2.016 Hydrodynamics Prof. A.H. Techet

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1 .016 Hydodynmics Reding #5.016 Hydodynmics Po. A.H. Techet Fluid Foces on Bodies 1. Stedy Flow In ode to design oshoe stuctues, suce vessels nd undewte vehicles, n undestnding o the bsic luid oces cting on body is needed. In the cse o stedy viscous low, these oces e stightowd. Lit oce, pependicul to the velocity, nd Dg oce, inline with the low, cn be clculted bsed on the luid velocity, U, oce coeicients, C D nd C, the object s dimensions o e, A, nd luid density, ρ. Fo L viscous lows the dg nd lit on body e deined s ollows 1 F Dg = ρu A C D (5.1) 1 F Lit = ρu A C L (5.) These equtions cn lso be used in quiescent (sttiony) luid o stedy tnslting body, whee U is the body velocity insted o the luid velocity, since U is still the eltive velocity o the luid with espect to the body. The dg oce ises due to viscous ubbing o the luid. The luid my be thought o s compised o sevel lyes which move eltive to one nothe. The lye t the suce o the body sticks to the suce due to the no-slip condition. The net lye o luid wy om the suce ubs ginst the lye below, nd this ubbing equies cetin mount o oce becuse o viscosity. One would epect tht in the bsence o viscosity, the oce would go to zeo. Jen Le Rond d'alembet ( ) peomed seies o epeiments to mesue the dg on sphee in lowing luid, nd on the bsis o the potentil low nlysis he epected tht the oce would ppoch zeo s the viscosity o the luid ppoched zeo. Howeve, this ws not the cse. The net oce seemed to convege on non-zeo vlue s the viscosity ppoched zeo. Hence, the vnishing o the net oce in the potentil low nlysis is known s d'alembet's Pdo. vesion.0 updted 8/0/ A. Techet

Lit oce, pependicul to the velocity, nd Dg oce, inline with the low, cn be clculted bsed on the luid velocity, U, oce coeicients, C D nd C, the object s dimensions o e, A, nd luid density, ρ.

2 .016 Hydodynmics Reding #5 D Allembet s Pdo o ied sphee in uniom inlow: Foce on sphee (dius ) in n unbounded STEADY moving luid with velocity U is eploed in the ollowing discussion. U z θ The coesponding D potentil unction o sphee in uniom inlow is simply: φ (,θ ) = U + cos θ (5.) The hydodynmic oce on the body due to the unstedy motion o the sphee is given s suce integl o pessue ound the body. Pessue omultion comes om the unstedy om o Benoulli. Foce in the -diection is F = ρ 1 + φ n ds (5.4) t B Hee, the time deivtive o the potentil is zeo since the low is stedy nd velocity is not unction o time. Since we wnt the oce cting on the body we need the velocity components on the sphee suce ( = ). In spheicl coodintes the velocity is ound by tking the gdient o the potentil unction s ollows: K 1 1 V = φ = V, V,V ϕ =, θ, (5.5) θ sinθ ϕ The velocity t the body, on =, cn only be tngentil to the body due to the kinemtic boundy condition ( KBC V = 0 ): K V = φ = (0, U sin θ, 0 ) (5.6) vesion.0 updted 8/0/ A. Techet

) The hydodynmic oce on the body due to the unstedy motion o the sphee is given s suce integl o pessue ound the body. Pessue omultion comes om the unstedy om o Benoulli.

3 .016 Hydodynmics Reding #5 The mgnitude o the velocity is simply: 1 φ = 9 U sin θ (5.7) 8 Using (5.7) in the omultion o F, om eqution (5.4), we get the hoizontl oce on the body: F = ρ )( ) cos 9 θ sin θ ( 8 U sin θ 0 d θ (5.8) F = 9 ρ U sin θ cos θ dθ (5.9) 4 0 This integl cn be integted by pts o substitution o vibles: p = sinθ ; dp = cosθ dθ (5.10) sin θ cos θ d θ = p dp 4 4 = p = sin θ (5.11) 4 4 sin 4 θ 9 F = 4 ρ U = 0 (5.1) 4 0 THERE IS NO FORCE ON A BODY IN A STEADY FLOW IN THE ABSENCE OF VISCOSITY! THIS IS D ALLEMBERT S PARADOX. The esolution becomes cle when we elize tht ny non-zeo viscosity, no mtte how smll, will esult in boundy lye nd the tngentil low velocity vnishing t the suce o the sphee. As we lowe the viscosity, the thickness o the boundy lye is educed, but the low velocity still dops to zeo coss tht lye (the "no-slip" condition). The esults o this boundy lye led to losses in the momentum o the lowing luid nd the tnseence o momentum to the sphee, i.e., to net unblnced oce. vesion.0 updted 8/0/ A. Techet

