Falls in the realm of a body force. Newton s law of gravitation is:

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1 GRAVITATION Falls in the ealm of a body foce. Newton s law of avitation is: F GMm = Applies to '' masses M, (between thei centes) and m. is =. diectional distance between the two masses Let ˆ, thus F = GMm ˆ Consequently, F masses.as, F is popotional to the squae of and vice vesa. the distance between the Let M be the mass of the eath and m be the mass of a small ai pacel, then the avitational foce pe unit mass (ecall that the equations of motion will be witten in tems of pe unit mass) of ai is: F GM = ˆ m *

2 Can we appoximate fo an ai pacel at heiht above the eath s suface? <<a, since: a=670 km < 15 km (toposphee) cente of the eath a Note that <<a, so =a + ~ a, and so ~ a. We fequently use this appoximation thouhout the couse. So, in the toposphee * GM a ˆ We can estimate Mass of the eath is M = of the eath. *, 4 π ρ, whee ρ is the aveae density of e e * km s a m k 4 πa ρe km N s = m = 9.8 s -11 s m k 4 π ( ) k m 550 m

3 The Viscous Foce (i.e. Fiction!) The popety of a fluid that enables it to develop and maintain an amount of sheain stess dependent upon the velocity of flow and then to offe continued esistance to flow. O similaly, Viscosity: The esistance of a liquid to flow is called its viscosity. The eate the viscosity, the 'moe slowly it flows ' Viscosity is a measue of the ease with which molecules move past one anothe. Viscosity depends on the attactive foce between the molecules. Viscosity deceases with inceasin tempeatue - the inceasin kinetic eney ovecomes the attactive foces and molecules can moe easily move past each othe. Extensive liteatue, lab and eseach wok in this aea! Examples: I think I m lactose intoleant

4 Some peliminay comments/divesions: The intenal atmosphee is not vey viscous (see below)! When scalin the equations of motion, the fiction tem scales as: ν = UL 1, Reynolds # wheeν = µ is ρ efeed to as the kinematic viscosity. Hee, µ is called the dynamic coefficient of viscosity - moe on this in a bit. U and L ae the chaacteistic velocity and lenth scales (fo lae-scale atmospheic flow these ae usually taken as 10 ms -1 and 10 km). Physically, the Reynolds # is the atio of the inetial-to-viscous tems in the fluid equations ms 10 m = 4 1 Re m s The point hee is, that the liteatue fequently efes to flows as hih o low Reynolds # flows. Because the fiction tem scales as the invese Reynolds # we have: Hih Reynolds # flows: low viscosity Low Reynolds # flows: hih viscosity The lowe the viscosity (Hih Reynolds #), the moe pone the flow is to tubulence! Note that the Reynolds tem is quite lae fo lae-scale atmospheic flow. Howeve, if one applies what they call no-slip bounday conditions (i.e. the fluid velocity oes to eo at the boundaies of the fluid basin), then one cannot completely nelect the fiction tems. OK, now back to the deivation of the fiction 11

5 foce fo ou fluid equations Viscous foce is a suface o contact foce A foce due to molecula fiction Molecules in a fluid undeo andom themal motion. Collisions between molecules esult in an exchane of eney, this exchane of eney esults in a chane of momentum of the molecules. Intenal foces due to intenal inteactions of the fluid itself that depend on the state of motion ae efeed to as stesses. Fom a molecula standpoint, andom themal motion (see the kinetic theoy of asses fo moe on this one), esults in a diffusion of momentum called viscosity. Fictional foces etad motion and the molecula conduction of heat mitiates tempeatue adients. Thus, in the absence of extenal focin and if these intenal foces ae allowed to expend themselves to thei natual conclusion, the fluid will each an equilbium state in which all motion has vanished.

6 Conside followin expeiment: fluid at est is contained between infinite paallel plates. Then, at t = 0, top plate stats movin at speed U 0 in the +x diection: U 0 x No slip condition: molecula fiction causes fluid to "stick" to solid objects/solid boundaies. Fluid at lowe plate neve moves since it's stickin to a nonmovin bounday (i.e. no-slip ). But fluid at top plate moves at speed U 0 since it's stickin to a movin bounday. No slip condition fom a molecula view point: Molecula fiction causes the movin fluid to pull alon fomely estin fluid beneath it. This momentum is impated to the fluid via the viscous foce. t = 1 min Lowe bounday U 0

7 t = min U 0 Afte awhile we et a steady-state, i.e. no chane in the velocity pofile at anytime o any point! The final (i.e. steady-state pofile) looks like: t = awhile U 0 This is a linea pofile! Why is this? Conside the x-component of the viscous foce (pe unit aea) exeted on a hoiontal aea at heiht by fluid above this level: u τ x = µ Aside: (because it eally seems as if this equation has come fom out of the blue!) This equation is actually a small component of a fa moe complicated elationship efeed to as the constitutive equation (e.., Newton, Stokes) which elates the

8 stess tenso to othe flow vaiables. Fo ou puposes, the fomation of the constitutive elationship assumes the followin: -tanslational/otational components of the flow do not contibute to stess on the pacel -only the expansion (diveence/conveence), sheain and stetchin of defomation (see Fi 5.8 Dutton), & the nomal pessue foce contibute to fluid stess -all scalas ae isotopic Note that the units of stess ae foce/aea What is the stess tenso? We won t cove it in this couse. Howeve, the stess tenso was deived usin postulates one fom a man named Cauchy who poposed: All foces exeted on a pacel by the est of the fluid can be epesented as a distibution of stess vectos applied to the boundin suface of the pacel The stess tenso includes the pessue foce as mentioned above. Thus the equations of motion can be expessed in tems of the stess tenso on the iht hand side of ou equations. Of couse these postulates upon which the stess tenso is deived ae subject to validation thouh expeimentation. It is eneally assumed that, outside of boundaies, chanes in the kinetic eney of the fluid due to fiction ae exactly balanced by chanes in the intenal eney. This manifests in a loss of KE and a ain in IE.