1 Introduction

Magnetic field assisted polishing (MFAP) methods have become one of research hotspots due to their flexibility, self-sharpening and self-adaptability for the recent years. MFAP methods are widely employed for polishing various hard-to-process materials, such as silicon, sapphire, glass, ceramic, superalloy. Barman et al. [1] investigated the influence of magnetorheological (MR) polishing fluids with different compositions on polishing performance. Two different MR fluids were utilized to polish the bio-titanium alloy to obtain hydrophilic and hydrophobic surfaces, respectively. Zhai et al. [2] proposed ultrasound assisted magnetorheological polishing (UAMP) on sapphire and established a material removal rate (MRR) model of UAMP based on the Preston equation. The feasibility of this model was experimentally verified. Luo et al. [3] conducted polishing experiments on zirconia ceramics using MRP method. Surface roughness Ra of zirconia ceramics decreased to 0.702 nm from initial 71.976 nm using diamond abrasive particles. The maximum material removal rate of 1 mg/min was attained. Yamaguchi et al. [4] employed magnetic abrasive finishing (MAF) method to improve the tool life of uncoated WC/Co cutting tool. It was concluded that the cutting tool life was improved by 150%. Li et al. [5] performed a hybrid post-processing of MAF and heat treatment (HT) on the surface of 3D-printed Inconel 718 alloy. The surface roughness Ra of the workpiece decreased to 0.15 μm from initial 2.7 μm. The hybrid post-treatment of MAF and HT demonstrated that the mechanical properties such as yield strength and elongation of the Inconel 718 alloy manufactured in additive form were significantly improved. Umehara et al. [6] designed and fabricated a novel magnetic floating polishing platform for large-size and large-volume silicon nitride (Si3N4) balls. The polishing process was divided into three stages i.e., roughing stage, intermediate stage and finishing stage. The surface roughness Ra of the polished silicon nitride (Si3N4) balls decreased to 6.70 nm from initial 937.14 nm.

For the MFAP method, a lot of numerical simulations were employed to understand the effects of processing parameters on the polishing performance before the experiments were conducted. This facilitated the simplification of tests and efficiency improvement. Kumar et al. [7] investigated the mechanism of magnetorheological polishing on miniature tooth contours where the forces on the surface of the micro-gear were simulated using COMSOL software. A surface roughness model of polished micro-gear was established using simulation results. Wang et al. [8] proposed a new magnetic field assisted mass polishing technology for efficient polishing of multiple free-form parts simultaneously. The effect of magnetic field on the material removal characteristics was analyzed by the finite element method. The factors affecting surface generation were investigated by polishing experiments. The results showed that magnetic field assisted mass polishing could effectively polish multiple free-form parts with nano-level surface finish. Liu et al. [9] simulated the three-dimensional profile of the polishing points during magnetorheological polishing. A material removal rate (MRR) model considering pressure and shear stress was developed using the simulation results, which was verified for accuracy. Sharma [10] simulated the magnetic field strength of the newly designed magnetorheological polishing equipment. A mathematical model of the relationship between surface roughness and magnetic field strength was established. The results showed that the surface roughness decreased with the increase of magnetic field strength. Grover et al. [11] designed a magnetorheological polishing equipment for polishing cylindrical inner surfaces using radially curved permanent magnets. Simulation of the magnetic field strength of this radially curved permanent magnet was conducted. A mathematical model for predicting surface roughness was developed based on the results of the simulation. After validation, the maximum error of the model was 8.06%. Guo et al. [12] designed a novel magnetorheological polishing equipment with Halbach array for quartz specimens. The magnetic field strength of Halbach array was simulated to verify its large area and high strength. The surface roughness of the quartz specimen eventually improved up to 0.544 nm after polishing.

An accurate material removal model is of benefit to provide theoretical guidance and reveal the polishing mechanism. Great efforts have been made to establish the predictive material removal model for the MFAP. Zhai et al. [2] developed a material removal model for magnetorheological polishing where the coupling effect of magnetic and vibrational fields was considered. It was verified that the relative error between the theoretical and experimental results was less than 20%. Chen et al. [13] established a two-dimensional material removal model for magnetorheological polishing based on hydrodynamic theory and the Preston equation. The rationality of the material removal model was verified by comparing the experimental results with the simulation results. The error between the model and the experimental results was 9.8%. Ming et al. [14] established a material removal model for a non-Newtonian fluid polishing method based on weak magnetorheological strengthening thickening effects. The model was developed using non-Newtonian fluid kinematics. The average error between the model and experimental results was 4.45%. Li et al. [15] established a material removal model for the magnetic abrasive finishing method using fluid abrasives in which the effect of magnetic fields on material removal was considered. This model agrees to the experimental results with an average error of 4.51%.

