3.1 Theoretical derivation
The results in Sec. II show that during the relaxation process of strong shear Couette flow, the vx distribution always evolves into an extreme nonequilibrium distribution with moments (σmax, S0, and Kmin) in the Kmin layer. Furthermore, under the conditions of small Lz, large uup (or γ), low Tw, and diffusereflection wall, the Kmin layer will present a specific relaxation state of platformshaped distribution. As discussed in Sec. II.B, the root of the platformlike phenomenon is qualitatively attributed to the nonsamewall molecular collision. This section will further deduce and analyze the physical mechanism of this phenomenon.
Extending the above conditions of the platformlike phenomenon to extreme situations (e.g., an very rarefied and strong shear flow with a tiny Tw) and ignoring some secondary factors will help us to analyze the physical mechanism of platformlike phenomenon. According to Kn = λ0/Lz, if λ remains unchanged, the smaller Lz is, the greater Kn is, and the flow is more rarefied. According to γ = uup/Lz, if uup remains unchanged, the smaller Lz is, the greater γ is, and the shear is stronger. Thus, to model the extremely rarefied and strong shear flow requires a very short spacing of walls (i.e., Lz). However, Lz should not be too small. Considering that the velocity of gas molecules will be reset randomly when they hit on and are diffusely reflected from the wall, the molecular movement process between two successive collisions can be regarded as an independent motion period. Qualitatively speaking, for a molecule at any velocity, the smaller the Lz is, the sooner it moves to hit the wall again (i.e., the shorter the motion period is), and the probability and number of potential collisions during this period are reduced accordingly. In extreme cases, when Lz reduces to some extent (e.g., Kn > 10), the flow becomes a free molecule flow. In this flow, collision, momentum transfer, energy dissipation, and nonequilibrium relaxation phenomenon will not occur, which makes the relaxation process meaningless. Therefore, molecular collisions (especially nonsamewall molecular collisions) are necessary to reproduce the relaxation process and nonequilibrium phenomenon under strong shear. Based on the above discussion, we assume that a gas molecule only experiences nonsamewall molecular collision once at most in a motion period. Considering the correlation between the number of collisions and Lz, this hypothetical flowfield naturally has a small wall spacing.
The molecules of nonsamewall molecular collisions may be relaxed molecules or reflected molecules from the opposite wall. Since a molecule in this hypothetical flow only experiences nonsamewall molecular collision once in a motion period, it will not collide again after it turns into the relaxed molecule by collision. In this case, the two molecules of the nonsamewall collision must be reflected from different walls, and such a collision is abbreviated as the dualwall molecular collision. Thus, the following discussion sets that the target molecule A is reflected from the lower wall, and the test molecule B is reflected from the upper wall.
The relative velocity (cr) of dualwall molecular collision is generally very high in this investigation. According to the variablehardsphere model, the collision crosssection [σT(cr)] is proportional to cr1–2ω (= cr−0.62, where ω = 0.81). Hence, with the increase of cr, the change rate of σT(cr) will decay, so the collision model is simplified to the ordinary hardsphere collision in the following discussion.
Suppose the wall temperature (Tw) is so small that the thermal velocity values of reflected molecules (vT) are almost negligible relative to uup. However, Tw and vT are tiny but should not be zero, because the diffusion of wall velocity information requires a zdirection thermal motion. Generally, the collision probability (P) is positively correlated with cr, i.e., within a Δt, the greater cr is, the larger the space volume swept by the HS molecule is, and P is higher. Thus, for the two molecules in a samewall molecular collision, their cr is determined by vT, so their cr must be tiny, and their collision probability (Psame) is negligible. However, for molecules A and B in a dualwall molecular collision, their velocity values (vA and vB) approach 0 and uup, respectively, so their cr ≈ uup, and their collision probability (Pdual) cannot be ignored. Based on the above assumptions, in a hypothetical Couette flow where Lz is small enough and Tw is tiny but nonzero, the elementary process of the relaxation of strong shear nonequilibrium can be modelled as the collision of molecule B against molecule A at the relative velocity uup (see Fig. 10).
The coordinate system in Fig. 10 is established at the centroid of molecule A, and the direction of uup is the positive x direction. R is the radius of HS of molecule A. Molecule B collides with molecule A at the collision point represented by the red dot. As shown in Fig. 10(a), the sharp angle between the xaxis and the radius passing through the collision point is defined as the collision angle (θ), where θ∈[0, π/2]. All potential collision points whose θ∈[θ, θ + dθ] (where dθ is an infinitesimal) form a ring belt on the sphere surface of molecule A (see the grey belt). Because only reflected molecules participate in this collision, and the reflected molecules A and B share the same thermal velocity and density, the collision will occur uniformly and randomly in space, i.e., the projection of collision point on the yz plane is uniformly distributed in the projected circle of the sphere of molecule A. Let F be the conditional probability under the condition that only such dualwall molecular collisions are considered. For example, the probability of the collisions whose θ ≤ θ0 can be denoted as F(θ ≤ θ0). The conditional probability [dF, see Eq. (1)] whose θ∈[θ, θ + dθ] is the ratio of the projected area of the corresponding ring belt (dS = πR2sin2θdθ) to that of the sphere of molecule A (S = πR2). It is easy to prove that the integral of the righthand term of Eq. (1) over the interval [0, π/2] is 1, i.e., F(θ ≤ π/2) = 1.
