Krantz W.B. Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation

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EXAMPLE PROBLEMS 73

U = U

−b

Figure 3.E.2-1 Laminar ﬂow of an incompressible Newtonian liquid that has constant

physical properties between two parallel circular plates that are approaching each other

slowly.

=0,u

=−

U(t)=−

−βt

at z =H(t) (3.E.2-5)

=0,u

U(t) =

−βt

at z =−H(t) (3.E.2-6)



−H

·2πr dz =−

(πr

H) =−2πr

= πr

−βt

= f

(z, t)

at r = R (3.E.2-7)

= 0,

∂u

∂r

= 0atr = 0 (3.E.2-8)

P = P

atm

at r = R (3.E.2-9)

The boundary condition given by equation (3.E.2-7) is a statement that the mass

squeezed out by the plates moving together must ﬂow out the periphery of the

circular plates. The function f

merely denotes that to solve these differential

equations, one would need to specify a boundary condition for u

at the periphery

of the circular plates. The simpliﬁcations justiﬁed by scaling analysis will obviate

the need to know f

Introduce the following scale factors, reference factor, and dimensionless vari-

ables (steps 2, 3, and 4):

∗

≡

; u

∗

≡

; P

∗

≡

P − P

; z

∗

≡

; r

∗

≡

;

∗

≡

(3.E.2-10)

Substitute these dimensionless variables into the describing equations and divide

through by the dimensional coefﬁcient of a term that must be retained in each

equation to obtain (steps 5 and 6)

∗

∂

∂r

∗

) +

∂u

∗

∂z

∗

= 0 (3.E.2-11)

74 APPLICATIONS IN FLUID DYNAMICS

ρz

μt

∂u

∗

∂t

∗

ρu

μr

∗

∂u

∗

∂r

∗

ρu

∗

∂u

∗

∂z

∗

=−

μu

∂P

∗

∂r

∗

∂

∂r

∗



∗

∂

∂r

∗

)



∂

∗

∂z

∗2

(3.E.2-12)

ρz

μt

∂u

∗

∂t

∗

ρu

μr

∗

∂u

∗

∂r

∗

ρu

∗

∂u

∗

∂z

∗

=−

μu

∂P

∗

∂z

∗

∂

∂r



∗

∂

∂r

∗



∗

∂u

∗

∂r

∗



∂

∗

∂z

∗2

(3.E.2-13)

∗

= 0,u

∗

=−

−βt

∗

at z

∗

H(t)

(3.E.2-14)

∗

= 0,u

∗

−βt

∗

at z

∗

=−

H(t)

(3.E.2-15)



H/z

−H/z

∗

−βt

∗

= f

∗

) at r

∗

(3.E.2-16)

∗

= 0,

∂u

∗

∂r

∗

= 0atr

∗

= 0 (3.E.2-17)

∗

atm

− P

at r

∗

(3.E.2-18)

The dimensionless group in equation (3.E.2-14) is set equal to 1 to obtain the

axial velocity scale u

. The geometric ratio groups in equations (3.E.2-14) and

(3.E.2-16), when set equal to 1, provide the axial and radial length scales z

and

. Equation (3.E.2-11) then provides the radial velocity scale u

. Since pressure

causes the radial ﬂow, P

is obtained by setting the dimensionless group multiply-

ing the pressure term in equation (3.E.2-12) equal to 1. The reference pressure P

is obtained from equation (3.E.2-18). Finally, the time scale t

is dictated by the

approach velocity of the two parallel plates and is obtained by setting the dimen-

sionless group in the argument of the exponential in equation (3.E.2-14) equal to 1.

This results in the following scale and reference factors (step 7):

; u

; P

μU

; P

= P

atm

; z

= H ;

= R; t

∗

(3.E.2-19)

Note in this case that several of our scales are time-dependent, owing to the presence

of H in their deﬁnition.

