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This paper proposes a novel computationally efficient method of modeling rarefied gas flow in microchannels based on the newly discovered and mathematically proven Ballistic Principle of the Property Balance in Space (BPPBS). The mechanism of influence of the effect of rarefication on the gas flow is specifically investigated. Also, a differential form of the momentum balance equation governing gas flow in the channel between two parallel plates due to the pressure gradient along the channel and its exact implicit solution in the form of an integral equation have been derived. The theory does not use the generalized concept of viscosity based on the variable mean free path (MFP) in the Knudsen layer (KL). Comparing the normalized flow rate as a function of the inverse Knudsen number according to the current theory and the experimental data shows good agreement in the range of the inverse Knudsen number from 0.01 to about 40. The correlation factor is found to be about 0.995. The results show that our approach based on the BPPBS offers substantial and practical advantages in modeling and simulation of rarefied gases. The validity of the widely disseminated claim of the geometry-dependent MFP in the KL was analyzed.

The modeling gas flow in microfluidic devices for microelectromechanical systems (MEMS) is critically important. MEMS devices based on parallel-plate structures have often been used as the capacitive sensing, electrostatic actuation mechanisms in micro gyroscopes, accelerometers, switches, mirrors, pressure sensors [

A well-known method of obtaining analytical solutions for isothermal gaseous flow with slip boundary conditions is based on the locally fully developed flow assumption. The following Navier-Stokes equation governing gas flow in a two-dimensional channel (with x-axis along the channel and y-axis across the channel between two spaced at distance H parallel plates) is expressed as:

∂ 2 u ∂ y 2 = 1 η d p d x , (1)

and applying the second-order velocity slip boundary conditions (Maxwell-type assumption) in the following form [

u − u w = ± C 1 λ f ∂ u ∂ y − C 2 λ f 2 ∂ 2 u ∂ y 2 , (2)

where η is the fluid dynamic viscosity, u is the gas slip velocity near the wall, u_{w} is the tangential velocity of the wall, and λ f is the mean free path, C_{1} and C_{2} are constants. It is well-known from the gas kinetic theory the following relationship:

η = 1 2 m n v T λ f , (3)

where n is gas-particle density, m is gas-particle mass, v_{T} is the magnitude of the thermal velocity, and η is the fluid dynamic viscosity in one-dimensional gas space. However, according to [

A rarefied gas flow is characterized by using the dimensionless Knudsen number (Kn), which is defined as the MFP ratio to a characteristic length scale of the gas flow. Keeping in mind that gas transport takes place through intermolecular collisions separated by the ballistic motion of molecules characterized by the molecular free paths, one can conclude that the MFP is an important parameter in gas dynamics. The KL, adjacent to a gas-solid interface region of several mean free paths thick, is another important rarefaction parameter. It is defined as the region of local nonequilibrium extending a thickness of a few MFPs from the wall in gas microflows [

The variety of recent studies considers the collisions between the freely moving gas molecules and wall atoms since they believed that these collision events lead to the momentum change, and show employing molecular dynamics simulation (MD) that the MFP varies near surfaces [

In this approach, a wall is included in the system so that some molecules will hit the wall, and the wall will terminate their flight paths. Consequently, the MFP of all the molecules in the system may be smaller than the MFP in the unconfined gas space because of the boundary limiting effect [

f ( r ) = 1 λ 0 e − r / λ 0 , (4)

where λ 0 is the MFP in an unbounded space.

Second, he defined the expectation E ( r 0 ) “of the length r of the free flight until a collision either with another molecule, or, if none happens, with the obstacle upon covering the distance r_{0}”. It is resulted in finding that

λ ( r 0 ) = E ( r 0 ) = λ 0 ( 1 − e − r 0 / λ 0 ) . (5)

Finally, Abramov [

λ ( d ) = λ 0 ( 1 − 1 2 ( e − d / λ 0 − d λ 0 E 1 ( d λ 0 ) ) ) , (6)

where is a standard notation for the exponential integral

E 1 ( d λ 0 ) = ∫ d λ 0 ∞ e − y y d y . (7)

Hadjiconstantinou [

The authors of this study admitted “erroneous definition and calculation practices.” They also highlighted that “variations in MFP as a function of the surface confinement, which disagrees with the kinetic theory and leads to wrong physical interpretations of nanoscale gas flows. This controversy occurs due to erroneous definition and calculation practices, such as consideration of gas wall collisions, using local bins smaller than a MFP, and utilizing time frames shorter than a mean collision time in the MFP calculations”. They also stated that the collisions between the moving gas molecules and the solid wall should be ignored when evaluating the individual free paths. Finally, they highlighted that “[l]ocal MFP variations are physically admissible only if there are local density and temperature variations in the system”.

Another approach consists in revisiting “the problem of micro-channel compressible gas flows and show[ing] that the axial diffusion of mass engendered by the density (pressure) gradient becomes increasingly significant with increased Knudsen number compared to the pressure-driven convection [

Returning to the critical notes in [

Our recent publications [_{T}, are constant. We have verified that, in each point of space between confining parallel plates, the mass and momentum balance are conserved. It implies that any deviation from the constant particle density or the MFP will violate the BPPBS and violate the physical principles.

