# Investigation on Computing Method of Martian Dust Fluid Based on the Energy Dissipation Method

Authors: Tianxiang Ding; Xuyan Hou; Man Li; Guangyu Cao; Jixuan Liu; Xianlin Zeng; Zongquan Deng
Journal: International Journal of Aerospace Engineering
 (2020)

---

## Abstract

In this paper, an initiative Martian dust fluid simulating research based on the energy dissipation method was developed to simulate the deposition process of Martian dust fluid which was caused by surface adhesion between particles and Martian rovers. Firstly, an energy dissipation model of particles based on the Discrete Element Method (DEM) was established because of the characteristics of Martian dust particles such as tiny size and viscoelasticity. This model is based on the existing DMT model to analyze the collision deposition of dust fluid particles, including particle-spacecraft collision and particle-particle collision. Secondly, this paper analyzed the characteristics of particles after their first collision, then, established the stochastic model of critical wind speed for the particle deposition process. Finally, a series of simulations of the Martian dust fluid particle deposition process were done based on DEM-CFD. The results verified the accuracy of the energy dissipation model and the stochastic model, which could also verify the feasibility and effectiveness of the computing method of Martian dust fluid based on the DEM-CFD technology.


---

## Body

## 1. Introduction

A global dust storm (GDS) occurred in Mars year (MY) 34 (2018). According to the observation from the MSL Curiosity rover, the daily maximum UV radiation in the Gale Crater decreased by 90% from sols 2075 (opacity~1) to sols 2085 (opacity~8.5) [1, 2]. Figure 1(a) are two views from NASA’s Curiosity rover showing that dust has significantly increased over three days from June 7th to June 10th. Figure 1(b) shows a series of views from the Opportunity rover with darkening Martian sky, while the right side was Opportunity’s worst visibility in the MY34 GDS.Figure 1
Views of the MY34 GDS from Martian rovers: (a) Curiosity’s view of the MY34 GDS; (b) Opportunity’s view of the MY34 GDS.
(a)(b)Due to the low solar radiation, the solar-powered Opportunity rover (MER-B) was forced to shut down since June 10th of 2018, and eventually led to the end of the MER mission. In fact, local and regional dust storms are ubiquitous on Mars. Every few Mars years, regional dust storms grow and merge to become a global dust storm (GDS). Lisano and Bernard created a timeline of dust storm observations, which is shown in Figure2. The timeline includes all major planet-encircling and regional dust storms, which were observed by Mars orbiters, by Mars landers or by telescopes from the Earth, from 1971 to 2013 (the MY9 to the MY31) [3–7]. According to limited GDS samples, a GDS occurred every three Mars years on average.Figure 2
The Dust Storm Historic Timeline from the MY9 to the MY31 [3]Martian dust particles lifted by Martian aeolian activities may cause serious challenge to Martian rovers. These fine powders, which are mainly crushed basaltic materials, may cause performance degradation and other issues of optics to the rovers [8–16]. The Opportunity rover was hit by the MY28 GDS in 2007 when it carried on exploring operations in the Victoria Crater. During the peak of the storm, power output from the solar panels was reduced over 80%, which led to two weeks of minimal operations, including several days with no contact from the rover to save power [17]. Figure 3 shows the comparison of Opportunity’s solar panels before and after the dust storm. The suspended dust dropped onto solar panels and polluted solar panels seriously, which caused a serious challenge to the rover’s power supply [18].Figure 3
Comparison of Opportunity’s solar panels before and after the MY28 GDS.The deposition process of Martian dust is a typical two-phase flow process, but it is difficult to solve the equations of these phenomena directly. Therefore, the numerical approach has become an effective tool to study the gas-solid two-phase flow. In recent years, the numerical approach based on DEM-CFD is widely used for the industrial two-phase flow process, such as pneumatic conveying and gas-solid fluidization [19–32]. Thus, the numerical approach based on DEM-CFD is applicable for the research on the effect of the Martian dust storm. In this paper, an initiative Martian dust fluid simulating research based on the energy dissipation method was developed to simulate the deposition process of Martian dust fluid which was caused by surface adhesion between particles and Martian rovers. Importantly, this study provides a theoretical basis for Martian dust protection and the Martian exploration mission.