dF = dS/S = 2sinθcosθdθ. (1)
As shown in Fig. 10(a), after the collision, molecule A bounces at vA = vn = uupcosθ along the normal direction of sphere surface at the collision point, and molecule B bounces at vB = vt = uupsinθ along the tangential direction. As shown in Fig. 10(b), the angle measured clockwise from the positive yaxis to the radius passing through the projection of the collision point is defined as the azimuth angle (φ). The projections (vAz and vBz) of vA and vB on the zaxis are
v Az = 0.5uupsin2θsinφ, vBz = – 0.5uupsin2θsinφ. (2)
As previously assumed, vT is much less than uup, and then the component of vT in the z direction (vTz) is even tinier. According to Eq. (2), as long as sin2θsinφ does not approach 0, vAz and vBz are in the same order of magnitude as uup and far greater than vTz. Because collision points are uniformly distributed in the projected circle, and the collision points satisfying sin2θsinφ = 0 are only distributed on the y axis and the edge line of the projected circle, vAz and vBz are indeed much greater than vTz in most collisions. To summarize, for the reflected molecule in this hypothetical flow, either it has not experienced a dualwall molecular collision and maintains a tiny vTz, or it turns into a relaxed molecule after such a collision and scatters at a much higher vz. As long as Lz is small enough, the relaxed molecules with large vz will pass through the flowfield and hit the wall in a short time without colliding again.
The projections (vAx and vBx) of vA and vB on the xaxis are
v Ax = uupcos2θ, vBx = uupsin2θ. (3)
It is clear that vAx and vBx are independent of φ. The total differentials of vAx and vBx are
dvAx = – 2uupcosθsinθdθ, dvBx = 2uupcosθsinθdθ. (4)
Solving Eq. (1) and Eq. (4) simultaneously, we obtain
– dF/dvAx = dF/dvBx = 1/uup. (5)
Eq. (5) illustrates that when the random variable of the conditional probability is vx, for the case where vAx (or vBx) after the collision is any specific value, its absolute value of conditional probability density is always 1/uup. It should be noted that dF/dvAx (= − 1/uup) is negative. According to Eq. (3), when θ = 0, the dualwall molecular collision is a central collision. After this collision, molecules A and B exchange their velocities, and vAx equals uup. Thus, as θ increases, vAx decreases, and dF/dvAx is negative. Because the derivations of Eqs. (3–5) are independent of φ, Eqs. (3–5) are always valid around x axis, and the vx of molecule A (or B) is uniformly distributed in the range [0, uup] after the collision. Therefore, the reflected molecules after dualwall molecular collisions will form a uniform vx distribution in the hypothetical flow.
3.2 Simulation verification
The deduction in Sec. III.A proves that for the hypothetical flow, if each reflected molecule only occurs one dualwall molecular collision in one motion period, their vx after collisions must be uniformly distributed. In real flow, besides the collisions of samewall/dualwall reflected molecules, there are recollisions of relaxed molecules. According to the conclusion in Sec. III.A, the conditions of uniform distribution are that vT is small and cr is large. However, the vT of relaxed molecules is large, and the cr of samewall molecules is small, so these nondualwall molecular collisions are not the root cause leading to the platformshaped distribution. In order to verify this conclusion, the multipliers β∈[0,1] are set to adjust the collision probability of different collision objects. When β = 0, the original probability of variable hard sphere model multiplied by β also equals 0, i.e., the collision will not happen. When β = 1, the original probability remains unchanged. The specific steps are as follows:
 Mark the molecule reflected from the upper (or lower) wall as “+1 (or –1)”;
 If a reflected molecule occurs a nonsamewall molecular collision, it becomes a relaxed molecule and marks “0”;
 If the relaxed molecule hits on the wall and reflects again, its mark shall be reset to be “+1/–1”;
 Set a multiplier β0 (or β1) to adjust the probability of recollisions (Pre) [or samewall molecular collisions (Psame)].
For example, for a collision pair including two molecules, if the product of their marks is “–1”, meaning that the two molecules are reflected from different walls and have not experienced a dualwall molecular collision. In this simulation, Pdual will not be adjusted because the dualwall molecular collision is the physical root of shear nonequilibrium. After this collision, they are regarded as relaxed molecules, and their marks are changed to “0”. If the product of their marks is “0”, there must be at least one relaxed molecule in this collision pair, and this collision belongs to a recollision. Now, multiple their Pre by β0. Suppose the product of their marks equals “+1”, which means that the collision is a samewall molecular collision. Then, multiple their Psame by β1. Within one Δt, the frequency of collisions is determined by P, so the total number of collisions (Ntotal) can be scaled down by adjusting β, that is:
N total = Ndual + β0Nre + β1Nsame, (6)
where Ndual, Nre, and Nsame are the numbers of dualwall molecular collision, recollision, and samewall molecular collision, respectively. Thus, Ntotal increases with β, and the increase of Ntotal will result in further relaxation. Table 6 gives the settings of cases in group 6 with different β.
Table 6
Group 6

