3.1. Evaluation of the Proposed Device Compared to the Case of No Thin Plate
Let us first consider the case when there is no thin plate mounted on the substrate at the uppermost boundary.
Figure 3a,b show the temperature and pressure distributions with the flow distribution obtained in the DSMC simulation for such a case, and
Figure 4a shows the flow and pressure distributions near the beam for the same case. Note that the pressure is normalized by
, which is the pressure spatially averaged in the whole region of the gas. In the present DSMC simulation, the gas pressure in the cell
is obtained by
, where
is the temperature of gas in the cell
,
is the number density of gas molecules in the cell
and given by
,
is the number of real molecules in the cell
, and
is the volume of the cell
. Here, the spatially averaged pressure
is defined as
, where
represents the summation over all the DSMC cells. Since we define the spatially averaged number density and the spatially averaged temperature as
and
, respectively, the relation
is satisfied. In the present simulations, the spatially averaged number density will not differ from the initial value, i.e., the setting value, because the number of real molecules will not vary during the simulations, and, hence,
is constant and common for all the considered cases. Therefore, the spatially averaged pressure
is proportional to the spatially averaged temperature
. On the other hand, the spatially averaged temperature
is determined as a result of DSMC simulation and depends on the ratio of the area of hot surface to the area of the cold surface. Since this ratio depends on the cases, the resultant spatially averaged temperature varies depending on the cases. Due to the variation in
, the spatially averaged pressure
also varies depending on the cases. Under the condition where the surrounding pressures are different among the cases, the pressure and the stress cannot be properly compared among the cases by using their absolute values. In order to avoid this problem, in the present study, we compare the pressure and the stress normalized by the spatially averaged pressure
instead of their absolute values.
According to Sone, Y [
41], in the case when an object of uniform temperature with sharp edges is immersed in a gas with a different temperature, the temperature field induced around the edges of the object drives a flow from a colder region to a hotter region near the edges. This flow is called thermal edge flow. The thermal edge flow should be considered as a pumping mechanism rather than a usual flow. For example, the driving mechanism of a pump using a thermal edge flow was proposed [
42]. In the present study, the thermal edge flows are induced near the beam corners in the directions from the central region of each surface of the beam to the corners. Since the thermal edge flows play a role as pump, these flows move gas molecules away from the corners. As a result, low pressure is induced around the beam, as shown in
Figure 3b. The low-pressure area on the bottom surface of the beam, which is induced by the thermal edge flow, sucks in gas from the wide bulk region of the gas below the beam due to the induced pressure difference; thus, the strong upward flow from the bulk to the beam is induced. This upward-pressure-driven flow supplies molecules to the low-pressure area on the bottom surface of the beam. In other words, this pressure-driven flow compensates the decrease in molecules in the central region of the bottom surface of the beam in such a manner that the supply of molecules is balanced with the removal of molecules due to the thermal edge flows in the steady state. Due to this compensation of the decrease in molecules by the upward-pressure-driven flow, the low pressure induced by the thermal edge flow is partly cancelled out, but remains in the central region of the bottom surface of the beam in the steady state, as shown in
Figure 3b and
Figure 4a. This is because if the low pressure is fully canceled out, the pressure-driven flow will disappear, and the thermal edge flow will pump out molecules from the considered region again and, hence, the flow cannot become steady.
The same is true for the low-pressure area on the top surface of the beam. The low-pressure region on the top surface of the beam sucks in gas from the narrow region of the gas above the beam. Thus, the downward-pressure-driven flow from the narrow region to the beam is induced. However, this downward flow to the top surface of the beam is much weaker than the upward flow to the bottom surface of the beam. The reason for the weaker downward flow is because the low-pressure area on the top surface can gather only a fewer number of molecules because of a smaller space between the beam and the upper substrate compared with the low-pressure area on the bottom surface. The decrease in the number of molecules near the bottom surface is more efficiently compensated by the strong upward molecular flow than that near the top surface. This causes relatively high pressure on the bottom surface compared with that on the top surface, as shown in
Figure 3b and
Figure 4a. This pressure difference can be seen in the stress distribution exerted on the beam surface shown in
Figure 4b.
