上海交通大学：《船舶流体力学》课程教学资源（课件讲稿）第4章 流体静力学

上游充通大学 Shanghai Jiao Tong University 第四章 流体静力学 流体静力学是研究船舶浮性、 稳性、抗沉性的主要理论基础

Shanghai Jiao Tong University 第四章 流体静力学 流体静力学是研究船舶浮性、 稳性、抗沉性的主要理论基础

浒充通大￥ 4.1 Shanghai Jiao Tong University Indroduction to Hydrostatics M GM W G C F B W B Water Water H Hinge 6m Hinge R 3m

Shanghai Jiao Tong University 4.1 Indroduction to Hydrostatics

上浒充通大￥ 4.1 Indroduction to Hydrostatics Shanghai Jiao Tong University Hydrostatics is the study of pressures throughout a fluid at rest and the pressure forces on finite surfaces. As the fluid is at rest,there are no shear stresses in it.Hence the pressure at a point on a plane surface always acts normal to the surface,and all forces are independent of viscosity. The pressure variation is due only to the weight of the fluid. (a) (b) (c)

Shanghai Jiao Tong University 4.1 Indroduction to Hydrostatics Hydrostatics is the study of pressures throughout a fluid at rest and the pressure forces on finite surfaces. As the fluid is at rest, there are no shear stresses in it. Hence the pressure at a point on a plane surface always acts normal to the surface, and all forces are independent of viscosity. The pressure variation is due only to the weight of the fluid

上浒充通大￥ 4.2 Indroduction to Pressure Shanghai Jiao Tong University Pressure always acts inward normal Pressure to any surface. Body Pressure is a normal stress,and hence has dimensions of force per unit area,or [ML-1T-2]. In the Metric system of units,pressure is expressed as "pascals"(Pa)or N/m2. △Fn Standard atmospheric pressure is 101.3 kPa. Surface Pressure is formally defined to be △A lim △Fn △A→0△A where AF is the normal compressive force acting on an infinitesimal area4

Shanghai Jiao Tong University 4.2 Indroduction to Pressure Pressure always acts inward normal to any surface. Pressure is a normal stress, and hence has dimensions of force per unit area, or [ML-1T-2]. In the Metric system of units, pressure is expressed as "pascals" (Pa) or N/m2. Standard atmospheric pressure is 101.3 kPa. 0 lim n A F p Δ → A Δ = Δ Pressure is formally defined to be where ΔF n is the normal compressive force acting on an infinitesimal area . ΔA

上浒充通大￥ 4.3 Pressure at a Point Shanghai Jiao Tong University By considering the equilibrium of a small triangular wedge of fluid extracted from a static fluid body,one can show that for any wedge angle the pressures on the three faces of the wedge are equal in magnitude: ps=py=p: independent of 0 This result is known as Pascal's law,which states that the pressure ,6r6 at a point in a fluid at rest,or in motion,is independent of direction 6 as long as there are no shear stresses present. y 2

Shanghai Jiao Tong University 4.3 Pressure at a Point By considering the equilibrium of a small triangular wedge of fluid extracted from a static fluid body, one can show that for any wedge angle θ, the pressures on the three faces of the wedge are equal in magnitude: independent of syz ppp = = θ This result is known as Pascal's law, which states that the pressure at a point in a fluid at rest, or in motion, is independent of direction as long as there are no shear stresses present

上游充通大学 4.3 Pressure at a Point Shanghai Jiao Tong University Pressure at a point has the same magnitude in all directions,and is called isotropic. Free surface Air p=0 gage Liquid h z=-h

Shanghai Jiao Tong University 4.3 Pressure at a Point Pressure at a point has the same magnitude in all directions, and is called isotropic

上游充通大睾 4.4 Pressure Variation with Depth Shanghai Jiao Tong University Consider a small vertical cylinder of fluid in equilibrium,where positive is pointing vertically upward.Suppose the origin Z-0 is set at the free surface of the fluid.Then the pressure variation at a depth=-h below the free surface is governed by (p+△p)A+W=pA → △pA+PgA△z=0 0 → △p=-P8△2 h → dp P+Ap dz -P8 cross sectional area=A or dp dz =一Y (as△z→0) p Therefore,the hydrostatic pressure increases linearly with depth at the rate of the specific weight y=pg of the fluid

Shanghai Jiao Tong University 4.4 Pressure Variation with Depth Consider a small vertical cylinder of fluid in equilibrium, where positive z is pointing vertically upward. Suppose the origin z = 0 is set at the free surface of the fluid. Then the pressure variation at a depth z = -h below the free surface is governed by ( ) 0 or (as 0) p p A W pA pA gA z p gz dp g dz dp z dz ρ ρ ρ γ +Δ + = ⇒ Δ + Δ= ⇒ Δ =− Δ ⇒ =− =− Δ → Δz p p+Δp W cross sectional area = A 0 h Therefore, the hydrostatic pressure increases linearly with depth at the rate of the specific weight γ ≡ ρ g of the fluid

上浒充通大￥ Shanghai Jiao Tong University 4.4 Pressure Variation with Depth Homogeneous fluid:p is constant. By simply integrating the above equation: ∫p=-∫Pgdk→p=-pg+C where C is an integration constant.When =0(on the free surface),p=C=po (the atmospheric pressure).Hence, p=-pgz+po P1=Patm ① 又 The equation derived above shows that when the density is constant,the pressure in a liquid at rest increases linearly with depth P2 Patm+pgh from the free surface

Shanghai Jiao Tong University 4.4 Pressure Variation with Depth Homogeneous fluid: ρ is constant. By simply integrating the above equation: dp gdz p gz C = − ⇒ =− + ρ ρ ∫ ∫ where C is an integration constant. When z = 0 (on the free surface), 0 pC p = = (the atmospheric pressure). Hence, 0 p g = − + ρ z p The equation derived above shows that when the density is constant, the pressure in a liquid at rest increases linearly with depth from the free surface

上游充通大学 4.4 Pressure Variation with Depth Shanghai Jiao Tong University The pressure is the same at all points with the same depth from the free surface regardless of geometry,provided that the points are interconnected by the same fluid. However,the thrust due to pressure is perpendicular to the surface on which the pressure acts,and hence its direction depends on the geometry. Patm Water PA=PB=PC=PD=PE=PF=PG=Patm+pgh Mercury PH≠P, H

Shanghai Jiao Tong University 4.4 Pressure Variation with Depth The pressure is the same at all points with the same depth from the free surface regardless of geometry, provided that the points are interconnected by the same fluid. However, the thrust due to pressure is perpendicular to the surface on which the pressure acts, and hence its direction depends on the geometry

上游充通大学 Shanghai Jiao Tong University 4.5 Hydrostatic Pressure Difference Between Two Points For a fluid with constant density, Pbelow=Pabove+Pg△z As a diver goes down,the pressure on his ears increases.So,the pressure "below"is greater than the pressure "above

Shanghai Jiao Tong University 4.5 Hydrostatic Pressure Difference Between Two Points As a diver goes down, the pressure on his ears increases. So, the pressure "below" is greater than the pressure "above." For a fluid with constant density, below above p p gz = + Δ ρ