Additionally, as we referenced above, the flow of a dilute gas near a solid surface is usually explained in various theories by non-continuum effects assumed to exist in the Knudsen layer. In contrast, our approach does not use the concept of the Knudsen layer at all. Particularly, using basic principles of Newton’s mechanics and the kinetic theory of gases, we have derived an analytical representation of the gas flow velocity profile in the microchannel formed by two being at rest infinite parallel plates spaced at distanceH (see Equation (116) of [

u x ( y ) = − H m n v T d P d x 1 K n ( − y 2 H 2 + y H + 2 − σ σ K n + 2 K n 2 ) , (8)

where n is the particle density, m is the mass, v_{T} is the magnitude of the thermal velocity of the particle/molecule, σ is the momentum accommodation coefficient, which is the probability, for an incident particle, to accommodate momentum from the gas-solid interface and to scatter back in the model gas as a

diffuse particle, K n = λ f H is the Knudsen number, λ f is the mean free path, and d P d x is the pressure gradient forcing gas flow along the channel with the

velocity distribution u x ( y ) across the channel. Remarkably, the derived tangential slip velocity coefficient (the third right-hand term in parenthesis) does contain the term being proportional to 2 − σ σ . Its appearance is not the result of using the semi-empirical Maxwell-type assumptions or the Knudsen layer’s notion but the result of the application of the BPPBS.

Also, we compared the non-dimensional flow rates calculated by Equation (110) of [

This paper further promotes the newly discovered BPPBS and its application to describe steady-state fluid flows in a microchannel. It specifically investigates the mechanism of influence of the Knudsen number on the rarefied gas flow. It offers a new way of computing a steady-state rarefied gas flow in a microchannel, which does not use the generalized concept of viscosity based on the variable MFP in the KL.

In Section 2 of this paper, we further analyze the validity of the currently widely disseminated claim of the MFP variation normal to the nano-channel surfaces as a function of the confinement level. We applied validation test 4.1, “Determining the total rate of collisions per unit area on a surface being in contact with the gas” of [

Section 3 describes the properties and features of the model gas, the Ballistic Model’s physical principles, and the general physical principles of constructing the property balance for one-dimensional steady-state gas flow.

In Section 4, we provide integro-differential forms of mass balance and momentum balance equations formulated for the gas space bounded by two infinite parallel plates.

In Section 5, we demonstrate the effect of ballistic bouncing of gas particles between confining plates on rarefied gas flows.

Section 6 shows a way to obtaining a differential form u_{x}-momentum balance equation governing gas flow in the channel between two parallel plates due to the pressure gradient along the channel.

Finally, in Section 7, we present the discussion and conclusions and highlight the advantages of the proposed approach for modeling rarefied-gas flows or gas flows in MEMS.

In our recent publications [

In a semi-sphere filled with the incompressible gas over the being at rest surface A_{s} having a directional vector n → , each particle having an instantaneous randomly directed vector of the thermal velocity of magnitude, v_{T} may have the instant vector-velocity component directing a particle toward the surface A_{s}. In _{s}. The angle between the instant vector n → i 0 and the directional vector n → is labeled as θ. The ends of the directionally random vector-velocity of magnitude v_{T} form spherical surface 102.

Adopting Equation (34) of [_{s}, is given as:

Z = − 1 4 π ∭ V Z V ( r → ′ ) | r → − r → ′ | 2 Q i 0 ( r → , r → ′ ) n → i 0 ⋅ n → d V ′ , (9)

where

n → i 0 = r → − r → ′ | r → − r → ′ | (10)

and Z_{V} is the rate of collisions per unit volume. In [_{V} by substitution of Equation (25) in Equation (77) of [_{V} is expressed as

Z V = 2 3 n P c v T , (11)

where n is particle density, v_{T} is the magnitude of the thermal velocity, and P_{c} is the number of particles placed within a collision tube of a unit length in an unbounded gas space.

Recognizing the similarity of the geometry of computation in [

d ≡ y ; r 0 ≡ ρ ′ = y ′ cos ( θ ) ; λ 0 ≡ 1 P c . (12)

In Equation (12), the left-hand terms are from [

P c ( y ′ , θ ) = 1 E ( r 0 ) = 1 λ 0 ( 1 − e − r 0 / λ 0 ) ≡ P c ( 1 − e − P c y ′ cos ( θ ) ) . (13)

The number of particles placed within a collision tube of a unit length along trajectory 103 is a function of both the distance from the wall y and the angle of the incidence θ. In Equations (9) and (11), we will use P c ( y ′ , θ ) defined in the equation above as a replacement for P_{c}. Still, the probability of free path traveling along the ballistic trajectory 101 is expressed as

Q i 0 ( r → , r → ′ ) = exp ( − 4 3 P c ( y , θ ) y ′ cos ( θ ) ) = exp ( − 4 3 P c ( 1 − e − P c y / cos ( θ ) ) y ′ cos ( θ ) ) (14)

Finally, using the geometry illustrated in

| r → − r → ′ | = ρ ′ = y ′ / cos ( θ ) (15)

and

n → i 0 ⋅ n → = − cos ( θ ) , (16)

and substituting Equations (11), (13), (15) and (16) in Equation (9), in which d V ′ = ρ ′ 2 sin ( θ ) d θ d φ d ρ ′ , we obtain:

Z ≡ Z β = 2 3 n P c v T 1 4 π ∫ 0 π 2 ∫ 0 2 π ∫ 0 ∞ exp ( − 4 3 P c ( 1 − β e − P c y ′ / cos ( θ ) ) y ′ cos ( θ ) ) ( 1 − β e − P c y ′ / cos ( θ ) ) sin ( θ ) d θ d φ d y ′ , (17)

where β is the model parameter. For the BM (compare with Equation (81) of [

Z β = 0 = 2 3 n P c v T 1 4 π ∫ 0 π 2 ∫ 0 2 π ∫ 0 ∞ exp ( − 4 3 P c y ′ cos ( θ ) ) sin ( θ ) d θ d φ d y ′ = n v T 4 (18)

