§3.4 The basic equation for fluid motion

The basic equation for steady flow are included continuity equation, motion differential equation for idea flow and motion differential equation for viscosity flow.

3.4.1 Continuity equation for fluid motion

1．The general form of basic equation for steady flow

Take a six-sided infinitesimal as control body in the flow field, which side length are dx, dy, dz.Control body is a designated space in the flow field, its shape, position fixed so that fluid may not be affected.For example, analyzing the quality control in the body in x direction.

By taking the first-order trace with taylor series expansion,

left surface velocity is:

right surface velocity is:

Quality flow from micro area in dt:

ΔM=ρuΔAdt

Poor quality of inflow and outflow from control body in the x direction:

Similarity

According to the law of conservation of mass, we have

The general form of basic equation for steady flow is:

or

The differential equations of continuity is derivated without introducing any constraints.Therefore, there is no limit for the scope of equation.

2．Different scope of application of the form

(1)Differential constant flow continuity

or

The scope:ideal, practical, compressible and incompressible constant flow.

(2)Incompressible fluid continuity differential equation

Becauseρ=constant, then

or

The physical meaning of incompressible fluid continuity differential equation is that the volume of fluid flowing into the unit space equal to the volume of outflow in unit time.

3.4.2 Differential equation of ideal fluid motion

1．Stress analysis

As can be seen in Fig.3-12, take a differential element hexahedral with a center(x, y, z)in ideal fluid.Its side length are dx, dy, dz, and its center of pressure is p(x, y, z).Pressure of M, N points are through the center of pressure by Taylor series expansion to take the amount of the first order.

(1)Surface force

Since ideal fluid has no viscosity, shear stress of each side are zero in infinitesimal

hexahedron.And because px=py=pz=p, then

left force in x direction is:

right force in x direction is:

(2)Mass force

Unit mass forces on each component of the coordinates are fx, fy, fz.So mass force in x direction is fxρd x d y d z.

2．Motion analysis

Newton's second law is applied in x direction∑Fx =max, then

Both sides of the body weight are divided by the infinitesimalρdxdydz.

Similarity

This formula is differential equation of ideal fluid.

If the acceleration are zero, this equation can be transformed into Euler equilibrium equations.

3.4.3 Differential equations of viscous fluid motion

Viscous fluid, also known as the actual fluid.Derivation of differential equations of motion of a viscous fluid the same as above, but due to the presence of viscous stress state than the more complex an ideal fluid.Here will not be detailed derivation, only the dynamics concept as outlined.

1．The area of the force of the viscous fluid

Area viscous fluid force includes compressive stress and shear stress caused by viscosity.

(1)shear stress caused by friction within the generalized Newton's law (Newton friction law in the promotion of ternary stream).

Where, τ represents shear stress, The first subscript indicates the role of the surface normal direction, the second index indicates the effect of shear stress direction, shown in Fig.3-13.

(2)the actual flow of the fluid dynamic pressure at any point, due to the presence of viscous shear stress, each of the dynamic pressure of varying sizes, namely pxx≠pyy≠pzz.The dynamic pressure of any point.

Where the subscript p and shear stress of τhave the same meaning.

Hydrodynamic theory and analysis shows that each of incompressible fluid pressure to move the following relationship:

2．The differential equations of motion of incompressible actual fluid

Also take a six-sided as infinitesimal stress analysis, surface forces in the x direction as shown in Fig.3-13.

Over the same force and fluid quality.Thus, in the x direction, according to Newton's second law∑Fx =max , You can write

The relationship between the expression of shear stress and principal stress, and incompressible fluid continuity differential

Substituting simplification, available incompressible viscous fluid equations of motion, namely

Similarly

Or represented by a vector

where ∇ is Laplace operator,

Incompressible viscous fluid motion equations, also known as Wiener-Stokes equations.This is one of the fundamental equations of the theory of fluid mechanics. Becauseρis constant, differential equations and continuity of the NS may be composed of differential equations containing four unknowns.Combined with the boundary conditions, in theory, the solution of NS equations exist.However, to describe directly solve complex and subtle viscous fluid motion, it is very complicated and difficult, research in this area is still the problem of frontier.