Fluid Mechanics (ME 201) 1

Transcription

1 Fluid Mechanics (ME 201) 1 9 Forces on Submered Bodies 9.1 Net Force on a Submered Plane Surfaces In this section, we will compute the forces actin on a plane surface. We consider a surface oriented at an anle θ with the free surface as shown in the fiure below. θ h ξ C Force on the top surface of the bod shown in the fiure can be computed as, F = pnd, where d = dd is the differential area and p is the pressure. The direction of the force, n, is normal to the plate. Substitutin the pressure at a depth h as p = p atm + ρ l h, and usin the transformation h = ξsin(θ), we et: F = (p atm + ρ l h)d = p atm + ρ l sin(θ) ξd. Here, is the total area of the plate. Usin ξd = ξ C, the net force can be written as: F = (p atm + ρ l sin(θ)ξ C ) = (p atm + ρ l h C ). Note that the pressure at the center of ravit, C, is iven b, p C = (p atm + ρ l h C ) and therefore, F = p C. The direction of the force is perpendicular to the plane of the surface and into the bod. 1 c aurav Tomar(2016) 17

2 0 It is instructive to look at the center of pressure. If we were to substitute the distributed load due to the pressure usin a sinle point force F, the location of application of this force on the plate in order to et the same moment about the center of ravit is known as the center of pressure. To obtain the center of pressure, we choose C a local coordinate sstem (, ) with the oriin at the center of ravit, i.e. C = C = 0 as shown in the adjacent fiure. The direction is allined with the ξ ais. Total moment about the center of ravit (0, 0) is iven b: M = r F = ( CP î + CP ĵ) ( F ˆk) = CP F ĵ CP F î, where ˆk is normal to the surface pointin outwards. For the distributed load, we can write: CP F = p d and CP F = p d The first equation ields: CP F = (p atm + ρξsin(θ))d = ρ ξsin(θ)d, where p atmd = C p atm = 0, since C = 0. Writin, ξ = ξ C, we et: CP F = ρsin(θ) (ξ C )d = ρsin(θ) 2 d = CP = ρsin(θ)i F where F = p C is the net force and I = 2 d is the moment of inertia about the -ais. The above epression depends on θ, i.e., the inclination of the surface relative to the free surface of the liquid. If the surface of the bod is horizontal, θ = 0, center of pressure would coincide with the center of ravit. Note that CP 0 (0 θ π/2) which implies that it is alwas below the center of ravit. lso note that, it is inversel proportional to p C, which implies that with increase in p C with depth, CP would move towards the center of ravit (i.e. as p C = CP 0). The coordinate of the center of pressure, CP, can also be obtained in the similar manner: CP = ρsin(θ)i p C. If I = d > 0 (more area in first and fourth quadrant of plane), CP < 0. If the bod is smmetricall placed, I = 0. C CP alues of area moment of inertia and product of inertia for some of the common shapes are iven below. 18

3 Rectanle: l/2 l/2 Circle: R rea = R π 2 4 I = πr /4 I = 0 b/2 b/2 3 rea = bl, I = bl, I = 0 12 s Trianle: Semi Circle: 2l/3 l/3 R b/2 b/2 3 2 rea = bl/2, I = bl /36, I = b(b 2s)l /72 rea = πr 2 /2 I =0.1098R I = 0 4R/3π Forces on Curved Surfaces Most bodies of practical interest do not have plane surfaces. In case of an irreular surface (see the sketch below), we can proceed in the followin wa. The vertical force on the surface is essentiall actin due to the component of pressure in the vertical direction and can essentiall be obtained b considerin the weiht of the fluid above the surface. H c p atm d W 1 F 1 F 1 p b a F W F H e C The net weiht of the fluid (marked b the reion abcde) alon with the weiht of the air i.e. the atmospheric pressure actin on the projected area H ields the net verticall downwards force, F, actin on the curved surface: F = W 1 + W 2 + p atm H For the horizontal force, the pressure on the vertical projection of the area ields: F H = pd = p C 19

4 9.3 Buoanc and Stabilit From our eperience we know that objects feel lihter in a water pool. rchimedes showed that, in fluids, submered bodies feel a verticall upward force which is equal to the weiht of the displaced fluid. This upward force is known as Buoanc force and can be understood in one of the followin was. Lets consider the solidification or freezin principle of Stevin (1634). In the adjacent fiure a pool of liquid in equilibrium is shown. If we freeze a portion of the fluid, with the same densit of course, then that frozen bod is oin to be in equilibrium with the surroundin fluid. Thus, we can conclude that the weiht of the bod, W l (actin verticall downwards), is balanced b the net surface force due to the pressure Freezin W l on the bod from all sides and there is no component of the bod force on the frozen portion in the horizontal direction. Thus, the (a) net horizontal force due to the pressure is zero and (b) net vertical force due to pressure is W l actin verticall upwards. This upward force is known as Buoanc. Now, if we replace the solidified portion with a solid of different densit with the net weiht, sa, W s, then the net force on the bod will be iven b: We can also write the same as: F net = W s W l F net = (ρ s ρ l ) where ρ s and ρ l are the densities of the solid and liquid, respectivel, and is the volume of the submered bod. Thus, if ρ s > ρ l bod sinks, whereas if ρ s < ρ l, it rises. The risin lihter bod finds equilibrium at the free surface, where the submered portion is smaller and thus the Buoanc force is less and eactl balances the weiht of the solid bod: W s = ρ l submered where submered < is the volume of the submered portion of the bod. Of course, there are several was to arrive at the result of Buoanc force. One can consider the weiht of the fluid above the upper surface of the bod which acts downwards and minus the force actin on the lower curved surface of the bod that acts upwards. The difference ields the weiht of the fluid bounded b the upper and lower surfaces and the net force is upward. Mathematicall, we can look at the Buoanc force as followin. The net force is: F net = p( n)ds where n is the local normal pointin outwards to the bod. 20

5 z Usin diverence theorem, for the fluid bounded b the surface, we can write: pnds = pd Interatin the pressure equation ( p = ρ l ˆk) for equilibrium over the entire volume of the bod: pd = ρ l d ˆk = ρ l ˆk Substitutin the above epression in the equation for F net, we et: pn F net = ρ l ˆk which indicates a net upward Buoanc force. The total downward force on the bod can be written as: F total = ρ l ˆk ρ s ˆk. 9.4 Stabilit of Floatin Bodies Floatin bodies, dependin upon the distribution of mass (such as in ships and boats), ma be unstable due to unbalanced moments. Investiation of stabilit of floatin bodies can be carried out b perturbin the bod b an small anle and then checkin if the net moment enerated is restorin or overturnin. Consider the fiures (a), (b) and (c) below. Stable Unstable M B M Restorin Overturnin B B fter tiltin the floatin bod b a small amount, the center of Buoanc would shift. Intersection of the vertical line drawn throuh the new center of Buoanc, B, with the ais of smmetr ields the Metacenter. If metacenter is above the center of ravit, this would result in a restorin moment that would brin the sstem back to the initial position (initial confiuration is stable), whereas if the the metacenter is below the center of ravit, an overturnin moment acts that takes the bod further awa from the initial confiuration (unstable). The new center of Buoanc, B, in tilted bodies, can be computed b considerin the chane in the confiuration of the submered portion of the bod. 21