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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PRACTICE PROBLEMS 113

(a) Write the appropriate form of the simpliﬁed equations of motion for this

ﬂow.

(b) Write the boundary conditions required for the above differential equations.

ignoring the effect of the gravitational body force on the ﬂow.

(d) Develop a dimensionless criterion for assuming that the radial pressure gra-

dient is much less than the axial pressure gradient.

(e) Develop a dimensionless criterion for assuming that the circumferential

velocity is much less than the axial velocity.

3.P.5 Laminar Flow Between Converging Flat Plates

In Section 3.3 we developed the criteria for invoking the lubrication-ﬂow approxi-

mation for laminar ﬂow between two converging ﬂat plates. However, these criteria

break down near the upstream region of this ﬂow. Use scaling analysis to determine

the thickness of the region of inﬂuence in which the lubrication-ﬂow approximation

breaks down.

3.P.6 Laminar Flow Between Diverging Flat Plates

Consider the pressure-driven steady-state one-dimensional laminar plug ﬂow of

an incompressible Newtonian ﬂuid with constant physical properties and constant

velocity U

impinging on two nonparallel inﬁnitely wide diverging ﬂat plates as

shown in Figure 3.P.6-1.

(a) Write the appropriate form of the simpliﬁed equations of motion for this

ﬂow.

(b) Write the boundary conditions required for the differential equations above.

making the lubrication-ﬂow approximation.

(d) Solve the resulting describing equations appropriate to lubrication ﬂow for

the x-andy-component velocity proﬁles.

Figure 3.P.6-1 Steady-state developing laminar ﬂow of an incompressible Newtonian ﬂuid

with constant physical properties between two inﬁnitely wide diverging ﬂat plates; only the

local axial velocity proﬁle is shown.

114 APPLICATIONS IN FLUID DYNAMICS

Figure 3.P.7-1 Pressure-driven steady-state laminar ﬂow of an incompressible Newtonian

ﬂuid with constant physical properties through a diverging nozzle of length L and a circular

cross-section with radii R

and R

3.P.7 Laminar Flow in a Diverging Nozzle

Consider the pressure-driven steady-state laminar ﬂow of an incompressible New-

tonian ﬂuid with constant physical properties through a diverging nozzle of length

L and a circular cross section as shown in Figure 3.P.7-1. Assume plug ﬂow at

z = 0 with a constant velocity U

(a) Write the appropriate form of the equations of motion.

(b) Write the boundary conditions that are necessary to solve the describing

equations.

lubrication ﬂow.

(d) Solve the resulting simpliﬁed lubrication-ﬂow equations for the z-and

r-velocity and axial pressure proﬁles. Note that the unspeciﬁed axial pres-

sure gradient can be shown to be a constant and can be obtained from an

integral mass balance and the known inlet velocity.

3.P.8 Steady-State Flow Between Parallel Circular Disks

Two parallel disks of outer radius R

are separated by a distance H as shown in

Figure 3.P.8-1. An incompressible Newtonian liquid with constant physical prop-

erties is injected through a porous tube of radius R

located concentric with the

axis of symmetry of the two disks.

(a) If the liquid is injected at a constant volumetric ﬂow rate Q, write the

appropriate form of the equations of motion.

(b) Write the boundary conditions required to solve the equations of motion.

lubrication ﬂow.

PRACTICE PROBLEMS 115

Figure 3.P.8-1 Steady-state radial ﬂow of an incompressible Newtonian liquid with con-

stant physical properties between two parallel disks of radius R

separated by a distance H

due to injection of liquid at a volumetric ﬂow rate Q through a porous tube of radius R

(d) Solve the lubrication-ﬂow equations to obtain the velocity proﬁle as a func-

tion of r and z.

(e) Determine the pressure drop P

− P

necessary to inject this liquid at the

constant volumetric ﬂow rate Q.

3.P.9 Unsteady-State Flow Between Parallel Circular Disks

Consider the radial ﬂow of an incompressible Newtonian liquid with constant phys-

ical properties between two parallel disks as shown in Figure 3.P.8-1. Assume

that the liquid injection through the porous cylindrical tube is time-dependent and

described by

Q = Q

−αt

, where Q

and α are constants (3.P.9-1)

(a) Write the boundary conditions required to solve the equations of motion.

(b) Scale the describing equations to determine the criterion necessary to assume

that the ﬂow is quasi-steady-state.

proﬁle as a function of r, z,andt.

3.P.10 Steady-State Flow Between Spinning Parallel Circular Disks

Consider the steady-state radial ﬂow of an incompressible Newtonian liquid with

constant physical properties between two parallel disks, as shown in Figure 3.P.8-1.