## 2. Materials and Methods

### 2.1. Energy Dissipation of Martian Dust Collision Process

The energy dissipation of collision is the major cause of the deposition process of Martian dust particles. The deposition process can be divided into two categories: the collision with the Martian rover’s surface and the collision with the grain bed, which is shown in Figure4. In this paper, the energy dissipation of collision is defined as the combined effect of adhesion energy loss and damping energy loss, and the contact status of Martian dust particles is analyzed based on the DMT contact model [33].Figure 4
Deposition process of Martian dust particles.According to the DMT contact model, the contact process is divided into two parts: the loading process and the unloading process. The loading process starts from the generation of the normal displacement until the normal displacement reaches the maximum. The unloading process starts from the maximum normal displacement until the normal displacement goes back to 0. However, due to the effect of the Van der Waals force, the Martian dust particle is applied to a negative contact force in the unloading process when the normal displace is 0. Figure5 shows the contact force in the loading-unloading process, while the shaded part in Figure 5 is the adhesion energy loss of the collision process.Figure 5
Comparison of Opportunity’s solar panels before and after the MY28 GDS.According to the DMT contact model [34, 35], the particle contact force in the loading process meets the following rule.
(1)a3=3R∗4E∗F.In equation (1), F is the normal contact force, a is the contact surface radius, and δ is the normal displacement.The particle contact force in the unloading process meets the following rule.(2)a3=3R∗4E∗F+2πΔγ∗R∗.While the contact surface radiusa and the normal displacement δ meet the following rule.
(3)δ=a2R∗.In the above equations,R∗=1/R1+1/R2−1 is the combined radius of contacted spheres, E∗=1−v12/E1+1−v22/E2−1 is the combined elastic modulus of two contacted spheres, where vi and Ei are Poisson’s ratios and elastic modulus of two particles, respectively.While analyzing the collision between the Martian dust particle and the Martian rover’s surface,R1=r1 is the curvature radius of the contact point A, R2 is the curvature radius of the surface. Thus,
(4)R1∗=R1R2R1+R2R1=r1,R2=+∞=r1.According to equation (2), the adhesion energy loss is not related to the magnitude and direction of the particle velocity, but related to the contact surface energy Δγ∗, the combined elastic modulus E∗, and the combined radius of contacted spheres R∗. Thus, the adhesion energy loss at the contact point A of the first collision (as shown in Figure 4) can be calculated as follows.
(5)WAv=∫δ0δAmax4E∗3r1a3∂δ∂ada+∫δAmaxδ04E∗3r1a3−2πΔγ∗r1∂δ∂ada=2πΔγ∗r13πr12Δγ∗2EA∗2/3=17.66Δγ∗5/3EA∗−2/3r14/3.According to the collision theory, the damping force during the collision is proportional to the instantaneous velocity of the particle. If we only consider the damping energy loss, the damping force is greater due to the higher particle velocity at the beginning of the collision process, which causes a lower deceleration rate during the collision process. If we only consider the adhesion energy loss, the adhesion force is smaller due to the smaller normal displacement at the beginning of the collision process, which causes a higher deceleration rate during the collision process. Considering the coupling effect of the adhesion energy loss and the damping energy loss, the relationship between the particle velocity and the normal displacement can be simplified as a linear curve, which is shown in Figure6.Figure 6
Simplification of the velocity-normal displacement curve.Thus, the damping energy loss at the first contact point A can be calculated as follows.(6)WAd=∫δ0δAmaxCVidδ=12CV03πr12Δγ∗2EA∗2/3=1.41CΔγ∗2/3EA∗−2/3r11/3V0.In equation (6), C is the damping coefficient, and V0 is the initial particle velocity during the collision process.According to equation (5) and equation (6), the total energy dissipation of the Martian dust particle at the first contact point A can be calculated as follows.
(7)WA=WAv+WAd=17.66Δγ∗5/3EA∗−2/3r14/3+1.41CΔγ∗2/3EA∗−2/3r11/3V0.According to Figure4, the particle collides with the Martian rover’s surface at the second contact point B after rotating a certain angle. R1=r2 is the curvature radius of the contact point B, and R2 is the curvature radius of the surface. Thus,
(8)R1∗=R1R2R1+R2R1=r2,R2=+∞=r2.The adhesion energy loss at the contact point B of the second collision can be calculated as follows.(9)WBv=∫δ0δBmax4E∗3r2a3∂δ∂ada+∫δBmaxδ04E∗3r2a3−2πΔγ∗r2∂δ∂ada=2πΔγ∗r23πr22Δγ∗2EA∗2/3=17.66Δγ∗5/3EB∗−2/3r24/3.The damping energy loss at the second contact point B can be calculated as follows.(10)WBd=∫δ0δBmaxCVidδ=12CVB03πr22Δγ∗2EB∗2/3=2CΔγ∗2/3EB∗−2/3r21/3m−1/2ΔWB1/2.In equation (10), ΔWB=1/2mV02−WA is the initial kinetic energy at the contact point B.According to equation (9) and equation (10), the total energy dissipation of the Martian dust particle at the second contact point B can be calculated as follows.
(11)WB=WBv+WBd=17.66Δγ∗5/3EB∗−2/3r24/3+2CΔγ∗2/3EB∗−2/3r21/3m−1/2ΔWB1/2.Therefore, the total energy dissipation of the particle-Martian rover collision is(12)WAB=WA+WB.While analyzing the collision between the Martian dust particle and the grain bed,R1=rC1 is the curvature radius of the depositing particle at the first contact point C, and R2=rC2 is the curvature radius of the grain bed particle at the contact point C. Thus,
(13)RC∗=R1R2R1+R2R1=rC1,R2=rC2=rC1rC2rC1+rC2.The adhesion energy loss at the contact point C of the first collision can be calculated as follows.(14)WCv=∫δ0δCmax4E∗3RC∗a3∂δ∂ada+∫δCmaxδ04E∗3RC∗a3−2πΔγ∗RC∗∂δ∂ada=2πΔγ∗RC∗3πRC∗2Δγ∗2Ep∗2/3=17.66Δγ∗5/3Ep∗−2/3RC∗4/3.The damping energy loss at the first contact point C can be calculated as follows.(15)WCd=∫δ0δCmaxCVidδ=1.41CΔγ∗2/3Ep∗−2/3RC∗1/3V0.According to equation (14) and equation (15), the total energy dissipation of the Martian dust particle at the first contact point C can be calculated as follows.
(16)WC=WCv+WCd=17.66Δγ∗5/3Ep∗−2/3RC∗4/3+1.41CΔγ∗2/3Ep∗−2/3RC∗1/3V0.According to Figure4, the particle collides with the grain bed at the second contact point D after rotating a certain angle. The adhesion energy loss WDv and the damping energy loss WDd at the second contact point D can be calculated as follows.
(17)WDv=∫δ0δDmax4E∗3RD∗a3∂δ∂ada+∫δDmaxδ04E∗3RD∗a3−2πΔγ∗RD∗∂δ∂ada=2πΔγ∗RD∗3πRD∗2Δγ∗2Ep∗2/3=17.66Δγ∗5/3Ep∗−2/3RD∗4/3,(18)WDd=∫δ0δDmaxCVidδ=2CΔγ∗2/3EP∗−2/3RD∗1/3m−1/2ΔWD1/2.In equation (18), ΔWD=1/2mV02−WC is the initial kinetic energy at the contact point D.According to equation (17) and equation (18), the total energy dissipation of the Martian dust particle at the second contact point D can be calculated as follows.
(19)WD=WDv+WDd=17.66Δγ∗5/3Ep∗−2/3RD∗4/3+2CΔγ∗2/3EP∗−2/3RD∗1/3m−1/2ΔWD1/2.Therefore, the total energy dissipation of the particle-grain bed collision is(20)WCD=WC+WD.


