Chapter 3: Partial Differential Equations in Mechanics 1

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Partial differential equationsof the first-order

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L Although the major focus of these introductory volumes is on linear partialdifferential equations of second- and higher-order, it is instructive to examinesome aspects of first-order partial differential equations. These can arise in anumber of engineering applications including the study of waves in shallowwater, traffic flow, gas dynamics, isothermal plug flow reactors in chemicalengineering and in the analysis of the thermal efficiency of heat exchangers.

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3.1 General concepts

We focus attention on the quasi-linear first'-order partial differential equa-tion which has the form

-<..

OU OU

A (x, y, u) ox + B (x, y, u) oy = c (x, y, u)(3.1)

where x and y are the independent variables. The solution of (3.1) is a sur-face in the (x, y, u) space. Let us consider a point P (x, y, u) on the solutionsurface u = u (x, y) and we move along a direction given by the vector t ={A(x,y,u),B(x,y,u),C(x,y,u)}. However, at any point on the surface,the direction ofthe normal is given by (Figure 3.1) n = {oulox,ouloy,-I}.

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88 3. Partial differential equations of the first-order

From these it is clear that nand tare orthogonal. Therefore any solutionsurface must be tangent to a vector with components {A, B, C} and sincethis vector is tangential we do not leave the solution surface. Also, the totaldifferential du is given by .

ou oudu = - . dx + - .dy

ox oy(3.2)

u

y

u = u(x,y)

x

Figure 3.1: Solution surface for u (x, y).

Consequently, from (3.1) and (3.2)

{A, B, C} =={dx, dy, du} (3.3)

The solution to equation (3.1) can be obtained by using the following the":orem.

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3.1 General concepts 89

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~

THEOREM 3.1

The general solution of the quasi-linear PDE

au auA (x,y, u) ox + B (x,y,u) ay = C (x,y, u) (3.4)

can be written in the form

F(F,G) = 0 (3.5)

where F is an arbitrary function and F (x, y, u) = k1, and G (x, y, u) = k2form a solution of the equations

'--"

~- dy -~A(x,y,u) - B(x,y,u) - C(x,y,u)

(3.6)

PROOF

The equations (3.6) consist of a set of two independent ODE's (i.e. a twoparameter family of curves in space). Also, one set can be written as

dY_B(x,y,u)dx - A(x,y,u)

(3.7)

which is referred to as a "characteristic curve".

(i) If A = A (x, y) and B = B (x, y) then (3.7) is a function in the (x, y)space. This is referred to as a base curve.

(ii) When A and B are constants (3.7) defines a set of parallel lines in the(x, y) space. -

/

..

"-;;;;;,.

..-...

IL

-90 3. Partial differential equations of the first-order

In both cases (i) and (ii) (3.7) can be evaluated without a knowledge ofu (x, y). In the quasi-linear case (3.7) cannot be evaluated until u (x, y) isknown. .

Returning to the equations (3.1) and (3.2) we can write

[

AB

] [

au/ox] [

c]dx dy au/ ay - du (3.8)

Both equations must hold on the solution surface and one can interpret eachequation as a plane element, where they intersect on a line along whichau/ax and au/ay may exist; i.e. au/ax and au/ay are themselves indeter-minate along the intersection line but they are related to or determinate toeach other since (3.8) must hold.

Using a principle in linear algebra, if a square coefficients matrix of a setof n simultaneous equations has a vanishing determinant, a necessary con-dition for the existence of finite solutions is that when the right hand sideis substituted for any column, the determinants of the resulting coefficientsmatrices must also vanish, i.e.

l

A B

I

-I

A C

I

-I

C B

I

-odx dy - dx du - du dy - (3.9)

Evaluating the determinants we have

~- dy -~A(x,y,u) - B(x,y,u) - C(x,y,u)

(3.10)

3.2 Examples involving first-order equations

In this section we shall examine some basic first-order equations and theircharacteristic curves.

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3.2 Examples involving first-order equations 91

Example 3.1J'

Examine the characteristic curves for the simplest possible first-order partialdifferential equation

ou =0ox (3.11)

Solution

The general solution of equation (3.11) is

'u=f(y) (3.12)

"'- where f (y) is a completely arbitrary function of y. The characteristic curvesin the x, y plane are straight lines (Figure 3.2).

y

~ y = constant.....0 x

Figure 3.2: Characteristic curves for oulox = o.