The radial ﬂow is caused by liquid injection through the porous cylindrical tube

at a constant volumetric ﬂow rate Q. Assume now that both disks are spun at a

constant angular velocity ω (radians per second).

116 APPLICATIONS IN FLUID DYNAMICS

(a) Write the appropriate form of the unsteady-state equations of motion assum-

ing that the lubrication-ﬂow approximation is applicable.

(b) Write the boundary conditions required to solve the equations of motion.

the effect of the rotating disks on the pressure proﬁle.

3.P.11 Lubrication-Flow Approximation for a Hydraulic Ram

In Example Problem 3.E.3 we considered lubrication ﬂow in a hydraulic ram as

shown in Figure 3.E.3-1 and used scaling to determine the criterion that must be

satisﬁed in order to ignore curvature effects on the ﬂow. However, we did not

justify the lubrication-ﬂow approximation that was made.

(a) Write the appropriate form of the equations of motion; however, do not

make the lubrication-ﬂow simpliﬁcations.

(b) Write the boundary conditions required to solve the equations of motion.

the lubrication-ﬂow approximation.

3.P.12 Flow in a Rotating Disk Viscometer

Consider an incompressible Newtonian liquid with constant physical properties that

ﬁlls a cylindrical container of radius R to a depth H . A circular plate contacts the

liquid at its upper surface but does not contact the sidewalls of the container, as

shown in Figure 3.P.12-1. By rotating the upper circular plate it is possible to

obtain the viscosity of the liquid by measuring the torque, which is the force times

the radial distance from the axis of rotation, on the upper plate. Operation of this

instrument involves accelerating the upper plate from rest to a constant angular

rotation rate of ω radians per second.

(a) Write the appropriate simpliﬁed form of the equations of motion for this

unsteady-state fully developed ﬂow; do not ignore the edge effects due to

the presence of the sidewall of the container.

(b) Write the initial and boundary conditions required to solve the equations of

motion.

ﬂow conditions can be assumed.

(d) Scale the describing equations to determine a criterion for when the effects

of the sidewall of the cylindrical container on the ﬂow can be neglected.

(e) Solve the steady-state describing equations in the absence of sidewall effects

for the angular velocity u

as a function of z and r.

(f) Use the velocity proﬁle that you obtained in part (e) to obtain an equation

for the torque exerted on the upper rotating plate by the liquid.

PRACTICE PROBLEMS 117

No contact between rotating

circular plate and sidewall

of cylindrical container

Rotating horizontal

circular plate

Viscous Newtonian liquid with

constant physical properties

Figure 3.P.12-1 Flow of a viscous Newtonian liquid with constant physical properties in

a rotating disk viscometer.

3.P.13 Flow in an Oscillating Disk Viscometer

Consider the viscometer shown in Figure 3.P.12-1, which is ﬁlled entirely with

an incompressible Newtonian liquid with constant physical properties. Another

way to operate this viscometer is to oscillate rather than rotate the upper circular

plate. Assume that the upper plate is oscillated continuously at an angular rate

of ω

sin 2πft radians per second, where ω

is the amplitude of the oscillation in

radians per second and f is the frequency of the oscillation in cycles per second.

(a) Write the appropriate simpliﬁed form of the equations of motion for this

unsteady-state, fully developed ﬂow for which edge effects due to the side-

wall can be ignored.

(b) Write the initial and boundary conditions required to solve the equations of

motion.

steady-state ﬂow conditions can be assumed.

(d) Consider the describing equations for the special case of very high fre-

quency oscillations for which the effects of the oscillating upper circular

plate are conﬁned to a thin boundary layer near the upper plate. Use scal-

ing to obtain an equation that can be used to estimate the thickness of the

viscous boundary layer at the upper oscillating circular plate.

(e) Based on your scaling in part (d), develop a criterion for ignoring the effect

of the bottom surface of the container on the oscillating ﬂow.

3.P.14 Falling Needle Viscometer

The falling needle viscometer is useful for obtaining viscosity measurements when

only small quantities of the liquid are available for measurement purposes. The

118 APPLICATIONS IN FLUID DYNAMICS

viscosity is obtained using this viscometer by measuring the time it takes for a

long cylindrical needle that is falling at its terminal velocity to pass between two

known reference planes. This viscometer can be modeled quite well by considering

the hydrodynamics to be steady-state fully developed laminar ﬂow in the annular

region between two cylinders, the inner of which is moving downward at a constant

velocity U

and the outer of which is stationary; that is, end effects are ignored.