The resulting scaled dimensionless describing equations then are given by

∗

∂

∂r

∗

) +

∂u

∗

∂z

∗

= 0 (3.E.2-20)

EXAMPLE PROBLEMS 75

ρβH

∂u

∗

∂t

∗

+ Reu

∗

∂u

∗

∂r

∗

+ Reu

∗

∂u

∗

∂z

∗

=−

∂P

∗

∂r

∗

∂

∂r

∗



∗

∂

∂r

∗

)



∂

∗

∂z

∗2

(3.E.2-21)

ρβH

∂u

∗

∂t

∗

+ Reu

∗

∂u

∗

∂r

∗

+ Reu

∗

∂u

∗

∂z

∗

=−

∂P

∗

∂z

∗

∂

∂r



∗

∂

∂r

∗



∗

∂u

∗

∂r

∗



∂

∗

∂z

∗2

(3.E.2-22)

∗

= 0,u

∗

=−e

−t

∗

at z

∗

= 1 (3.E.2-23)

∗

= 0,u

∗

−t

∗

at z

∗

=−1 (3.E.2-24)



−1

∗

= r

∗

−t

∗

= f

∗

) at r

∗

= 1 (3.E.2-25)

∗

= 0,

∂u

∗

∂r

∗

= 0atr

∗

= 0 (3.E.2-26)

∗

= 0atr

∗

= 1 (3.E.2-27)

where Re≡ρU

H/2μ is the Reynolds number. We see that equations (3.E.2-20)

through (3.E.2-27) can be greatly simpliﬁed if the following conditions hold (step 8):

Re =

ρU

2μ

 1 ⇒ creeping ﬂow (3.E.2-28)

 1 ⇒ lubrication ﬂow if Re  1 as well (3.E.2-29)

ρβH

 1 ⇒ quasi-steady-state ﬂow (3.E.2-30)

All of these conditions are satisﬁed for sufﬁciently close approach distances H

between the two plates. Note that our scaling ensured that the dimensionless radial

pressure gradient was

◦

(1); this scaling does not necessarily ensure that the axial

pressure derivative is also

◦

(1). In fact, the condition given by equation (3.E.2-29)

implies that the dimensionless axial pressure gradient in equation (3.E.2-22) is of

the following order:

∂P

∗

∂z

∗

∼

∂

∗

∂z

∗2

◦





 1 (3.E.2-31)

This implies that the axial pressure gradient is essentially zero for the case of

lubrication ﬂow; this in turn implies that the radial pressure gradient is not a

function of z.

76 APPLICATIONS IN FLUID DYNAMICS

If the conditions given by equations (3.E.2-28) through (3.E.2-30) are satisﬁed,

equations (3.E.2-20) through (3.E.2-27) simplify to

0 =−

∂P

∗

∂r

∗

∂

∗

∂z

∗2

(3.E.2-32)

∂P

∗

∂z

∗

∼

0 (3.E.2-33)

∗

= 0,u

∗

=−e

−t

∗

at z

∗

= 1 (3.E.2-34)

∗

= 0,u

∗

= e

−t

∗

at z

∗

=−1 (3.E.2-35)



−1

∗

= r

∗

−t

∗

= f

∗

) at r

∗

= 1 (3.E.2-36)

∗

= 0atr

∗

= 1 (3.E.2-37)

Equations (3.E.2-36) and (3.E.2-37) are required to obtain the radial pressure dis-

tribution. This simpliﬁed set of describing equations can be solved analytically in

closed form.

3.E.3 Design of a Hydraulic Ram

Consider the operation of a hydraulic ram shown in Figure 3.E.3-1. This device

consists of a piston of radius R

that is free to slide within a cylinder of inner

radius R

containing a viscous oil that can be assumed to be an incompressible

Newtonian liquid with constant physical properties. An applied force causes the

piston to be pushed into the cylinder, which in turn causes a high pressure P

to be

generated within the oil conﬁned between the closed end of the cylinder and the

piston head. The difference between this high pressure and the ambient pressure

atm

then causes oil to ﬂow in the gap between the piston and the cylinder. The

force that must be applied to push the piston into the cylinder is equal to the sum

of the force exerted by P

on the piston head and the drag force caused by the

atm

Figure 3.E.3-1 Hydraulic ram consisting of a cylindrical piston of radius R

that slides

within a cylindrical housing of radius R

; the cylindrical housing and piston assembly

contains a viscous oil that can be considered to be an incompressible Newtonian liquid with

constant physical properties; a time-dependent force is applied to the piston, causing it to

move into the cylindrical housing at a constant velocity U

; the instantaneous axial velocity

proﬁle in the annular gap is sketched in the ﬁgure.

EXAMPLE PROBLEMS 77

oil being pushed through the annular gap. Let us assume that the piston is being

pushed in at a constant velocity U

. We ignore any end effects and assume that the

lubrication-ﬂow approximation can be made for the ﬂow in the thin annular gap.

