A gas flow in a square cavity driven by a lid sliding in the direction of its line of contact with the cavity wall is considered. The steady behavior of the gas is numerically investigated based on the linearized Bhatnagar–Gross–Krook kinetic equation and the diffuse reflection boundary condition. When one applies the Stokes equation and the no-slip boundary condition to the system considered here, the flow velocity becomes multivalued at the corner between the lid and the cavity wall, and the shear stress diverges at the corner inversely proportionally to the distance from there, which is known as the so-called corner singularity. In the present work, the behavior of the gas near the corner is examined based on numerical results obtained from the kinetic theory. Although the range of the flow velocity value in the kinetic solution is limited due to the significant velocity slip near the corner, the flow velocity is, nevertheless, multivalued at the corner. The shear stress varies inversely proportionally to the distance from the corner up to the position that is a few tens of mean free paths away from there. The increase in the stress is suppressed at positions closer to the corner and its magnitude remains bounded. Thus, the total forces acting on the lid and the side cavity walls are bounded as well. Due to the distinctive behavior of the stress near the corner, the resulting nondimensional total forces behave with an unconventional rate $ Kn \u2009 ln \u2009 Kn$ for small Knudsen numbers $ Kn$.

## REFERENCES

*Lectures on Topological Fluid Mechanics*

*The Scientific Papers of Sir Geoffrey Ingram Taylor*

*Molecular Gas Dynamics*

*An Introduction to Fluid Dynamics*

*Rarefied Gas Dynamics*

*Rarefied Gas Dynamics*

*Kinetic Theory and Fluid Dynamics*

*Recent Advances in Kinetic Equations and Applications, Springer INdAM Series*

*J*

_{1}tend to 0 as $\n\n\sigma \n+\n\u2192\n+\n0$ and $\n\psi \n\u2192\n\n\theta \nL\n(\nx\n)$, care should be taken in taking the limit $\n\n\sigma \n+\n\u2192\n+\n0$. Fortunately, the result in (34) can be justified, for example, as follows. To begin with, $\n\n\theta \nL\n(\nx\n)$ can be expressed as $\n\n\theta \nL\n=\n\pi \n\u2212\n\alpha $ with $\n\alpha \n=\n\n\ntan\n\n\u2212\n1\n(\n\n\sigma \n+\n\u2009\nsin\n\u2009\n\n\phi \n+\n/\n(\n1\n\u2212\n\n\sigma \n+\n\u2009\ncos\n\u2009\n\n\phi \n+\n)\n)$. We can take another angle $\n\beta \n=\n\n\ntan\n\n\u2212\n1\n(\n\n\n\n\sigma \n+\n\u2009\nsin\n\u2009\n\n\phi \n+\n/\n(\n1\n\u2212\n\n\sigma \n+\n\u2009\ncos\n\u2009\n\n\phi \n+\n)\n)$, which is larger than

*α*and tends to 0 as $\n\n\sigma \n+\n\u2192\n+\n0$. Then, we may split the integral as $\nb\n(\nx\n)\n=\n\n\pi \n\n\u2212\n1\n\n\n\u222b\n\n\n\phi \n+\n\n\pi \n\u2212\n\alpha \n\n\nJ\n1\nd\n\psi \n=\n\n\pi \n\n\u2212\n1\n\n\n\u222b\n\n\n\phi \n+\n\n\pi \n\u2212\n\beta \n\n\nJ\n1\nd\n\psi \n+\n\n\pi \n\n\u2212\n1\n\n\n\u222b\n\n\pi \n\u2212\n\beta \n\n\pi \n\u2212\n\alpha \n\n\nJ\n1\nd\n\psi $. Thanks to the choice of

*β*, in the first integral, where $\n\n\phi \n+\n<\n\psi \n<\n\pi \n\u2212\n\beta $, the argument of

*J*

_{1}approaches 0 uniformly as $\n\n\sigma \n+\n\u2192\n+\n0$. Thus, it approaches the result in (34). On the other hand, we see that the second integral tends to 0 because its magnitude is bounded from above by $\n\n\pi \n\n\u2212\n1\n\nJ\n1\n(\n0\n)\n(\n\beta \n\u2212\n\alpha \n)$ [ $\n\nJ\n1\n(\n0\n)$ is the maximum of the function

*J*

_{1}] and $\n\alpha \n,\n\beta \n\u2192\n0$ as $\n\n\sigma \n+\n\u2192\n+\n0$.

*k*in the range is larger than 0.22026 and less than 0.22072.