Fluid Mechanics Exam Formula Sheet

p = ρRT ∂u x τ yx = µ D Dt ∫ ηρ dV = Vsystem d dt ∫ ηρ dV + ∫ η ( ρ u rel CV CS dN ⋅ dA ) or dt = system ∂ d ∫ ρdV + CS∫ (ρ urel ⋅d A) = 0 or ∂t ∫CV ρ dV + dt CV  ∫ e ρ dV + ∫ ( h + CS CV 1 2 g   ∂ ∫ ηρ dV + ∫ ηρV ⋅ d A ∂t CV CS  ∫ ρV ⋅ d A= 0 CS d ∂ ∫ u xyz ρdV + CS∫ u xyz ( ρu rel ⋅ d A ) = FS + FB − CV∫ a xyz / XYZ ρdV or ∂t dt CV d dt z dp = −ρg dz ∂y  ∫V CV       ρ dV + ∫ Vxyz ρVxyz ⋅ d A = FS + FB − ∫ arf ρ dV xyz CS CV  where e = u + 12 V2 + gz and h = u + p V 2 + gz ρu rel ⋅ dA = Qinto CV + W on CV ρ )( ) z g p V2 + + z = constant (for inviscid flow) ρ g 2g 1/ 2 2 ⎡⎛ x ∂ R ⎞ 2 ⎛ x ∂ R ⎞ 2 ⎛ x ∂R ⎞ ⎤ u R = ± ⎢⎜ 1 u1 ⎟ + ⎜ 2 u 2 ⎟ + ⋅⋅⋅ + ⎜ n un ⎟ ⎥ R ∂x 1 ⎠ ⎝ R ∂x 2 ⎠ ⎝ R ∂x n ⎠ ⎦⎥ ⎣⎢⎝ ⎛ p ⎞ ⎛ p ⎞ V2 V2 +α +z ⎟ =⎜ +α + z ⎟ − H L,1 →2 + H S,1 →2 ⎜ 2g 2g ⎝ρ g ⎠2 ⎝ ρ g ⎠1 z g ⎛p ⎞ ⎛p ⎞ V V + gz ⎟ = hL,1 →2 − hS,1 →2 ⎜ + α + gz ⎟ − ⎜ + α 2 2 ⎝ρ ⎠1 ⎝ ρ ⎠2 2 2 where WS ,1→2 Vi Δp ⎛L⎞ ⎛L⎞ ; K≡1 ; Kmajor = f ⎜ ⎟ ; K = f ⎜ e ⎟ ; H S ,1→2 = ; h = gH  mg ρ 2 2g ⎝D⎠ ⎝ D⎠ 2 V 2 H L ,1→ 2 = ∑ Ki i ∂V ⎤ ∂p ∂z ⎡ ∂V +V = − − ρg ∂ s ⎦⎥ ∂s ∂s ⎣ ∂t ρ⎢ n ∂p ∂z V2 ρ = − + ρg ∂n ∂n R Page 1 of 8 z g g = 9.81 m/s2 = 32.2 ft/s2 ⎫⎪ ⎬ Water µ = 1.00 × 10−3 kg/ (m ⋅ s) ⎭⎪ ρ = 1000 kg/m3 ⎪⎫ ⎬ Air (STP) µ = 1.79 × 10−5 kg/ (m ⋅ s) ⎭⎪ ρ = 1.23 kg/m3 Properties of air (treated as a perfect gas) N⋅ m ⋅m k = 1.4 R = 287 N c p = 1005 kg kg ⋅K ⋅K R = 53.3 ft ⋅lb f lb m⋅¡ R c p = 187 cv = 718 ft ⋅lb f lb m⋅¡ R c v = 133 Specific gravity, SG = ρ/ρH O (at 4 oC) 2 Kinematic viscosity, ν ≡ µ/ρ 1 Pa = N/m2; 1 atm = 101 kPa = 14.7 psia 1 ft = 0.305 m; g = 9.81 m/s2 = 32.2 ft/s2 1 lbm = 0.454 kg; 1 slug = 32.2 lbm 1 lbf = 32.2 lbm⋅ft/s2 = 1 slug⋅ft/s2 1 ft3 = 7.48 gal; 1 m3 = 103 L; 1000 L = 1 m3 1 hp = 550 ft⋅lbf/s = 746 W 1 Btu = 778 ft⋅lbf = 1.06 kJ 1 rpm = 0.1047 rad//s Page 2 of 8 N⋅ m kg ⋅K ft ⋅lb f lb m⋅¡ R !"