### 2.2. Critical Wind Speed for Martian Dust Particle Deposition Process

If the initial energy of the Martian dust particle is greater than the energy dissipation during the collision process, the falling particle will bounce from the Martian rover’s surface or the grain bed after the collision process. In this paper, we define the critical wind speed for the deposition process is the maximum velocity for the particle that rests on the Martian rover’s surface or the grain bed after the collision process. Thus, the critical wind speed for the deposition process on the Martian rover’s surfaceVc1 should meet the following rules.
(21)WA=WAv+WAd=17.66Δγ∗5/3EA∗−2/3r14/3+1.41CΔγ∗2/3EA∗−2/3r11/3Vc1,ΔWB=12mVc12−WA=WB,12mVc12=WA+17.66Δγ∗5/3EB∗−2/3r24/3+2CΔγ∗2/3EB∗−2/3r21/3m−1/2ΔWB1/2.We assumeg1=17.66Δγ∗5/3EA∗−2/3r14/3, g2=1.41CΔγ∗2/3EA∗−2/3r11/3, g3=1/2m, g4=17.66Δγ∗5/3EB∗−2/3r24/3, g5=2CΔγ∗2/3EB∗−2/3r21/3m−1/2, then, Vc1 can be calculated as follows.
(22)Vc1=g2+g22+4g3g1+ΔWB2g3,ΔWB=14g5+4g4+g522.In the above equations,g1 is the coefficient of the adhesion energy loss at the contact point A, g2 is the coefficient of the damping energy loss at the contact point A, g3 is the coefficient of the kinetic energy, g4 is the coefficient of the adhesion energy loss at the contact point B, and g5 is the coefficient of the damping energy loss at the contact point B.Similarly, the critical wind speed for the deposition process on the grain bedVc2 should meet the following rules.
(23)WC=WCv+WCd=17.66Δγ∗5/3Ep∗−2/3RC∗4/3+1.41CΔγ∗2/3Ep∗−2/3RC∗1/3Vc2,ΔWD=12mVc22−WC=WD,12mVc22=WC+17.66Δγ∗5/3Ep∗−2/3RD∗4/3+2CΔγ∗2/3EP∗−2/3RD∗1/3m−1/2ΔWD1/2.We assumeh1=17.66Δγ∗5/3Ep∗−2/3RC∗4/3, h2=1.41CΔγ∗2/3EP∗−2/3RC∗1/3, h3=1/2m, h4=17.66Δγ∗5/3Ep∗−2/3RD∗4/3, h5=2CΔγ∗2/3Ep∗−2/3RD∗1/3m−1/2, then, Vc2 can be calculated as follows.
(24)Vc2=h2+h22+4h3h1+ΔWD2h3,ΔWD=14h5+4h4+h522.In the above equations,h1 is the coefficient of the adhesion energy loss at the contact point C, h2 is the coefficient of the damping energy loss at the contact point C, h3 is the coefficient of the kinetic energy, h4 is the coefficient of the adhesion energy loss at the contact point D, and h5 is the coefficient of the damping energy loss at the contact point D.According to equation (22) and equation (24), the critical wind speed for the deposition process is the function of the combined radius of particles and the curvature radius of contact points. Due to the irregularly shaped characteristics of Martian dust particles, the combined radius and the curvature radius are diverse for different particles. Thus, the critical wind speed for the deposition process should be the random variable related to the combined radius and the curvature radius.Vc1 is the function of the combined radius R∗, the curvature radius of the falling particle at the contact point A r1 and the curvature radius of the falling particle at the contact point B r2. X=R∗,r1,r2T is defined as the random vector of Vc1, f is the probability density function, Z=Vc1,R∗,r1,r2T, thus,
(25)fZZ=fXX1JZX,fXX=fR∗R∗fr1r1fr2r2.In equation (25), fXX is the joint probability density of random independent variables, JZX is the Jacobian determinant of the random vector Z which is related to the random vector X. JZX can be calculated as follows.
(26)JZX=∂Vc1∂R∗∂Vc1∂r1∂Vc1∂r2∂r1∂R∗∂r1∂r1∂r1∂r2∂r2∂R∗∂r2∂r1∂r2∂r2.Thus, the probability density ofVc1 is
(27)fVc1Vc1=∫∫fXXJZXdr1dr2=∫∫fR∗R∗fr1r1fr2r2∂Vc1/∂R∗dr1dr2,∂Vc1∂R∗=∂Vc1∂g3∂g3∂m∂m∂R∗+∂Vc1∂ΔWB∂ΔWB∂g5∂g5∂m∂m∂R∗.Vc2 is the function of the combined radius R∗, the curvature radius of the falling particle at the contact point C rC1, the curvature radius of the grain bed particle at the contact point C rC2, the curvature radius of the falling particle at the contact point D rD1, and the curvature radius of the grain bed particle at the contact point D rD2. Thus, the probability density of Vc2 can be calculated as follows.
(28)fVc2Vc2=∬∫∫fR∗R∗frC1rC1frC2rC2frD1rD1frD2rD2∂Vc2/∂R∗drC1drD1drC2rD2,∂Vc2∂R∗=∂Vc2∂h3∂h3∂m∂m∂R∗+∂Vc2∂ΔWD∂ΔWD∂h5∂h5∂m∂m∂R∗.


