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 133

(f) Solve the simpliﬁed form of the equations of motion for the velocity proﬁle

assuming that the effect of the solid boundaries can be ignored.

(g) Show how the solution for the velocity proﬁle that you derived in part

3.P.28 Gravity-Driven Film Flow over a Saturated Porous Medium

In Example Problem 3.E.9 we considered the steady-state fully developed ﬂow of

an incompressible Newtonian liquid ﬁlm over an inclined liquid-saturated porous

medium due to a gravitationally induced body force as shown in Figure 3.E.9-1.

We used scaling analysis to determine when the ﬂow through the porous media has

a negligible effect on the ﬂow of the liquid ﬁlm. We introduced one set of scales for

the velocity and the length variables in the liquid ﬁlm and another set of scales for

velocity and length variables in the porous medium. We stated that if we had not

done this, we would have arrived at a contradiction in our scaled equations. How-

ever, the forgiving nature of scaling would then indicate that we had not scaled some

quantity so that it was

◦

(1). To understand better what is meant by the forgiving

nature of scaling, rework this problem while assuming (incorrectly!) that the veloc-

ity and the length scales are the same in both the liquid ﬁlm and the porous medium.

3.P.29 Radial Flow from a Porous Cylindrical Tube

Consider the steady-state laminar ﬂow of a Newtonian liquid with constant physical

properties that is caused by ﬂuid emanating radially at a uniform volumetric ﬂow

rate Q from a cylindrical tube having length L, outer radius R, and porous walls

that is immersed in an inﬁnite pool of the same liquid as shown in Figure 3.P.29-1.

Figure 3.P.29-1 Steady-state laminar ﬂow of a Newtonian liquid with constant physical

properties that is caused by ﬂuid emanating radially at a uniform volumetric ﬂow rate Q

from a cylindrical tube of length L, outer radius R, and porous walls that is immersed in

an inﬁnite pool of the same liquid; the radial velocity proﬁle at the surface of the cylinder

is shown at which the pressure is P

; the pressure far removed from the cylinder is P

∞

134 APPLICATIONS IN FLUID DYNAMICS

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

exterior to the cylinder, allowing for the fact that the ﬁnite length of the

cylinder implies that there will be end effects; note that this ﬂow is caused

by the pressure gradient P

− P

∞

generated between the outer surface of

the porous cylindrical tube and inﬁnity.

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

effects on the radial velocity proﬁle; note that the radial velocity scale is

determined from the known volumetric ﬂow rate.

(d) Solve the resulting simpliﬁed describing equations to obtain the radial veloc-

ity proﬁle in the liquid exterior to the cylinder.

3.P.30 Entry-Region Flow in a Tube with a Porous Annulus

In this problem we explore the idea that one might be able to decrease the entry

region length for steady-state laminar pipe ﬂow by lining the wall of the pipe

with an annular region of porous medium, as shown in Figure 3.P.30-1. The outer

impermeable wall of the pipe is at R

. The porous medium is conﬁned in the annular

region deﬁned by radii R

and R

,whereR

. Assume that ﬂuid enters the pipe

in plug ﬂow; that is, u

= U

at z = 0for0 r  R

. The ﬂow within the pipe,

including the porous annular region, is caused by an applied pressure difference

over the length of the pipe L; the downstream pressure is known and denoted by

; however, the upstream pressure is not speciﬁed. This ﬂuid may be assumed to

be Newtonian and to have constant physical properties. The pipe is assumed to be

horizontal such that gravitational body forces can be ignored.

(a) Write the appropriately simpliﬁed continuity and equations of motion for

the ﬂow within the pipe; in writing these equations, denote the velocity

components as u

,andu

(b) Write the appropriately simpliﬁed continuity and equations of motion for

the ﬂow within the porous pipe wall; in writing these equations, denote the

velocity components as u

, u

,andu

you obtained in part (a).