The result of integration in Equation (18) is identical to the rate of collisions per unit area of an ideal gas according to the kinetic theory of gases, thus supporting the Ballistic Model’s validity. However, modification of the scheme of calculation by incorporation of the corresponding expression for the MFP derived in [

For clarity and simplicity of quantitative evaluation, we provide below the analysis of a one-dimensional gas system. _{T} may have the instant vector-velocity component n i 0 = − 1 directing a particle toward the plate 201. In

Upon modification of Equation (9) to applying to the one-dimensional configuration, the total rate of collisions per unit area on plate 201 (schematically shown by arrow 203) is given as:

Z b = − 1 2 ∫ 0 ∞ Z V ( y ′ ) Q i 0 ( y ′ , 0 ) n i 0 n d y ′ , (19)

where Z_{V} is the rate of collisions per unit volume and Q i 0 is the probability of free path traveling from the control volume C V ′ at y ′ to the plate 201. In [_{V} for one-dimensional incompressible gas as follows

Z V = 1 2 n P c v T , (20)

where n is gas-particle density, v_{T} is the magnitude of the thermal velocity, and P_{c} is the number of particles placed within a collision tube of a unit length in an unbounded gas space. The survival probability Q i 0 ( y ′ , 0 ) that a particle will have traveled along the ballistic trajectory in incompressible model gas at the uniform temperature from y ′ to y = 0 is calculated as

Q i 0 ( y ′ , 0 ) = exp ( − P c y ′ ) (21)

Substituting Equations (20) and (21) in Equation (19) and executing integration, we obtain

Z = 1 4 n P c v T ∫ 0 ∞ exp ( − P c y ′ ) d y ′ = 1 4 n v T . (22)

The result of integration in Equation (22) is identical to the derivation of the rate of collisions per unit area of an ideal gas according to the kinetic theory of gases, thus supporting the Ballistic Model’s validity.

We use the same approach to evaluate the rate of collisions per unit area of the gas, in which spatial variation of MFP near a wall [

P c ( y ′ , θ = 0 ) = P c ( y ′ ) = P c 1 − e − P c y ′ . (23)

In Equations (20) and (21), we will use P c ( y ′ ) defined in the equation above as a replacement for P_{c}. Substituting Equations (20) and (21) modified by Equation (23) in Equation (19), we obtain

Z = 1 4 n v T ∫ 0 ∞ P c 1 − e − P c y ′ exp ( − P c 1 − e − P c y ′ y ′ ) d y ′ . (24)

Limiting integration within the KL of three MFP, the impact from each is the most significant and in which P c y ′ ≪ 1 , the equation above is approximated as follows

Z ≅ 1 4 n v T ∫ 0 3 P c 1 y ′ exp ( − 1 ) d y ′ = 1 4 n v T exp ( − 1 ) ln y ′ | 0 3 P c → ∞ (25)

One can see that the integration above does not produce a meaningful value for the rate of collisions per unit area of the incompressible gas. We expect similar negative results of validation for the incompressible gas if we try other forms of MFP distribution functions, such as the power-law distribution function investigated by Dongari et al. [

This section describes the BM’s physical principles in one-dimensional incompressible steady-state gas flow at the uniform temperature, a lack of external field of the force, and low flow velocity (with the Mach number less than 0.3). These principles will be further applied to describe the steady-state gas flow in a micro-channel.

Here we assign the following properties of the model gas, which are adapted to the steady-state gas flow from the originally proposed in [

1) The model gas enables a distant transport of one or more properties, including one or more of mass, momentum, and energy by particles being in a constant state of mostly random motion and interaction by collisions.

2) Each gas particle is assigned to travel by obeying a ballistic trajectory governed by a law of motion in free space. It overcomes a distance between any of the two points of the ballistic trajectory with a certain survival probability.

3) Each gas particle is adapted to transport a combination of one or more properties, comprising mass, momentum, and energy between a point of initial collision and a point of ending collision.

4) Each point within the space occupied by the model gas is treated as a point of collisions for converging particles, each following a ballistic trajectory with the same ending point simultaneously.

5) Each point of collisions is treated as either a point source for diverging ballistic particles or a point sink for converging ballistic particles.

6) Each of the particles moving from the point source to the point sink is treated as a property carrier. The property carrier is created in the point source by obtaining one or more properties of specific values being intrinsic to the model gas surrounding the point source. It is ended in the point sink by transferring one or more properties of specific values in the point sink.

7) The velocity of a point source equals the mass flow velocity of the model gas flow in a corresponding point of the initial collisions.

One can note from the above that the model gas properties differ from the properties typically assigned to the ideal gas.

Here we assign the following additional unique properties to the model gas being in contact with a gas-solid interface revealing mixed diffuse and specular scattering of particles [

1) Each collision on a gas-solid interface of the model gas system, which has resulted in the scattering of the diffuse particles from the gas-solid interface, is treated as an act of interaction involving a property transport from the gas-solid interface to the scattered particle.

2) Each point of the diffuse particle scattering on the gas-solid interface is treated as a heterogeneous point source for each of the scattered particles.