In our previous studies, magnetorheological shear thickening polishing (MRSTP) was proposed as a novel magnetic field assisted polishing process which combined the magnetorheological effect with the shear thickening effect together [16,17,18,19]. As one of loosed abrasive polishing methods, MRSTP possesses excellent adaptability and processing efficiency. It has a magnetorheological effect that enables the rapid formation of magnetically enhanced particle clusters in the polishing media under the excitation of a magnetic field [16]. These clusters exert much stronger holding force on the abrasive particles, which makes the polishing process more efficient than the traditional MFAP. Additionally, MRSTP has a shear thickening effect during polishing, which results in larger shear stress on the workpiece surface than the conventional MRF. Due to its high adaptability and efficiency, MRSTP has been used to polish various difficult-to-machine materials [16,17,18,19]. MRSTP media was prepared by mixing a certain proportion of polyethylene glycol (PEG 200), SiO2, abrasives, carbonyl iron powder (CIP) and chemical additives [16]. Simulation of the magnetic field strength of the magnetic field generation device was performed to verify the rationality of the structure [17]. The magnetic field generator and the modified magnetic abrasive were used to polish the titanium alloy Ti-6Al-4 V. Surface roughness of workpiece reduced from 1.17 μm to 54 nm. Also, in a case MRSTP media were specially developed for titanium alloy under the action of a four-pole rotating magnetic field [18]. The surface roughness of titanium alloy Ti-6Al-4 V was reduced from 169 to 61 nm and the effectiveness of this polishing media was verified. A new magnetic field generation device was designed which was capable of generating alternating magnetic fields for the additive manufacturing of 361L rectangular micro-fabricated structures[19]. Material removal rate models were developed and processing experiments were conducted. The results showed that the top and bottom of the rectangular microstructure were uniformly removed from the material. The feasibility of machining additively manufactured workpieces with MRSTP media was verified.

Nevertheless, recent studies on MRSTP were focused on processing experiments and simulation of magnetic fields. The coupling simulation of magnetic and flow fields in the MRSTP process with CFD methods was not explored. Therefore, in this work, CFD simulations of the designed MRSTP apparatus were carried out using Maxwell and Fluent modules in ANSYS software. It aimed to investigate the pattern of influence of polishing parameters on the polishing force applied to the workpiece surface. A mathematical model between MRR and polishing parameters was developed using the output of the normal forces from the CFD simulation results. The accuracy of the MRR model and the validity of the simulation results were verified by experiments.

2 Principles

Figure 1 shows the principle of MRSTP. MRSTP media are distributed inside polishing bowl which is fixed on the rotary table of either CNC machine or industrial robot. The magnetic field generation device is mounted below the polishing bowl. The workpiece is clamped onto the spindle bottom of either CNC machine or industrial robot. Under the action of the magnetic field, the MRSTP media form magnetic-force-reinforced flexible abrasive clusters. As the magnetic field has adsorption effect on the CIP, the shear rate of the MRSTP media will decrease when they flow through the magnetic field. This is macroscopically expressed as an increase in the apparent viscosity of the MRSTP media. The MRSTP media move in a circular motion driven by the rotary polishing bowl. The CIP and abrasive grains in the magnetic-force-reinforced flexible abrasive clusters are constantly renewed and pass through the immersed workpiece. The relative velocity between the MRSTP media and the workpiece is easily controlled by the spindle and the rotary table of either CNC machine or industrial robot.

Fig. 1
figure 1

Principle of the MRSTP

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3 Design of magnetic field generation device

Figure 2 shows the three-dimensional model and two-dimensional model of the magnetic field generation device. The device comprises of two magnetic pole mounts and twelve magnetic poles. The two pole mounts were made of Steel 45 with twelve holes (i.e., each mount had six poles) for holding the magnetic poles. The six poles were divided into three sets i.e., the top set, the middle set and the bottom set. The middle set were designed at an angle of 120° to the bottom sets. The material of all twelve poles was chosen as N38. The distribution of the magnetic field was simulated using Maxwell software. The materials defined in the model for the magnetic field generation device were same as those used in the actual polishing process. Three lines were planned for the measurement of magnetic field strength as shown in Fig. 2. The magnetic field intensity along the design path was measured using a Gauss meter (Model: GM 500).

