CEE 3310 – Hydrostatics 12.6 Review• Compressible fluids ⇒ Isothermal, adiabatic• The U.S. standard atmosphere is based on the concept of adiabatic conditions(γ = γ(θ)) for the troposphere (0 < z < 11.0 km) and isothermal conditions(γ = γ0) for the stratosphere (11.0 < z < 20.1).• Pressure measurement – absolute, relative to a vacuum; gage, relative to local at-mosphere; vacuum/suction, the negative of gage.2.7 Hydrostatic Force on a Plane SurfaceNote: The approach presented here is different than the approach presentedin the text but I feel it is more powerful and makes more sense!Consider the force on a horizontal plate due to a fluid of depth HWe know t hat over the entire plate (with surfae area A) the pressure is constant (sincewe do not change our z position) hence we can write FR= P A where FRis known asthe resultant force, as the net of the pressure acting on the plane surface is equivalentto the resultant force applied at a particular location (which can be shown to be thecentroid of the a r ea). Recall that for the case shown (a rectangular fish tank) P = γh⇒ FR= γhA = γ∀ = the weight of the fluid! Thus the pressure force on the bottom ofthe tank is just the weight of the fluid which makes sense!What a bout a surface inclined to the horizontal? Consider2We start with the differential form of our statement above about the fo rce on an area,namely:dFR= P dA which we integra te to find FR=ZAP dAIn a hydrostatic fluid we expect no variation of P with position in the x direction butwe do expect P to vary as the depth changes (e.g., as we move in z).If we consider a 2-D Cartesian coordinate system with its origin in the plane of our surface(S), the y coo rdinate oriented vertically along the surface (e.g., maximally aligned withgravity – note we are now using y as the vertical coordinate along the surface a nd it isin the direction of gravity, as oppo sed to z which is in the opposite direction of gravity,and, in particular, is still truly vertical) and the x direction normal to the y direction onthe surface, we can write thisFR=Z ZSP (x, y) dxdy (2.1)In stating the above all we have assumed is:1. The fluid is hydro static (u = constant).2. The surface S is planar.Thus for flat surfaces in hydrostatic fluids the above is always true.If we now allow ourselves one more assumption, namely:3. The density over the surface S is constant.CEE 3310 – Hydrostatics 3Then we may writeFR= PCA (2.2)wherePC=1AZ ZSP (x, y) dxdy (2.3)hence PCis just the average pressure acting over the surface S which we see is mathe-matically the same as the pressure acting at the centroid of the surface S and A is thearea of the surface S. Thus our only restriction in applying equation 2.2 is that the fluidmust have a constant density wherever it contacts S. Setting equations 2.1 and 2.2 equalyields equation 2.3 confirming that the pressure acting at the centroid (PC) is equivalentto the average pressure acting over the surface S.Therefore the magnitude of the force depends on• PC– the pressure acting at the centr oid of the surface S.• A – the area of the surface S.Note the above is a bit different than most books present this material. If we make twomore assumptions4. The column of fluid above the surface S is exposed to atmospheric pressure.5. The density of the entire fluid, from the deepest part of the surface S all the wayto the free surface o f the fluid, is constant.Then we arrive at the form our book (and most books) present, namelyFR= γ sin θZAy dAwhere θ is the angle between vertical a nd our surface S (e.g., θ = 0 for a vertical surface).But from mechanics we recognize1ARAy dA as the first moment of the area A with respect4to the x axis which we will denote ycsince this is the position along the y axis of thecentroid of the area. Hence we haveyc=1AZAy dA ⇒ and henceFR= γ sin θ ycA ⇒ but we can write hc= sin θ ychenceFR= γhcAwhere hcis the depth of the centroid (e.g., now in a direction para llel to gravity).But this assumes assumptions 4 and 5 are true! Examples of problems thatviolate these assumptions will appear in the problem sets and Lab #2 soproceed with caution if you like the books approach.2.7.1 Where is the force located on the surface?Why doesn’t the resultant force just act at the centroid of the area?O.K., let’s rigo r ously determine where the force acts. Let’s define our 2-D coordinatesystem on the surface such that the origin is at the centroid of the surface S with ydirected positively downward (e.g., aligned maximally with gravity) . We can define x bythe right-hand-rule where what would be the positive third axis points in the directionof the surface normal (e.g., equal and opposite to FR). Now, we define θ as the anglebetween the vertical and the y axis (e.g., θ = 0 is a vertical surface).The location that the force acts on the surface is known as the center of pressure and isCEE 3310 – Hydrostatics 5denoted yRas it is the location that the resultant force, FR, acts on the surface.Based on the definition of pressure, the force is clearly directed perpendicular to thesurface S, and from the fluid toward the surface.Now, recalling from mechanics that the moment of the resultant force about any pointis equal to the sum of the moments of the component forces about the same point, wecan writeyRFR=ZAyP dA =ZAy(PC+ γy cos θ) dA = PCZAy dA + γ cos θZy2dABut the first term on the right-hand-side is just the first moment of A which occurs atthe centroid and hence in our coordinate system this is just = 0! HenceyRFR= γ cos θZy2dArecalling that FR= PCA and solving for yRyR=γ cos θPCAZy2dAThe integral in the above expression is the second moment of the area with respect to thex axis through the centroid (horizontal axis, or the line perpendicular to gravity throughthe centroid of the surface S) and is denoted Ixc, hence we arrive at our final expressionfor yRyR=Ixcγ cos θPCA(2.4)whereIxc=Zy2dA (2.5)We make two import ant findings based on this last equation:1. The above is positive definite hence we find that yRis always positive which in ourcoordinate syst em means that it always acts below the centroid (which is locatedat y = 0 by definition.62. As PCgets larger f or a given surface S with area A, the value of yRgets smallerindicating that the deeper we go the closer the center of pressure is to the centroidof the surface S.2.8 Review• Manometer Pdown= Pup+ γ|∆z|• Force on Plane Surfaces– In general FR=R RSP