(d) Write the boundary conditions required to solve the differential equations

you obtained in part (b). Assume that the radial velocity proﬁle within

the porous medium becomes pluglike within a region of inﬂuence that has

an unspeciﬁed thickness δ

whose value will be estimated in the scaling

process; that is, assume that the ﬂow within the porous medium departs

from being pluglike only very near the boundaries at r = R

and r = R

(e) Determine the scale and reference factors for the ﬂow in the pipe; estimate

the thickness of the boundary layer or region of inﬂuence δ

PRACTICE PROBLEMS 135

SIDE VIEW

Impermeable outer wall

Porous media

END VIEW

Porous media

Figure 3.P.30-1 Side and end views of entry-region ﬂow of an incompressible Newtonian

ﬂuid with constant physical properties in a pipe consisting of an open region of radius R

and an annular region of porous medium between R

and R

(f) Determine the scale and reference factors for the ﬂow within the porous

medium; in particular, estimate the thickness of the region of inﬂuence δ

(g) What is the criterion for ignoring the effect of the porous medium on ﬂow

in the pipe?

(h) What are the criteria for ignoring the effect of the curvature on the equations

of motion for ﬂow in both the nonporous and porous regions of the pipe?

(i) Assess the merits of using this outer annular region of the porous medium

to reduce the entrance length required to achieve fully developed laminar

ﬂow in the pipe.

3.P.31 Steady-State Laminar Flow of a Compressible Gas

In Section 3.9 we considered the steady-state laminar ﬂow of a compressible gas

in a cylindrical tube as shown in Figure 3.9-1. We scaled the describing equations

to determine the criterion for assuming that this ﬂow was incompressible. This

criterion was that the Mach number for the ﬂow must be much less than 1. In

scaling the describing equations we introduced a scale for the radial derivative of

the pressure since we did not anticipate that this derivative would scale in the same

136 APPLICATIONS IN FLUID DYNAMICS

way as the axial pressure gradient. We raised the question as to how one knows

whether to scale a derivative as the ratio of some dependent variable scale divided

by some independent variable scale or to introduce a separate scale for the entire

derivative. We indicated that the answer to this rhetorical question was contained

in the forgiving nature of scaling. To better understand what is meant by this, let

us assume (incorrectly!) that the radial pressure derivative scales as the pressure

scale P

divided by the radial length scale r

. Show that this leads to inconsistency

in the resulting dimensionless equations in that one term becomes much greater

than 1, with no other terms balancing it.

3.P.32 Velocity Proﬁle Distortion Effects Due to Fluid Injection and

Withdrawal

Flow-ﬁeld-ﬂow fractionation is a technique for separating small particles such as

proteins and viruses from a carrier ﬂuid such as water by combining a longitudinal

laminar ﬂow with a transverse ﬂow. The latter can be imposed by making the

closely spaced parallel lateral walls of the horizontal ﬂow channel consist of two

permeable membranes. Inﬂow and outﬂow at a constant velocity V

of the same

carrier ﬂuid (without any particles) occurs through the upper and lower mem-

branes, respectively, as shown in Figure 3.P.32-1. This drives the particles, which

are injected as a pulse in the axially ﬂowing ﬂuid, toward the lower membrane and

thereby provides a means of separating them. Field-ﬂow fractionation is consid-

ered in more detail in Chapter 5 when scaling is applied to mass transfer. In this

problem we are concerned with developing a criterion for determining when the

longitudinal velocity proﬁle can be assumed to be unaffected by the transverse ﬂow.

Figure 3.P.32-1 Flow-ﬁeld-ﬂow fractionation showing transverse injection of a ﬂuid with

a velocity V

into a longitudinal ﬂow of the same ﬂuid that is assumed to be initially in

fully developed laminar ﬂow; the distortion of the velocity proﬁle that can occur due to

transverse injection is shown schematically.