Fig. 2
figure 2

Model of magnetic field generating device. (a) 3D model, (b) 2D model

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Figure 3 shows the results of the magnetic field simulation. It can be seen that the magnetic force lines uniformly distributed in the polished area while magnetic field intensity were still strong around the workpiece. The maximum magnetic field strengths of 525.7 mT and 527.1 mT were obtained at the starting points of lines 1 and 3, respectively. Along line 1 and line 3, the magnetic field strength decreased with the increase of the distance. A maximum magnetic field strength of 183.3 mT was obtained at 3 mm away from the starting point of the line 2. As shown in Fig. 3 (a), since the magnetic force lines at the starting point of line 2 was not close to the magnetic field generating device, the magnetic field strength along line 2 firstly increased and then decreased with the distance increasing from the starting point.

Fig. 3
figure 3

Maxwell simulation results. (a) Distribution of magnetic force lines, (b) Distribution of magnetic field intensity

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Figure 4 shows the comparisons of magnetic field intensity between the simulation and experimental results. It can be seen that the measured magnetic field strengths are consistent with the simulation values. The measured values at the starting point of the line 1 and line 3 were 315.4 mT and 367.0 mT, respectively. The measured value was 125.3 mT at 3 mm away from the starting point of the line 2. The validity of the simulation was verified.

Fig. 4
figure 4

Comparisons of magnetic field intensity between simulations and experiments

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4 Numerical simulation of flow field

4.1 Computational fluid dynamics methods

The flow field of the MRSTP process was numerically simulated using Ansys Fluent. Mixture models including STFs, CIP, SiC and other multiphase blends were used. The output from Maxwell was coupled using the MHD module in Fluent to simulate the magnetic field environment in the flow field. The following assumptions were made during this numerical simulation: (i) The workpiece was an ideal rigid material. (ii) The processing environment temperature was controlled at 25℃. In the MRSTP process, it was assumed that the fluid was incompressible and operated in a constant temperature environment. Hence, the continuity and momentum equations for incompressible viscous fluids were used to describe the motion state of the MRSTP media [20]:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$$
(1)
$$\left\{\begin{array}{c}\frac{\partial u}{\partial t}+\left(V\cdot \nabla \right)u={f}_{x}-\frac{1}{\rho }\frac{\partial p}{\partial x}+\frac{1}{\rho }\left(\frac{\partial {\sigma }_{xx}}{\partial x}+\frac{\partial {\tau }_{yx}}{\partial y}+\frac{\partial {\tau }_{zx}}{\partial z}\right)\\ \frac{\partial v}{\partial t}+\left(V\cdot \nabla \right)v={f}_{y}-\frac{1}{\rho }\frac{\partial p}{\partial y}+\frac{1}{\rho }\left(\frac{\partial {\sigma }_{yy}}{\partial y}+\frac{\partial {\tau }_{zy}}{\partial z}+\frac{\partial {\tau }_{xy}}{\partial x}\right)\\ \frac{\partial w}{\partial t}+\left(V\cdot \nabla \right)w={f}_{z}-\frac{1}{\rho }\frac{\partial p}{\partial z}+\frac{1}{\rho }\left(\frac{\partial {\sigma }_{zz}}{\partial z}+\frac{\partial {\tau }_{xz}}{\partial x}+\frac{\partial {\tau }_{yz}}{\partial y}\right)\end{array}\right.$$
(2)

where V is the velocity of the fluid. u, v, and w are the components of the fluid velocity in the x, y, and z directions. ρ is the density of the fluid. p is the pressure of the fluid. f is the mass force acting on the fluid. σ and τ are stress components.