#$%#&%$'()*&+$%"#( rectangular coordinates (x, y, z) ∂ρ ∂ ( ρu x ) ∂(ρ u y ) ∂ (ρu z ) + + + =0 ∂z ∂y ∂x ∂t cylindrical coordinates (r, θ, z) ∂ρ 1 ∂( ρru r ) 1 ∂( ρuθ ) ∂( ρu z ) + + + =0 ∂z ∂t r ∂r r ∂θ ,$-)..($)#."-(/"01"#)#$.(2"-(+(3)4$"#%+#(25&%6( rectangular coordinates (x, y, z) ⎡ ∂ux 2 ⎤ − (∇ ⋅u )⎥ ⎣ ∂x 3 ⎦ ⎡ ∂uy 2 ⎤ − ( ∇ ⋅u )⎥ σ yy = − p + µ ⎢ 2 ⎣ ∂y 3 ⎦ 2 ⎡ ∂u ⎤ σ zz = − p + µ ⎢2 z − (∇ ⋅ u )⎥ ⎣ ∂z 3 ⎦ σ xx = − p + µ ⎢2 ⎡ ∂ ux ∂u y ⎤ + ∂x ⎥⎦ ⎣ ∂y ⎡ ∂u ∂u ⎤ σ xz = σ zx = µ ⎢ x + z ⎥ ∂x ⎦ ⎣ ∂z ⎡ ∂u y ∂uz ⎤ + σ yz =σ zy = µ ⎢ ⎥ ∂y ⎦ ⎣ ∂z σ xy =σ yx = µ ⎢ ∇⋅u = ∂ u x ∂u y ∂u z + + ∂x ∂y ∂z cylindrical coordinates (r, θ, z) ⎡ ∂ur 2 ⎤ − ( ∇ ⋅ u) ⎥ ⎣ ∂r 3 ⎦ ⎡ ⎛ 1 ∂ uθ ur ⎞ 2 ⎤ = −p + µ ⎢ 2⎜ + ⎟ − (∇ ⋅ u )⎥ r ⎠ 3 ⎣ ⎝ r ∂θ ⎦ σ rr = − p + µ ⎢ 2 σ θθ ⎡ ∂u z 2 ⎤ − (∇ ⋅ u )⎥ ⎣ ∂z 3 ⎦ ⎡ ∂ ⎛ uθ ⎞ 1 ∂u r ⎤ = µ ⎢r ⎜ ⎟ + ⎥ ⎣ ∂r ⎝ r ⎠ r ∂θ ⎦ 1 ∂u z ⎤ ⎡ ∂u =µ ⎢ θ + ⎣ ∂z r ∂θ ⎥⎦ σ zz = − p + µ ⎢ 2 σ rθ = σ θ r σ θz = σ z θ ∂ uz ∂ ur ⎤ + ∂z ⎥⎦ ⎣ ∂r σ zr = σ rz = µ ⎡⎢ ∇⋅u= ∂u z 1 ∂(ru r ) 1 ∂uθ + + r ∂r r ∂θ ∂z 3+7%)-8,$"9).()*&+$%"#.(2"-(+(3)4$"#%+#(25&%6(4%$:(/"#.$+#$(6)#.%$'(;ρ <(+#6(6'#+0%/(7%./".%$'(;µ <( rectangular coordinates (x, y, z): ∂u x ∂u ∂u x ⎞ ⎡∂ 2 u x ∂ 2 u x ∂ 2 u x ⎤ ⎛ ∂u x ∂p +u x +u y x + u z ⎟⎟ = − ∂ x + µ ⎢ ∂x 2 + ∂y 2 + ∂ z 2 ⎥ + ρf x ∂ ∂ ∂ ∂ z y x t ⎝ ⎠ ⎣ ⎦ ρ ⎜⎜ ⎛ ∂u y ρ ⎜⎜ ⎝ ∂t + ux ∂u y ∂x +uy ∂uy ∂y + uz ⎡ ∂2 u y ∂ 2 u y ∂ 2 u y ⎤ ∂u y ⎞ ∂p ⎟⎟ = − + µ⎢ 2 + + + ρfy 2 2 ⎥ ∂z ⎠ ∂y ∂y ∂ z ⎦⎥ ⎣⎢ ∂x ∂u ∂u ⎛ ∂u ∂u ⎞ ∂p ∂ 2u ⎡∂ 2u ∂2 u ⎤ ρ ⎜⎜ z + u x z + u y z + u z z ⎟⎟ = − + µ ⎢ 2z + 2 z + 2z ⎥ + ρf z ∂x ∂y ∂z ⎠ ∂z ∂y ∂z ⎦ ⎝ ∂t ⎣ ∂x cylindrical coordinates (r, θ, z): ⎛ ∂ ur ⎡ ∂ ⎛1 ∂ ∂ u u ∂ u u2 ∂u ⎞ ∂p ∂2 ∂2 ∂ ⎤ + ur r + θ r − θ + u z r ⎟ = − + µ ⎢ ⎜ (ru r )⎞⎟ + 12 u2r + u2r − 22 uθ ⎥ + ρ f r ∂r ∂z ⎠ ∂r ∂z r ∂θ r r ∂θ ⎦ ⎠ r ∂θ ⎝ ∂t ⎣ ∂r ⎝ r ∂r ⎡ ∂ ⎛1 ∂ ∂u ∂ u u ∂u u u ∂u ∂2 ∂2 ∂ ⎤ 1 ∂p ρ ⎛⎜ θ + u r θ + θ θ + r θ + u z θ ⎞⎟ = − + µ⎢ ⎜ ( ru θ ) ⎞⎟ + 12 uθ2 + u2θ + 22 ur ⎥ + ρ fθ ∂r ∂z ⎠ ∂z r ∂θ r r ∂θ r ∂θ ⎦ ⎝ ∂t ⎠ r ∂θ ⎣ ∂r ⎝ r ∂r ρ⎜ ∂u u ∂uz ∂u ⎞ ∂p ⎡ 1 ∂ ⎛ ∂ u z ⎞ 1 ∂2 u z ∂2 u z ⎤ ⎛ ∂uz + ur z + θ + uz z ⎟ = − + µ ⎢ + 2 ⎥ + ρ fz ⎜r ⎟+ 2 2 ∂r ∂z ⎠ ∂z ∂z ⎦ r ∂θ ⎝ ∂t ⎣ r ∂r ⎝ ∂r ⎠ r ∂θ ρ⎜ Page 3 of 8 =+$:(>"-0&5+.(( d dx d dx d dx d dx d dx d tan x = sec2 x dx d csc x = − csc x cot x dx d 1 ln x = dx x ( cos x) = − sin x ( ( ) ) ( ) ∫ sin x dx = x − sin 2x ∫ cos x dx = x + sin 2x ∫ tan x dx = tan x − x ∫ sin x dx = − (2 + sin x ) cos x ∫ cos x dx = ( 2 + cos x) sin x ∫ tan x dx = tan x + ln cos x 2 (sin x ) = cos x 1 2 1 4 2 (sec x ) = sec x tan x 3 1 3 2 1 2 4 2 1 3 1 2 3 (cot x ) = − csc 1 2 3 2 2 x ⎣⎡exp ax ⎦⎤ = aexp ax ( ) ( ) ( ) cos ( α + β ) = cosα cos β − sinα sin β tan α + tanβ tan (α + β )= 1− tan α tan β sin α + β = sin αcos β + cos α sin β ⎛θ⎞ 1 sin2 ⎜ ⎟ = 1− cosθ ⎝ 2⎠ 2 ( ) ⎛θ ⎞ 1 cos2 ⎜ ⎟ = 1+ cosθ ⎝ 2⎠ 2 ( ) =%./)55+#)"&.(7)/$"-("1)-+$%"#.((;?#($:)($+@5)(@)5"4A((Ν (%.(+(./+5+-(+#6(#(%.(+(7)/$"-B<( rectangular coordinates (x, y, z) cylindrical coordinates (r, θ, z) ∇N = ∂N ∂N ∂N öe + eö + eö ∂x x ∂ y y ∂z z ∇⋅n = ∇N = ∂nx ∂ny ∂nz + + ∂x ∂ y ∂z ∇⋅n = ∂n 1∂ ( rn )+ 1 θ + ∂n∂zz r ∂r r r ∂θ ⎛ 1 ∂nz ∂nθ ⎞ − ∇ × n =⎜ eö + ⎝ r ∂θ ∂z ⎟⎠ r ⎛ ∂n ∂n ⎞ ∇ × n = ⎜ z − y ⎟ eö x + ⎝ ∂ y ∂z ⎠ ∇2 N = 1 ∂N ∂N ∂N öe öe + öe + ∂r r r ∂θ θ ∂z z ⎛ ∂nx ∂nz ⎞ − ⎜ ⎟ eö y + ⎝ ∂z ∂x ⎠ ⎛ ∂nr ∂nz ⎞ eö + − ⎝⎜ ∂z ∂r ⎠⎟ θ ⎛ ∂ny ∂nx ⎞ öe ⎜ ∂x − ∂ y ⎟⎠ z ⎝ ∂n ⎞ 1⎛ ∂ ( rnθ ) − ∂ θr ⎟⎠ öe z r ⎜⎝ ∂r ∂2 N ∂2 N ∂2 N + + ∂x 2 ∂ y 2 ∂z 2 ∇2 N = C+D-+#D%+#(;+9+(0+$)-%+5E(.&@.$+#$%+5<(F//)5)-+$%"#( rectangular coordinates (x, y, z) Dux ∂ux ∂u ∂u ∂u = + ux x + uy x + uz x Dt ∂t ∂x ∂y ∂z ∂u y ∂u y ∂u y Du y ∂u y = +ux +uy + uz ∂t ∂x ∂y ∂z Dt Duz ∂uz ∂u ∂u ∂u = +u x z + u y z +u z z Dt ∂t ∂x ∂y ∂z 2 2 1 ∂ ⎛ ∂N ⎞ 1 ∂ N ∂ N + 2 r 2 ⎟+ ∂z r ∂r ⎝⎜ ∂r ⎠ r 2 ∂θ cylindrical coordinates (r, θ, z) ∂ u u ∂ u u2 ∂u Dur ∂ ur = + ur r + θ r − θ + uz r ∂t ∂r ∂z Dt r ∂θ r Duθ ∂uθ ∂u u ∂ u u u ∂u = + ur θ + θ θ + r θ + u z θ ∂t ∂r ∂z Dt r ∂θ r ∂u u ∂u ∂u Duz ∂ uz = +ur z + θ z +uz z ∂t ∂r ∂z Dt r ∂θ Page 4 of 8 Page 5 of 8 F7)-+D)(G"&D:#)..