3) The velocity of each of the heterogeneous point sources on the gas-solid interface equals the velocity of the gas-solid interface of corresponding points of diffuse particle scattering.

4) The point source strength of the heterogeneous point sources on the gas-solid interface is directly proportional to a property accommodation coefficientσ in a corresponding point of diffuse particle.

Note: Diffuse scattering is an act of interaction involving property transfer from a gas-solid interface to a scattered particle. Specular scattering does not involve property exchange between the gas-solid interface and a scattered particle.

_{x}-momentum) transport in the one-dimensional configuration of a Newtonian model gas flow between infinite parallel plates spaced at a distance H. Ballistic trajectories 303 and 304 show the trajectories of particles from the control volume at y' into the control volume at y. Ballistic trajectories 305 and 306 show traces of ballistic particles, which are initiated from the preceding collisions in the gas space (such as the ballistic trajectories 303 and 304, respectively), into targeting point y.

For diffuse scattering with no adsorption effects after colliding with the surface, the particle flux per unit time diffusively scattered back from the gas-solid

interface into the model gas system equals to the particle flux per unit time incident on the surface of the boundary with a negative sign. Here we have also accepted that particles diffusively scattered back from the gas-solid interface will accommodate a thermal velocity and other properties in a point of contact with the boundary’s surface. Also, we recognize that only half of the particles near the scattering surface had a recent departure from the surface with the magnitude of velocity equal to the magnitude of thermal velocity corresponding to the gas-solid interface’s temperature. Ballistic trajectories shown by arrow 307 indicate traces of ballistic particles initiated from the preceding collisions in the gas space from y = H to y = 0 and incident on targeting plate 2 at y = 0 (point A). Ballistic trajectories 308 and 309 show traces of ballistic particles initiated from the diffuse scatterings on plate 2 but have a different destination. Particles following trajectory 308 have a chance to collide in the CV, while particles following trajectory 309 can travel by free path to the opposite plate 1. Analogously, ballistic trajectories shown by arrow 310 indicate traces of ballistic particles, which are initiated from the preceding collisions in the gas space from y = 0 to y = H and incident on targeting plate 1 at y = H (point B). Ballistic trajectories 311 and 312 show traces of ballistic particles initiated from the diffuse scatterings on plate 1 (point B). Particles following trajectory 311 have a chance to collide in the CV, while particles following trajectory 312 have a chance to travel by free path to the opposite plate 1. Ballistic trajectories shown by arrows 313 and 314 indicate the tracks of diverging ballistic particles, which are originated from the preceding collisions in the control volume surrounding point y.

The mentioned above trajectories were initially introduced in our publications [

In the microscopic scale, the model gas flow is characterized by the group of particles of mass m, which move randomly and interact by collisions with effective collision cross-section σ_{c}. In each of the points in space, the gas properties such as particle density n, the magnitude of thermal velocity v_{T}, and the vector of mass flow velocity u quantify the model gas. According to the BM, each point in space occupied by the model gas may serve as both a sink accumulating property delivered by converging ballistic particles from the entire model gas system and a source dispersing into the surrounding property by diverging ballistic particles. In the interests of simplicity, the particles are considered to have a unit mass unless otherwise stated.

Based on the BPPBS, we expect maintenance of an overall property balance in each collisions’ points within the model gas system. In a steady-state gas flow, the BPPBS is formulated as follows: in each nonmoving point y, the net rate of property influx per unit volume, B i n Ψ _ F S , formed the converging ballistic particles (each traveling along a ballistic trajectory with certain survival probability) from the model gas system is equated to the net rate of property efflux per unit volume, B o u t Ψ _ F S , formed the diverging ballistic particles. This statement is expressed symbolically as

B i n Ψ _ F S ( y ) = B o u t Ψ _ F S ( y ) (26)

The table of the model parameters associated with defining the net rate of total property influx per unit volume and the net rate of total property efflux per unit volume is presented in

Referring to

B i n Ψ _ F S ( y ) = B i n Ψ _ F ( y ) + B i n Ψ _ B D ( y ) (27)

Modifying Equation (38) of [

B i n Ψ _ F ( y ) = − 1 2 ∂ ∂ y ∫ y b 2 y b 1 Q i ( y ′ , y ) Z V ( y ′ ) v ( y ′ , y ) v T ( y ′ ) Ψ ( y ′ ) d y ′ , (28)

where

Q i ( y ′ , y ) = exp ( − P c | y − y ′ | ) , (29)

Z V = 1 2 n P c v T , (30)

v ( y ′ , y ) = v T n i + u ( y ′ ) , (31)

and where n i is a unit vector of arbitrary direction:

n i = y − y ′ | y − y ′ | . (32)

Modifying Equations (40), (41), (49), and (50) of [

B i n Ψ _ B D 1 ( y ) = Z b ∂ ∂ y Q i b 1 ( y , H ) Ψ b 1 , (33)

Parameters | Short description |
---|---|

∂ ∂ y | the divergence operator in a one-dimensional configuration |

y | position of the ending point of the converging particle |

y ′ | position of the starting point of the converging particle |

u ( y ) | mass flow velocity in the point y |

v T ( y ) | the average magnitude of the thermal velocity of converging particle in point |

Z V ( y ′ ) | the rate of collisions per unit volume at the point of the collision y ′ |

v → ( y ) | velocity vector in the ending point y |

Q i ( y ′ , y ) | the probability of free path traveling along the ballistic trajectory of the converging ballistic trajectory starting at y ′ and ending y |