## 2.1. Energy Dissipation of Martian Dust Collision Process

The energy dissipation of collision is the major cause of the deposition process of Martian dust particles. The deposition process can be divided into two categories: the collision with the Martian rover’s surface and the collision with the grain bed, which is shown in Figure4. In this paper, the energy dissipation of collision is defined as the combined effect of adhesion energy loss and damping energy loss, and the contact status of Martian dust particles is analyzed based on the DMT contact model [33].Figure 4
Deposition process of Martian dust particles.According to the DMT contact model, the contact process is divided into two parts: the loading process and the unloading process. The loading process starts from the generation of the normal displacement until the normal displacement reaches the maximum. The unloading process starts from the maximum normal displacement until the normal displacement goes back to 0. However, due to the effect of the Van der Waals force, the Martian dust particle is applied to a negative contact force in the unloading process when the normal displace is 0. Figure5 shows the contact force in the loading-unloading process, while the shaded part in Figure 5 is the adhesion energy loss of the collision process.Figure 5
Comparison of Opportunity’s solar panels before and after the MY28 GDS.According to the DMT contact model [34, 35], the particle contact force in the loading process meets the following rule.
(1)a3=3R∗4E∗F.In equation (1), F is the normal contact force, a is the contact surface radius, and δ is the normal displacement.The particle contact force in the unloading process meets the following rule.(2)a3=3R∗4E∗F+2πΔγ∗R∗.While the contact surface radiusa and the normal displacement δ meet the following rule.
(3)δ=a2R∗.In the above equations,R∗=1/R1+1/R2−1 is the combined radius of contacted spheres, E∗=1−v12/E1+1−v22/E2−1 is the combined elastic modulus of two contacted spheres, where vi and Ei are Poisson’s ratios and elastic modulus of two particles, respectively.While analyzing the collision between the Martian dust particle and the Martian rover’s surface,R1=r1 is the curvature radius of the contact point A, R2 is the curvature radius of the surface. Thus,
(4)R1∗=R1R2R1+R2R1=r1,R2=+∞=r1.According to equation (2), the adhesion energy loss is not related to the magnitude and direction of the particle velocity, but related to the contact surface energy Δγ∗, the combined elastic modulus E∗, and the combined radius of contacted spheres R∗. Thus, the adhesion energy loss at the contact point A of the first collision (as shown in Figure 4) can be calculated as follows.
(5)WAv=∫δ0δAmax4E∗3r1a3∂δ∂ada+∫δAmaxδ04E∗3r1a3−2πΔγ∗r1∂δ∂ada=2πΔγ∗r13πr12Δγ∗2EA∗2/3=17.66Δγ∗5/3EA∗−2/3r14/3.According to the collision theory, the damping force during the collision is proportional to the instantaneous velocity of the particle. If we only consider the damping energy loss, the damping force is greater due to the higher particle velocity at the beginning of the collision process, which causes a lower deceleration rate during the collision process. If we only consider the adhesion energy loss, the adhesion force is smaller due to the smaller normal displacement at the beginning of the collision process, which causes a higher deceleration rate during the collision process. Considering the coupling effect of the adhesion energy loss and the damping energy loss, the relationship between the particle velocity and the normal displacement can be simplified as a linear curve, which is shown in Figure6.Figure 6
Simplification of the velocity-normal displacement curve.Thus, the damping energy loss at the first contact point A can be calculated as follows.(6)WAd=∫δ0δAmaxCVidδ=12CV03πr12Δγ∗2EA∗2/3=1.41CΔγ∗2/3EA∗−2/3r11/3V0.In equation (6), C is the damping coefficient, and V0 is the initial particle velocity during the collision process.According to equation (5) and equation (6), the total energy dissipation of the Martian dust particle at the first contact point A can be calculated as follows.
(7)WA=WAv+WAd=17.66Δγ∗5/3EA∗−2/3r14/3+1.41CΔγ∗2/3EA∗−2/3r11/3V0.According to Figure4, the particle collides with the Martian rover’s surface at the second contact point B after rotating a certain angle. R1=r2 is the curvature radius of the contact point B, and R2 is the curvature radius of the surface. Thus,
(8)R1∗=R1R2R1+R2R1=r2,R2=+∞=r2.The adhesion energy loss at the contact point B of the second collision can be calculated as follows.(9)WBv=∫δ0δBmax4E∗3r2a3∂δ∂ada+∫δBmaxδ04E∗3r2a3−2πΔγ∗r2∂δ∂ada=2πΔγ∗r23πr22Δγ∗2EA∗2/3=17.66Δγ∗5/3EB∗−2/3r24/3.The damping energy loss at the second contact point B can be calculated as follows.(10)WBd=∫δ0δBmaxCVidδ=12CVB03πr22Δγ∗2EB∗2/3=2CΔγ∗2/3EB∗−2/3r21/3m−1/2ΔWB1/2.In equation (10), ΔWB=1/2mV02−WA is the initial kinetic energy at the contact point B.According to equation (9) and equation (10), the total energy dissipation of the Martian dust particle at the second contact point B can be calculated as follows.
(11)WB=WBv+WBd=17.66Δγ∗5/3EB∗−2/3r24/3+2CΔγ∗2/3EB∗−2/3r21/3m−1/2ΔWB1/2.Therefore, the total energy dissipation of the particle-Martian rover collision is(12)WAB=WA+WB.While analyzing the collision between the Martian dust particle and the grain bed,R1=rC1 is the curvature radius of the depositing particle at the first contact point C, and R2=rC2 is the curvature radius of the grain bed particle at the contact point C. Thus,
(13)RC∗=R1R2R1+R2R1=rC1,R2=rC2=rC1rC2rC1+rC2.The adhesion energy loss at the contact point C of the first collision can be calculated as follows.(14)WCv=∫δ0δCmax4E∗3RC∗a3∂δ∂ada+∫δCmaxδ04E∗3RC∗a3−2πΔγ∗RC∗∂δ∂ada=2πΔγ∗RC∗3πRC∗2Δγ∗2Ep∗2/3=17.66Δγ∗5/3Ep∗−2/3RC∗4/3.The damping energy loss at the first contact point C can be calculated as follows.(15)WCd=∫δ0δCmaxCVidδ=1.41CΔγ∗2/3Ep∗−2/3RC∗1/3V0.According to equation (14) and equation (15), the total energy dissipation of the Martian dust particle at the first contact point C can be calculated as follows.
(16)WC=WCv+WCd=17.66Δγ∗5/3Ep∗−2/3RC∗4/3+1.41CΔγ∗2/3Ep∗−2/3RC∗1/3V0.According to Figure4, the particle collides with the grain bed at the second contact point D after rotating a certain angle. The adhesion energy loss WDv and the damping energy loss WDd at the second contact point D can be calculated as follows.
(17)WDv=∫δ0δDmax4E∗3RD∗a3∂δ∂ada+∫δDmaxδ04E∗3RD∗a3−2πΔγ∗RD∗∂δ∂ada=2πΔγ∗RD∗3πRD∗2Δγ∗2Ep∗2/3=17.66Δγ∗5/3Ep∗−2/3RD∗4/3,(18)WDd=∫δ0δDmaxCVidδ=2CΔγ∗2/3EP∗−2/3RD∗1/3m−1/2ΔWD1/2.In equation (18), ΔWD=1/2mV02−WC is the initial kinetic energy at the contact point D.According to equation (17) and equation (18), the total energy dissipation of the Martian dust particle at the second contact point D can be calculated as follows.
(19)WD=WDv+WDd=17.66Δγ∗5/3Ep∗−2/3RD∗4/3+2CΔγ∗2/3EP∗−2/3RD∗1/3m−1/2ΔWD1/2.Therefore, the total energy dissipation of the particle-grain bed collision is(20)WCD=WC+WD.


