Equations of Motion for Incompressible Viscous Fluids

Table of Contents

1. Frontmatter

2. Chapter 1. Miscellanea of Analysis

   Tujin Kim, Daomin Cao

   Abstract

   In this chapter, we outline some knowledge of analysis: Banach space, fixed point, Lebesgue and Sobolev spaces, operator and operator-differential equations and convex functional, which will be used in the main part of this book. We do not describe the best results, but to help readers’ understanding sometimes we say more than necessary. The readers who are already acquainted with the elements of functional analysis can skip this chapter and may consult the necessary parts when reading main part of this book.

3. Chapter 2. Fluid Equations

   Tujin Kim, Daomin Cao

   Abstract

   In this chapter, we first show how the Navier-Stokes equations and the equations of motion for fluid under consideration of heat are derived. Next, we outline some boundary conditions for the Navier-Stokes equations, mainly being concerned with the ones dealt with in this book. Last, we consider three kind of bilinear forms for the Stokes and Navier-Stokes equations, variational formulations for the Navier-Stokes problems with mixed boundary conditions and establish the equivalence between the variational formulations and the original PDE problems.

4. Chapter 3. The Steady Navier-Stokes System

   Tujin Kim, Daomin Cao

   Abstract

   In this chapter, we are concerned with the steady Navier-Stokes systems with mixed boundary conditions involving Dirichlet, pressure, vorticity, stress and normal derivative of velocity together. As we have seen in Sect. 2.3.2, according to what kinds of bilinear forms for variational formulation are used, types of boundary conditions under consideration together are different. The variational formulations in Sect. 2.3.2 do not reflect, for example, the boundary conditions for stress and pressure together, but this case is important in practice.

5. Chapter 4. The Non-steady Navier-Stokes System

   Tujin Kim, Daomin Cao

   Abstract

   In this chapter we are concerned with the non-steady Navier-Stokes equations and Stokes equations with mixed boundary conditions including conditions for velocity, pressure, stress, vorticity and Navier slip condition together. As in Sect. 3.2, relying on the result in Sect. 3.1, we embed all these boundary conditions into variational formulations of problems.

6. Chapter 5. The Steady Navier-Stokes System with Friction Boundary Conditions

   Tujin Kim, Daomin Cao

   Abstract

   In this chapter we are concerned with the steady Navier-Stokes systems with mixed boundary conditions which may include Tresca slip condition, leak boundary condition, one-sided leak boundary conditions, velocity, pressure, vorticity, stress and normal derivative of velocity together. Relying on the results in Sect. 3.1 and using the strain bilinear form, we embed all these boundary conditions into variational formulations of corresponding problems.

7. Chapter 6. The Non-steady Navier-Stokes System with Friction Boundary Conditions

   Tujin Kim, Daomin Cao

   Abstract

   In this chapter we are concerned with the non-steady Navier-Stokes and Stokes problems corresponding to the steady problems in Chap. 5. In Sect. 6.1 relying on the results of Sect. 3.1, we embed all boundary conditions to variational formulations. We get variational inequalities with one unknown which are equivalent to the original PDE problems for the smooth solutions. In Sect. 6.2 we study the existence and uniqueness of solutions to the variational inequalities obtained in Sect. 6.1. In Sect. 6.3 using the results of Sect. 6.2, we get the existence, uniqueness and estimates of solutions to the Navier-Stokes and Stokes problems with the boundary conditions.

8. Chapter 7. The Steady Boussinesq System

   Tujin Kim, Daomin Cao

   Abstract

   In this chapter we are concerned with the steady Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak condition, one-sided leak conditions, velocity, pressure, vorticity, stress together and the conditions for temperature may include Dirichlet, Neumann and Robin conditions together. We will get variational formulations consisting of a variational inequality for velocity and a variational equation for temperature, which are equivalent to the original PDE problems for smooth solutions.

9. Chapter 8. The Non-steady Boussinesq System

   Tujin Kim, Daomin Cao

   Abstract

   In this chapter we are concerned with the non-steady Boussinesq problem corresponding to the steady problem in Chap. 7. The formulations consist of a non-steady variational inequality for velocity and a non-steady variational equation for temperature. For the problem with boundary conditions involving the static pressure and stress, it is proved that if the data of problem are small enough and compatibility conditions at the initial time for velocity and temperature are satisfied, then there exists a unique solution on the given interval. For the problem with boundary conditions involving the total pressure and total stress, the existence of a solution is proved without restriction on the data of problem.

10. Chapter 9. The Steady Equations for Heat-Conducting Fluids

    Tujin Kim, Daomin Cao

    Abstract

    In this chapter we are concerned with the equation for steady flow of heat-conducting incompressible Newtonian fluids with dissipative heating under mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak condition, one-sided leak conditions, velocity, pressure, vorticity, stress together and the conditions for temperature may include Dirichlet, Neumann and Robin conditions together. On the basis of results of Sect. 3.1, we get variational formulations consisting of a variational inequality for velocity and a variational equation for temperature, which are equivalent to the original PDE problems for smooth solutions.

11. Chapter 10. The Non-steady Equations for Heat-Conducting Fluids

    Tujin Kim, Daomin Cao

    Abstract

    In this chapter we are concerned with a non-steady system for motion of incompressible Newtonian heat-conducting fluids with mixed boundary conditions. The boundary condition for fluid is the case of total pressure and the boundary conditions for temperature may include Dirichlet, Neumann and Robin conditions together.

12. Backmatter