("2(!"00)-/%+5(H%1).( Material (new) Riveted steel Concrete Wood stave Cast iron Galvanized iron Asphalted cast iron Commercial steel or wrought iron Drawn tubing Plastic, glass ( ft 0.003-0.03 0.001-0.01 0.0006-0.003 0.00085 0.0005 0.0004 0.00015 0.000005 0.0 (smooth) mm 0.9-9.0 0.3-3.0 0.18-0.9 0.26 0.15 0.12 0.045 0.0015 0.0 (smooth) I+@5)("2(=%#"-(C"..(!")22%/%)#$.( Component a. Elbows Regular 90o, flanged Regular 90o, threaded Long radius 90o, flanged Long radius 90o, threaded Long radius 45o, flanged Regular 45o, threaded b. c. d. K Component 0.3 1.5 0.2 0.7 0.2 0.4 180o return bends 180o return bends, flanged 180o return bends, threaded 0.2 1.5 Tees Line flow, flanged Line flow, threaded Branch flow, flanged Branch flow, threaded 0.2 0.9 1.0 2.0 Union, threaded 0.06 e. f. g. h. Sudden Contraction/Expansion: Page 6 of 8 K Valves Globe, fully open Angle, fully open Gate, fully open Gate, ¼ closed Gate, ½ closed Gate, ¾ closed Swing check, forward flow Swing check, backward flow Ball valve, fully open Ball valve, 1/3 closed Ball valve, 2/3 closed 10 2 0.15 0.26 2.1 17 2 ∞ 0.05 5.5 210 Entrances Re-entrant Sharp-edged Slightly rounded Well rounded 0.8 0.5 0.2 0.04 Exits Re-entrant, sharp-edged, slightly rounded, well-rounded 1 H&01.( Ψ= gH Π= ω 2 D2 NS = W ρω 3 D5 Φ= Q ω D3 η= ρ QgH W 1 1 ω ( rpm) Q ( gpm) ωQ 2 Φ2 or Nsd ≡ = 3 3 3 4 Ψ 4 ( gH ) 4 ⎣⎡ H ( ft ) ⎦⎤ Nsd = 2733 [rpm⋅(gpm)1/2/(ft)3/4] Ns ⎛ p V2 ⎞ pv + NPSH = ⎜ ⎟ − ⎝ ρg 2g ⎠ S ρg δ δ D = δ = ∫ (1− * 0 CD = 1 2 δ u )dy U FD ρV 2 A CL = δM = θ = ∫ 0 1 2 FL ρV 2A Parameter 99% thickness, δ u u (1− )dy U U τw d dU (U2θ ) + δ * U = ρ dx