4.10 A gas is compressed from an initial volume of 0.42 m3 to a final volume of 0.12 m3. During
the quasi-equilibrium process, the pressure changes with volume according to the relation P =
aV + b, where a = -1200
1. A well-insulated rigid tank contains 3 kg of a saturated liquidvapor mixture of
water at 200 kPa. Initially, three-quarters of the mass is in the liquid phase. An
electric resistance heater placed in the tank is no
0.5-lbm of a saturated vapor is converted to a saturated liquid by being cooled in a
weighted piston- cylinder device maintained at 50 psia. During the phase conversion,
the system volume decreases
An insulated pistoncylinder device contains 0.8 L of saturated liquid water at a
constant pressure of 120 kPa. An electric resistance heater inside the cylinder is
turned on, and electrical work is done on the water in the amount of 1400 kJ.
Why is it necessary to learn the complex variable theory ?
Useful for certain types of integrals (the residue theorem).
1 + x6
1 + sin 2
Useful for 2-D uid mechanics/solid mechanics problems 1 .
u = v =
Important little integral
z n dz =
2i n = 1
For n = 0, 1, 2, 3, . . ., z n is analytic everywhere, hence
z n dz = 0
thanks to the Cauchy theorem.
For n = 1,
= ln e2i ln e0
If f (z) has (isolated) singular points, z1 , z2 , z3 , . . . zn inside a closed loop, C, it follows
f (z)dz = 2i
Res(f, zj ).
Application of Residue Theorem
There are three types of integrals where you can apply the residue
The function, f (z) = 1/(1 + z), can be expanded by geometric series around z = 0 as
= 1 z + z2 z3 + . . .
From the ratio test 1 of convergence, the series above is convergent if |z| < 1. However, the same function
can be also exp
Dierentiability (Cauchy-Riemann relation)
f (z) = z = x iy
dierentiable at z = 0 ?
The denition of dierentiability is such that
f (z + z) f (z)
must exist and converge to a unique value regardless of the way z approaches to 0. To s
Examples of MWR Applied to S-L systems
Trigonometric functions (sine)
1. S-L system:
Ly = y,
(y + y = 0),
y(0) = y(1) = 0.
2 sin n x,
n = (n)2 ,
(yn , ym ) = nm .
3. Galerkin method
Ly = y,
Review of eigenvalues/eigenvectors in linear algebra
When a square matrix, A, is symmetrical, i.e.
A = AT
(Ax, y) = (x, Ay),
the following properties are held:
1. All the eigenvalues, i s, are real.
2. All the eigenvectors, en , are orthogonal, i.e
The diusion equation is a parabolic partial dierential equation and is exemplied by the following system
(after setting all the physical constants as unity):
Lu(x, t) =
Initial condition: u(x, 0) = f (x)
(0 < x < 1)
Finite Dierence Method Diusion equation
Denote the step size in the x direction as x, in time as t, and an approximation of
u(mx, nt) as um,n . Approximate partial dierentiation by dierence1 as
um+1,n 2um,n + um1,n
This topic corresponds to Sections 17.6 and 17.7 in the textbook.
Definitions of vector space
A set S is called a vector space or a linear space if the following conditions (axioms) are satised:
1. a + b = b + a (commutative law)
2. a + (b +
Application of Complex Variable Theory to Fluid Mechanics
For an incompressible ow,
y = ,y
x = ,x ,
and is written as
From eq.(1), u and v can be satised if
and from equation eq.(2), u and v can be satised
f (z , ln z)dz
The residue theorem can be used to evaluate singular integrals if
1. f (z) is a rational function.
2. |z 2 f (z)| < (for convergence)
3. is non-integer.
The function to be integrated (integra
Classification of singular points
If the Laurent series expansion of f (z) in the immediate neighborhood of z = a is expressed as
f (z) =
+ . . . + c0 + c1 (z a) + . . .
we say z = a is an m-th order pole.
f (z) =
Method of Weighted Residuals
Our objective is to solve general linear equations in the form
Lu = c,
where L is a linear operator (dierential operators, matrices etc . . . ), u is the unknown function and c is a
An approximate solution
The set of simultaneous equations to obtain the unknown coecients, ci , for the least square method (best
c (f , e )
(e1 , e1 ) (e2 , e1 ) . . . (en , e1 )
(f , e2 )
(e1 , e2 ) (e2 , e2 ) . . .
sin (1 + i).
Exponential and logarithmic functions
To evaluate complex exponential functions and logarithmic functions, it is necessary to express a complex
number in polar form. Wi
Consider a linear system of
Lu = c,
L = I + M,
where I is the identity operator. If the norm of M is small such that
|M | < 1,
then the solution, u, is expressed as a convergent series as
(I + M )1 c
(I M + M 2 M 3 +
For the correct pronunciation of Liouville, see the footnote on page 887 and
The Sturm-Liouville system is a boundary value problem dened by
Ly = y
(a < x < b)
+ q(x) ,
where w(x)is a we