Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 35

If the normal derivative of the dependent variable u is known on the boundary S ,
the boundary condition takes the form
∂

S = ϕ ( S = ϕ (

u ∂ u

, ) ) .

M t or M (1.80)

∂ n ∂ n

Such boundary conditions are called boundary conditions of the second kind .

Boundary Conditions of the Third Kind

In some applications, boundary conditions are given by neither the value of depen-
dent variable nor its derivative, but by a linear combination of the two.
Consider the longitudinal vibration of a rod of length l

( 0 ≤ x ≤ l ) and cross area

S ,fixedat x = 0 and connected to a spring of elasticity k at x = l . A force balance at

x = l yields the boundary condition

YS ∂

u ∂ u k

YS u

x = −

ku or

x +

= 0 ,

∂ ∂

is the longitudinal displacement.
Therefore a linear combination of the dependent variable u and its derivative is
specified at the endpoint x

where Y is the Yung's modulus and u = u ( x , t )

l .
Consider heat conduction in a solid body of boundary surfaces S surrounded by
a fluid of temperature u 1 (Fig. 1.2). Let u and h 1 respectively be the body temper-
ature and the convective heat transfer coefficient between S and the surrounding
fluid. An energy balance for the differential element

=

S of S leads to the bound-
ary condition, by using the Fourier law of heat conduction and the Newton law of
cooling,

Δ

S = ∂

hu S = ϕ ( S ∂

u u

− k Δ h 1 Δ S ( u − u 1 ) | S or

n +

M , t ) , M ∈ S ,

∂ n ∂

where n is the outward-drawn normal of S and k is the thermal conductivity of the
body ( k

> 0), h = h 1 / k > 0, ϕ ( M , t )= h 1 u 1 / k .

Fig. 1.2 Energy balance for differential element Δ S