Unfortunately, it is difﬁcult to drop the needle exactly along the centerline of the

outer cylinder. Hence, in general, the cylinders are not concentric, as shown in

Figure 3.P.14-1. We seek to simplify the describing equations to permit a tractable

solution.

It is convenient to use a moving cylindrical coordinate system located

with its axis concentric with that of the falling cylinder.

(a) Consider the steady-state fully developed ﬂow of an incompressible Newto-

nian liquid with constant physical properties in the annular region between

two cylindrical tubes whose centers are displaced by a distance ε, as shown

in Figure 3.P.14-1. Recall that this ﬂow is driven by the moving boundary of

Side view

w(q)

Cross-sectional view

Needle of radius R

and length L falling at

a constant velocity U

Velocity

profile in

stationary

coordinate

system

Figure 3.P.14-1 Falling needle viscometer showing a small inner cylinder having radius

being dropped at a constant velocity U

a distance ε off-center in a cylindrical tube of

radius R

This effect has been analyzed by D. B. Thiessen and W. B. Krantz, Rev. Sci. Instrum., 63(9),

4200–4204 (1992).

PRACTICE PROBLEMS 119

the inner cylinder (the needle), which is falling at its terminal velocity dic-

tated by a balance between the gravity force on its volume and the drag

force on its surface area. Show that the appropriate form of the equations of

motion in cylindrical coordinates is given by

0 =

∂P

∂r

(3.P.14-1)

0 =

∂P

∂θ

(3.P.14-2)

0 =−

∂P

∂z

+ μ

∂

∂r



∂u

∂r



+ μ

∂

∂θ

+ ρg (3.P.14-3)

(b) Show that equation (3.P.14-3) simpliﬁes to

0 =−

P

+ μ

∂

∂r



∂u

∂r



+ μ

∂

∂θ

+ ρg (3.P.14-4)

where P ≡ P

− P

is the pressure drop across the length of the falling

needle for which P

is the pressure at z = 0andP

is the pressure at z = L.

equations of motion. Since the axial pressure drop across the length of the

needle P is unknown, an auxiliary condition is needed. This is determined

from a force balance across the falling needle that involves the gravitational,

viscous drag, and pressure forces. It will be helpful in specifying the no-slip

condition at the outer cylinder to recall the law of cosines, which permits

relating the local gap thickness w(θ) to the radii of the inner and outer

cylinders, needle displacement ε, and local angular coordinate θ in the form

w(θ) = ε cos θ − R

+ (R

− ε

sin

θ)

1/2

∼

ε cos θ

+ (R

− R

) if ε  R

(3.P.14-5)

(d) Scale the describing equations to determine the criterion for assuming that

the derivatives of the axial velocity in the circumferential direction can be

ignored. Discuss the physical implications of this criterion.

(e)UsescalingtoobtainanestimateofP , the pressure driving force that

causes ﬂow in the annular gap between the inner cylindrical needle and the

outer stationary tube wall.

(f) Use scaling and the result you obtained in part (e) for P to obtain an

estimate of the velocity U

of the falling needle.

3.P.15 Leading-Edge Considerations for Laminar Boundary-Layer Flow

In Section 3.4 we consider laminar boundary-layer ﬂow over a semi-inﬁnite ﬂat

plate and determined that the hydrodynamic boundary-layer approximation can

120 APPLICATIONS IN FLUID DYNAMICS

be made when the criterion given by equation (3.4-38) is satisﬁed. However, as

discussed in Section 3.4, the hydrodynamic boundary-layer approximation always

breaks down near the leading edge of the plate. Use scaling analysis to estimate the

region of inﬂuence wherein the criterion given by equation (3.4-38) is no longer

satisﬁed.

3.P.16 Laminar Boundary-Layer Flow with Blowing

Consider a uniform plug ﬂow of an incompressible viscous Newtonian liquid with

constant physical properties and velocity U

∞

intercepting a stationary semi-inﬁnite

inﬁnitely wide horizontal ﬂat plate such as that considered in Section 3.4. Assume

that the horizontal ﬂat plate is porous such that there is a constant blowing velocity

along its length, as shown in Figure 3.P.16-1. Blowing is used to increase heat

and mass transfer in boundary-layer ﬂows since it causes the local Reynolds number

to increase, which in turn can cause a transition to turbulent boundary-layer ﬂow.

(a) Use scaling to determine the condition for which the blowing effect on the

boundary-layer ﬂow can be neglected.

(b) Provide a physical interpretation of the condition on the dimensionless group

that you obtained in part (a).

(x)

∞

Figure 3.P.16-1 Uniform plug ﬂow with velocity U

∞

of an incompressible viscous New-

tonian ﬂuid with constant physical properties intercepting a stationary semi-inﬁnite inﬁnitely

wide horizontal porous ﬂat plate along which there is a blowing velocity V

; the blowing

causes the boundary-layer thickness, δ

, to increase.

