§2.4 Relative balance of liquid

The liquid in a container makes relative movement with the earth as a reference, and there is no relative movement between different parts of the liquid or between the liquid and the container.Thus, liquid is in a state of equilibrium relative to the coordinate system if the coordinate system was built in the container.For the fluids in a relative balance, mass forces include gravity and inertia force. According to D'Alembert's principle, the relative motion of a liquid can be transformed into an energy balance problem in form and equation(2-2)can be applied to analyze the relative motion of the liquid if inertia force is included in mass force.

2.4.1 Relative balance of the liquid in a uniformly accelerated rectilinear motion

As shown in Fig.2-14, the liquid makes a uniformly accelerated rectilinear motion along with the container.The central point of the original liquid surface is designated as the origin.z axis is vertically upward and the included angle between the acceleration and the x axis isα.Unit mass force includes gravity and inertial force of acceleration. The unit mass force at an arbitrary point M(x, y, z)is:

fx=-acosα, fy=0, fz=-g-asinα

The following equation can be obtained by substituting the equations above into equation(2-2).

dp=ρ[-acosαdx-(g+asinα)dz]

After being integrated, the following equation can be arrived at:

p=-ρ[acosα·x+(g+asinα)z]+C

On the liquid surface, x=z=0, and p=p0, which was substituted into the equation above, C=p0can be obtained.

Then, we can write

This is general expression for the law of pressure distribution when the liquid in uniformly accelerated rectilinear motion along with the container is in a state of relative equilibrium.

Based on equation(2-11), isobaric equation can be obtained as:

Thus, equipressure plane is an inclined plane at an angleθto the horizontal plane.θ can be calculated as:

The value of vertical coordinate of liquid surface can be calculated as:

The following equation can be obtained by substituting equation (2-13)into equation(2-11):

p=p0+ρ(g+αsinα)(z0-z)

where z0-z is the submergence depth of the particle below the liquid level, denoted as h.So that

which indicates the variation of the pressure at an arbitrary point with water depth is linear for the liquid in uniformly accelerated rectilinear motion.

Equation(2-14)also suggests the pressure is p0at an arbitrary point of the liquid and the pressure is zero at an arbitrary of the liquid in an open container if the liquid goes into free fall along with the container.

2.4.2 Relative balance of rotating liquid

In the case of a upright cylindrical container filled with liquid rotating at the uniform angular speedωas shown in Fig.2-15, liquid close to the wall of the container rotates with the container at the beginning, and subsequently all liquid rotates with the cylindrical container at the uniform angular speedω.A state of relative equilibrium can be achieved.In this case, free surface of the liquid transforms from a horizontal plane to a paraboloid of revolution.A coordinate axis is built in the rotating cylinder, with origin coinciding with the vertex of rotating paraoloid and z axis vertically upward.The centripetal acceleration of any particle A in a rotating obj ect is -ω2r, and the component on the x and y axis is -ω2 x and -ω2 y, respectively.Unit mass force of gravity is fx=fy=0, and fz=-g.Unit mass force of inertial force is fx=ω2 x, fy=ω2 y, and fz=0. Thus, the total unit mass force is fx=ω2 x, fy=ω2 y, and fz=-g.

The following equation can be obtained by substituting the equations above into Euler equations:

dp=ρ(ω2xdx+ω2ydy-gdz)

After being integrated, the equation above can be transformed as:

At the origin point (x=y=z=0), p=p0, so C=p0.Then, we can write

This is the general expression for the law of pressure distribution of the liquid at relative equilibrium in a vertical container rotating at the uniform angular speed.

The equation of isobaric cluster including free surface is:

which indicates isobaric cluster is a cluster of rotating paraoloids with central axis.

On the free surface of liquid, p=pa=p0.The equation of free surface can be expressed by relative pressure:

where z0is vertical coordinate of free surface.The following equation can be obtained by substituting equation(2-17)into equation(2-15):

p=p0+ρg(z0-z)

where z0-z is the depth of the particle A below the free surface of liquid, denoted as h. Then we can write

Equation(2-18)indicates the variation of pressure at an arbitrary point with the depth of the liquid is linear in the rotating liquid at a relative equilibrium.However, the piezometric head at an arbitrary is not a constant in a rotating container.