Page 492 - Introduction to Continuum Mechanics

476 Current Configuration as Reference Configuration

Part B Nonlinear Viscoelastic Fluid

8.5 Current Configuration as Reference Configuration
Let x be the position vector of a particle at current time t, and let x' be the position vector
of the same particle at time T. Then the equation

defines the motion of a continuum using the current time t as the reference time. The
subscript t in the function x' t(x, r) indicates that the current t is the reference time and as such
x',(x, r) is also a function oft
For a given velocity field v = v(x, t}, the velocity at the position x' at time T is v = v(x', r).
On the other hand, for a particular particle (i.e., for fixed x and t), the velocity at time r is given

Equation (8.5.2) allows one to obtain the pathline equations from a given velocity field, using
the current time t as the reference time.

Example 8.5.1
Given the velocity field of the steady unidirectional flow

describe the motion of the particles by using the current time / as the reference time.
Solution. From the given velocity field, we have, the velocity components at the position
(xi',X2,.£3') at time r:

Thus, with x' = x'fo, Eq. (8.5.2) gives

But, at r = t, X2 = *2> therefore, for all r