EXAMPLE: Water Flow in a Pipe

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1 EXAMPLE: Water Flow in a Pipe P 1 > P 2 Velocity profile is parabolic (we will learn why it is parabolic later, but since friction comes from walls the shape is intuitive) The pressure drops linearly along the pipe. Does the water slow down as it flows from one end to the other? Only component of velocity is in the x-direction. v = v x i Incompressible Continuity: v y = v z = 0 v x x v x x + v y y + v z z = 0 = 0 and the water does not slow down.

2 EXAMPLE: Flow Through a Tank V = constant (always full) Integral Mass Balance: ( v n)da = 0 S v 1 A 1 = v 2 A 2 Q Constant volumetric flow rate Q. EXAMPLE: Simple Shear Flow v y = v z = 0 v x = v x (y) satisfied identically v v x x + v y y + v z z = 0

3 NAVIER-STOKES EQUATIONS (p. 1) (in the limit of slow flows with high viscosity) ρ = density η = viscosity v = typical velocity scale D = typical length scale Reynolds Number: R e ρvd η (1-62) For R e 1 have laminar flow (no turbulence) ρ v t = P + ρ g + η 2 v Vector equation (thus really three equations) The full Navier-Stokes equations have other nasty inertial terms that are important for low viscosity, high speed flows that have turbulence (airplane wing).

4 NAVIER-STOKES EQUATIONS (p. 2) ρ v t = P + ρ g + η 2 v ρ v t = v t = acceleration ρ = force unit volume mass unit volume ( F = m a) Newton s 2 nd Law Navier-Stokes equations are a force balance per unit volume What accelerates the fluid? P = Pressure Gradient ρ g = Gravity η 2 v = Flow (fluid accelerates in direction of increasing velocity gradient. Increasing v 2 v > 0

5 GENERAL FLUID MECHANICS SOLUTIONS v r = constant r Already know the way velocity varies with position, and have not used the Navier-Stokes equations! Navier-Stokes equations + Continuity + Boundary Conditions Four coupled differential equations! Always look for ways to simplify the problem! EXAMPLE: 2D Source Flow Injection Molding a Plate Continuity equation v = 1 r 1. Independent of time 2. 2-D v z = 0 3. Symmetry Polar Coordinates 4. Symmetry v θ = 0 d (rv dr r) = 0 rv r = constant

6 EXAMPLE: Poiseuille Flow between Parallel Plates (important for injection molding) (P. 1) Independent of time v y = v z = 0 Cartesian coordinates Continuity: Navier-Stokes equation: v x x = 0 v x = v x (y) P = P (x) P x + µ 2 v x y 2 = 0 P y = P z = 0 v x = v x (y) P x = v x µ 2 y 2 How can f(x) = h(y)? Each must be constant! P x = C 1 P = C 1 x + C 2 B.C. x = 0 P = P 1 C 2 = P 1 x = L P = P 2 C 1 = P/L where : P P 1 P 2 P = P 1 P x L

7 EXAMPLE: Poiseuille Flow between Parallel Plates (important for injection molding) (P. 2) µ 2 v x y 2 = C 1 = P/L 2 v x y 2 = P µl v x y = P µl y + C 3 v x = P 2µL y2 + C 3 y + C 4.C. NO SLIP top plate y = d/2 v x = 0 bottom plate y = d/2 v x = 0 0 = P 8µ L d2 + C 3 d 2 + C 4 0 = P 8µ L d2 C 3 d 2 + C 4 v x = P 2µl C 3 = 0 [ d 2 4 y2 ] C 4 = P d2 8µL Parabolic velocity profile

8 EXAMPLE: Poiseuille Flow between Parallel Plates (important for injection molding) (P. 3) Where is the velocity largest? Maximum at vx = 0 = P y y µl maximum at y = 0 centerline What is the average velocity? A v ave = v xda da = 1 v x da A A A v ave = 1 z d/2 v x dydz = 1 d/2 zd 0 d/2 d d/2 v ave = P [ ] d 2 d/2 2µLd 4 y y3 3 d/2 For constant P, µ, L: double d quadruple v A = zd [ ] P d 2 2µL 4 y2 dy = P d2 12µL

9 EXAMPLE: Poiseuille Flow in an Annular Die (important for blow molding) (P. 1) P 1 > P 2 Independent of Time Cylindrical Coordinates v r = v θ = 0 v z = v z (r) Continuity: vz z = 0 Navier-Stokes equation: P z = µ [ 1 r f(z) = g(r) = a constant Split into two parts - Pressure Part: P z = C 1 P = C 1 z + C 2 r ( r v )] z r B.C. z = 0 P = P 2 C 2 = P 2 z = L P = P 1 C 1 = P/L where : P P 1 P 2 P = P 2 + P L z P = P 2 + P z analogous to Poiseuille flow between parallel plates. L

10 EXAMPLE: Poiseuille Flow in an Annular Die (important for blow molding) (P. 2) [ 1 µ r r ( r v )] z = P r L r v z r = P 2µL r2 + C 3 v z r = P 2µL r + C 3 r v z = P 4µL r2 + C 3 ln r + C 4 B.C. NO SLIP at r = R i, v z = 0 at r = R 0, v z = 0 0 = P 4µL R2 i + C 3 ln R i + C 4 subtract 0 = P 4µL R2 0 + C 3 ln R 0 + C 4 ( ) 0 = P 4µL (R2 0 Ri 2 R ) + C 3 ln 0 R i C 3 = P (R2 0 Ri 2 ) 4µL ln(r 0 /R i ) C 4 = P [ ] R0 2 (R2 0 Ri 2 ) ln R 0 4µL ln(r 0 /R i )

11 EXAMPLE: Poiseuille Flow in an Annular Die (important for blow molding) (P. 3) v z = P [ r 2 (R2 0 Ri 2 ) 4µL ln(r 0 /R i ) ln r R2 0 + (R2 0 Ri 2 ) ln(r 0 /R i ) ] [ 1 + r2 v z = P R2 0 4µL r < R 0 always, so v z < 0 R 2 0 (R2 0 R 2 i ) ln(r 0 /R i ) ln(r/r 0) Leading term is parabolic in r (like the flow between plates) but this one has a logarithmic correction. What is the volumetric flow rate? Q = π P R4 0 8µL Q = [ A 1 + v z da = ( Ri R 0 R0 R i v z 2πrdr ) ] 4 + (1 (R i/r 0 ) 2 ) 2 ln(r 0 /R i ) ]

12 GENERAL FEATURES OF NEWTONIAN POISEUILLE FLOW Parallel Plates: Q = P d3 W 12µL Circular Tube: Q = π P R4 8µL Annular Tube: Q = π P R4 0 8µL f(r i/r 0 ) Rectangular Tube: All have the same general form: Q = P d3 w 12µL Q P Q 1/µ Weak effects of pressure, viscosity and flow length Q 1/L Q R 4 or d 3 w Strong effect of size. In designing and injection mold, we can change the runner sizes.

13 NON-NEWTONIAN EFFECTS EXAMPLE: Poiseuille Flow in a Circular Pipe Newtonian Velocity Profile: v z = P R2 4µL [ 1 (r/r) 2 ] Shear Rate: γ = v z r = P r 2µL Apparent Viscosity: where γ is higher Viscosity is lower Real Velocity Profile: Lower η a increases v z non-parabolic

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