Ψ o u t ( y ) | property content of a diverging point y particle |

Ψ i n ( y ′ ) | property content of a diverging point y ′ particle and heading to y |

n | incompressible gas particles density |

n e f ( y ′ ) | effective gas-particle density in point y ′ associated with particles carrying transported property |

m | particle mass |

σ c | the cross-section of collisions |

P c = σ c n | the number of particles placed within a collision tube of a unit length |

y b 1 | a position of the upper plate of the micro-channel |

y b 2 | a position of the lower plate of the micro-channel |

V | the volume of integration over space occupied by the model gas |

v + ( y , y ′ ) | the velocity vector of the diverging particle in point y ′ |

Q + ( y , y ′ ) | the probability of free path traveling along the ballistic trajectory from y to y ′ |

B i n Ψ _ F S ( y ) | the net rate of property influx per unit volume from the model gas system |

B i n Ψ _ F ( y ) | the net rate of property influx per unit volume from the surrounding gas (trajectories 303 and 304) |

B i n Ψ _ B D ( y ) | the net rate of property influx per unit volume from the boundaries (trajectories 315 and 316, diffuse component) |

Q 21 = exp ( − P c H ) | the probability of free path traveling along the ballistic trajectory between confining plates in incompressible gas |

and

B i n Ψ _ B D 2 ( y ) = − Z b ∂ ∂ y Q i b 2 ( y , 0 ) Ψ b 2 , (34)

where B i n Ψ _ B D 1 ( y ) is the net rate of property influx per unit volume from the boundary at y b 1 = H (trajectory 308) and B i n Ψ _ B D 2 ( y ) is the net rate of property influx per unit volume from the boundary at y b 2 = 0 (trajectory 307), and Z_{b} is the rate of collisions per unit area on plate 1 or plate 2. In the equations above, Q i b 1 ( y , H ) is the survival probability that a particle will have traveled along the ballistic trajectory in incompressible gas at the uniform temperature from y b 1 = H to y

Q i b 1 ( y , H ) = exp ( − P c ( H − y ) ) (35)

Q i b 2 ( y , 0 ) is the survival probability that a particle will have traveled along the ballistic trajectory in incompressible gas at the uniform temperature from y b 1 = 0 to y

Q i b 2 ( y , 0 ) = exp ( − P c y ) , (36)

and Ψ b 1 and Ψ b 2 are property values acquired by diffusively scattered particles from plate 1 and plate 2, respectively.

We have recognized that the linear dimensions of the main control volume surrounding point y need to be sufficiently small to prevent two and more consecutive collisions of the same particle with other particles within the main control volume CV [

Modifying Equation (73) of [

B o u t Ψ _ F S ( y ) = 1 2 n { ∂ ∂ y [ Q + ( y , y ′ ) v + ( y , y ′ ) Ψ o u t ( y ) ] } y ′ → y , (37)

where

Q + ( y , y ′ ) = exp ( − P c | y ′ − y | ) , (38)

for incompressible gas.,

v + ( y , y ′ ) = v T n + + u ( y ) , (39)

and where n + is a unit vector of arbitrary direction:

n i = y ′ − y | y ′ − y | . (40)

The integro-differential form of property balance equation is formulated by Equation (41) given below, which is obtained by substitution of Equations (28), (33), (34), and (37) in Equation (26):

− 1 2 ∂ ∂ y ∫ 0 H Q i ( y ′ , y ) Z V ( y ′ ) v ( y ′ , y ) v T ( y ′ ) Ψ i n ( y ′ ) d y ′ + Z b ∂ ∂ y Q i b 1 ( y , H ) Ψ b 1 − Z b ∂ ∂ y Q i b 2 ( y , 0 ) Ψ b 2 + Q ˙ Ψ = 1 2 n { ∂ ∂ y [ Q + ( y , y ′ ) v + ( y , y ′ ) Ψ o u t ( y ) ] } y ′ → y (41)

where Q ˙ Ψ is a term representing a generation of a transported quantity within the model gas system, resulting from transmitting property across the surfaces of inlet and outlet openings. Suppose the property transport is the momentum transport. In that case, the generation term in a confined model gas system results from forces acting on the model gas in the form of the external pressure force, P, applied across the surface bounding the system. More specifically, normal gradient pressure force acting on the model gas and prompting model gas flow represents the effect of surroundings on the momentum change. Here point y is not included in integration for converging ballistic particles. Substituting Equations (29), (30), (31), (35), (36), (38), and (39) in the equation above, and executing differentiation, we obtain the following general integral form of the property balance equation:

1 2 Z V P c ∫ 0 H exp ( − P c | y − y ′ | ) Ψ i n ( y ′ ) d y ′ + 1 2 Z V P c ∫ 0 H exp ( − P c | y − y ′ | ) n i u ( y ′ ) v T Ψ i n ( y ′ ) d y ′ + Z b P c exp ( − P c ( H − y ) ) Ψ b 1 + Z b P c exp ( − P c y ) Ψ b 2 + Q ˙ Ψ = 1 2 n { ∂ ∂ y [ exp ( − P c | y ′ − y | ) ( v T n + + u ( y ) ) Ψ o u t ( y ) ] } y ′ → y (42)

To formulate a general integro-differential form of mass balance equation in each nonmoving point of space occupied by the gas, we modify Equation (42) by assigning