We employ scaling analysis to determine when curvature effects in the describing

equations for the ﬂow in the annular gap can be neglected; that is, the conditions

for which the describing equations in cylindrical coordinates reduce to those in

rectangular coordinates.

The describing equations are obtained by simplifying appropriately the equations

of motion in cylindrical coordinates given by equations (C.2-1), (D.2-10), and

(D.2-12) in the Appendices (step 1). In writing these describing equations we

place our coordinate system on the moving piston. In doing so, it appears that the

cylinder is moving in the direction opposite to that of the piston.

0 =−

∂P

∂z

+ μ





(3.E.3-1)

= 0atr = R

(3.E.3-2)

= U

at r = R

(3.E.3-3)

P = P

atm

at z = L(t) (3.E.3-4)

where P

atm

is the prevailing atmospheric pressure and L(t) is the wetted length of

the piston; that is, the instantaneous length of the piston that is in contact with the

ﬂowing oil. Note that we retain the partial derivative ∂P/∂z in equation (3.E.3-1)

because the pressure gradient is also a function of time, since the wetted length

increases as the piston is pushed into the cylinder.

Introduce the following scale factors and dimensionless variables (steps 2, 3,

and 4):

∗

≡

; P

∗

≡

P − P

; r

∗

≡

r − r

; z

∗

≡

(3.E.3-5)

Note that we have introduced a reference factor for the radial coordinate since it

is not naturally referenced to zero within the region wherein the ﬂow is occurring;

that is, within the annular gap. We have also introduced a reference pressure since

the pressure is also not naturally referenced to zero at either end of the annular gap.

Substitute these dimensionless variables into the describing equations and divide

through by the dimensional coefﬁcient of a term that must be retained in each

equation to obtain (steps 5 and 6)

0 =−

μu

∂P

∗

∂z

∗



∗

+ 1



∗



∗

+ 1



∗



(3.E.3-6)

The lubrication-ﬂow approximation for this ﬂow can also be justiﬁed using scaling analysis; this is

considered in practice Problem 3.P.11.

78 APPLICATIONS IN FLUID DYNAMICS

∗

= 0atr

∗

− r

(3.E.3-7)

∗

at r

∗

− r

(3.E.3-8)

∗

atm

− P

at z

∗

(3.E.3-9)

The dimensionless radial coordinate can be bounded between zero and 1 if

we set the dimensionless group in equation (3.E.3-7) equal to zero and that in

equation (3.E.3-8) equal to 1, thereby obtaining r

= R

and r

= R

− R

.The

velocity scale is obtained by setting the dimensionless group in equation (3.E.3-8)

equal to 1 to obtain u

= U

. The axial length scale can be obtained by setting

the dimensionless group in equation (3.E.3-9) equal to 1 to obtain z

= L(t).The

dimensionless pressure can be referenced to zero by setting the dimensionless group

in equation (3.E.3-9) equal to zero to obtain P

= P

atm

. Finally, since pressure

causes the ﬂow in the axial direction, P

can be obtained by setting the dimen-

sionless group in equation (3.E.3-6) equal to 1 to obtain P

= μU

L/(R

− R

)

(step 7).

The resulting scaled dimensionless describing equations then are given by

0 =−

∂P

∗

∂z

∗

[(R

− R

)/R

∗

+ 1

∗



− R

∗

+ 1



∗



(3.E.3-10)

∗

= 0atr

∗

= 0 (3.E.3-11)

∗

= 1atr

∗

= 1 (3.E.3-12)

∗

= 0atz

∗

= 1 (3.E.3-13)

We see that if the dimensionless group (R

− R

)/R

◦

(0.01), equation

(3.E.3-10) reduces to (step 8)

0 =−

∂P

∗

∂z

∗

∗2

(3.E.3-14)

This is the same equation that would apply to steady-state pressure-driven lubri-

cation ﬂow between two parallel ﬂat plates. Hence, we conclude that curvature

effects can be ignored in problems such as that considered here if the following

condition is satisﬁed:

− R

 1 ⇒ curvature effects can be neglected (3.E.3-15)

A closed-form analytical solution for the axial velocity proﬁle can easily be

obtained for equation (3.E.3-14) subject to the boundary conditions given by

EXAMPLE PROBLEMS 79

equations (3.E.3-11) through (3.E.3-13). The unknown axial pressure gradient in

the solution for the velocity proﬁle can be obtained from the known volumetric

ﬂow rate corresponding to the volume swept out by the piston moving in at a

constant velocity U

. The force required to push the piston in at a constant velocity

can be obtained by a force balance on the piston that includes the pressure force

at each end of the piston and the drag force exerted by the ﬂowing oil.