9) 4 0 This integl cn be integted by pts o substitution o vibles: p = sinθ ; dp = cosθ dθ (5.10) sin θ cos θ d θ = p dp 4 4 = p = sin θ (5.11) 4 4 sin 4 θ 9 F = 4 ρ U = 0 (5.

4 .016 Hydodynmics Reding #5. Unstedy Motion nd Added Mss. Do this t home: A) Wve you hnd in the i. Feel the oce it tkes to ccelete you hnd. F=m. B) Fill you bthtub with wte. Run you hnd though (with the plm cing owd) t slow, constnt speed. Feel the dg on you hnd. Notice tht the wte must move to low ound you hnd. B1.5) Run you hnd though t nothe constnt, ste speed. Notice tht it tkes moe oce. Recll the dg oce is popotionl to U^. Notice tht the wte now moves t some constnt, ste speed ound you hnd. C) Ty nd ccelete you hnd om the slow speed to the st speed. It s hd, huh?! Notice tht the wte lowing pst you hnd hs to ccelete s you hnd cceletes. Since some mss o wte must ccelete, you hnd eels hevie. We cptue this ide with the concept o dded mss. Beyond stedy low, especilly in the pesence o ee suce wves, we must conside unstedy, time dependent motions o both the luid nd the body nd the luid inetil oces ise, dding to the totl ocing on body. Tke the cse o n unstedy moving body, Ub( t), in n unbounded inviscid, iottionl luid ( µ = 0 ) with zeo velocity, U = 0. The time-dependent oce on the body is diectly popotionl to the body cceletion: t Ft () = m du () b (5.1) dt whee m, is the system dded mss, depends on the body geomety nd diection o motion. This is n dded inetil oce o dded mss oce on the body. By compison, in n inviscid stedy low, by D Alembet s Pdo, the oce on the body would be zeo. vesion.0 updted 8/0/ A. Techet

Notice tht it tkes moe oce. Recll the dg oce is popotionl to U^. Notice tht the wte now moves t some constnt, ste speed ound you hnd. C) Ty nd ccelete you hnd om the slow speed to the st speed.

5 .016 Hydodynmics Reding #5 Unstedy Moving Body Sttiony Fluid: Foce on sphee (dius ) cceleting in n unbounded quiescent (non-moving) luid. U = U () t is the unstedy body velocity. z U(t) θ The Kinemtic Boundy Condition on the sphee, gunteeing no luid low though the body suce, is = = U() t cos θ. (5.14) The potentil unction o moving sphee with no ee stem (still luid) is simply φ = Ut () cos θ. (5.15) You cn double check this solution o the velocity potentil by substituting φ into the Kinemtic Boundy Condtion (eq. (5.14)) to mke sue this potentil woks t the boundy o the sphee. The hydodynmic oce on the body due to the unstedy motion o the sphee is given s suce integl o pessue ound the body. Pessue omultion comes om the unstedy om o Benoulli. Foce in the -diection is F = ρ 1 + φ n ds t B (5.16) Since we wnt the oce cting on the body we need the velocity components on the sphee suce ( = ). In spheicl coodintes the velocity is ound by tking the gdient o the potentil vesion.0 updted 8/0/ A. Techet

14) The potentil unction o moving sphee with no ee stem (still luid) is simply φ = Ut () cos θ. (5.