3.P.17 Laminar Boundary-Layer Flow with Suction

Consider a uniform plug ﬂow of an incompressible viscous Newtonian liquid

with constant physical properties and velocity U

∞

intercepting a stationary semi-

inﬁnitely long inﬁnitely wide horizontal ﬂat plate such as that considered in Sec-

tion 3.4. Assume that the horizontal ﬂat plate is porous so that a constant suction

PRACTICE PROBLEMS 121

(x)

∞

Figure 3.P.17-1 Uniform plug ﬂow with velocity U

∞

of an incompressible viscous New-

tonian ﬂuid with constant physical properties intercepting a stationary semi-inﬁnitely long

inﬁnitely wide horizontal porous ﬂat plate along which there is a suction velocity V

;the

suction causes the boundary-layer thickness, δ

, to decrease.

velocity V

can be applied along its length, as shown in Figure 3.P.17-1. Suction

is used to decrease the boundary-layer thickness to decrease the local Reynolds

number and thereby delay the transition to turbulence. It is also used to increase

heat and mass transfer in the laminar boundary-layer ﬂow regime by decreasing the

thickness of the boundary-layer region, which provides the controlling resistance

to conduction or diffusion.

(a) Write the appropriate forms of the equations of motion applicable to this

boundary-layer ﬂow; it is not necessary here to justify the form of these

equations by scaling.

(b) Write the boundary conditions required to solve the equations of motion.

this problem, the boundary-layer thickness might ultimately become con-

stant rather than grow without bound as it does for a boundary layer on a

semi-inﬁnitely long ﬂat plate without suction or with blowing. Use scaling

analysis to determine the criterion for obtaining a constant boundary-layer

thickness; express your answer in terms of a dimensionless group that must

be very small.

(d) For the constant boundary-layer condition obtained in part (c), determine

the x-andy-velocity component proﬁles.

3.P.18 Entry-Region Laminar Flow in a Cylindrical Tube

Figure 3.P.18-1 shows a schematic of pressure-driven steady-state laminar entry-

region ﬂow of a viscous Newtonian ﬂuid with constant physical properties in a

cylindrical tube of radius R. The ﬂow velocity at the entrance is assumed to be a

constant U

. This is assumed to be a high Reynolds number ﬂow for which the

inertia terms cannot be ignored in the entry region. Hence, in the entry region the

122 APPLICATIONS IN FLUID DYNAMICS

(z)

Figure 3.P.18-1 High Reynolds number steady-state pressure-driven entry-region laminar

ﬂow of a viscous Newtonian ﬂuid with constant physical properties in a cylindrical tube of

radius R, showing the developing boundary layer δ

(z) and entrance length L

required for

the initial uniform ﬂow U

to rearrange so as to become fully developed

action of viscosity will be conﬁned to a region of inﬂuence near the tube wall

denoted by the boundary-layer thickness δ

(z). The latter will increase axially;

when it reaches the center of the tube, the ﬂow is fully developed. Note that since

this is a conﬁned ﬂow, the ﬂuid in the center of the tube must accelerate as the

boundary layer grows.

(a) Write the appropriate simpliﬁed form of the equations of motion for this

steady-state, developing ﬂow.

(b) Write the boundary conditions required to solve the equations of motion.

the hydrodynamic boundary-layer approximation; that is, for which the cou-

pling between the axial and radial components of the equations of motion

can be ignored and for which the axial diffusion of vorticity can be ignored.

(d) Use your scaling analysis to estimate the axial distance required to attain

fully developed ﬂow; reconcile your result with that obtained from the

approximate analytical solution of Langhaar given by

= 0.227

ρU

(3.P.18-1)

3.P.19 Pressure-Driven Flow in an Oscillating Tube

Consider the unsteady-state laminar ﬂow of an incompressible Newtonian ﬂuid

with constant physical properties through a horizontal cylindrical tube of radius R

whose length L is sufﬁciently long to ensure that entrance and exit effects can be

ignored; that is, the tube can be assumed to be essentially inﬁnitely long. Initially,

there is no ﬂow. At time t = 0 a constant pressure gradient (P

− P

)/L = P /L

is impressed across this tube. Simultaneously, the wall of this tube is oscillated at

a velocity u

= U

cos ωt,whereω is the angular frequency of the oscillation in

radians per second. A schematic of this ﬂow problem is shown in Figure 3.P.19-1.

H. L. Langhaar, Trans. ASME, 64, A55 (1942).