The magnetic force on the fluid in the MHD model can be calculated by the following equation:

$$\left\{\begin{array}{c}{\overrightarrow{f}}_{m,x}={\mu }_{0}\left({\overrightarrow{M}}_{x}\frac{\partial {H}_{x}}{\partial x}+{\overrightarrow{M}}_{y}\frac{\partial {H}_{x}}{\partial y}+{\overrightarrow{M}}_{z}\frac{\partial {H}_{x}}{\partial z}\right)\\ {\overrightarrow{f}}_{m,y}={\mu }_{0}\left({\overrightarrow{M}}_{x}\frac{\partial {H}_{y}}{\partial x}+{\overrightarrow{M}}_{y}\frac{\partial {H}_{y}}{\partial y}+{\overrightarrow{M}}_{z}\frac{\partial {H}_{y}}{\partial z}\right)\\ {\overrightarrow{f}}_{m,z}={\mu }_{0}\left({\overrightarrow{M}}_{x}\frac{\partial {H}_{z}}{\partial x}+{\overrightarrow{M}}_{y}\frac{\partial {H}_{z}}{\partial y}+{\overrightarrow{M}}_{z}\frac{\partial {H}_{z}}{\partial z}\right)\end{array}\right.$$
(3)

where μ0 is the vacuum permeability. Mx, My and Mz are the components of the magnetization intensity in the x, y, and z directions. Hx, Hy and Hz are the component of the magnetic field intensity in the x, y, and z directions.

The motion pattern of the fluid can be obtained from the calculation formula of Reynolds number Re:

$$Re=\frac{\rho \bullet v\bullet {d}_{re}}{\mu }$$
(4)

where ρ is the density of the fluid, v is the velocity of the fluid, μ is the dynamic viscosity of the fluid, dre is the characteristic length of the fluid. The flow state of the MRSTP media was determined to be laminar based on the experimental parameters calculated as Re < 2320. The SIMPLE algorithm was used to solve the coupled velocity–pressure problem. Since the diffusion effect in the MRSTP media flow process could be neglected, it was a pure convection problem and the discrete format for establishing the discrete equations adopted the second order windward format.

4.2 Geometric model for numerical simulation

Figure 5 shows the geometric model for the numerical simulation in the software of Ansys Fluent. In order to reduce the computational effort and facilitate the computational convergence, the circular polishing bowl of MRSTP was designed as a linear polishing trough. This linear polishing trough had a velocity inlet on one side and a pressure outlet on the other side. The wall of the polishing trough was defined as non-slip wall surface. The wall of the workpiece was set up as rotational wall surface. As the top surface of the polishing trough was MRSTP medium, it was defined as symmetric boundary. In order to improve the computational accuracy, the polished trough was unstructured meshed. The grid size was chosen as 1 mm while the grid was encrypted near the workpiece with size of 0.5 mm. The magnetic field simulation results from the Sect. 3 were further processed and loaded into the software of Ansys Fluent. The loading results are shown in Fig. 5 (c). There was not any distortion observed in the magnetic field and the loading results were accurate. Two lines, AB and CD, were set up on the workpiece surface as the output locations of pressure and shear force in the simulation results, respectively.

Fig. 5
figure 5

Simulation models. (a) 3D model of a linear polishing trough, (b) Grid division, (c) Results of magnetic field loading, (d) Position of the force measurement on the surface of the workpiece

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4.3 Input parameters for numerical simulation

The simulation input parameters are listed in Table 1. The effects of with the rotational speed of workpiece and the flow velocity of the MRSTP media on the normal pressure and shear force on the workpiece surface were investigated. The maximum magnetic field strength was obtained along line 2 at 3 mm from the starting point as shown in Fig. 4 (a). Considering the existence of excitation gap and the thickness of polishing bowl in the actual processing, the gap between the workpiece and the bottom of the polishing trough was set up as 1 mm.

Table 1 Input parameters used for simulation study
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The rheological profiles of the STFs were measured in order to give shear thickening effect to the MRSTP media during the numerical simulation [16]. When the shear rate is less than the critical shear rate, STFs show shear thinning action. When the shear rate is between the critical shear rate and the maximum shear rate, the STFs show shear thickening action [21,22,23]. When the shear rate is greater than the maximum shear rate, the STFs show a shear thinning action. The viscosity of the STFs was defined in FLUENT using this rheological curve via programming with C language (Fig. 6).

Fig. 6
figure 6

Variation of STFs viscosity with shear rate

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4.4 Analysis of simulation results

4.4.1 Variation of normal pressure

In the Preston equation, the material removal rate during polishing is associated to the normal pressure on the surface of the workpiece and the velocity of the relative motion between the workpiece and the polishing fluid [24]. Therefore, it is essential to investigate the normal pressure distribution and variation on the workpiece surface through numerical simulation of the flow field.