## 2.2. Critical Wind Speed for Martian Dust Particle Deposition Process

If the initial energy of the Martian dust particle is greater than the energy dissipation during the collision process, the falling particle will bounce from the Martian rover’s surface or the grain bed after the collision process. In this paper, we define the critical wind speed for the deposition process is the maximum velocity for the particle that rests on the Martian rover’s surface or the grain bed after the collision process. Thus, the critical wind speed for the deposition process on the Martian rover’s surfaceVc1 should meet the following rules.
(21)WA=WAv+WAd=17.66Δγ∗5/3EA∗−2/3r14/3+1.41CΔγ∗2/3EA∗−2/3r11/3Vc1,ΔWB=12mVc12−WA=WB,12mVc12=WA+17.66Δγ∗5/3EB∗−2/3r24/3+2CΔγ∗2/3EB∗−2/3r21/3m−1/2ΔWB1/2.We assumeg1=17.66Δγ∗5/3EA∗−2/3r14/3, g2=1.41CΔγ∗2/3EA∗−2/3r11/3, g3=1/2m, g4=17.66Δγ∗5/3EB∗−2/3r24/3, g5=2CΔγ∗2/3EB∗−2/3r21/3m−1/2, then, Vc1 can be calculated as follows.
(22)Vc1=g2+g22+4g3g1+ΔWB2g3,ΔWB=14g5+4g4+g522.In the above equations,g1 is the coefficient of the adhesion energy loss at the contact point A, g2 is the coefficient of the damping energy loss at the contact point A, g3 is the coefficient of the kinetic energy, g4 is the coefficient of the adhesion energy loss at the contact point B, and g5 is the coefficient of the damping energy loss at the contact point B.Similarly, the critical wind speed for the deposition process on the grain bedVc2 should meet the following rules.
(23)WC=WCv+WCd=17.66Δγ∗5/3Ep∗−2/3RC∗4/3+1.41CΔγ∗2/3Ep∗−2/3RC∗1/3Vc2,ΔWD=12mVc22−WC=WD,12mVc22=WC+17.66Δγ∗5/3Ep∗−2/3RD∗4/3+2CΔγ∗2/3EP∗−2/3RD∗1/3m−1/2ΔWD1/2.We assumeh1=17.66Δγ∗5/3Ep∗−2/3RC∗4/3, h2=1.41CΔγ∗2/3EP∗−2/3RC∗1/3, h3=1/2m, h4=17.66Δγ∗5/3Ep∗−2/3RD∗4/3, h5=2CΔγ∗2/3Ep∗−2/3RD∗1/3m−1/2, then, Vc2 can be calculated as follows.
(24)Vc2=h2+h22+4h3h1+ΔWD2h3,ΔWD=14h5+4h4+h522.In the above equations,h1 is the coefficient of the adhesion energy loss at the contact point C, h2 is the coefficient of the damping energy loss at the contact point C, h3 is the coefficient of the kinetic energy, h4 is the coefficient of the adhesion energy loss at the contact point D, and h5 is the coefficient of the damping energy loss at the contact point D.According to equation (22) and equation (24), the critical wind speed for the deposition process is the function of the combined radius of particles and the curvature radius of contact points. Due to the irregularly shaped characteristics of Martian dust particles, the combined radius and the curvature radius are diverse for different particles. Thus, the critical wind speed for the deposition process should be the random variable related to the combined radius and the curvature radius.Vc1 is the function of the combined radius R∗, the curvature radius of the falling particle at the contact point A r1 and the curvature radius of the falling particle at the contact point B r2. X=R∗,r1,r2T is defined as the random vector of Vc1, f is the probability density function, Z=Vc1,R∗,r1,r2T, thus,
(25)fZZ=fXX1JZX,fXX=fR∗R∗fr1r1fr2r2.In equation (25), fXX is the joint probability density of random independent variables, JZX is the Jacobian determinant of the random vector Z which is related to the random vector X. JZX can be calculated as follows.
(26)JZX=∂Vc1∂R∗∂Vc1∂r1∂Vc1∂r2∂r1∂R∗∂r1∂r1∂r1∂r2∂r2∂R∗∂r2∂r1∂r2∂r2.Thus, the probability density ofVc1 is
(27)fVc1Vc1=∫∫fXXJZXdr1dr2=∫∫fR∗R∗fr1r1fr2r2∂Vc1/∂R∗dr1dr2,∂Vc1∂R∗=∂Vc1∂g3∂g3∂m∂m∂R∗+∂Vc1∂ΔWB∂ΔWB∂g5∂g5∂m∂m∂R∗.Vc2 is the function of the combined radius R∗, the curvature radius of the falling particle at the contact point C rC1, the curvature radius of the grain bed particle at the contact point C rC2, the curvature radius of the falling particle at the contact point D rD1, and the curvature radius of the grain bed particle at the contact point D rD2. Thus, the probability density of Vc2 can be calculated as follows.
(28)fVc2Vc2=∬∫∫fR∗R∗frC1rC1frC2rC2frD1rD1frD2rD2∂Vc2/∂R∗drC1drD1drC2rD2,∂Vc2∂R∗=∂Vc2∂h3∂h3∂m∂m∂R∗+∂Vc2∂ΔWD∂ΔWD∂h5∂h5∂m∂m∂R∗.