'~j:r

1

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Example 3.2

- Consider the first-order linear homogeneous partial differential equation

au + au = 0ox ay

(3.13)

Examine the characteristic curves associated with this equation.

Solution

Comparing this with (3.1) we have A = 1 and B = 1; and (3.10) gives

dx = dy (3.14)

Integrating (3.14) we have

x - y = const (3.15)

Consequently, the general form of the solution is

u=F(x-y) (3.16)

where F is an arbitrary function. Therefore if y = x or y = x + .1, thenu = const. Hence there is a characteristic direction in the x, y plane alongwhich u is constant (Figure 3.3).

j

T

j~.

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'I

t

j...1

~p

"-'

~

-

3.2 Examples involving first-order equations

y

~

x

Figure 3.3: Characteristic curves for ouj ox + ouloy = o.

The general solution (3.16) can be made specific by imposing additionalconditions.

e.g. If u (x, y) = 'f}X,where'f} is a constant, when y = 0, the specific solutionfor (3.16) is

u(x,Y)='f}(x-y) (3.17)

Example 3.3

Generalize the results presented in connection with Example 3.2 to includethe case

ou OUAo-+Bo- =0

ox oy (3.18)

where Ao and Bo are both constants.

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Solution

We can prove that the characteristic curves are still straight lines exceptthat they are inclined at an angle a = tan-l (Bo/Ao) to the x-axis (Figure3.4). Avoiding details, the general solution of (3.18) takes the form

-(x Y )u (x, y) = F Ao - Bo

(3.19)

- -where F is an arbitrary function. Again specific forms for F can be obtainedby assigning additional conditions.

y

Ao = Rx

Bo = Ry

x

Figure 3.4: Characteristic curves for Ao (au/ax) + Bo (au/ay) = O.

Example 3.4

Consider the first-order partial differential equation (3.18) which can be fur-ther generalized to the following form

au auA (x,y) ox + B (x,y) ay = 0 (3.20)

--,

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tI

IIihr

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f

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--3,2 Examples involving first-order equations 95

J?xamine the characteristic curves associated with the partia! differential

~q,!ation

Solution

Considering (3.7) we have

dy - B (x, y)dx - A (x, y)

(3.21)

Since A(x,y) and B(x,y) are arbitrary functions, the orientations of thetangent vector R to the characteristic lines are still parallel to the x, y plane(Le.independent of u) but has a variable orientation in the x, y plane (Figure3.5).

y

~ Ry = B(x,y)Rx = A(x,y)

x

il

'I

,1

Figure 3.5: Characteristic curves A (x, y) (8uI8x) + B (x, y) (8uI8y) = O.

We can rewrite (3.21) as

~= dy =dsA(x,y) B(x,y)

(3.22)

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Using (3.2) and (3.22) we have

8u 8u

[

8u 8u

]du = 8x dx + 8y dy = ds A (x, y) 8x + B (x, y) 8y = 0(3.23)

The result (3.23) confirms that along a characteristic curve du = 0, irrespec-tive of A (x, y) and B (x, y).

Example 3.5

Examine a specific case of the general first-order homogeneous partial dif-ferential equation (3.20); -

.!. 8u + .!.8u = 0x 8x y 8y

Obtain a solution such that u(x,O) = f-LX4where f-Lis a constant.

Solution

A comparison with (3.20) gives

1 1A(x,y)=- ; B(x,y)=-

x y

and using (3.25) in (3.21) and integrating we have

Jdx

Jdy

A(x,y) = B(x,y)

or

(3.24)

(3.25)

l'

rtJ ,~~ _'m


11] X dx = J y dy

X2 - y2 = -CO (3.26)

Co is a constant. Therefore the most general functional form of thesolution of (3.24) is

u{x, y) = F* (X2 - y2) (3.27)

Substituting (3.27) in (3.24) it is evident that the latter is satisfied for allchoices of F*. Again, a specific form of (3.27) can be obtained by imposingadditional conditions on u (x, y). e.g. If u (x, 0) = J.LX4is it evident that thespecific solution of u (x, y) is,

u (x, y) = J.L(x2 - y2)2 (3.28)

where J.Lis a constant.