Ψ i n = Ψ o u t = Ψ b 1 = Ψ b 2 = 1 , Q ˙ Ψ = 0 (43)

executing differentiation in the right-hand of the resulting equation followed by executing limit y ′ → y , and rearranging the terms. Then, we obtain the following general integro-differential form of the mass balance equation

Z V + 1 2 n ∂ ∂ y u = 1 2 Z V P c ∫ 0 H exp ( − P c | y − y ′ | ) d y ′ + 1 2 Z V P c ∫ 0 H exp ( − P c | y − y ′ | ) n i u ( y ′ ) v T d y ′ + Z b P c exp ( − P c ( H − y ) ) + Z b P c exp ( − P c y ) (44)

To formulate a general integro-differential form of momentum (along with y-axis) balance equation in each nonmoving point of space occupied by the gas, we modify Equation (42) by assigning

Ψ i n = v T n i + u ( y ′ ) ; Ψ o u t = v T n + + u ( y ) ; Ψ b 1 = − v T ; Ψ b 2 = + v T ; Q ˙ Ψ = 0 (45)

executing differentiation in the right-hand of the resulting equation followed by executing limit y ′ → y , and rearranging the terms. Then, we obtain the following general integro-differential form of the momentum balance equation

2 Z V u + 1 2 n ∂ ∂ y u 2 = 1 2 Z V P c v T ∫ 0 H exp ( − P c | y − y ′ | ) n i d y ′ + Z V P c ∫ 0 H exp ( − P c | y − y ′ | ) u ( y ′ ) d y ′ + 1 2 Z V P c 1 v T ∫ 0 H exp ( − P c | y − y ′ | ) u 2 n i d y ′ − Z b P c v T exp ( − P c ( H − y ) ) + Z b P c v T exp ( − P c y ) (46)

According to the BPPBS, the mass-balance and momentum-balance shall be maintained at every point of the gas space [_{b}, one can solve a system of the above-mentioned equations and obtain a solution that is u = 0 and

Z b = Z V 2 P c = n v T 4 . (47)

The fact that the result of the derivation of Z_{b} from the mass- and momentum-balance equations is identical to the result of the derivation of the rate of collisions per unit area of an ideal gas according to the kinetic theory of gases supports the Ballistic Model’s validity.

The functional relationship expressed by the equation above may also be obtained by analyzing fluxes formed by the ballistically traveling particles. Referring to

J F S 1 = − J F S 2 = N 21 1 + N 2 1 , (48)

where J F S 1 and J F S 2 are the particle fluxes per unit time on the gas-solid interfaces of plate 1 and plate 2, respectively, N 21 1 is the particle flux per unit time on the gas-solid interface of plate 1, which is associated with the incident on the gas-solid interface of ballistic particles from the gas space following ballistic trajectories 307, and N 2 1 is the particle flux per unit time on the gas-solid interface of plate 1, which is associated with an incident on the gas-solid interface of ballistic particles from originated from diffuse scattering on the gas-surface interface of plate 4. Using Equations (43) and (42) of [

N 21 1 = 1 2 ∫ 0 H Q i ( y ′ , y ) Z V d y ′ = 1 2 P c Z V [ 1 − exp ( − P c H ) ] (49)

and

N 2 1 = Z b exp ( − P c H ) , (50)

respectively. Referring again to

Z b = − J F S 1 n b 1 = − ( N 21 1 + N 2 1 ) n b 1 = − J F S 2 n b 2 . (51)

Substituting Equations (48)-(50) in Equation (51) and taking n b 1 = − 1 and performing algebraic operations, we finally obtain

Z b = Z V 2 P c = n v T 4 , (52)

which is identical to Equation (47). The above implies that Z_{b} is constant at any Knudsen number.

Here we shall admit that flux N 2 1 produces an impact on the rate of collisions on the gas-solid interface, but it does not directly involve the collisional property exchange within the gas volume. Considering the above, the effective rate of collision per unit area, which is involved directly in the collisional property exchange within the gas volume, is defined from the equation given below

Z b e f = Z b + N 2 1 n b 1 = Z b ( 1 − exp ( − P c H ) ) . (53)

Referring to _{x} momentum transport. This group of active gas particles and associated with it the effective rate of collision per unit volume, Z V e f , in the CV at y is evaluated from Equation (44), in which u = 0 , Z_{V} is substituted by Z V e f , and Z_{b} is substituted by Z b e f :

Z V e f ( y ) = 1 2 P c ∫ 0 H Z V e f ( y ′ ) exp ( − P c | y − y ′ | ) d y ′ + Z b e f P c exp ( − P c ( H − y ) ) + Z b e f P c exp ( − P c y ) (54)

Resolving the equation above with respect to Z V e f , we obtain:

Z V e f = 2 P c Z b e f = Z V ( 1 − exp ( − P c H ) ) . (55)

Recognizing that the rate of collisions per unit length, P_{c}, is constant for incompressible gas, we may introduce the effective density of particles currying u_{x} momentum, n_{eff}, by expressing Z V e f f as

Z V e f f = 1 2 P c n e f f σ c v T . (56)

Now substituting Equations (30), (55), and (56) in Equation (54) and executing integration and algebra operations and replacing P c H by K n − 1 , we may define the normalized effective density,k_{ef}, as follows:

k e f = n e f n = Z V e f Z V = 1 − exp ( − 1 K n ) , (57)

where Kn is the Knudsen number defined as the ratio of the mean free path λ f = 1 / P c and the representative length scale H. Results for the normalized effective density profile between two parallel plates are presented in

We intentionally provided a graphical interpretation of Equation (54) to highlight that the normalized effective density is constant within the gas space and depends only on the Knudsen number. As the Knudsen number’s value increases from 0.2 to 20, the effective normalized density decreases from 1 to 0.048.