Figure 7 shows the variation of normal pressure distribution versus different workpiece rotational speeds with the increase of working distance at the MRSTP media velocity of 0.8 m/s. It can be seen that the normal pressure distribution on the surface of the workpiece decreased along the x-direction and almost unchanged along the y-direction. This main reason was because the MRSTP media flowed along the x-direction. When the MRSTP media flowed under the workpiece, the height of the flow channel suddenly became narrow that made the MRSTP media to gather over one side of the workpiece and generated great pressure on the workpiece surface. Meanwhile, due to the high kinetic energy of the MRSTP media, the strong impact produced a large normal pressure on the surface of the workpiece when the media got to touch the workpiece surface. As the kinetic energy of the MRSTP media was gradually consumed with the flow of MRSTP media over the workpiece surface, the normal pressure on the workpiece surface slightly decreased.

Fig. 7
figure 7

Variation of normal pressure distribution versus workpiece rotational speed

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The normal pressure on the workpiece surface gradually decreased as the workpiece rotational speeds increased from 1000 to 2000 rpm. As the rotation speeds of the workpiece increase, the MRSTP media underneath the workpiece will be driven by the workpiece to increase the flow rate. In this case, the normal pressure on the surface of the workpiece decreased, which was in accordance with the description of Bernoulli’s principle.

Figure 8 shows the variation of normal pressure distribution versus the MRSTP media flow velocity of 0.8 m/s, 1.2 m/s and 1.6 m/s with the increase of working distance at the workpiece rotational speed of 2000 rpm. With the increase of flow velocity of MRSTP media, the normal pressure on the surface of the workpiece gradually increased as the kinetic energy became large.

Fig. 8
figure 8

Variation of normal pressure distribution versus flow velocity of MRSTP media

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4.4.2 Variation of shear stress

Shear stress is mainly caused by the fluid viscosity when the fluid media flow through the workpiece contact surface. In the ultraprecision polishing with non-newton fluid, material removal from the workpiece surface is not only related to the normal pressure but also the shear stress on the workpiece. Therefore, it is also important to investigate the effect of processing parameters on the shear stress.

Figure 9 shows the variation of shear stress distribution on the workpiece surface along the y-direction versus different workpiece rotational speed with the increase of working distance the MRSTP media flow velocity of 0.8 m/s. It was seen that the minimum shear stress value appeared near the low side of the workpiece. This was attributed to the combined effects between workpiece rotation and MRSTP media flow. During the numerical simulation, the flow direction of the MRSTP media was set along x-direction and the workpiece rotational speed was defined as counter clockwise as labelled in Fig. 9. Hence, along the y-axis, the linear velocity of the workpiece and the flow velocity of the MRSTP media gradually varied from the same direction to the opposite direction. This caused that the minimum shear rate in the polishing process occurred close to the point C. When the workpiece rotational speed increased, the shear stress near the D point increased. Meanwhile, the location of the minimum shear stress on the workpiece surface gradually moved upward.

Fig. 9
figure 9

Variation of shear stress versus workpiece rotational speed

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Figure 10 shows the variation of shear stress distribution versus MRSTP media flow velocity of 0.8 m/s, 1.2 m/s, and 1.6 m/s with the increase of working distance at the workpiece rotational speed of 2000 rpm. As the MRSTP media flow velocity increased, the shear stress near the D point increased. The location of the minimum shear stress on the workpiece surface gradually shifted downward.

Fig. 10
figure 10

Variation of shear stress versus flow velocity of MRSTP media

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It was found that the maximum normal pressure on the workpiece surface occurred at the point A while the maximum shear stress occurred at the point D in the MRSTP as shown in Figs. 9 and 10. With the increase the workpiece rotational speed, the normal pressure on the workpiece surface decreased while the shear stress increased. It might lead to nonuniform polishing due to the upward shift of the minimum shear stress location. As the MRSTP media flow velocity increased, both the normal pressure and shear forces on the workpiece surface became large. Due to the downward shift of the minimum shear stress location, the polishing uniformity of the workpiece surface was positively affected. Hence, in the actual MRSTP process, the polishing efficiency and uniformity would be improved with low workpiece rotational speed and high the MRSTP media flow velocity.

