Exact Solutions to the Navier-Stokes Equation

Unsteady Flow in a Tube

 The boundary condition is

(40)

with the initial conditions

(41)

Let

(42)

Equation (39) reduces to

(43)

The boundary and initial conditions (40) and (41) now become

At   .

(44)

At   .

(45)

To determine the solution, the method of separation of variables is used. That is let

(46)

Equation (43) then becomes

.

(47)

From Equation (47), it follows that

,

(48)

.

(49)

The solutions to Equations (48) and (49) are given as

(50)

(51)

where  are Bessel function of first and second kind of zeroth order.

The boundary conditions are

,

(52)

and

.

(53)

Equation (53) is a characteristic equation. The corresponding eigenvalues, , are given as

,

(54)

The general solution for Equation (43) then is given by

.

(55)

Using the initial condition

(55)

then

(56)

or

(57)

Hence,

(58)

and

(59)

Variation of the velocity profile in the pipe is shown schematically in Figure 5.

Figure 5. Variations of velocity field in a tube subject to a step change in pressure.