04_FlowInVariedGeometries

7/28/2019 04_FlowInVariedGeometries

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EXAMPLE: Water Flow in a Pipe

P1 > P2 Velocity profile is parabolic (we will learn why it is paraboliclater, but since friction comes from walls the shape is intu-itive)

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 = vxi

vy = vz = 0

Incompressible Continuity:

vxx

+vyy

+vzz

= 0

vxx = 0 and the water does not slow down.


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EXAMPLE: Flow Through a Tank

V = constant (always full)

Integral Mass Balance: S(v n)dA = 0v1A1 = v2A2 Q

Constant volumetric flow rate Q.

EXAMPLE: Simple Shear Flow

vy = vz = 0 vx = vx(y)

v vxx

+vyy

+vzz

= 0

satisfied identically


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NAVIER-STOKES EQUATIONS (p. 1)(in the limit of slow flows with high viscosity)

Reynolds Number: Re vD

(1-62)

= density = viscosityv = typical velocity scaleD = typical length scale

For Re 1 have laminar flow (no turbulence)

vt

= P + g + 2v

Vector equation (thus really three equations)

The full Navier-Stokes equations have other nasty inertial terms that areimportant for low viscosity, high speed flows that have turbulence (airplanewing).


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NAVIER-STOKES EQUATIONS (p. 2)

v

t= P + g + 2v

v

t= acceleration

=mass

unit volume

v

t =

force

unit volume ( F = ma) Newtons 2nd

Law

Navier-Stokes equations are a force balance per unit volume

What accelerates the fluid?

P = Pressure Gradient

g = Gravity

2

v = Flow (fluid accelerates in direction of increasing velocity gradient.Increasing v 2v > 0


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GENERAL FLUID MECHANICSSOLUTIONS

Navier-Stokes equations + Continuity + Boundary Conditions

Four coupled differential equations!

Always look for ways to simplify the problem!

EXAMPLE: 2D Source FlowInjection Molding a Plate

1. Independent of time

2. 2-D vz = 0

3. Symmetry Polar Coordinates

4. Symmetry v = 0

Continuity equation v = 1rddr

(rvr) = 0

rvr = constant

vr =constant

r

Already know the way velocity varies with position, and have not usedthe Navier-Stokes equations!


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EXAMPLE: Poiseuille Flow between Parallel Plates(important for injection molding) (P. 1)

Independent of time

vy = vz = 0

Cartesian coordinates

Continuity:

vx

x = 0 vx = vx(y)Navier-Stokes equation:

P

x+

2vxy2

= 0P

y=

P

z= 0

P = P(x) vx = vx(y)

P

x=

2vxy2

How can f(x) = h(y)? Each must be constant!Px

= C1 P = C1x + C2

B.C. x = 0 P = P1 C2 = P1x = L P = P2 C1 = P/L where : P P1 P2P = P1

PxL


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2vxy2

= C1 = P/L

2vxy2

= P

L

vxy

= P

Ly + C3

vx = P2L

y2 + C3y + C4

.C. NO SLIP top plate y = d/2 vx = 0bottom plate y = d/2 vx = 0

0 =

P8L d2 + C3d2 + C

4

0 =P

8Ld2 C3

d

2+ C4

C3 = 0 C4 =P d2

8L

vx =P

2l

d2

4 y2

Parabolic velocity profile


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Where is the velocity largest?Maximum at vx

y= 0 = P

Ly

maximum at y = 0 centerline

What is the average velocity?

vave =

A

vxdA

A

dA=

1

A

A

vxdA A = zd

vave = 1zd

z0

d/2d/2

vxdydz = 1d

d/2d/2

P2L

d2

4 y2

dy

vave =P

2Ld

d2

4y

y3

3

d/2d/2

=P d2

12L

For constant P, , L: double d quadruple v


9/13

EXAMPLE: Poiseuille Flow in an Annular Die(important for blow molding) (P. 1)

P1 > P2

Independent of Time

Cylindrical Coordinates

vr = v = 0

vz = vz(r)

Continuity: vzz

= 0

Navier-Stokes equation:

P

z=

1

r

r

r

vzr

f(z) = g(r) = a constant

Split into two parts - Pressure Part:Pz

= C1 P = C1z+ C2

B.C. z = 0 P = P2 C2 = P2z = L P = P1 C1 = P/L where : P P1 P2P = P2 +

PL

z

P = P2 +PL

z analogous to Poiseuille flow between parallel plates.


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1

r

rr

vz

r =

P

L

rvzr

=P

2Lr2 + C3

vzr

=P

2Lr +

C3r

vz =P4L

r2 + C3 ln r + C4

B.C. NO SLIP at r = Ri, vz = 0at r = R0, vz = 0

0 = P4L

R2i + C3 ln Ri + C4

0 =P

4LR20

+ C3 ln R0 + C4

subtract 0 = P4L

(R20 R2i ) + C3 lnR0Ri

C3 = P(R2

0 R2i )

4L ln(R0/Ri)

C4 = P

4L R20 (R2

0 R2i ) ln R0

ln(R0/Ri)


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vz =P

4L

r2

(R20

R2i )

ln(R0/Ri)ln r R2

0+

(R20

R2i )

ln(R0/Ri)

vz =P R2

0

4L

1 +

r2

R20

(R20

R2i )

ln(R0/Ri)ln(r/R0)

r < R0 always, so vz < 0

Leading term is parabolic in r (like the flow between plates) but this onehas a logarithmic correction.

What is the volumetric flow rate?

Q =

A

vzdA =

R0Ri

vz2rdr

Q = P R4

0

8L

1 +

RiR0

4+ (1 (Ri/R0)

2

)2

ln(R0/Ri)


12/13

GENERAL FEATURES OF NEWTONIANPOISEUILLE FLOW

Parallel Plates: Q =P d3W

12L

Circular Tube: Q =P R4

8L

Annular Tube: Q =P R40

8Lf(Ri/R0)

Rectangular Tube: Q = P d3

w12L

All have the same general form:

Q PQ 1/

Q 1/LWeak effects of pressure, viscosity and flow length

Q R4 or d3w Strong effect of size.

In designing and injection mold, we can change the runner sizes.


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04_FlowInVariedGeometries