## 3. Results and Discussion

### 3.1. Influence Factors of Critical Wind Speed for Martian Dust Particles Deposition Process

While analyzing the collision between the rover’s surface and the Martian dust particle, we use the Matlab to plot theVc1 curves in different combined radii R∗. Parameters of Martian dust materials and Martian rover materials are shown in Table 1. The coefficient of energy loss η0≤η≤1 is defined as the ratio of the particle velocity after collision to the particle velocity before the collision. It is obvious that the particle has a deposition process while η=0.Table 1
Parameters of Martian dust materials and Martian rover materials.
ParametersValueCurvature radius of the contact pointr10-7 mCombined radius of Martian dust particlesR∗5~100μmShear modulus of Martian dust particles20 GPaPoisson’s ratio of Martian dust particles0.25Shear modulus of Martian rover materials70 GPaPoisson’s ratio of Martian rover materials0.3Surface energy between the particles and the Martian roverΔγ∗0.04 J/m2The effect of the combined radiusR∗ to the critical wind speed Vc1 is shown in Figure 7. It can be seen from Figure 7 that the Vc1 grows substantially with the declining R∗. The coefficient of energy loss η shows a truncation effect around the related critical wind speed.Figure 7
The effect of the combined radiusR∗ to the critical wind speed Vc1.The effect of the curvature radiir1, r2 at the contact point A and the contact point B to the critical wind speed Vc1 are shown in Figure 8. It can be seen from Figure 8 that the Vc1 rises with the growing curvature radius, while the adhesion energy loss is greater during the collision process.Figure 8
The effect of the curvature radii to the critical wind speedVc1: (a) The effect of r1 to the Vc1; (b) The effect of r2 to the Vc1.
(a)(b)Figures9 and 10 are the surface diagram and the contour map of the Vc1, respectively. According to these two figures, we can estimate and compare the Vc1 in different combined radii R∗ and curvature radii r.Figure 9
Surface diagram of the critical wind speedVc1.Figure 10
Contour map of the critical wind speedVc1.While analyzing the collision between the grain bed and the Martian dust particle, we use the Matlab to plot theVc2 curves in different combined radii R∗ and curvature radii r. Parameters of Martian dust materials are shown in Table 2, while the effects of the contact parameters to the Vc2 are shown in Figure 11. According to Figure 11, the combined radius of the particle shows a dominant influence on the critical wind speed Vc2 while the curvature radii show similar influence on the Vc2.Table 2
Parameters of Martian dust materials.
ParameterValueCurvature radius of the contact pointr10-7 mCombined radius of Martian dust particlesR∗5~100μmShear modulus of Martian dust particles20 GPaPoisson’s ratio of Martian dust particles0.25Surface energy between the falling particles and the grain bed particlesΔγ∗0.04 J/m2Figure 11
Effects of the contact parameters to the critical wind speedVc2.Figures12 and 13 are the surface diagram and the contour map of the Vc2, respectively. According to these two figures, we can estimate and compare the Vc2 in different combined radii R∗ and curvature radii r.Figure 12
Surface diagram of the critical wind speedVc2.Figure 13
Contour map of the critical wind speedVc2.


### 3.2. Simulation of Martian Dust Deposition Process Based on DEM-CFD

A series of simulations of the Martian dust deposition process are performed by using the coupling module of EDEM-Fluent. Table3 shows the CFD simulation parameters of Martian airflow in Fluent. The velocity inlet and the pressure outlet are selected in the Fluent, and the simulation temperature is constant. The turbulence model is the RNG k—ε model. The mesh for the simulation is shown in Figure 14.Table 3
CFD simulation parameters of Martian airflow in Fluent.
Simulation parametersValueOperating pressure690 paGravity3.73 m/s2Temperature220 KDensity0.0167 kg/m3Viscosity1.35×10−5kgm−1s−1Inlet velocity magnitude0.5~10 m/sGauge pressure0 paFigure 14
Mesh for the simulation of the Martian dust deposition process.Martian dust particles are modeled in the Discrete Element Method (DEM) software EDEM, and simulation parameters of these particles are shown in Table4. Since Martian dust particles have characteristics of irregular shape and tiny size, 4 kinds of particle models are modified with the fractal dimension of 2.2, 2.3, 2.4, and 2.5, respectively, which are shown in Figure 15 [36]. All these particles have the curvature radii of 1.78 μm~4.58 μm.Table 4
DEM simulation parameters of Martian dust particles in EDEM.
Simulation parametersValuePoisson’s ratio of the particlesυ10.25Shear modulus of the particlesG12×1010paDensity of the particlesρ11300 kg/m3Poisson’s ratio of the Martian probeυ20.3Shear modulus of the Martian probeG27×1010paDensity of the Martian probeρ22719 kg/m3Coefficient of friction between the particles and the Martian probef0.5Coefficient of rolling friction between the particles and the Martian probesfR0.01Recovery coefficient between the particles and the Martian probek00.1Surface energy between the particles and the Martian probeγ0.04 J/m2Figure 15
DEM models of Martian dust particles: (a) Fractal dimension 2.2; (b) Fractal dimension 2.3; (c) Fractal dimension 2.4; (d) Fractal dimension 2.5.
(a)(b)(c)(d)A series of simulations for the particle-rover collision and particle-particle collision are carried out for the study on the Martian dust deposition process, as shown in Figure16. The particle factory is modified as the dynamic factory in order to generate new falling particles during the simulation process. In the simulation of particle-particle collision, particles are preplaced on the Martian rover’s surface to simulate the grain bed, while these particles have a speed of less than 10-6 m/s.Figure 16
Simulation model for Martian dust deposition process: (a) Simulation model for particle-rover collision; (b) Simulation model for particle-particle collision.
(a)(b)The particles of fractal dimension 2.2 are used for analyzing the effect of the combined radiusR∗ to the critical wind speed Vc1 and Vc2. The collision process between the 20 μm sized particle and the Martian rover’s surface is shown in Figure 17. According to the simulation result shown in Figure 17(a), the particle bounced back to the airflow after the collision process while the wind speed is 0.75 m/s. It can be seen from Figure 17(b) that the particle deposited on the spacecraft surface after the collision process while the wind speed is 0.65 m/s. Therefore, the critical wind speed Vc1 of this particle should be between 0.65 m/s and 0.75 m/s. Based on this, we carried out a series of simulations with different wind speed conditions (from 0.65 m/s to 0.75 m/s). Hence, the critical wind speed was defined as the maximum of these wind speed conditions based on DEM-CFD simulation results, in which the falling particle did not bounce back to the airflow. According to the simulation results, the critical wind speed Vc1 of this particle is 0.68 m/s, which has a 7% error with the theoretical result (0.73 m/s).Figure 17
The collision process between the 20μm sized particle and the Martian rover’s surface: (a) Bouncing process of the collision; (b) Deposition process of the collision.
(a)(b)A series of simulations are carried out for analyzing the deposition process on the Marian rover’s surface by using the fractal dimension 2.2 particles with the combined radii of 5μm, 10 μm, 40 μm, 60 μm, 80 μm, and 100 μm, respectively. The simulation results and the theoretical results of the Vc1 are shown in Figure 18. According to Figure 18, the critical wind speed Vc1 drops significantly with the increasing combined radius, which proves that the smaller particles are more likely to be deposited on the Martian rover’s surface. Besides, the errors between the simulation results (discrete data point) and the theoretical results (continuous curve) are less than 10%, which proves the reliability of the theory.Figure 18
The simulation results and the theoretical results of the critical wind speedVc1.The simulation results (discrete data point) and the theoretical results (continuous curve) for analyzing the deposition process on the grain bed by using the different sized fractal dimension 2.2 particles are shown in Figure19. The results of the Vc2 has a similar trend with the results of Vc1, which also proves the reliability of the theory.Figure 19
The simulation results and the theoretical results of the critical wind speedVc2.The 20μm sized particles in different shapes are used for analyzing the effect of different curvature radii to the critical wind speed Vc1. Since curvature radii of the fractal dimension 2.2 particle and the fractal dimension 2.4 particle are too close, the fractal dimension 2.3 particle and the fractal dimension 2.5 particle are used for the simulation. The distribution probabilities of the Vc1 of these two particles are shown in Tables 5 and 6, respectively. It can be seen from Tables 5 and 6 that particles with a higher fractal dimension had a more dispersed probability of critical wind speed.Table 5
Distribution probability of theVc1 of the fractal dimension 2.3 particle.
NumberCurvature radius (μm)Critical wind speed (m/s)Probability (%)11.80.673622.50.783533.10.8629Table 6
Distribution probability of theVc1 of the fractal dimension 2.5 particle.
NumberCurvature radius (μm)Critical wind speed (m/s)Probability (%)11.80.531422.50.671532.70.792143.20.9635