3.E.4 Rotating Disk Flow

Consider a disk rotating at a constant angular frequency ω (radians per second) in

an inﬁnite ﬂuid having constant physical properties, as shown in Figure 3.E.4-1.

The rotation of the disk causes both an angular ﬂuid velocity and ﬂow in the radial

direction. This in turn causes ﬂow toward the disk in the axial direction. Inﬁnitely

far into the ﬂuid the velocity must be purely axial toward the disk. This ﬂow

geometry is of considerable practical value since the rotating disk is referred to

as a uniformly accessible surface. By this we mean that the heat- or mass-transfer

ﬂux to the surface of a rotating disk is invariant with position along its surface. For

example, this makes the rotating disk an ideal geometry for determining reaction

kinetics in electrochemical systems for which the rotating disk can serve as one

of the electrodes.

The rotating disk was ﬁrst analyzed by von K

arm

;amore

−U

∞

Figure 3.E.4-1 Flow created by a ﬂat disk rotating at an angular velocity ω (rad/s) in an

unbounded ﬂuid; the rotational motion of the disk draws ﬂuid toward the disk; the axial

velocity inﬁnitely far removed from the disk is U

∞

The rotating disk electrode apparatus was ﬁrst exhibited at the Brussels World’s Fair in 1958. The

application of scaling analysis for using the rotating disk to study mass transfer is considered in

Chapter 5.

T. von K

arm

an, Z. Angew. Math. Mech., 1, 244–247 (1921).

80 APPLICATIONS IN FLUID DYNAMICS

complete analysis was done by Cochran

; an overview of the analysis and use

of this instrument is given by Levich.

This problem can be solved analytically

for laminar ﬂow conditions in the absence of free convection effects. However,

under proper operating conditions, the hydrodynamics take on a boundary-layer

character such that the change in velocity components occurs within a thin region

of inﬂuence near the rotating disk. Establishing a thin boundary layer is important

for heat- or mass-transfer characterization using this apparatus since it ensures that

container boundary effects are minimized. We use scaling analysis to ascertain the

conditions for which the boundary layer will be thin.

A surprising aspect of laminar rotating disk ﬂow is that there is no radial pressure

gradient and the radial and angular velocities are directly proportional to the radial

position, whereas the axial velocity depends only on the axial position. Classical

treatments of this ﬂow begin by recognizing these considerations intuitively and

then proceeding to develop the rigorous analytical solution for this ﬂow. However,

these considerations can be ascertained via simple integral mass and momentum

balances. Consider a mass balance on a control volume of arbitrary radius r extend-

ing from the surface of the disk far into the ﬂuid, where there is only an axial

velocity component given by u

=−U

∞

. Note that in practice U

∞

is unknown;

however, it can be determined from the solution for the hydrodynamics and a

speciﬁed disk rotation rate. A mass balance on this control volume then yields

∞

πr

= 2πr



∞

dz ⇒ U

∞



∞

dz ⇒ u

= rf

(z) (3.E.4-1)

If the general form of the radial velocity proﬁle given by equation (3.E.4-1) is sub-

stituted into the continuity equation given by equation (C.2-1), we can conclude

that the axial velocity is independent of r:

∂

∂r

(ru

∂u

∂z

=0 ⇒ 2f

(z)+

∂u

∂z

=0 ⇒ u

=−2



(˜z) d ˜z =f

(z)

(3.E.4-2)

where ˜z denotes a dummy integration variable. Hence, we conclude that u

is a

function only of z. We can now prove that there is no radial pressure gradient

by considering an integral z-momentum balance on a control volume of arbitrary

radius r extending from an arbitrary height z into the ﬂuid far from the rotating

disk where u

=−U

∞

ρU

∞

πr

−





ρu

· 2π ˜rd˜r





− P

∞

πr





P · 2π ˜rd˜r









∞

· 2πrd˜z





−





· 2π ˜rd˜r





= 0 (3.E.4-3)