Lz/λ0

uup/a0

γλ0/a0

Tw(K)

α

β0

β1

Case 6.1

6

15

2.500

300

1

0.0

0.0

Case 6.2

6

15

2.500

300

1

1.0

0.0

Case 6.3

6

15

2.500

300

1

0.0

1.0

Case 6.4

6

15

2.500

300

1

1.0

1.0

Case 6.5

6

15

2.500

300

1

0.2

0.2

Case 6.6

6

15

2.500

300

1

0.5

0.5

Figure 11 shows the variation of flux vx distributions in cases 6.1–6.4, where the flux vx distribution records the vx distribution in an upward molecular flux that is defined as all the molecules crossing the interface between two statistical layers along the positive z direction (i.e., vz > 0). This statistical way was usually used in the study of gassolid interaction [34] to observe the v distribution of incident/reflected molecular flux at the wall. Here, it is used to observe the v distribution after molecular collisions. For a certain molecule in a statistical layer, no matter how long it stays, how many times it collides, or how its velocity changes in this layer, it will only cross the adjacent upper interface at the velocity after its last collision, and only this velocity will be recorded in the flux distribution at this interface. Thus, the flux vx distribution at a certain interface is just equivalent to the vx distribution of the last collisions of molecules before passing through this interface. It should be noted that the vx distribution mentioned in Sec. II counts the vx of all molecules staying in the space of a statistical layer, so it can be named the spatial vx distribution to distinguish from the flux vx distribution. The longer the molecules stay in a statistical layer, the more times they will be recorded in the spatial distribution. Even if there is no collision, the molecular velocity may be repeatedly recorded in the spatial distribution. Therefore, the statistical method based on the molecular flux is more convenient for observing the vx distribution after collisions.
The blue dotted line in Fig. 11 represents the result of flux vx distribution at the interface featured with Kmin, and this interface is denoted as the Kmin interface. Figure 11(a) shows that in case 6.1, the flux vx distribution at interface z = 0.1λ0 (see the red long dashed line) is still the reflected molecular peak at vx/a0 = 0. The flux vx distribution at interface z = 3λ0 (see the green short dashed line) is the superposition of a reflected molecular peak and a platformshaped distribution, which means that some reflected molecules have undergone dualwall molecular collisions. Because in case 6.1, β = 0 (i.e., β0 = β1 = 0), a molecule will only occur one dualwall molecular collision between two successive reflections. Then, after all the molecules reflected from the lower wall have already collided once, their flux vx distribution will remain the platformshaped distribution in their subsequent motion. It can be seen that the flux vx distribution at interface z = 5.5λ0 almost coincides with that at the upper wall (z = 6λ0), and their K(v) both approximate Kmin (= 1.81) with deviations less than 10− 2. Thus, the interface z = 5.5λ0 is selected as the Kmin interface with a typical platformshaped distribution. As shown in Figs. 11(b), when β0 = 1, recollisions will occur, and the increased number of collisions will lead to further relaxation. Hence, the flux vx distribution at the Kmin interface deviates from the typical platformshaped distribution and transforms into an ordinary unimodal structure with Kmin = 2.20. When β1 = 1, the samewall molecular collision will happen. However, the result shown in Fig. 11(c) [or 11(d)] is only slightly different from that shown in Fig. 11(a) [or 11(b)]. The results shown in Fig. 11 confirm the conclusions in Sec. III.A, i.e., vx is uniformly distributed after the dualwall molecular collision, the samewall molecular collision has little effect on the vx distribution, and the vx distribution after the nondualwall molecular collision is nonuniform.