Considering the effect of the confining plates on the density of -momentum carriers in the channel’s gas space, we analyze the velocity profile generated in the model gas due to the pressure gradient along the channel. We use a similar approach that has been initially introduced in our publications [

Concerning _{x}-momentum, so that the fluxes associated with trajectories 308 and 311 are zeroed. Here we also consider that the magnitude of the thermal velocity of the particles is much higher than the magnitude of the mass flow velocity vector along the x-direction, | u x | ≪ v T . To formulate a general integro-differential form of u_{x}-momentum balance equation in each nonmoving point of space occupied by the gas, we modify Equation (42) by assigning

Ψ i n = m u x ( y ′ ) ; Ψ o u t = m u x ( y ) ; Ψ b 1 = 0 ; Ψ b 2 = 0 ; Q ˙ Ψ = − d P d x (58)

executing differentiation in the right-hand of the resulting equation followed by executing limit y ′ → y , and rearranging the terms. Then, we obtain the following general integro-differential form of the momentum balance equation:

m Z V u x = − k e f d P d x + 1 2 P c m Z V e f ∫ 0 H exp ( − P c | y − y ′ | ) u x ( y ′ ) d y ′ (59)

One can note that the equation above, in which k e f f = 1 and Z V e f = Z V , is identical to Equation (112) of our paper [_{ef} upon the following analysis. We have recognized that some of the particles initiated from collisions in the gas space and carrying u_{x} momentum may travel by free path to a wall, then after accommodation with the gas-solid interface of a wall diffusively scatter back in the gas space. Here we shall recite that “the cause of the thermalization and “diffuse” escape of particles from the surface lies not in the trapping of molecules by the surface, as assumed by Maxwell himself and repeated in practically all the literature published on this question, but in the relaxation, in interaction with the phonon subsystem” [^{−11} - 10^{−10} sec, the thermalized particle scatters back from a gas-solid interface almost immediately. All the particles scattered from the walls do not carry u_{x}-momentum. Some of these scattered particles may even travel to the opposite wall by free path. Such group of the particles will not participate in u_{x} momentum transport anymore. This effect was not taken into consideration in our publications [

Here we also understand that the flow through the channel is motivated by the surface force exerted by the surroundings on the CV through the pressure force, and the pressure is directly proportional to gas particles density. We suggest that, in the gas of particle density n and the effective density n_{eff}, the effective pressure force applied to the differential CV at y through the active gas particles may be expressed as

F s = − k e f d P d x = − d P d x [ 1 − exp ( − P c H ) ] (60)

Here we need to admit that while evaluating the effective pressure force applied to the differential CV, we used the effective density n_{ef} of active particles, which is represented in the right-hand terms of Equation (59) by the coefficient k_{ef}. However, in the left-hand term and the second right-hand term of Equation (59), we shall use the density of the incompressible gas, n. Such differentiation may be explained by the fact that eventually, all particles of the incompressible gas system are needed to satisfy the laws of mass balance and momentum balance (see the system of Equations (44) and (46) and Equation (47) as the solution of the system). As such, all particles of the incompressible gas system are involved in u_{x} momentum balancing in each point of the gas space according to Equation (59). Substituting k_{ef} taken from the equation above and Equation (30) in Equation (59), we finally obtain:

u x = − 2 H n P c H v T 1 m d P d x [ 1 − exp ( − P c H ) ] + 1 2 P c ∫ 0 y exp ( − P c ( y − y ′ ) ) u x ( y ′ ) d y ′ + 1 2 P c [ 1 − exp ( − P c H ) ] ∫ y H exp ( − P c ( y ′ − y ) ) u x ( y ′ ) d y ′ (61)

Normalizing Equation (61) to

u max = ( − H m n v T d P d x ) (62)

and assigning

U x ( y ) = u x ( y ) u max , (63)

we obtain:

U x ( y ) = 2 P c H [ 1 − exp ( − P c H ) ] + 1 2 P c exp ( − P c y ) [ 1 − exp ( − P c H ) ] ∫ 0 y exp ( P c y ′ ) U x ( y ′ ) d y ′ + 1 2 P c exp ( P c y ) [ 1 − exp ( − P c H ) ] ∫ y H exp ( − P c y ′ ) U x ( y ′ ) d y ′ (64)

where U x ( y ) = u x ( y ) u max is a non-dimensional velocity of the gas flow in the

channel. The equation above’s explicit solution is obtained numerically by sequential approximation described previously in [

Further integration of U x ( y ) along y-direction and normalization by H will result in finding a normalized volume flow rate U x A :

U x A = 1 H ∫ 0 H U x ( y ) d y . (65)

Then we can obtain the normalized velocity profile as:

U x N ( y ) = U x ( y ) U x A = U x ( y ) 1 H ∫ 0 H U x ( y ) d y . (66)

In

_{2}, He, CO_{2}, freon 12, and air”, obtained from [

It is also worth noting that the normalized volume flow rate U x A as a function of the inverse Knudsen number, according to the current theory, has the so-called “Knudsen minimum” at δ ≅ 0.9 . It is in good agreement with Knudsen’s observation of the minimum at about K n ~ 1 [