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Let us now consider the inhomogeneous first-order partial differential equa-tion posed by a reduced form of (3.1). i.e.

au auA(x,y) ax +B(x,y) ay = C(x,y)

(3.29),j

The characteristic curves occupy the (x, y, u) space except that a character-istic line does not lie in a plane u = const. Consider (3.10) and (3.22)

-

-

~ = dy = du = dsA(x,y) B(x,y) C(x,y)

(3.30)


Of these, two equations can be taken to be independent, i.e.

dx dy

A(x,y) - B(x,y)~-~

B(x,y) - C(x,y)(3.31)

F(x,y)=C1; G(X,y)=C2 (3.32)

and as stated previously, if these equations are integrable, they can be eval-uated in the form

and the general solution of the partial differential equation (3.29) can bewritten in the form

Q (F, G) = 0 (3.33)

Example 3.6

As a specific case of (3.29), consider the non-homogeneous partial differen-tial equation

Iou Iou 1--+--=-x ox y ay y

Develop a solution such that u = ~ (3 - x2) when y = 1.

Solution

Comparing (3.29) and (3.34) we have

r'lIi:'"


1

= ~ ; B (x, y) = -yX

1

C(X,y)=-y(3.35)

Xhe assQciated ordinary differential equations corresponding to (3.31) are""" :;!t""'"

dx -~ . ~-~- (1/y) , (1/y) - (1/y)

(3.36)

I]ltegrating (3.36) we have," £ "

f == (x2 - y2) = Cl ; G = (u - y) = C2 (3.37)

e-rid,~thegE)peral form of the function is

n[(X2-y2),(U-Y)] =0 (3.38)

Hence for y = 1, (3.38) gives

j

n [(x2 - 1) , (u - 1)] = 2u - 3 + x2 = 2 (u - 1) + (x2 - 1) (3.39)

Therefore the general relationship equivalent to (3.33) is obtained by replac-ing (x2 - 1) by (x2 - y2) and (u - 1) by (u - y) in (3.39), Le.

I'~

n [(u - y) , (x2 - y2)] = 2 (u - y) + (x2 - y2) (3.40)t'I

!

which gives

u(x,y) = ~[2y - (X2 - y2)]2 (3.41):,!

1~~~

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It can be verified by substitution that (3.41) satisfies the first-order partialdifferential equation (3.34) subject to the additional condition (3.39) wheny=1.

3.3 Advective transport in reactor column

In the preceding, we have examined the various types of first-order partialdifferential equations in a very abstract sense without reference to any engi-neering application. We now apply the general theory concerning the first-order non-homogeneous partial differential equation to a problem which hasapplications in both environmental engineering and

Continuoussupplyofchemicalat "

cO}lcentrationCo at time t =0

r xControl element

--pressuresupply

-------- ----.~ x

Outflowof ~chemical atConcentration, C*C*(O,t)

et 1.0

0

Figure 3.6: Advective transport in column.

chemical engineering. The problem deals with the so called" advective trans-port" of a species (such as a chemical, biological medium, effluent, etc.) alonga reactor column. The reactor column is basically a tube with a circular crosssectional area which contains either a reactive or non-reactive porous solid(Figure 3.6). A chemical species is introduced into the flow regime and the"concentration" of the chemical or species is maintained constant at the startof the experiment. The objective of the analysis is to determine the variationof the concentration of the species along the tube as a function of time.

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3.3 Advective transport in reactor column 101

verning equation - one dimensional case

pting to formulate the governing equation, it is necessary to.rameters governing the problem. The porous medium is char-bOI1.tinuousflow paths within the void space. A measure of thethe scalar quantity porosity (n *) . This is defined by

0}.~.,

y()lume of voids - Vv..total volume - V

(3.42)

the volume of voids and V is the total volume.

_o~.which results in advection of the species takes place in the void'rhe average local velocity of flow in the void space at any cross sec-

efined by v. We define an average advective velocity over the entire:.ectionas v. In the case of one dimensional flow, both these velocities

a be aligned in the x-direction (Figure 3.6). For conservation of massequire.