5 Modeling of material removal rate

5.1 Forces in the MRSTP process

Figure 11 shows illustration of the acting force of a single abrasive grain on workpiece surface during MRSTP process. Under the action of the normal pressure Fn, the abrasive grain generates an indentation depth d on the workpiece surface. Because of the relative movement between the workpiece and the media fluid, the abrasive grain is subjected to the shear force Fs. When the reacting force FR is less than the shear force Fs, the relative motion of the abrasive grain occurs to achieve the material removal on the workpiece [25, 26].

Fig. 11
figure 11

Illustration of acting force of a single grain on workpiece surface

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According to the Preston equation [27,28,29] as expressed as below.

$$MRR=k\cdot p\cdot v$$
(5)

where k is the Preston coefficient, p is the normal pressure applied to the workpiece surface, and v is the instantaneous relative velocity of the workpiece surface at the point of contact with the media, the material removal rate is a function of the normal pressure on the workpiece surface and the relative velocity between the workpiece and the media. The normal pressure on the workpiece surface during MRSTP consist of the three components i.e. the fluid dynamic pressure Pd, the floating force of the polishing media Pf and the magnetic pressure Pm [14]:

$$p={p}_{d}+{p}_{f}+{p}_{m}$$
(6)

As both the rheological properties of the STFs and the magnetic field on the kinematic state of the MRSTP media were considered in the numerical simulation (Sect. 4. 2), the pressure in the numerical simulation was the sum of the mentioned the three different pressures viz: Pd, Pf and Pm.

According to the simulation results, the normal pressure distribution on the workpiece surface was almost unchanged along the y-axis direction while the normal pressure on the workpiece surface decreased along the x-axis direction. To simplify the calculation, the surface of the workpiece was divided into N bar areas as shown in Fig. 12.

Fig. 12
figure 12

Area division of the workpiece surface

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When N is infinite, all these N regions could be considered as rectangular regions. At this point, the normal pressure in the region Di (i = 1, 2, 3…N) are equal at each point, the normal pressure was defined as pi (i = 1, 2, 3… N). Therefore, the total pressure on the region Di can be calculated as:

$${p}_{\text{i}}={\overline{p} }_{i}\cdot {S}_{Di}$$
(7)
$${S}_{Di}=\frac{50}{N}\cdot \sqrt{{25}^{2}-{\left(25-\frac{25}{N}i\right)}^{2}}$$
(8)

where SDi is the area of region Di. Thus, when N tended to infinity, the total pressure on the workpiece surface can be expressed as:

$$p=\underset{N\to +\infty }{\mathrm{lim}}\sum_{i=1}^{N}{p}_{i}$$
(9)

Substituting Eq. (5), (6) into Eq. (7), it was yielded as:

$$p=\underset{N\to +\infty }{\mathrm{lim}}\sum_{i=1}^{N}{\overline{p} }_{i}\cdot \frac{1250}{N}\cdot \sqrt{\frac{i}{N}\left(2-\frac{i}{N}\right)}$$
(10)

5.2 Relative velocity in the MRSTP process

Figure 13 shows a simplified kinematic model of the MRSTP. O1 is the rotation center of the spindle and O2 is the rotation center of the rotary table. The xO1y coordinate system is established with O1 as the coordinate origin. ρ is the radius of the polishing bowl and ρ0 is the eccentric distance between the spindle and the rotary table. It is assumed that the workpiece rotated around the spindle with angular velocity ω1 and the polishing bowl rotated around the rotary table with angular velocity ω2. The linear velocity of any point P on the surface of the workpiece is v1 about O1 and v2 about O2. Hence, the sum of v1 and v2 is the instantaneous relative velocity v at the point of contact between the workpiece surface and the MRSTP media. The bit vectors of point P about O1 and O2 are defined as r1 and r2, respectively. The included angles of the line between P and O1 and the line between P and O2 in the positive direction of the x-axis were θ1 and θ2, respectively.