We have simulated gas flow in a plane channel using the BPPBS and obtained the gas flow velocity profile in the implicit form shown by Equation (64). It would also be interesting to obtain and analyze a differential form of the equation governing such gas flow in the channel. Below we show a way to obtaining a differential form u_{x} momentum balance equation governing gas flow in the channel between two parallel plates due to the pressure gradient along the channel, like the presented in Step 2 of Section 4.4.3 of our paper [

Applying to Equation (64) the method of differentiation for integral equations (ones, twice, and so on) with subsequent elimination of the terms belonging to the original equation [

1 P c 2 d 2 d y 2 U x ( y ) = U x ( y ) exp ( − 1 K n ) − 2 K n [ 1 − exp ( − 1 K n ) ] . (67)

The equation above is different from the classical Navier-Stokes equation applicable to incompressible Newtonian flow. Here we will not further proceed to obtain an explicit analytical solution of the differential equation above, as we did in Section 4.4.3 of our paper [

1 P c 2 d 2 d y 2 U x ( y ) = − 2 K n , (68)

which, if it is multiplied by u max , is identical to Equation (104) of [

U x ( y ) ≅ 2 . (69)

Here we consider the collision-dominated flow regime, in which the relative change of any property value (velocity along the channel) is insignificant on the

length scale of the average distance between the model gas particles, 1 P c (see

Equation (61) of [

U x A = 2 . (70)

Remarkably, the normalized volume flow rate reaches a constant value for n → ∞ , which agrees with Knudsen’s finding [

The recently discovered and mathematically proven BPPBS [_{x}-momentum from the Knudsen number. It has been found that the effective normalized density is constant within the gas space and depends only on the Knudsen number. Considering the effect of the confining plates on the density of u_{x}-momentum carriers in the channel’s gas space, we have formulated an integral form expressing implicitly the velocity profile generated in the model gas due to the pressure gradient along the channel. The implicit integral form of the velocity profile has been solved numerically by the sequential approximation. Comparing the normalized flow rate as a function of the inverse Knudsen number according to the current theory and the experimental data shows good agreement in the range of the inverse Knudsen number from 0.01 to about 40. The correlation factor is found to be about 0.995. Also, a differential form of the u_{x}-momentum balance equation governing gas flow in the channel between two parallel plates due to the pressure gradient along the channel has been derived. It was noted that the differential form of the u_{x}-momentum balance equation derived from the implicit integral form of the velocity profile is different from the classical Navier-Stokes equation applicable to incompressible Newtonian flow. However, the governing equation derived in the present theory is reduced, for K n → 0 , to the Navier-Stokes equation for well-known general Couette flow. In turn, for K n → ∞ , the governing equation is reduced to a constant value, thus representing the free-molecular regime. It implies that both the integral and the differential forms, according to our theory, are applicable to describe gas flow in the microchannel in the range of Knudsen numbers from zero to infinity.

We also analyzed the validity of the currently widely disseminated claim of the MFP variation in the KL as a function of the distance from the gas-solid interface. For this purpose, we compared the rate of collisions per unit area on a gas-solid for the incompressible gas at the uniform temperature in a semi-infinite space for our and other models based on the variable MFP in the KL region. First, we found that the collision rate per unit area obtained by applying the BM and the corresponding characteristic known from the kinetic theory of gases are correlated and analytically identical. These findings support our theory based on the BM and the BPPBS. Second, we incorporated in the equation expressing the total rate of collisions per unit area on the solid surface, which we have introduced earlier in [

Based on the results above, we may conclude the following: fictitious

1) “Knudsen layer” seems to be a useful abstraction to employ specific boundary conditions providing reasonable approximate solution beyond the Knudsen layer (by using Navier-Stokes equations or a lattice Boltzmann (LB) method to model a rarefied gas flow). However, gas particles’ microscopic and physical behavior in the KL is not well understood, leading to a pure prediction of gas flow characteristics.

2) The assumption that the MFP varies as a function of the surface confinement (Knudsen number), widely disseminated in the literature, is erroneous. Such an assumption disagrees with the kinetic theory of gases. As we found, the interaction by collisions of the gas with variation in MFP does not produce a meaningful value for the rate of collisions per unit area. As a result, such an assumption leads to wrong physical interpretations of nanoscale gas flows.

3) We offer a new way of computing a steady-state rarefied gas flow in a microchannel based on recently discovered and mathematically proven by us the Ballistic Principle of the Property Balance in Space. We specifically investigated the mechanism of influence of the Knudsen number on the rarefied gas flow. Our theory does not use the generalized concept of viscosity based on the variable MFP in the KL. Comparing the normalized flow rate as a function of the inverse Knudsen number according to the current theory and the experimental data shows good agreement in the range of the inverse Knudsen number from 0.01 to about 40. The correlation factor is found to be about 0.995.

4) Based on the BPPBS, our theory is advantageous over any available methods or theories for modeling rarefied gas flow. It allows the formulation of the integro-differential or integral form of the property balance (mass, momentum, energy, or any other property) and obtaining an exact solution of the rarefied gas problem. Our approach based on the BPPBS is computationally efficient. It offers substantial and practical advantages in modeling and simulation of rarefied gases.

The author declares no conflicts of interest regarding the publication of this paper.

Kislov, N. (2021) Effect of Ballistic Bouncing of Gas Particles across a Microchannel on Rarefied Gas Flows. Journal of Applied Mathematics and Physics, 9, 779-808. https://doi.org/10.4236/jamp.2021.94054