PRACTICE PROBLEMS 137

The entering carrier ﬂuid can be assumed to be a Newtonian ﬂuid with constant

physical properties and to have a velocity proﬁle given by

= 2U



−



(3.P.32-1)

where

U is the average axial velocity.

(a) Write the appropriate form of the equations of motion that describe this

ﬂow.

(b) Write the boundary conditions that would be necessary to solve the equations

of motion.

through the upper and lower membrane boundaries on the solution for the

x-component of the velocity if the latter is to be used to determine quantities

such as the volumetric ﬂow rate or average velocity.

(d) If one is interested in determining quantities in the vicinity of the mem-

brane, such as the drag on its surface, one cannot ignore the effect of the

permeation on the ﬂow in this wall region (i.e., since the nonzero perme-

ation velocity can have a signiﬁcant inﬂuence on the small axial velocity

near the membrane surface). Use scaling to determine the thickness of the

region of inﬂuence near the lower membrane boundary wherein the effects

of the permeation can never be ignored.

3.P.33 Flow Between Parallel Impermeable and Permeable Flat Plates

Consider the steady-state laminar ﬂow of a Newtonian ﬂuid with constant physical

properties through a horizontal channel due to a pressure driving force P

− P

applied over the length L as shown in Figure 3.P.33-1. The upper surface of this

channel at y = H consists of an impermeable solid plate. The lower surface of this

channel at y = 0 consists of a permeable membrane; the permeation velocity V

through this membrane is given by

= k

[P(x)−P

atm

] (3.P.33-1)

where k

is the permeability of the membrane, P(x)− P

atm

is the pressure drop

across the membrane in which P(x) is the local pressure on the high-pressure side

of the membrane at y = 0, and P

atm

is the constant pressure on the low-pressure

side of the membrane, which is assumed to be atmospheric pressure. Note that, in

general, P

− P

atm

> 0, to ensure that permeation occurs over the entire length L

of the membrane. Ignore body forces and lateral edge effects (i.e., those in the z-

direction perpendicular to the plane of Figure 3.P.33-1). Also assume that the ﬂow

is fully developed when it enters this membrane module at x = 0; note, however,

that we are not given any information on the average or maximum velocity of the

velocity proﬁle at x = 0.

138 APPLICATIONS IN FLUID DYNAMICS

Impermeable solid plate

Permeable membrane

atm

(atmospheric pressure)

Figure 3.P.33-1 Steady-state laminar ﬂow of a Newtonian ﬂuid with constant physical

properties through a horizontal channel due to a pressure driving force P

− P

applied

over the length L; the upper surface at y = H consists of an impermeable solid plate; the

lower surface at y = 0 consists of a permeable membrane through which permeation occurs

at a constant velocity V

(a) Write the appropriate form of the equations of motion that describe this

ﬂow.

(b) Write the boundary conditions that would be necessary to solve the equations

of motion.

mentum).

(d) Determine the criteria necessary to assume that this ﬂow is essentially fully

developed within the region 0  x  L.

(e) For the simplifying assumptions appropriate to parts (c) and (d), determine

the solution for the dimensionless pressure and axial velocity proﬁles.

(f) For the simplifying assumptions appropriate to parts (c) and (d), derive an

equation for determining the y-component of the velocity; it is not necessary

to solve this equation.

(g) In part (e) you determined the solution for the dimensionless pressure proﬁle

P(x). In fact, the pressure will have a slight dependence on y as well. For

the simplifying assumptions appropriate to parts (c) and (d), determine the

equations necessary to obtain the complete dimensionless pressure proﬁle

P(x,y); that is, including the y-dependence as well. It is sufﬁcient to express

your result for the pressure proﬁle in terms of the two velocity components

and u

(h) Determine the criterion necessary to ignore the effect of the permeation

through the membrane on the solution for the x-component of the velocity

if the latter is to be used to determine quantities such as the volumetric ﬂow

rate or average velocity.