Fig. 13
figure 13

A simplified kinematic model of the MRSTP

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The relative velocity of the workpiece and MRSTP media at point P can be expressed as:

$$\overset\rightharpoonup {v}={\overset\rightharpoonup {v}}_1+{\overset\rightharpoonup {v}}_2$$
(11)
$$\left\{\begin{array}{c}\overset\rightharpoonup {v}_{1}=-\overset\rightharpoonup {r}_{1}\cdot {\omega }_{1}\\ \overset\rightharpoonup {v}_{2}=\overset\rightharpoonup {r}_{2}\cdot {\omega }_{2}\end{array}\right.$$
(12)

The velocities v1 and v2 along the x and y axes are decomposed, respectively. Meanwhile, according to the formula for relative velocity v and the geometric relationship as shown in Fig. 13, the formulas are deduced as below.

$$v=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}$$
(13)
$${r}_{2}=\sqrt{{\left({r}_{1}\mathrm{sin}{\theta }_{1}\right)}^{2}+{\left({\rho }_{0}-{r}_{1}\mathrm{cos}{\theta }_{1}\right)}^{2}}$$
(14)
$$\mathrm{cos}\left({\theta }_{1}-{\theta }_{2}\right)=\frac{{r}_{1}^{2}+{r}_{2}^{2}-{\rho }_{0}^{2}}{2{r}_{1}{r}_{2}}$$
(15)

The instantaneous relative velocity of the contact point between the workpiece surface and the MRSTP media is obtained as below [30]:

$$v=\sqrt{{r}_{1}^{2}{\left({\omega }_{1}-{\omega }_{2}\right)}^{2}+{\rho }_{0}^{2}{\omega }_{2}^{2}+2{r}_{1}{\rho }_{0}\left({\omega }_{1}-{\omega }_{2}\right){\omega }_{2}\mathrm{cos}{\theta }_{1}}$$
(16)

5.3 Material removal rate model in MRSTP process.

Based on the Preston equation, the theoretical material removal rate for a planar workpiece surface is expressed as:

$$MR{R}_{t}=\underset{D}{\iint }k\cdot p\cdot vd\sigma$$
(17)

Substituting Eq. (8) and Eq. (14) into Eq. (15), the Eq. (16) is yielded as below.

$$\begin{array}{c}MR{R}_{t}=k\underset{N\to +\infty }{\mathrm{lim}}\sum\limits_{i=1}^{N}{\overline{p} }_{i}\cdot \frac{2\cdot {25}^{2}}{N}\cdot \sqrt{\frac{i}{N}\left(2-\frac{i}{N}\right)}\cdot \\ \underset{D}{\iint }\sqrt{{r}_{1}^{2}{\left({\omega }_{1}-{\omega }_{2}\right)}^{2}+{\rho }_{0}^{2}{\omega }_{2}^{2}+2{r}_{1}{\rho }_{0}\left({\omega }_{1}-{\omega }_{2}\right){\omega }_{2}\mathrm{cos}{\theta }_{1}}d\sigma \end{array}$$
(18)

where k is the Preston coefficient, which is a constant determined by the processing conditions including magnetic field. N is the number of workpiece surface division areas. I (i = 1, 2, 3…N) is the serial number of the area. pi is the pressure at any point on the area i, obtained from the simulation results. r1 is the distance from any point P on the workpiece to the center of rotation of the spindle. ω1 and ω2 are the rotational speeds of the spindle and rotary table, respectively. ρ0 is the eccentric distance between the spindle and the rotary table. θ1 is the angle between the line connecting the point P and the center of rotation of the spindle with the positive direction of the x-axis. Equation (16) showed that the MRR is related to the MRSTP geometry configuration and process variables. The key process variables are the rotational speeds of the spindle and rotary table and working gaps.

6 Experimental details

To verify the accuracy of the material removal model and the validity of the simulation results, a series of experiments was carried out. The experimental setup was built up on a CNC machining center (VKN640), as shown in Fig. 14. The magnetic field generation device was installed on a vertical slipway to adjust the working gap. The vertical slipway was held to the CNC worktable by a magnetic base. The workpiece was fixed on the spindle. The polishing bowl was installed on the rotary table. The MRSTP media were contained in the polishing bowl. Magnetic-enhanced particle clusters were formed under the action of magnetic field. The workpiece relative motion was generated with the magnetic-enhanced particle cluster to achieve material removal via the rotation of the workpiece and bowl [31].