W. G. Cochran, Proc. Cambridge Philos. Soc., 30, 365–375 (1934).

V. G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ 1962, pp. 60–78.

EXAMPLE PROBLEMS 81

where ˜r and ˜z again denote dummy integration variables. When the components

of Newton’s constitutive equation given by equations (D.2-6) and (D.2-9) in the

Appendices are substituted, we obtain

ρU

∞

−





ρu

· 2˜rd˜r





− P

∞





P · 2˜rd˜r





−





∞

∂u

∂ ˜z

· 2rd˜z









∂u

∂z

· 4˜rd˜r





= 0

(3.E.4-4)

Substitute the functional forms for u

and u

into equation (3.E.4-4) to obtain

ρU

∞

− ρf

− P

∞

+ 2





P ·˜rd˜r





+ 2μr

+ 2μ

= 0

(3.E.4-5)

Hence, we conclude that



P ·˜rd˜r =−ρU

∞

+ ρf

+ P

∞

− 2μf

− 2μ

⇒ P = f

(z)

(3.E.4-6)

When the functional forms for u

,andP given by equations (3.E.4-1),

(3.E.4-2), and (3.E.4-6) are substituted into the radial component of the equations

of motion given by equation (D.2-10) in the Appendices, we can conclude that

= rf

(z);thatis,

ρrf

−

+ f

=−



+μ

∂

∂r



∂

∂r

)





 

⇒ u

= rf

(z)

(3.E.4-7)

Hence, we conclude that for rotating disk laminar ﬂow, u

,P,ru

,andu

/r are

functions only of the axial coordinate z.

The prior considerations are rigorous for a rotating disk, causing laminar ﬂow in

an inﬁnite ﬂuid. However, in practice the rotating disk is placed in a ﬁnite container,

which implies that the velocity might not be purely axial far above the rotating disk,

due to recirculation. When one is using the rotating disk to characterize some heat-

or mass-transfer process, one would like to minimize this ﬁnite container effect.

This will be minimized when there is a thin boundary layer adjacent to the rotating

disk across which the radial and angular velocities components decay. Hence, we

use scaling to ascertain the criteria required to assure that this boundary layer is thin.

In view of the considerations for a rotating disk in an inﬁnite ﬂuid, the equations

of motion in cylindrical coordinates given by equations (C.2-1), (D.2-10), (D.2-11),

82 APPLICATIONS IN FLUID DYNAMICS

and (D.2-12) in the Appendices simplify to (step 1)

ρu

∂u

∂r

− ρ

+ ρu

∂u

∂z

= μ

∂

∂z

(3.E.4-8)

ρu

∂u

∂r

+ ρ

+ ρu

∂u

∂z

= μ

∂

∂z

(3.E.4-9)

ρu

=−

+ μ

(3.E.4-10)

∂

∂r

(ru

) +

∂u

∂z

= 0 (3.E.4-11)

The corresponding boundary conditions are given by

= 0,u

= ωr at z = 0 (3.E.4-12)

=−U

∞

= 0,u

= 0,P= P

∞

as z →∞ (3.E.4-13)

= 0,u

= 0atr = 0 (3.E.4-14)

Note that equations (3.E.4-8) through (3.E.4-14) are rigorous for a rotating disk in

an unbounded ﬂuid.

Introduce the following dimensionless variables containing appropriate reference

and scale factors (steps 2, 3, and 4):

∗

≡

; u

∗

≡

; u

∗

≡

θs

; P

∗

≡

P − P

; r

∗

≡

;

∗

≡

(3.E.4-15)

Substituting these dimensionless variables into equations (3.E.4-8) through

(3.E.4-14) and dividing through by the dimensional coefﬁcient of one of the prin-

cipal terms in each equation then yields the following dimensionless describing

equations (steps 5 and 6):

ρu

μr

∗

∂u

∗

∂r

∗

−

ρu

θs

μu

∗2

∗

ρu

∗

∂u

∗

∂z

∗

∂

∗

∂z

∗2

(3.E.4-16)

ρu

μr

∗

∂u

∗

∂r

∗

ρu

μr

∗

ρu

∗

∂u

∗

∂z

∗

∂

∗

∂z

∗2

(3.E.4-17)

ρu

∗

=−

μu

∗

∗2

(3.E.4-18)