PRACTICE PROBLEMS 139

(i) If one is interested in determining quantities in the vicinity of the membrane,

such as the drag on its surface, one cannot ignore the effect of the permeation

on the ﬂow in this wall region (i.e., since the nonzero permeation velocity can

have a signiﬁcant inﬂuence on the small axial velocity near the membrane

surface). Use scaling to determine the thickness of the region of inﬂuence

wherein the effects of the permeation can never be ignored.

3.P.34 Flow in an Annulus with Fluid Injection and Withdrawal

Consider the steady-state fully developed ﬂow of an incompressible Newtonian

ﬂuid with constant physical properties in the annular region between radii R

and

, as shown in Figure 3.P.34-1. The ﬂow in the axial direction is caused by a

constant-pressure driving force given by P ≡ P

− P

applied across the length

L. Both the inner wall at R

and the outer wall at R

are permeable membranes.

Fluid is injected at a constant radial velocity V

into the annular region through

the inner wall at R

and withdrawn at some constant unspeciﬁed velocity (not

necessarily equal to V

) from the annular region through the outer wall at R

Injection of fluid at inner wall

at constant velocity V

SIDE VIEW

Withdrawal of fluid at outer wall

END VIEW

Axial and radial flow in

this annular region

Inner wall at R

Outer wall at R

Figure 3.P.34-1 Steady-state fully developed laminar ﬂow of a Newtonian ﬂuid with con-

stant physical properties through an annulus due to a pressure driving force P

−P

applied

over the length L; ﬂuid injection occurs at constant velocity V

through the inner permeable

wall at r = R

and ﬂuid withdrawal occurs at a constant unspeciﬁed velocity at the outer

permeable wall at r = R

140 APPLICATIONS IN FLUID DYNAMICS

This injection and withdrawal of ﬂuid is done under conditions that maintain fully

developed ﬂow throughout the annulus.

(a) Simplify the continuity equation in cylindrical coordinates.

(b) What are the implications of the continuity equation for the case of fully

developed ﬂow for the axial component of velocity? Explore these math-

ematically by integrating the appropriate term in the continuity equation

that contains the axial velocity. Be careful in considering the integration

constant(s) you obtain in your partial integration.

developed ﬂow for the radial component of velocity? Explore these mathe-

matically by integrating the appropriate term in the continuity equation that

contains the radial velocity. Apply an appropriate boundary condition to this

ﬁrst-order differential equation for the axial velocity and obtain a equation

for the radial velocity proﬁle. Note in curvilinear coordinates that fully

developed ﬂow does not necessarily mean that the radial velocity component

does not change in the radial direction, however, it does imply something

about the radial variation of the radial velocity.

(d) Simplify the axial component of the equations of motion in cylindrical coor-

dinates for this ﬂow. Note that the axial pressure gradient is constant, as

indicated in the problem statement.

(e) Write the appropriate boundary conditions needed to solve the differential

equation that you derived in part (d).

(f) Simplify the radial component of the equations of motion in cylindrical

coordinates for this ﬂow; you may ignore the small effect of the gravitational

body force.

(g) Scale the differential equations and boundary conditions that you derived

for this ﬂow in parts (d) and (e). Your answer should include specifying

the scale and reference factors needed to ensure that your dependent and

independent variables are bounded of order one.

(h) Use your scaling analysis in part (g) to determine the criterion for ignoring

the effect of the ﬂuid injection and withdrawal at the walls on the solution

for the axial velocity.

(i) Use your scaling analysis in part (g) to determine the criterion for assuming

that the inner wall is essentially at r = 0.

(j) Use your scaling analysis in part (g) to determine the criterion for ignoring

the curvature effects. That is, what is the criterion for assuming that this is

essentially the same as ﬂow between two parallel ﬂat plates at which there

is injection at the lower plate and withdrawal at the upper plate?

(k) Solve the differential equation that you derived in part (f) to obtain an

analytical solution for the pressure proﬁle as a function of r and z;in

carrying out this integration, do not forget that ∂P/∂z =−P /L since it

will help you determine the integration “constant” that you obtain in your

radial integration.