VA = vAv (3.43)

For most porous media with an isotropic or direction independent fabric orstructure,

Av Vv *-"'-=nA-V (3.44)

We define the concentration C(x, t) as the mass of the species per unit vol-'l!:...meof the fluid conveying the species. We can also define a concentrationC(x, t) as the mass of the species per unit total volume of the entire porousmedium. If the porous medium is saturated, the volume of the fluid is iden-tically equal to the volume of the voids. From these definitions we have,

C( )mass of species mass of species mass of speciesx t - - -

, - volume of fluid - volume of voids - Vv

~


or

mass of species = C(x, t)lIv = n*C(x,t) = C(x, t)total volume V (3.45)

Consider the one-dimensional flow in the tube, where, prior to the intro-duction of the species, a pressure gradient induces a uniform average fluidvelocity of v. The cross sectional area of the tube is ao. The mass of speciestransported in the tube by "advection", per unit cross sectional area per unittime is V'n*C(x, t).

The total mass of the species which enters the control element of cross sec-tional area ao and length dx is

Fx =ao,n*C(x, t) (3.46)

The total mass of the species leaving the control element

oFx=Fx+-.dxOx (3.47)

The conservation of mass equation for the control element can be stated asfollows:

[

net rate of change

]of mass of specieswithin the element [

flux of species

]out of the element

[

flux of species

]- into the element

[

loss or gain of the

]:I: mass of .the species

to reactIOns(3.48)

--,

1j

~

l

t,.j/

'ft~.I"

ti,i'1'l.,i...J

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3.3 Advective transport in reactor column 103

have

h*Caodx) = :x (vn*Cao) dx::!:~n*Caodx (3.49)

i§,th~ rate of accumulation (+) or loss (-) of chemicals within theel~ment due to reactions. We can reduce (3.49) to the form

(3.50)

he simple one-dimensional advective flow problem, which includes aHon term, the governing first-order differential equation is identical inl.to (3.1). The boundary and initial conditions governing (3.50) can be

tten as follows:

(3.51)

(3.52)

In the instance where there is loss of the chemical concentration due to re-active processes we take the negative component of the right hand side of(3.50).Furthermore we introduce a change of variables such that

C

f} = C ; 8 = ~t. - x~0 ' T-- V (3.53)

t Consequently we can write

Of} of}-+-=-f}08 OT

(3.54)

.~

........

C=o , t = 0 ; x2':O

C=Co ; t> 0 ; x=O


with the appropriate boundary conditions being

(3.55)

(3.56)

The characteristic equations are

dO d7 d1]

T = T = -~ (3.57)

which gives

(3.58)

(3.59)

These linear ordinary differential equations can be integrated to give thesolution

C

Co = 0 ; x > vi (3.60)

and

C - -xfJv ; x<vt--eCo

(3.61)

In this simple example we have considered only "advective transport" pro-cesses of the species in the porous medium. There is a further mechanism

,

I.

1

i'it

ff.

f

I

I; ""

W

1] = 0 ; 0=0 ; 7;:::0

1]= 1 ; 0> 0 ; 7=0

d70;:::0 ;dO = 1 ;

7;:::0

d1] 1]= 0 ; 0 = 0 ; 7;:::0d7 = -1] ; 1]= 1 ; 0> 0 ; 7=0

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3.3 Advective transport in reactor column

Iledti/diffusivetransport" which accounts for transport by virtue of a gra-&redt1:>fthe concentration of the species at a point. These processes can6CCl1f§inmltaheOuslyand the degree to which one process is dominant will

'd!3p€)ndon the yelocity of flow and the mechanical properties which governthe diffu$iohprocesses. These will be discussed in a later chapter.

3;3.2 Governing equation - generalized formulation

The formulation of the advective transport problem presented in the previ-oJIssection is useful from the point of view of illustrating the distinct factorsthat should be taken into consideration in the development of the final gov-erning equation)n one-dimension. It is also possible to present a generalizedformlllation of the advective transport problem applicable to three dimen-sions.

Thefundamental definitions required for the three dimensional formulationwill now be briefly reviewed. Consider an arbitrary region V of a porousmediumwith surface S. (Figure 3.7)

n

JFm

"-'If,

s

elementalsurface area, dStotal

volume. V

Figure 3.7: Advective transport in a porous medium.

The porosity measure n* defined by (3.42) and (3.44) is equally applicableto the three dimensional case. The concentration of the species per unit vol-ume of the fluid is defined by C(x, t); indicating that the concentration is afunction of the position vector )C. $imill:\,rlythe concentration of the speciesper unit total volume of the porous medium is defined by C(x, t). Again at

105

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Chapter 3: Partial Differential Equations in Mechanics 1