• Communications of the Korean Mathematical Society
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• v.14 no.4
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• pp.833-842
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• 1999

In this paper, we construct a fully discrete solution of the incompressible Navier Stokes equations using implicit Runge kutta method. We prove the existence of the fully discrete solution.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.819-839
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• 2019

We study the stabilization of 2D g-Navier-Stokes equations in bounded domains with no-slip boundary conditions. First, we stabilize an unstable stationary solution by using finite-dimensional feedback controls, where the designed feedback control scheme is based on the finite number of determining parameters such as determining Fourier modes or volume elements. Second, we stabilize the long-time behavior of solutions to 2D g-Navier-Stokes equations under action of fast oscillating-in-time external forces by showing that in this case there exists a unique time-periodic solution and every solution tends to this periodic solution as time goes to infinity.

• Journal of the Korea institute for structural maintenance and inspection
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• v.22 no.5
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• pp.13-22
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• 2018

In this study, a governing differential equation for the bending problem of orthotropic rectangular plate is drived. It's exact solution for various boundary conditions is presented. This solution follows traditional method like Navier's solution or Levy's solution that transforms the governing differential equation into an algebraic equation by using trigonometric series. To obtain a solution by Levy's method, it is required that two opposite edges of the plate be simply supported. And the boundary conditions, for which the Navier's method is applicable, are simply supported edge at all edges. In this study, it overcomes the limitations of the previous Navier's and Levy's methods.This solution is applicable for any combination of boundary conditions with simply supported edge and clamped edge in x, y direction. The plate could be subjected to uniform, partially uniform, and line loads. The advantage of the solution is that it is the exact solution as well as it overcomes the limitations of the previous Navier's and Levy's methods. Calculations are presented for orthotropic plates with nonsymmetric boundary conditions. Comparisons between the result of this paper and the result of Navier, Levy and Szilard solutions are made for the isotropic plates. The deflections were in excellent agreement.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.537-558
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• 2008

Consider a weak solution u of the Navier-Stokes equations in the class $L^2((0,T);X_1(\mathbb{R}^d)^d)$. We establish a new approach to treat the regularity problem for the Navier-Stokes equation in term of the multiplier space $X_1(\mathbb{R}^d)$.

• Journal of the Korean Chemical Society
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• v.40 no.6
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• pp.409-414
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• 1996

The limit of applicability of Navier-Stokes equation to stationary plane shock-waves is examined by using the principle of minimum entropy production of linear irreversible thermodynamics. In order to obtain analytic results, the equation is linearized near the equilibrium of downstream. Results show that the solution of Navier-Stokes equation which fits the boundary condition of far downstream flow is consistent with the thermodynamic requirement within the first order when the solution is expanded around the M=1, where M is the Mach number of upstream speed.

• Transactions of the Korean Society of Mechanical Engineers
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• v.17 no.9
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• pp.2329-2338
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• 1993

An accurate analysis procedure to solve the flow about a flat plate at various incidences has been developed. The Navier-Stokes equations of stream function and vorticity form are solved in a sufficiently large computational domain, in which the grid lines are mutually orthogonal. The details of the flow near the singularity at the tip of the plate is well captured by the analytic solution which is asymptotically matched to the numerically generated outer solution. The solution for each region is obtained iteratively : the solution of one (inner or outer) region uses that of the other as the boundary condition after each cycle. The resulting procedure is accurate everywhere and also computationally efficient as the singularity has been removed. It is applied to the flat plate for a wide range of Re : the results agree very well with the existing computation and experiment.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.2
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• pp.51-67
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• 1999

The numerical solution of the Navier-Stokes equations in a constricted stepped channel problem has been obtained using a fourth order numerical method. Transformations are made to have a fine grid near the sharp corner and a long channel downstream. The derivatives in the Navier-Stokes equations are replaced by fourth order central differences which result a 29-point computational stencil. A procedure is used to avoid extra numerical boundary conditions near the solid walls. Results have been obtained for Reynolds numbers up to 1000.

• Communications of the Korean Mathematical Society
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• v.29 no.4
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• pp.505-518
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• 2014

Considered here is the first initial boundary value problem for the two-dimensional g-Navier-Stokes equations in bounded domains. We first study the long-time behavior of strong solutions to the problem in term of the existence of a global attractor and global stability of a unique stationary solution. Then we study the long-time finite dimensional approximation of the strong solutions.

• Journal of the Korean Mathematical Society
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• v.50 no.4
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• pp.771-795
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• 2013

In this paper we consider the Navier-Stokes equations in the half space. Our aim is to construct a mild solution for initial data in $B^{-\alpha}_{{\infty},{\infty}}(\mathbb{R}^n_+)$, 0 < ${\alpha}$ < 1. To do this, we derive the estimate of the Stokes flow with singular initial data in $B^{-\alpha}_{{\infty},q}(\mathbb{R}^n_+)$, 0 < ${\alpha}$ < 1, 1 < $q{\leq}{\infty}$.

• Transactions of the Korean Society of Mechanical Engineers
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• v.15 no.1
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• pp.120-126
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• 1991

An expanded scheme of a direct numerical solution method for solving the Navier-Stokes equation considering the modified boundary conditions for gas lubrication with ultra low clearance at high .LAMBDA. region is presented. Many examples are calculated by this scheme and their results are compared to the previous solutions using P$^{2}$H$^{[-992]}$ . This scheme has the advantages of fast calculation time and stable convergence in high .LAMBDA. region, and gives very good results in the case of fluid film thickness discontinuity.