PRACTICE PROBLEMS 141

(l) Solve the differential equation and appropriate boundary conditions that

you derived in parts (d) and (e) to obtain an analytical solution for the axial

velocity proﬁle as a function of r.

3.P.35 Flow Between Parallel Permeable Membranes

Consider the parallel permeable membranes shown in Figure 3.P.35-1, which are

open at the downstream end at x = L to atmospheric pressure P

atm

, but closed at

the upstream end at x = 0. All along the semipermeable walls of the parallel mem-

branes, an incompressible Newtonian liquid with constant physical properties ﬂows

in at a constant velocity V

. The ﬂuid that ﬂows in through the semipermeable walls

ultimately exits to the atmosphere from the open end of the parallel membranes.

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

(b) Write the boundary conditions that are required to solve the equations of

motion.

lubrication-ﬂow approximation.

(d) Solve for the axial velocity proﬁle as a function of y and x; be certain that

you express your answer entirely in terms of known quantities; that is, you

must eliminate the pressure gradient from the equation that you obtain for

the axial velocity proﬁle.

(e) Solve for the pressure proﬁle at any axial position along the ﬂow.

Figure 3.P.35-1 Steady-state laminar ﬂow of an incompressible Newtonian liquid with

constant physical properties between parallel permeable membranes owing to a constant

radial velocity V

through the permeable wall at y =±H ; the end between the parallel

membranes at x = 0 is closed and impermeable; the end at x = L is open to atmospheric

pressure P

atm

142 APPLICATIONS IN FLUID DYNAMICS

(f) Solve the simpliﬁed describing equations that you obtained in part (c) for

the y-component of velocity as a function of y and x.

3.P.36 Dimensional Analysis for Flow Around a Falling Sphere

In Section 3.10 we used the scaling approach for dimensional analysis to develop a

correlation for the terminal velocity U

of a spherical particle having radius R and

density ρ

falling, owing to gravitational acceleration g through an incompressible

Newtonian liquid having density ρ and viscosity μ, as shown in Figure 3.10-1.

(a) Use the scaling analysis approach to dimensional analysis to develop a

correlation for the viscous drag force on the spherical particle.

(b) Consider how the correlation that you developed in part (a) simpliﬁes for

the special case of creeping ﬂow.

3.P.37 Dimensional Analysis for Impulsively Initiated Laminar Tube Flow

In Example Problem 3.E.7 we considered the impulsively initiated laminar ﬂow of

an incompressible Newtonian ﬂuid with constant physical properties in a cylindrical

tube, as shown in Figure 3.E.7-1.

(a) Use the scaling approach to dimensional analysis to determine the dimen-

sionless groups needed to correlate the instantaneous local velocity u

(r, t).

Isolate u

into just one dimensionless group.

(b) Use the scaling approach to dimensional analysis to determine the dimen-

sionless groups needed to correlate the instantaneous viscous drag force on

the wall. Be certain to write all the equations you would solve in order to

obtain the viscous drag force. Isolate u

into just one dimensionless group.

special case of fully developed ﬂow?

3.P.38 Dimensional Analysis for Flow in an Oscillating Tube

In Practice Problem 3.P.19 we considered the unsteady-state ﬂow of an incom-

pressible Newtonian ﬂuid with constant physical properties through a horizontal

cylindrical tube of radius R and length L due to both an axial pressure gradient

and an oscillating wall, as shown in Figure 3.P.19-1. Consider the special case of

unsteady-state ﬂow in this tube that is caused only by tube wall that is oscillated

at a velocity u

= U

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

radians per second.

(a) Use the scaling approach to dimensional analysis to determine the dimen-

sionless groups needed to correlate the instantaneous local velocity u

(z, t);

isolate u

into one dimensionless group, t into another, and the viscosity

into a third dimensionless group; note that there may be more than three

dimensionless groups.