## 3.1. Influence Factors of Critical Wind Speed for Martian Dust Particles Deposition Process

While analyzing the collision between the rover’s surface and the Martian dust particle, we use the Matlab to plot theVc1 curves in different combined radii R∗. Parameters of Martian dust materials and Martian rover materials are shown in Table 1. The coefficient of energy loss η0≤η≤1 is defined as the ratio of the particle velocity after collision to the particle velocity before the collision. It is obvious that the particle has a deposition process while η=0.Table 1
Parameters of Martian dust materials and Martian rover materials.
ParametersValueCurvature radius of the contact pointr10-7 mCombined radius of Martian dust particlesR∗5~100μmShear modulus of Martian dust particles20 GPaPoisson’s ratio of Martian dust particles0.25Shear modulus of Martian rover materials70 GPaPoisson’s ratio of Martian rover materials0.3Surface energy between the particles and the Martian roverΔγ∗0.04 J/m2The effect of the combined radiusR∗ to the critical wind speed Vc1 is shown in Figure 7. It can be seen from Figure 7 that the Vc1 grows substantially with the declining R∗. The coefficient of energy loss η shows a truncation effect around the related critical wind speed.Figure 7
The effect of the combined radiusR∗ to the critical wind speed Vc1.The effect of the curvature radiir1, r2 at the contact point A and the contact point B to the critical wind speed Vc1 are shown in Figure 8. It can be seen from Figure 8 that the Vc1 rises with the growing curvature radius, while the adhesion energy loss is greater during the collision process.Figure 8
The effect of the curvature radii to the critical wind speedVc1: (a) The effect of r1 to the Vc1; (b) The effect of r2 to the Vc1.
(a)(b)Figures9 and 10 are the surface diagram and the contour map of the Vc1, respectively. According to these two figures, we can estimate and compare the Vc1 in different combined radii R∗ and curvature radii r.Figure 9
Surface diagram of the critical wind speedVc1.Figure 10
Contour map of the critical wind speedVc1.While analyzing the collision between the grain bed and the Martian dust particle, we use the Matlab to plot theVc2 curves in different combined radii R∗ and curvature radii r. Parameters of Martian dust materials are shown in Table 2, while the effects of the contact parameters to the Vc2 are shown in Figure 11. According to Figure 11, the combined radius of the particle shows a dominant influence on the critical wind speed Vc2 while the curvature radii show similar influence on the Vc2.Table 2
Parameters of Martian dust materials.
ParameterValueCurvature radius of the contact pointr10-7 mCombined radius of Martian dust particlesR∗5~100μmShear modulus of Martian dust particles20 GPaPoisson’s ratio of Martian dust particles0.25Surface energy between the falling particles and the grain bed particlesΔγ∗0.04 J/m2Figure 11
Effects of the contact parameters to the critical wind speedVc2.Figures12 and 13 are the surface diagram and the contour map of the Vc2, respectively. According to these two figures, we can estimate and compare the Vc2 in different combined radii R∗ and curvature radii r.Figure 12
Surface diagram of the critical wind speedVc2.Figure 13
Contour map of the critical wind speedVc2.


