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