Fig. 14
figure 14

MRSTP experimental setup

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A circular Aluminum alloy 6061 with diameter of 25 mm and thickness of 5 mm was used as the workpiece. The initial (Sa) value of the workpiece surface was 337.63 nm. The working gap between the workpiece and the polishing bowl was chosen as 1.0 mm. The concentration of STF base fluid was 20 wt% in the MRSTF media. For comparison with the simulation results, the flow velocity of the MRSTP media in the simulation parameters were converted into the rotational speed of the rotary table. The detailed experimental parameters and their values are shown in Table 2.

Table 2 Experimental parameters
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7 Results and discussion

7.1 Verification of material removal rate model

The theoretical material removal rate (MRRt) of MRSTP was compared with the experimental material removal rate (MRRe). MRRt was calculated by Eq. (16). MRRe was calculated by Eq. (17) as below:

$$MR{R}_{e}=\frac{{M}_{i}-{M}_{p}}{S\cdot \rho \cdot t}$$
(19)

where, Mi was the initial mass of the workpiece, Mp was the mass of the workpiece after MRSTP, S was the area of the polished plane of the workpiece, ρ was the density of the workpiece, t was the polishing time of MRSTP.

The error Er between MRRt and MRRe could be calculated by Eq. (18) [32]:

$$Er=\left|\frac{MR{R}_{t}-MR{R}_{e}}{MR{R}_{e}}\right|\times 100\%$$
(20)

The effect of different spindle rotational speeds on MRR was investigated, as shown in Fig. 15. As the spindle rotational speed increased, the MRR increased approximately as a power function. This phenomenon was consistent with the expression of the Preston equation. As the relative velocity of the workpiece and the media increased with the increase of spindle rotational speed, the MRR became large. Although the workpiece surface pressure decreased as the spindle rotational speed increased (see Fig. 7), the reduction in normal pressure played less role in the MRR than the relative velocity.

Fig. 15
figure 15

Influence of spindle rotational speed on MRR

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Figure 16 shows the effects of rotational speeds of the rotary table on the MRR. It was seen that the MRR also increased with the increase of rotational speed of the rotary table. According to Eq. (15), the relative velocity of the workpiece and MRSTP media became large with the increase of rotational speeds of rotary table. Meanwhile, according to Fig. 8, the normal workpiece pressure also increased as rotational speed of the rotary table increased. This resulted in large MRR.

Fig. 16
figure 16

Influence of rotational speeds of rotary table on MRR

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According to Fig. 15 and Fig. 16, the MRRt of the MRSTP aligned well with the MRRe. The average error was only 4.1%. The accuracy and validity of the MRR model was verified.

7.2 Surface morphology after polishing

Figure 17 shows the 3D morphology of the workpiece (Aluminum alloy 6061) surface before and after MRSTP. It was observed that there were obvious scratches on the workpiece surface before polishing. After polishing for 1 h, the scratches on the workpiece surface were completely removed. The surface roughness (Sa) of the workpiece surface decreased from 337.6 nm to 79.0 nm, which improved by 77%. The results demonstrated that MRSTP was effective to obtain excellent polishing performance for the Aluminum alloy 6061 workpiece.

Fig. 17
figure 17

3D topography of workpiece surface before and after MRSTP

Full size image

8 Conclusions

In this work, CFD method was employed to investigate the effect of polishing parameters on normal and shear stress in the MRSTP process. The MRR model of MRSTP process was established based on the CFD pressure output and the Preston equation. The accuracy and validity of the MRR model was verified. The effect of polishing parameters on the MRR was investigated. The main conclusions were drawn as follows:

  1. (1).

    The CFD simulation results showed that the pressure on the workpiece surface decreased with the increase of the rotational speed of the workpiece while it increased with the increase of the flow velocity of the MRSTP media. A continuous increment in shear stress on the workpiece surface was observed with the increase of the rotation speed of the workpiece and the flow velocity of the MRSTP media.

  2. (2).

    The MRR model indicated that the MRR had relation with the MRSTP geometry configuration and process variables. The key process variables were the rotational speeds of the spindle and rotary table and working gaps.

  3. (3).

    The accuracy and validity of the simulated results and MRR model were verified. The average error between theoretical and experimental results was 4.1%.

  4. (4).

    The workpiece surface roughness (Sa) decreased to 79.0 nm from initial 337.6 nm with 77% improvement after MRSTP. The scratches on the workpiece surface were completely removed after polishing for 1 h.