After the upward-pressure-driven flow impinges on the bottom surface of the beam, it is directed sideways. As a result, it forms two vortices of opposite rotation, such that it satisfies the equation of continuity. However, the formation of two vortices seen here can be considered to be caused by the limited region of the computational domain in the present study. If the domain is large enough, i.e., if the lower substrate and the periodic boundaries on the right and left sides are far enough from the beam, it can result in just a jet of gas impinging on a plate, which is converted into two opposing flows along the plate. In the same manner, two opposing vortices are induced also near the top surface of the beam, although they are much weaker than those appearing near the bottom surface. This is because the downward-pressure-driven flow to the top surface of the beam is much weaker than the upward-pressure-driven flow to the bottom surface of the beam, as mentioned above.
Let us now consider the case when a thin plate is mounted on a flat substrate.
Figure 5a,b show the temperature and pressure distributions with the flow distribution obtained from DSMC for such a case, and
Figure 6a shows the flow and pressure distributions near the beam for the same case. In this case, except the region near the right side of the top surface of the beam, the obtained flow field is similar to the case of no thin plate, while the right vortex near the top surface of the beam naturally disappears because it is obstructed by the thin plate. Since the thin plate is hotter than the surrounding gas, a thermal edge flow is induced along the plate from the gap region between its tip and the top surface of the beam towards its root. Thus, similar to the thermal edge flow appearing around the corners of the beam, the thermal edge flow appearing around the edge of the thin plate acts like a pump that sucks in gas from the region near the top surface of the beam. This results in a reduction in pressure on the top surface of the beam located near the thin plate, as seen in
Figure 6a. As a result, this increases the pressure difference between the bottom and top surfaces of the beam. Furthermore, since this hot thin plate is located near the right top corner of the cold beam, the temperature gradient near the right top corner is increased, as seen in
Figure 5a. Due to this, the upward thermal edge flow along the right side of the beam is enhanced, as seen in
Figure 5a and
Figure 6a. This enhanced thermal edge flow along the right side of the beam carries molecules out of the region on the right side and, hence, induces the reduction in pressure on the right side of the beam, as seen in
Figure 6a, whereas the normalized pressure on the left side is the same as that for the case of no plate shown in
Figure 4a. In the case of no thin plate, the pressures on the right and left sides are the same because of symmetry in the field and, hence, no horizontal force appears. On the other hand, in the case when the thin plate is mounted and deviated rightward from the center of the beam, a rightward force is induced by the above-mentioned pressure difference appearing on the right and left sides of the beam.
Usually, in a gas, absolute values for the normal components of stress are much larger than those of its tangential components. This is because the normal component of stress includes the contribution of ambient pressure. If we draw a figure of stress exerted on the beam surface, such as
Figure 4b, using the absolute stresses exerted on it, the tangential components of stresses are not visible since their absolute values are much smaller than those of normal components. Nevertheless, in general, if we integrate a surface force due to ambient pressure or some other constant pressure over a surface of an object, the net force exerted on the object due to such pressure is zero. Therefore, the contribution of ambient pressure or some other constant pressure can be excluded when we discuss the net force exerted on the object. Considering this fact, it is convenient for us to subtract some constant pressure comparable to ambient pressure from the normal components of the local stress when we discuss the net force exerted on the beam. Due to this subtraction, the magnitudes of the normal components of stress can be expressed in the same order as those of the tangential components of stress. Here, firstly, we calculate the average pressure exerted on the beam surface,
, which is obtained by averaging a pressure obtained by Equation (1) over the beam surface, and then divide this superficially averaged pressure
by the spatially averaged pressure
. Then, we subtract the obtained value from the normal component of the stress obtained by Equation (1), which is also normalized by
. Thus, we obtain local stresses relative to the superficially averaged pressure
exerted on the beam surface as follows:
Here, in order to differentiate the thus-obtained stress from absolute pressure or absolute stress, we call it “modified stress”, of which the concept is similar to “gauge pressure”. Furthermore, note that in the present study, “normal stress” is reckoned as positive when it corresponds to a state of compression.