Figure 12 compares the variations of vx distribution in the cases with different β (= 0, 0.2, 0.5, and 1) based on the two statistical ways. Compared with Fig. 11, Figs. 12(ad) show the more detailed variation of flux vx distributions at representative interfaces (especially the Kmin interface) caused by the increase of β. Figures 12(eh) show that spatial vx distributions in the Kmin layer (see blue dotted lines) and other representative statistical layers gradually change with β. For case 6.1 with β = 0, Fig. 12(e) also shows that the spatial vx distribution near the upper wall [its K(v) = 1.60] has two peaks at abscissas vx = 0 and uup, but they are not reflected molecular peaks. This is because, referring to Fig. 12(a), the reflected molecules have already occurred the dualwall molecular collision once. Thus, the two peaks are formed by the relaxed molecules with vx close to 0 and uup, respectively. This distribution is owing to the central collision. According to Eq. (3), if the collision angle (θ) is 0 (i.e., the central collision), the vx after the collision will equal 0 or uup. Two molecules after the central collision will exchange their molecular velocities (v) but not change their v distribution. Hence, the centralcollision molecules will preserve the wall velocity information and have a relatively small vz. For the molecule passing through any equidistant statistical layer (with the constant spacing of 0.1λ0), its travelling time is proportional to 1/vz. Then, the smaller its vz is, the longer it travels through this layer, and the more times its vx will be recorded in the spatial vx distribution in this layer. Therefore, compared with the molecules after eccentric collisions, centralcollision molecules will stay in the statistical layer for a longer time and be counted repeatedly to form the bimodal peaks. Qualitatively, this bimodal phenomenon is still similar to the result in case 2.1 [see Fig. 6(a)], i.e., the wall velocity information cannot be fully relaxed by very few collisions. As β and Ntotal increase, the further relaxation will cut the two peak values down and eliminate the wall velocity information, and the spatial vx distribution transforms into the platformshaped distribution [see Fig. 6(h), its Kmin = 2.03]. Therefore, although the nondualwall molecular collision is not the root cause of the platformlike phenomenon, it still plays an important role in the equivalent relaxation process that generates the platformshaped spatial vx distribution.
Figures 12(ad) show that when β increases, the Kmin interface gradually approaches the lower wall (i.e., from LKm = 5.5λ0 to 4.8λ0). A similar trend of the Kmin layer (i.e., from LKm = 5.8λ0 to 4.4λ0) can also be seen in Figs. 12(eh). This trend is mainly because the increased nondualwall molecular collisions accelerate the relaxation of wall velocity information. In the flow with β = 0, the reflected molecule after the dualwall molecular collision is deemed the relaxed molecule and no longer collides. Thus, for the molecules reflected from a certain wall, their number will be continuously reduced by dualwall molecular collisions when they fly away from this wall. Then, the further these reflected molecules fly away from this wall, the fewer their number remains, and the less chance the residual reflected molecules collide, which will slow down the relaxation of wall velocity information. From the perspective of symmetry, moving away from this wall also means approaching the opposite wall, so the relaxation rates near both walls are relatively low. However, in the flow with β ≠ 0, the relaxed molecules may recollide with reflected molecules, which increases the likelihood and number of collisions of all molecules. Therefore, with the increase of β, the more the collisions occur, the shorter the distance of reflected molecules relaxing from the wall to the relaxation state featured with Kmin is.
In order to explore the influence of flow parameters on the dualwall molecular collision, Table 7 gives the settings of flow cases with β = 0 in group 7.
Table 7
Group 7

Lz/λ0

uup/a0

γλ0/a0

Tw(K)