dx Laminar (Re < 500,000) δ x displacement thickness, δD or δ* = δ 0.382 = x Re x 5.0 1 Re x2 1 5 δ D 1.72 = x Re x δ M 0.664 = x Re x 0.664 τw = Cf = 1 ρU 2 Rex 2 1.328 D = = CD 1 ρU 2 LW Re L 2 1 2 momentum thickness, δM or Θ 1 2 friction coefficient, Cf 1 2 drag coefficient, CD 1 2 Page 7 of 8 Turbulent (Re > 500,000) δD 0.0478 = 1 x Re x 5 δ M 0.0371 = x Re x 0.0594 τw = Cf = 1 ρU 2 Re x 2 0.0742 D = = CD 1 ρU 2 LW Re L 2 1 5 1 5 1 5 ?6)+5(D+.(-)5+$%"#.( p = ρ RT k = du = cv dT cp dh = c p dT k 1 R cv = R k −1 k −1 ⎛ T2 ⎞ ⎛ν 2 ⎞ ⎛ T2 ⎞ ⎛ P2 ⎞ s 2 − s 1 = c v ln ⎜ ⎟ + R ln ⎜ ⎟ = c p ln ⎜ ⎟ − R ln⎜ ⎟ ⎝ T1 ⎠ ⎝ ν1 ⎠ ⎝ T1 ⎠ ⎝ P1 ⎠ c p = cv + R cv c = kRT cp = sinα=1/Ma F6%+@+$%/(-)5+$%"#.(2"-(+(1)-2)/$(D+.( T0 = T + 2 T ⎛ c ⎞ ⎛ k −1 2⎞ = ⎜ ⎟ = ⎜1 + Ma ⎟ T0 ⎝ c0 ⎠ ⎝ 2 ⎠ V2 2c p −1 ?.)#$-"1%/(-)5+$%"#.(2"-(+(1)-2)/$(D+.( p2 ⎛ T2 ⎞ =⎜ ⎟ p1 ⎝ T1 ⎠ k k ρ 2 ⎛ T2 ⎞ =⎜ ⎟ ρ1 ⎝ T1 ⎠ k −1 1 p2 ⎛ ρ 2 ⎞ =⎜ ⎟ p1 ⎝ ρ1 ⎠ k −1 k Po ⎛ k − 1 ⎞k −1 Ma 2 ⎟ = ⎜1+ P ⎝ 2 ⎠ k+1 1 ρo ⎛ k −1 ⎞ k −1 Ma 2 ⎟ = ⎜1 + ρ ⎝ 2 ⎠ 2 (k−1 ) k −1 ⎛ 2⎞ 1 ⎜ 1+ 2 Ma ⎟ A = ⎟ ⎜ A* Ma ⎜ 1+ k − 1 ⎟ ⎠ ⎝ 2 k+1 ⎛ k − 1 ⎞ 2 (1−k ) p0 mchoked = ⎜ 1+ 2 ⎟⎠ ⎝ M 0.00 0.20 0.40 0.60 0.80 1.00 1.50 2.00 2.50 3.00 T/To 1.0000 0.9921 0.9690 0.9328 0.8865 0.8333 0.6897 0.5556 0.4444 0.3571 k * A RT0 P/Po 1.0000 0.9725 0.8956 0.7840 0.6560 0.5283 0.2724 0.1278 0.05853 0.02722 A/A* ρ/ρo 1.0000 0.9803 0.9243 0.8405 0.7400 0.6339 0.3950 0.2301 0.1317 0.07623 ∞ 2.964 1.590 1.188 1.038 1.000 1.160 1.688 2.637 4.235 !"#6%$%"#.(+/-"..(+(#"-0+5(.:"/9(4+7)( Ma 22 = ( k −1) Ma21 + 2 2 kMa 12 − ( k −1 ) 2 ( k +1) Ma1 ρ2 V1 = = ρ1 V2 2 + (k −1 )Ma 12 p2 k −1 2k = Ma21 − p1 k +1 k +1 2kMa 21 −(k −1) T2 ⎡ = 2 + (k − 1) Ma12 ⎦⎤ 2 T1 ⎣ ⎣⎡( k + 1) Ma1 ⎦⎤ 2 p02 ρ02 A1* ⎡ ( k + 1) Ma1 ⎤ = = =⎢ ⎥ p01 ρ01 A2* ⎢ 2 + (k −1) Ma12 ⎥ ⎣ ⎦ k (k −1 ) ⎡ ⎤ k +1 ⎢ ⎥ 2 ⎢⎣ 2kMa1 − ( k −1) ⎥⎦ Page 8 of 8 1 (k −1 ) T02 =1 T01