§2.2 Differential balance equation

2.2.1 Differential balance equation of fluids-euler Equations

In order to analyze the law of fluid balance, an infinitesimally small cube was considered in a fluid at equilibrium as an obj ect of study, and each side of the cube is parallel to the corresponding Cartesian coordinate axis, respectively (see Fig.2-3).The pressure at the central point M of cube was set as p=p (x, y, z).According to the characteristics of continuous function, when the independent variables (coordinate position)even change slightly, the magnitude of function (pressure)would change accordingly.The magnitude of function can be calculated by a Taylor series with the ignorance of high-degree Taylor polynomials.Then, the pressure at the central point on the left and right surfaces can be determined asand, respectively.

Because each face of the cube is infinitesimally small, it can be approximated that the pressure applied at each point on the surface is the same as that at the central point on the surface.The surface force on the surface AA＇D＇D and B B＇C＇C can be inferred as follows:

The mass of the cube is ρdxdydz, and fx, fy, fz represents the component of unit mass force in the direction of the corresponding coordinate axis.The component of mass force in the direction of x-axis is fxρd x d y d z.

Under the equilibrium conditions, ∑Fx =0 in the direction of x-axis, i.e.

which can be rearranged as:

In the same way, the following equations can be obtained in the direction of y and z axes:

so

Equations(2-1)are the differential balance equations of fluids, which is also called Euler equations because these equations were firstly deduced by Euler, a Swiss scholar in 1755.These equations indicate for the fluid at equilibrium, the surface forcesacting on the unit mass fluid and the components of unit mass force are equal to each other.The variation rate of pressure along the axial directionis equal to the component of mass force acting on unit volume in the axial direction (ρfx, ρfy, ρfz).

2.2.2 Integrated form of the differential balance equations of fluids

If equation(2-1)was multiplied by dx, dy and dz, respectively and then added together, we can write

Hydrostatic pressure p is the function of coordinates, i.e.p=p(x, y, z), the total differentiation of which is:

When the two equations above were merged together, equation(2-2)can be obtained as

Equation(2-2)is called integrated form of the differential balance equations of fluids.When the mass forces exerted on the fluid are known, the pressure distribution in the fluid can be calculated.

The physical meaning of equation(2-2)can be described as follows.

For the fluid at equilibrium, ρis a constant.Make=ω.Because p=p(x, y, z), ω=ω(x, y, z).According to equation(2-2), we can write

Thus, it can be concluded that

If partial derivatives of a coordinate functionω(x, y, z)and the component of force in the force field in the corresponding coordinate axis are equal to each other, the function is called force function or potential function.The force is called potential force. Thus, the mass force in equation(2-2)is potential force.In other words, the fluid with a constant density can maintain the balance only under the action of potential mass.

Inference:Fluid can be at equilibrium under the action of gravity or inertia force, indicating gravity and inertia force are potential force.

2.2.3 Equipressure surface

1．The concept of equipressure surface

Equipressure surface is a surface comprised of the points, at which the pressure equals.For example, the interface between liquid and gas and the interface between two liquids at equilibrium are equipressure surface.

On equipressure surface, p=const, dp=0.Thus,

2．Characteristics of equipressure surface

Mass force at any point on equipressure surface is eternally orthogonal to equipressure surface, which can be proven as follows.

There is a fluid particle M on equipressure surface, as shown in Fig.2-4.Unit mass force of this fluid particle is:f=fxi+fyj+fzk, and the linear vector of this fluid particle is:ds=dxi+dyj+dzk.According to equation(2-3), we can write

f·ds=fx dx+fy dy+fz dz=0

indicating f is orthogonal to ds, θ=90°, where the direction of ds is arbitrary on equpressure surface.Thus, it can be concluded that the mass force exerted on any point in stationary fluid must be perpendicular to the equipressure surface passing through this point.This is an important characteristic of equipressure surface.According to this characteristic, the shape of equipressure surface can be determined if the direction of mass force is already known, or the direction of mass force can be determined if the shape of equipressure surface is already known.