop largest between pressure locations drop betweenO3 and locationsO4 among all the equally-spaced③ and ④ among locations, all the equally-spaced with a maximal locations, difference with a maximal of 419.84 difference Pa occurring of 419.84 at tPa= occurring4.89 s. It was evidentat t = that 4.89 thes. It systemwas evident was that eff ectivelythe system damped was effectively during damped all stages duri ofng theall stages earthquake of the earthquake and exhibited the expectedand exhibited characteristics the expected during characteristics both the during large bo andth the small large inlet and small pressure inlet variations.pressure variations. In a similar mannerIn a tosimilar the sinusoidalmanner to the pressure sinusoidal profile, pressure the profil modelse, the also models exhibited also exhibi similarted similar behavior behavior with with magnetic magnetic field strengths of B = 0.5 T and 1.5 T applied. Those results are omitted for conciseness. field strengths of B = 0.5 T and 1.5 T applied. Those results are omitted for conciseness. Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 17

1600 1.0T Inlet 1200 1.0T Location (1) 1.0T Location (2) 1.0T Location (3) 800 1.0T Location (4) 1.0T Location (5) 400 1.0T Location (6) 1.0T Outlet 0 Time (s)

Pressure (Pa) Pressure -400

-800

-1200

-1600 0 5 10 15 20 25 FigureFigure 12. Change 12. Change in staticin static pressure pressure at at various various measurementmeasurement locations locations with with respect respect to time to time under under an an inlet pressureinlet pressure profile profile based based on on the the 1994 1994 Northridge Northridge earthquakeearthquake and and an an externally externally applied applied magnetic magnetic field offield 1.0 of T. 1.0 T.

5. Conclusions In this study, a numerical model was developed to simulate the behavior of magnetorheological (MR) fluid dampers by utilizing a computational fluid dynamics approach in conjunction with the magnetohydrodynamics model. Ohm’s law and Maxwell’s equations were coupled with the Navier– Stokes transport equations through the inclusion of the Lorentz force as a source term. The simulations in this study were conducted based on a two-dimensional axisymmetric MR fluid damper geometry with a magnetized area representing the annular orifice. Three test cases were conducted to investigate the capability of the present model, including a constant pressure case, a sinusoidal pressure case, and a seismic pressure case. Various plots of pressure and velocity were generated against time and displacement, and the model was found to accurately capture the macroscopic characteristics of MR fluids. In the constant pressure case, the MR fluid was found to have a lower velocity when the damper was in the on-state. In this state, it was also observed that the MR fluid decelerated after entering the annular orifice and then accelerated after exiting the annular orifice. This was due to the increased apparent viscosity and viscoelasticity of the fluid created by the resisting Lorentz force in the annular orifice. It was found that MR fluids were most effective at damping constant pressure profiles with the highest pressure drop measured across the annular orifice. However, this was due to the high frequency of the sinusoidal profile and it is expected that the performance of the MR fluid between the two cases would be similar in real world conditions where the frequency of the load would be reduced. Despite this, in both the sinusoidal and seismic pressure cases, there were large pressure variations measured before the magnetized annular orifice, followed by significantly smaller ones after the annular orifice, signifying that the MR fluid dampers were effective at damping these pressure profiles. In conclusion, the presented numerical methodology and simulations in MR fluid modeling is beneficial in reducing the time and cost required in optimizing the design and performance of such devices. The present model is advantageous over the microscopic computational models, such as the discrete phase model, as it is significantly less computationally expensive to effectively model the behavior of MR fluid dampers in their full scale. Some limitations of the present study include the two-dimensional simplification of the MR fluid damper geometry. However, the underlying physical principles are identical between two-dimensional and three-dimensional simulations, thus the Appl. Sci. 2019, 9, 4189 14 of 16

5. Conclusions In this study, a numerical model was developed to simulate the behavior of magnetorheological (MR) fluid dampers by utilizing a computational fluid dynamics approach in conjunction with the magnetohydrodynamics model. Ohm’s law and Maxwell’s equations were coupled with the Navier–Stokes transport equations through the inclusion of the Lorentz force as a source term. The simulations in this study were conducted based on a two-dimensional axisymmetric MR fluid damper geometry with a magnetized area representing the annular orifice. Three test cases were conducted to investigate the capability of the present model, including a constant pressure case, a sinusoidal pressure case, and a seismic pressure case. Various plots of pressure and velocity were generated against time and displacement, and the model was found to accurately capture the macroscopic characteristics of MR fluids. In the constant pressure case, the MR fluid was found to have a lower velocity when the damper was in the on-state. In this state, it was also observed that the MR fluid decelerated after entering the annular orifice and then accelerated after exiting the annular orifice. This was due to the increased apparent viscosity and viscoelasticity of the fluid created by the resisting Lorentz force in the annular orifice. It was found that MR fluids were most effective at damping constant pressure profiles with the highest pressure drop measured across the annular orifice. However, this was due to the high frequency of the sinusoidal profile and it is expected that the performance of the MR fluid between the two cases would be similar in real world conditions where the frequency of the load would be reduced. Despite this, in both the sinusoidal and seismic pressure cases, there were large pressure variations measured before the magnetized annular orifice, followed by significantly smaller ones after the annular orifice, signifying that the MR fluid dampers were effective at damping these pressure profiles. In conclusion, the presented numerical methodology and simulations in MR fluid modeling is beneficial in reducing the time and cost required in optimizing the design and performance of such devices. The present model is advantageous over the microscopic computational models, such as the discrete phase model, as it is significantly less computationally expensive to effectively model the behavior of MR fluid dampers in their full scale. Some limitations of the present study include the two-dimensional simplification of the MR fluid damper geometry. However, the underlying physical principles are identical between two-dimensional and three-dimensional simulations, thus the present study provides a solid foundation for larger-scale, three-dimensional simulations. The focus of the present model was the fluid behavior within a MR fluid damper when subjected to pressure and external magnetic fields. It may be combined with structure-focused computational models, such as the finite element method [47,48], to form a fluid structure interaction (FSI) study for further examinations of the effectiveness of MR fluid dampers in protecting various structures against vibration. Lastly, in order to further validate the computational model presented, physical experiments are to be conducted in the future for direct quantitative comparison.

Author Contributions: Conceptualization, D.D.L., K.H., and G.H.Y.; methodology, D.D.L., K.H., and G.H.Y.; numerical simulations, D.D.L. and K.H.; formal analysis, D.D.L., D.F.K., and Q.N.C.; writing—original draft preparation, D.D.L. and D.F.K.; writing—review and editing, A.C.Y.Y., Q.N.C., and V.T.; visualization, K.H. and D.D.L.; supervision, V.T., C.M., and G.H.Y.; project administration, D.D.L. and A.C.Y.Y.; funding acquisition, V.T. and G.H.Y. Funding: This research was supported by the Australian Research Council (ARC Project ID DP150101065 and DP160100021). The support of NVIDIA Corporation with the donation of the Titan Xp and Quadro P6000 GPUs used for this research is also gratefully acknowledged. Conflicts of Interest: The authors declare no conflict of interest. Appl. Sci. 2019, 9, 4189 15 of 16

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