6 .016 Hydodynmics Reding #5 unction s ollows: K 1 1 V = φ = V, V,V ϕ =, θ, (5.17) θ sinθ ϕ such tht the gdient o φ() t o moving sphee is 1 φ = Ut () cos θ, Ut)sin ( θ,0 =. (5.18) To evlute the pessue ound the sphee we need the mgnitude o the velocity φ on the suce ( = ): ( 4 φ ) = U cos θ + 1 U sin θ; n = ê ; n = cosθ (5.19) Net, the time deivtive o the velocity potentil, evluted t the sphee suce, is = U () t cos θ = U t t 1 () cos θ (5.0) nd the suce integl cn be e-witten in spheicl coodintes s b ds = ( d θ )( sin θ ) (5.1) 0 Substituting (5.19), (5.0), nd (5.1) into (5.16) we cn solve o F, the dded mss oce on spheicl body moving with n unstedy cceletion: The volume o sphee is = s F () ρ = U t (5.) unit =Mss thus eqution (5.) is simply F = U () t ( 1 ρ s ) (5.) 4 vesion.0 updted 8/0/ A. Techet

18) To evlute the pessue ound the sphee we need the mgnitude o the velocity φ on the suce ( = ): ( 4 φ ) = U cos θ + 1 U sin θ; n = ê ; n = cosθ (5.

7 .016 Hydodynmics Reding #5 1 whee m = ρ s is the dded mss in the system. Unstedy Moving Fluid Sttiony Body: Foce on sphee (dius ) in n unbounded unstedy moving luid. U = U (t) is the unstedy luid velocity. U (t) z θ The coesponding potentil unction is simply: φ (, θ, t ) = U (t ) + cos θ (5.4) The velocity, t the body on, cn only be tngentil to the body due to the kinemtic boundy condition ( KBC V = 0 ) : The time deivtive o the potentil is K V = φ = (0, U sin θ, 0 ) (5.5) = U cos θ (5.6) t The mgnitude o the velocity is simply: 1 φ = 9 U sin θ (5.7) 8 Agin, using (5.6), (5.7), (5.1) in the omultion o F, om eqution (5.16), we get vesion.0 updted 8/0/ A. Techet

4) The velocity, t the body on, cn only be tngentil to the body due to the kinemtic boundy condition ( KBC V = 0 ) : The time deivtive o the potentil is K V = φ = (0, U

8 .016 Hydodynmics Reding #5 the hoizontl oce on the body: F = ρ )( ) 0 cos θ sinθ U cos θ + 8 U sin θ d θ 9 ( (5.8) 9 F = ρ U cos θ sin θ d θ + 4 ρ U sin θ cos θ dθ 0 0 (5.9) By pts the ight most tem in the integl in (5.9) educes to zeo, nd the integl om let tem is simply: cos θ sin θ dθ = (5.0) 0 so we e let with oce: F = ρ U = ρ U (5.1) 4 This cn be ewitten in tems o the sphee volume, s =, nd dded mss, m = 1 ρ s, om (5.) s ollows F = U (ρ + m ) (5.) The non-dded mss tem o eqution (5.) is due to the pessue gdient necessy to ccelete the luid ound the sphee. This is like buoyncy eect. Unstedy Moving Fluid; Unstedy Moving Body: Foce on moving sphee (dius ) vesion.0 updted 8/0/ A. Techet

1) 4 This cn be ewitten in tems o the sphee volume, s =, nd dded mss, m = 1 ρ s, om (5.) s ollows F = U (ρ + m ) (5.) The non-dded mss tem o eqution (5.

9 .016 Hydodynmics Reding #5 in n unbounded moving luid. U = U (t) is the unstedy luid velocity nd U b = U b (t) is the body velocity. U (t) z U b (t) θ The cse o the unstedy moving body nd luid cn be detemined by combining the esults om the pevious two cses. 1 F = U b ( ρ s ) F = U (ρ + Moving Body Moving Fluid Still Fluid Still Body Moving Body, Moving Fluid: m ) F = U b (m ) +U (ρ + m ) = U ρ + m (U U b ) (5.) So now, ll we hve to do is ind the dded mss! vesion.0 updted 8/0/ A. Techet

U (t) z U b (t) θ The cse o the unstedy moving body nd luid cn be detemined by combining the esults om the pevious two cses.

(Ch. 22.5) 2. What is the magnitude (in pc) of a point charge whose electric field 50 cm away has a magnitude of 2V/m?

(Ch. 22.5) 2. What is the magnitude (in pc) of a point charge whose electric field 50 cm away has a magnitude of 2V/m? Em I Solutions PHY049 Summe 0 (Ch..5). Two smll, positively chged sphees hve combined chge of 50 μc. If ech sphee is epelled fom the othe by n electosttic foce of N when the sphees e.0 m pt, wht is the