α

β

Case 7.1

10

6

0.600

300

1

0.0

Case 7.2

8

9

1.125

300

1

0.0

Case 7.3

6

12

2.000

300

1

0.0

Case 7.4

6

15

2.500

300

1

0.0

Figure 13 shows the flux vx distributions at the Kmin interfaces in group 7. The simulation results of the flows with β = 1 in Sec. II.B indicate that the larger γλ0/a0 is, the more typical the platformshaped distribution is. If γ is small, the flow is close to the quasiequilibrium flow, e.g., case 1.2 shown in Fig. 5(b). As shown in Fig. 7, if γ is large enough, there is still a platformshaped distribution even in case 3.1 with a relatively small uup. However, according to the inference in Sec. III.A, the vx distribution after the dualwall molecular collisions is only determined by cr (≈ uup) and is independent of γ. Thus, γ is meaningless in the flow with β = 0. For example, in case 7.1 with a small γ, there is still a platformshaped distribution. Then, the variation of the flux vx distribution at the Kmin interface (see Fig. 13) mainly reflects the influence of uup. It can be seen that with the increase of uup, LKm decreases from z = 9.6λ0 (case 7.1) to 5.5λ0 (case 7.4). This is because in the flows of group 7, the thermal velocity of reflected molecules (vTw) based on Tw is relatively small, and the magnitude of cr is mainly determined by uup. Then, according to the VHS model, for the dualwall molecular collisions ignoring vTw, their collision crosssections are roughly correlated with uup1–2ω (= uup−0.62), and their Pdual are correlated with the velocity (i.e., uup) that they swept, that is, proportional to uup × uup1–2ω = uup0.38. Hence, within a Δt, the higher uup is, the larger the space swept by upperwallreflected molecules are, the more likely the dualwall molecular collisions happen, and the shorter LKm is. This also explains the corresponding settings of Lz (from 10λ0 to 6λ0) in group 7, i.e., to ensure that all reflected molecules can experience one dualwall molecular collision to obtain a typical platformshaped distribution. The Kmin value changes from 2.05 (case 7.1) to 1.81 (case 7.4), which indicates that as uup increases, the flow is getting closer to the hypothetical flow mentioned in Sec. III.A (i.e., vT is far less than uup), and the platformshaped distribution approaches the standard uniform distribution too.
The decoupling study based on taking the dualwall molecular collision as an independent factor shows that uup and cr will affect LKm, which partly explains why the LKm of group 3 cases in Fig. 7 is not linearly with λ. It should be pointed out that the nondualwall molecular collision also plays a part in the strong shear nonequilibrium. However, we cannot decouple the nondualwall molecular collision from the dualwall molecular collision in the simulation of strong shear flow, because the dualwall molecular collision is the physical root of strong shear nonequilibrium. Thus, as mentioned in the discussion of group 3 cases, for any flow case with a platformlike phenomenon, the total effect of all collisions leading to the platformlike phenomenon can be qualitatively regarded as an equivalent relaxation process. This analogy may not be accurate enough, but it will help us analyze and understand the strong shear relaxation process in more general flows in Sec. IV.
In summary, the flux distribution reflects the speed information of the last collision of molecules before passing through an interface. Because the distribution of vx after the dualwall molecular collision is uniform, there is a typical platformshaped flux vx distribution at the Kmin interface in case 6.1 with β = 0. The spatial distribution reflects the ensemble average speed information of molecules existing in a statistical layer. The spatial vx distribution in the Kmin layer of case 6.1 is nonuniform due to the weighting relation shown in Eq. (7). This nonuniformity can be further relaxed and eliminated by more collisions, so a platformshaped spatial vx distribution is shown in case 6.3 with β = 1. In Fig. 12, the dominant factor of all platformlike phenomena based on different statistical ways is the dualwall molecular collision. In Fig. 2, the globalspace distribution is the superposition of two halfspace distributions that are correlatively symmetrical to each other. Hence, the physical root of the platformlike transition [see Fig. 2(a)] is still the dualwall molecular collision.
In the flow with β = 0, the wall velocity information is continuously relaxed by dualwall molecular collisions while diffusing along the z direction. The platformshaped distribution will appear when almost all reflected molecules have experienced one dualwall molecular collision. Therefore, the position of the platformlike phenomenon can flag the relaxation distance of the wall velocity information relaxing to a specific state. However, the nondualwall molecular collision also plays an important role in the relaxation process in the flow with β ≠ 0. Because it is difficult to exclude the dualwall molecular collision and study the effect of nondualwall molecular collision alone, we can only analyze the equivalent relaxation process causing the platformlike phenomenon by analogy with the relaxation of the dualwall molecular collision.
Someone may wonder whether the platformlike phenomenon is caused by the simplified modelling of intermolecular interaction (i.e., the VHS collision model) in DSMC. In fact, the appendix shows that the platformshaped distribution also exists in the MD result.