Fluid Statics
Fluid statics focuses on equilibrium problems of forces exerting on a motionless fluid and corresponding application in practical situations. In this chapter, we will first discuss Eulerian equilibrium equation of fluids and its integral. In addition, we
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Fluid Statics
Abstract
Fluid statics focuses on equilibrium problems of forces exerting on a motionless fluid and corresponding application in practical situations. In this chapter, we will first discuss Eulerian equilibrium equation of fluids and its integral. In addition, we will introduce pressure calculation and measurement. Finally, according to the principles of pressure distribution, we can calculate forces acting on a plate. Keywords
Surface force Eulerian equilibrium Pressure distribution Vacuum degree Pressure measurement
2.1
Absolute pressure
Forces on a Fluid
2.1.1 Mass Force Mass force is a noncontact force acting on all fluid particles leading to a certain force field [1, 2]. The magnitude of mass force is proportional to the fluid mass or fluid volume. It is also called body force. Unit mass force is defined as the mass force acting on unit mass, of which the unit is same with acceleration, m/s2 . If the fluid particle volume approaches zero, namely, DV ! 0, then the mass force can be written as dFm ¼ dm am ¼ dmðXi þ Yj þ ZkÞ;
© Metallurgical Industry Press, Beijing and Springer Nature Singapore Pte Ltd. 2018 H. Song, Engineering Fluid Mechanics, https://doi.org/10.1007/978-981-13-0173-5_2
ð2:1Þ
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Fluid Statics
where dm is unit mass, am is acceleration, X; Y; Z are the components of acceleration or unit mass force on x, y, z axes respectively. General mass force includes gravity and inertial force.
2.1.2 Surface Force Surface force is the force acting on the contact surface of fluid particles by the adjacent fluid layer or other fluid [3, 4]. Its value is proportional to contact area. According to the acting directions, surface forces can be divided into two categories: one is normal force which is perpendicular to acting surface such as pressure and normal stress, the other is shear force like shear stress which is parallel to acting surface.
2.2
Equilibrium Equation and Its Integral
2.2.1 Eulerian Equilibrium Equation As we might all agree, the static fluid is in equilibrium. The total force involving the mass force and surface force in fluid element should equal zero, which can be used to establish the equilibrium equation and obtain the static pressure distribution. As shown in Fig. 2.1, select a fluid cubic unit abdcc0d0b0a0 in a static fluid. Its length, width and height are dx; dy; dz, respectively. The pressure of the unit centroid Mðx; y; zÞ is pðx; y; zÞ. The forces acting on the fluid unit include (1) Surface force. Since the fluid pressure is a continuous function of the position. Therefore, the pressure acting on surface ad and a0 d0 in the x direction can be expanded as Taylor’s series. Neglecting high order terms, the pressures of 1 @p surface ad and a0 d0 are p þ 12 @p @x dx and p 2 @x dx, respectively. Similarly, the 1 @p 0 pressures of other surfaces are: ac0 : p 12 @p @y dy and bd : p þ 2 @y dy in the y
0 1 @p direction; a0 b: p þ 12 @p @z dz and c d: p 2 @z dz in the z direction. (2) Mass force. The components of the mass force on x, y, z
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