Figure 4b shows the distribution of the modified stress exerted on the beam surface in the case of no thin plate. In this case, the ratio
was
. Here, the normal component of normalized modified stress has a value in the range
to
as shown in
Figure 4b. In the case of negative value, the modified stress is drawn as if it pulls the beam outward like a tension in
Figure 4b. However, the magnitudes of normalized modified stresses are much smaller than the subtracted normalized pressure of
. Therefore, the tensile stresses in
Figure 4b should not be considered real tension. Actually, they represent a positive but low pressure, less than
. On the other hand, the compressive stresses in
Figure 4b represent a positive pressure higher than
. In
Figure 6b, we show the stress distribution such as
Figure 4b for different case, where the same subtracted value as that for the case of no thin plate in
Figure 4b, i.e.,
, was used. The same process is done for the stress distributions like
Figure 4b presented in
Section 3.2 and
Section 3.3. In
Figure 4b, we chose the superficially averaged pressure,
, as a subtracted pressure to visualize the effective force due to stress on the beam. However, for the purpose of examining the net force on the beam, we can choose subtracted pressure flexibly, in a way that is convenient for us. Therefore, here, we choose the same subtracted normalized pressure of
for all the cases. Thanks to this choice, we can directly compare the normalized modified stress over all the cases.
In the case when there is no thin plate mounted on the substrate, as shown in
Figure 4b, the stresses are symmetric in the
-direction; hence, zero net force in the
-direction can be obtained. However, the magnitude of the pressure exerted on the bottom surface of the beam is slightly larger than that exerted on the top surface of the beam; hence, a non-zero net force is directed upward in the y-direction. This pressure difference can be seen also in the pressure distribution in
Figure 3b and
Figure 4a, as mentioned above.
On the other hand, in the case when a thin plate is mounted on the substrate, as shown in
Figure 6b, the stresses are non-uniform and asymmetric horizontally and vertically. Specifically, as mentioned in the discussion for
Figure 6a, the stresses on the left and right sides of the beam are unequal, which result in a non-zero net force directed rightward. Furthermore, the presence of the thin plate causes an increase in the difference between the normal stresses exerted on the bottom and top surfaces of the beam, as mentioned above; hence, a stronger net force directed upward is obtained. Particularly, the upward force is stronger in the vicinity of the thin plate compared to the case of no thin plate shown in
Figure 4b. Here, note that the upward pulling force drawn in the vicinity of the thin plate represents not a real tension but a positive pressure, significantly smaller than the other parts of the beam surface, as mentioned above. These non-zero net forces are caused by the pressure differences on the surfaces of the beam, whose mechanism is described in the fifth paragraph of this section. The results show here that by mounting a thin plate on a flat substrate, a force that attracts the beam to the plate horizontally and vertically is induced.
3.2. Effect of the Position of the Thin Plate
Next, let us consider how the position of the thin plate relative to the beam affects the stress and the resultant net force exerted on the beam, by adjusting the deviation distance,
, while keeping all other parameters constant.
Figure 7a–d show the stress distributions on the beam surface for such cases.
Figure 8 shows the dependencies of the resultant net rightward and upward forces,
and
, exerted on the beam on the deviation distance,
. Here, the rightward and upward forces,
and
, are normalized by the forces exerted on the left side and the bottom surface due to the spatially averaged pressure
, i.e.,
and
, respectively.
Consider the case of no thin plate shown in
Figure 4b as a reference case. By mounting a thin plate on the substrate just above the center of the beam, as shown in
Figure 7a, the high pressure on the central region of the top surface of the beam shown in
Figure 4b is weakened, while the stresses exerted on the other surfaces are almost the same as those of the case of no thin plate. This results in the strong upward net force, as shown in
Figure 8a. This net upward force is 0.009-times as strong as the force due to the spatially average pressure
. In the case when the system is in the atmospheric air condition, a lift force of about
can be obtained. Assuming that the beam is of silicon material (2329
[
36]), this lift force is equivalent to the force that levitates the beam with a height of
.