## 3.2. Simulation of Martian Dust Deposition Process Based on DEM-CFD

A series of simulations of the Martian dust deposition process are performed by using the coupling module of EDEM-Fluent. Table3 shows the CFD simulation parameters of Martian airflow in Fluent. The velocity inlet and the pressure outlet are selected in the Fluent, and the simulation temperature is constant. The turbulence model is the RNG k—ε model. The mesh for the simulation is shown in Figure 14.Table 3
CFD simulation parameters of Martian airflow in Fluent.
Simulation parametersValueOperating pressure690 paGravity3.73 m/s2Temperature220 KDensity0.0167 kg/m3Viscosity1.35×10−5kgm−1s−1Inlet velocity magnitude0.5~10 m/sGauge pressure0 paFigure 14
Mesh for the simulation of the Martian dust deposition process.Martian dust particles are modeled in the Discrete Element Method (DEM) software EDEM, and simulation parameters of these particles are shown in Table4. Since Martian dust particles have characteristics of irregular shape and tiny size, 4 kinds of particle models are modified with the fractal dimension of 2.2, 2.3, 2.4, and 2.5, respectively, which are shown in Figure 15 [36]. All these particles have the curvature radii of 1.78 μm~4.58 μm.Table 4
DEM simulation parameters of Martian dust particles in EDEM.
Simulation parametersValuePoisson’s ratio of the particlesυ10.25Shear modulus of the particlesG12×1010paDensity of the particlesρ11300 kg/m3Poisson’s ratio of the Martian probeυ20.3Shear modulus of the Martian probeG27×1010paDensity of the Martian probeρ22719 kg/m3Coefficient of friction between the particles and the Martian probef0.5Coefficient of rolling friction between the particles and the Martian probesfR0.01Recovery coefficient between the particles and the Martian probek00.1Surface energy between the particles and the Martian probeγ0.04 J/m2Figure 15
DEM models of Martian dust particles: (a) Fractal dimension 2.2; (b) Fractal dimension 2.3; (c) Fractal dimension 2.4; (d) Fractal dimension 2.5.
(a)(b)(c)(d)A series of simulations for the particle-rover collision and particle-particle collision are carried out for the study on the Martian dust deposition process, as shown in Figure16. The particle factory is modified as the dynamic factory in order to generate new falling particles during the simulation process. In the simulation of particle-particle collision, particles are preplaced on the Martian rover’s surface to simulate the grain bed, while these particles have a speed of less than 10-6 m/s.Figure 16
Simulation model for Martian dust deposition process: (a) Simulation model for particle-rover collision; (b) Simulation model for particle-particle collision.
(a)(b)The particles of fractal dimension 2.2 are used for analyzing the effect of the combined radiusR∗ to the critical wind speed Vc1 and Vc2. The collision process between the 20 μm sized particle and the Martian rover’s surface is shown in Figure 17. According to the simulation result shown in Figure 17(a), the particle bounced back to the airflow after the collision process while the wind speed is 0.75 m/s. It can be seen from Figure 17(b) that the particle deposited on the spacecraft surface after the collision process while the wind speed is 0.65 m/s. Therefore, the critical wind speed Vc1 of this particle should be between 0.65 m/s and 0.75 m/s. Based on this, we carried out a series of simulations with different wind speed conditions (from 0.65 m/s to 0.75 m/s). Hence, the critical wind speed was defined as the maximum of these wind speed conditions based on DEM-CFD simulation results, in which the falling particle did not bounce back to the airflow. According to the simulation results, the critical wind speed Vc1 of this particle is 0.68 m/s, which has a 7% error with the theoretical result (0.73 m/s).Figure 17
The collision process between the 20μm sized particle and the Martian rover’s surface: (a) Bouncing process of the collision; (b) Deposition process of the collision.
(a)(b)A series of simulations are carried out for analyzing the deposition process on the Marian rover’s surface by using the fractal dimension 2.2 particles with the combined radii of 5μm, 10 μm, 40 μm, 60 μm, 80 μm, and 100 μm, respectively. The simulation results and the theoretical results of the Vc1 are shown in Figure 18. According to Figure 18, the critical wind speed Vc1 drops significantly with the increasing combined radius, which proves that the smaller particles are more likely to be deposited on the Martian rover’s surface. Besides, the errors between the simulation results (discrete data point) and the theoretical results (continuous curve) are less than 10%, which proves the reliability of the theory.Figure 18
The simulation results and the theoretical results of the critical wind speedVc1.The simulation results (discrete data point) and the theoretical results (continuous curve) for analyzing the deposition process on the grain bed by using the different sized fractal dimension 2.2 particles are shown in Figure19. The results of the Vc2 has a similar trend with the results of Vc1, which also proves the reliability of the theory.Figure 19
The simulation results and the theoretical results of the critical wind speedVc2.The 20μm sized particles in different shapes are used for analyzing the effect of different curvature radii to the critical wind speed Vc1. Since curvature radii of the fractal dimension 2.2 particle and the fractal dimension 2.4 particle are too close, the fractal dimension 2.3 particle and the fractal dimension 2.5 particle are used for the simulation. The distribution probabilities of the Vc1 of these two particles are shown in Tables 5 and 6, respectively. It can be seen from Tables 5 and 6 that particles with a higher fractal dimension had a more dispersed probability of critical wind speed.Table 5
Distribution probability of theVc1 of the fractal dimension 2.3 particle.
NumberCurvature radius (μm)Critical wind speed (m/s)Probability (%)11.80.673622.50.783533.10.8629Table 6
Distribution probability of theVc1 of the fractal dimension 2.5 particle.
NumberCurvature radius (μm)Critical wind speed (m/s)Probability (%)11.80.531422.50.671532.70.792143.20.9635


## 4. Conclusions

Based on the Discrete Element Method (DEM), the present study has established a contact model that includes the collision energy loss. This paper for the first time developed a method to study the energy dissipation of Martian dust particles after colliding with Martian rovers or the particle bed, which played an important role in the deposition process. Through the numerical approach, the combined radius of the particle showed a dominant influence on the critical wind speed for the particle deposition process, followed by curvature radii of contact points. Besides, particles with a higher fractal dimension had a more dispersed probability of critical wind speed. Importantly, this study provides a theoretical basis for Martian dust protection and the Martian exploration mission.



---

*Source: 2370385-2020-05-23.xml*