As the position of the thin plate is moved rightward away from the center of the beam, the region on the top surface of the beam where the pressure is reduced by the upward thermal edge flow towards the thin plate moves rightwards together with the position of the thin plate, as shown in
Figure 6b and
Figure 7a–d. Since the region of pressure reduction is gradually deviated from the center of the beam as the thin plate moves rightward from the center of the beam, the net upward force decreases with increasing
, as shown in
Figure 8a. Note that even in the case of no thin plate, the net upward force is non-zero. It is indicated by a horizontal solid line in
Figure 8a. In
Figure 8a, the position of the edge of the beam,
, i.e.,
, is also indicated by the vertical solid line. By mounting a thin plate on the flat substrate, the net upward force is enhanced, as shown in
Figure 8a. However, this is only true up to the point when the position of the thin plate is at
from the center of the beam, where the net upward force approaches that of the case of no thin plate. Note that in the case of
, the thin plate is located outside the region above the beam, i.e.,
.
Simultaneously, as the thin plate is moved towards the edge of the beam from its center, the pressure on the right side of the beam is gradually reduced, as shown in
Figure 6b and
Figure 7a–c. As a result, the net rightward force appears and increases with increasing
, up to the edge of the beam, i.e.,
, as shown in
Figure 8b. Similarly to
Figure 8a, the position of the edge of the beam,
, is indicated in
Figure 8b. The net rightward force obtained when the thin plate is located at the right edge of the beam is 0.0045-times as strong as the force due to the spatially average pressure
. In the case when the system is in the atmospheric air condition, a rightward force of about 450 Pa can be obtained. Assuming that the beam is of silicon material, this force is equivalent to the force that accelerates the beam with a width of 20 mm rightward with the earth’s gravitational acceleration. However, when the position of the thin plate is moved further away from the edge of the beam, the reduction in pressure on the right side of the beam is decreased, as shown in
Figure 7d. As a result, the net rightward force decreases as the thin plate is moved further away from the edge of the beam, i.e.,
, as shown in
Figure 8b.
Nevertheless, the results here show that in a range of , the net force is always directed upward and rightward, i.e., it always attracts the beam to the plate horizontally and vertically. Therefore, by using these thermally induced forces, the mechanism where the thin plate acts like tweezers that can trap the beam is obtained.
3.3. Effect of the Beam Height
Next, let us investigate the effect of the height of the beam,
, on the stress and the resultant net force exerted on the beam, by adjusting the beam height,
, while keeping all other parameters constant. For such cases,
Figure 9a and
Figure 10a show the flow and pressure distributions near the beam, and
Figure 9b and
Figure 10b show the stress distributions on the beam surface.
Figure 11 shows the dependencies of the resultant net rightward and upward forces,
and
, exerted on the beam on the height. Here, similar to what was shown in
Section 3.1 and
Section 3.2, the rightward and upward forces,
and
, are normalized by the forces exerted on the left side and the bottom surface due to the spatially averaged pressure
, i.e.,
and
, respectively.
Consider the case shown in
Figure 6, where
and
, as the reference case. Firstly, let us compare the case when the beam height is decreased to
as shown in
Figure 9, with the reference case. By decreasing the beam height, the thermal edge flows around the corners of the beam become stronger and the pressure on the top surface of the beam is decreased, as shown in
Figure 9a, compared to
Figure 6a. However, the high pressure on the central region of the bottom surface of the beam is almost the same as that of the reference case. As a result, the pressure difference between the bottom and top surfaces of the beam is increased, as seen in
Figure 6a,b and
Figure 9a,b. Therefore, a stronger upward net force is generated, as shown in
Figure 11a. Simultaneously, as the beam height is decreased, the pressures on both the right side and the left side of the beam are reduced, as shown in
Figure 6a and
Figure 9a. Although the pressure is decreased on both sides, the pressure reduction on the right side is slightly greater than that on the left side, as shown in
Figure 6a,b and
Figure 9a,b. Therefore, a stronger rightward net force is generated, as shown in
Figure 11b.
Secondly, let us compare the case when the beam height is changed to
, with the reference case. By increasing the beam height, as shown in
Figure 10a, the thermal edge flows around the corners of the beam become weaker and the reduction in the pressure on the top surface of the beam is decreased compared to
Figure 6a. However, the high pressure on the central region of the bottom surface of the beam remains almost the same as that of the reference case. As a result, the pressure difference between the bottom and top surfaces of the beam is decreased, as seen in
Figure 6a,b and
Figure 10a,b. Therefore, upward net force is weakened, as shown in
Figure 11a. Simultaneously, as the beam height is increased, the reductions in the pressures on both the right side and the left side of the beam are decreased, as shown in
Figure 6a and
Figure 10a. The decrease in the reductions in the pressures on both sides indicates that the pressures on both sides become closer to the average pressure and, hence, the pressure difference between both sides becomes smaller. Therefore, a rightward net force is still generated, but weaker, as shown in
Figure 11b. The results presented here and in the previous paragraph show that for a shorter beam height, the net upward and rightward forces are stronger.
3.4. Effect of Lower Temperature Differences
So far, only a temperature difference of
has been considered. Now, let us consider the effect of using temperature differences lower than
between the heated substrate (thin plate) and the colder object (beam), i.e.,
, on the net forces.
Figure 12 shows the distribution of the resultant net rightward and upward forces,
and
, exerted on the beam, for various temperature differences less than and equal to
at
. Here, the temperature difference
is changed while the middle value
is kept at
. Similar to what was described in
Section 3.2 and
Section 3.3, the rightward and upward forces,
and
, are normalized by the forces exerted on the left side and the bottom surface due to the spatially averaged pressure
, i.e.,
and
, respectively. It can be seen from
Figure 12 that as the temperature difference is decreased from 200 K, both the net upward and rightward forces decrease. Specifically, both the net upward and rightward forces are proportional to the temperature difference, and both vanish in the case of no temperature difference, i.e.,
. As discussed in previous sections, the reduction in pressure around the surface of the beam is due to the thermal edge pump effect. A low-temperature difference means that the strength of the thermal edge flows is weak. This results in smaller reductions in pressure around the beam and, hence, weaker net upward and rightward forces. In the case of the minimum temperature difference considered here at 25 K, under the atmospheric air condition, a lift force of about
and a rightward force of about 20 Pa can be obtained. Assuming that the beam of silicon material of the density of
, this lift force is equivalent to the force that levitates the beam with a height of
, and the rightward force is equivalent to the force that accelerates the beam with a width of
rightward with the Earth’s gravitational acceleration. Thus, the obtained force for low-temperature differences is still large enough to move a small object less than
. Therefore, even at the low temperature difference of 25 K, the mechanism proposed in this paper can still be achieved.
3.5. Effect of Knudsen Number
Up to this point, only a Knudsen number of
was considered. Now, let us consider the effect of different Knudsen number cases on the net forces.
Figure 13 shows the resultant net rightward and upward forces,
and
, exerted on the beam, for various Knudsen numbers. Here, the Knudsen number is adjusted by adjusting all the lengths of the setup considered here while keeping the gas mean free path
constant, i.e., keeping the reference gas pressure
constant. For example, the total gap distance
is adjusted as
. Similar to what was described in
Section 3.2,
Section 3.3 and
Section 3.4, the rightward and upward forces,
and
, are normalized by the forces exerted on the left side and the bottom surface due to the spatially averaged pressure
, i.e.,
and
, respectively. It can be seen from
Figure 13 that the net upward force and the net rightward force are both maximum around
, which is used as the reference Kn throughout this study, and both become small at low Knudsen numbers
and high Knudsen numbers
. Note that the Knudsen number here is evaluated based on the total gap distance,
, such that
.
As discussed in previous sections, the reduction in pressure around the surface of the beam is due to the thermal edge pump effect. Specifically, the low-pressure region on the top surface of the beam is increased by the thermal edge flow induced on the tip of the thin plate. Therefore, considering the local Knudsen number between the tip of the thin plate and the top surface of the beam, i.e., , is significant when discussing the effect of the Knudsen number on the net forces. Note that since . Using , we can restate the above-mentioned knowledge as follows: the net upward force and the net rightward force are both maximum around and both become small at low Knudsen numbers and high Knudsen numbers . This means that the thermally induced forces, which attract the beam to the thin plate horizontally and vertically, are optimum when the distance between the beam and the thin plate is in the order of one mean free path, as what is used throughout this study.