Similarity Solutions to Unsteady Pseudoplastic Flow Near a Moving Wall

It is shown that similarity solutions to a partial differential boundary value problem for power law fluids may be generated by the application of a multiparameter stretching group. Furthermore, the resulting ordinary differential boundary value problem arising from the initial group analysis is itself transformed to an initial value problem by the application of a further stretching (oneparameter) group transformation. The resultant initial value problem is solved by a standard forward marching method. Mathematics Subject Classification: 34B40, 35C06, 35G30, 76M55, 76M60


Introduction
In this paper, we present a similarity group analysis of the equation of motion of a semi-infinite body of pseudoplastic (power law) fluid occupying the halfspace 0  y [2,8], that is 1466 where ) , ( t y u is the x-component of velocity in a cartesian system.The fluid is set in motion at time 0  t by imparting a constant velocity U on the boundary ; 0  y since the effects of this velocity is expected to decay and eventually vanish as we move into the body of the fluid, equation (1.1) is to be solved with the boundary conditions In equation (1.1), m /


is the ratio of the fluid density  to the empirical constant , m characteristic of the fluid, defined via equation ( 2) of [2]; the parameter n also is characteristic of the fluid [2].The solution process proposed here for the partial differential equation (1.1) with boundary conditions (1.2), and where the novelty lies, involves the complete reduction of the original problem (1.1)/(1.2) to an initial value problem entirely by stretching group transformations.
The solution process and paper are organized as follows.The first part of the process, presented in section 2, involves the identification of a 3-parameter stretching group transformation that the original problem (1.1)/(1.2) is invariant under [5,6].This is achieved by demanding [5,6] that the original problem (1.1)/(1.2) be invariant under the more basic stretching [9] 5-parameter group Next, once the 3-parameter stretching group is determined, it is then used, in section 3, to transform the partial differential boundary value problem (1.1)/(1.2) into an ordinary differential boundary value problem.To do this, we assume the existence of an implicit solution [4] to the original problem with g an arbitrary function, which is to be invariant under the 3-parameter group of section 2. This invariance requirement on (1.4) leads to two systems of partial differential equations having identifiable [5,6]

Determining the r-parameter group
Following Moran [5] and Moran and Marshek [6], we will use the 5-parameter stretch transformation (with the group parameters to construct an r-parameter group transformation, which we write as a group, , r G and a subgroup, r S [5,6], that is It will be apparent that the r-parameter group (2.2) is simply another way of writing the 5-parameter group (2.1), that is, the 5 parameters are not independent and so .

 r
This means that equation (1.1) and the boundary conditions (1.2) will be invariant under the r-parameter group transformation a fortiori.
Applying the transformations (2.1) and the chain rule to equation (1.1), we find (subscripts on u or u implying partial differentiation)

W. Robin
If we require equation (1.1) to be invariant under the transformation (2.1), then from (2.3) we must have Also, if we demand that the boundary conditions (1.2) be invariant under the transformations (2.1), then we require From (2.4) and (2.5), we see that we may take 2 1 , a a and 3 a as the basic independent group parameters, with the transformation (2.2) now taking the form of a 3-parameter group transformation, which we write as a group, , 3 G and a subgroup, 3 S [5, 6], that is In the next section we show that by determining the group invariants associated with 3 S and 3 G [5, 6], that the partial differential boundary problem (1.1)/(1.2) can be transformed to a corresponding ordinary differential boundary problem, (3.3) below.

The 3-parameter group analysis
The next stage of the analysis, is the reduction of the partial differential boundary value problem ( a any (absolute [5,6]) group invariant for the group 3 G must satisfy the partial differential system (where we may drop the bars) That is, we have three independent linear homogeneous partial differential equations, in five variables, which has two solutions [6].
However, if we consider the subgroup , (where we drop the bars again) which has one solution [6].Apparently the sole solution  to (3.5) is also one of the two solutions to the 3 G equations (3.3).

W. Robin
From the general theory of the solution of equations such as (3.3) and (3.5) [6], we know that we may look for particular group invariants of the form The parameters As the invariants (3.7) are functionally independent, the general solution, , g of (3.3) is a (differentiable) function [6], f say, of  and , ĝ so that Solving the implicit relation (3.8) for g ˆ in terms of, ,  we see that we may rewrite (currently) an arbitrary function.

Finally, since ), ( UF u 
we can now reduce the original partial differential equation (1.1), with boundary conditions (1.2), to a boundary value problem in one variable only.That is, an ordinary differential equation with appropriately transformed boundary conditions.Indeed, under the similarity transformation (3.9), equations (1.1) and (1.2) are transformed into the following ordinary differential boundary value problem where the dash denotes differentiation with respect to . The relations (3.9) and (3.10) are essentially those obtained by Bird [2] from ad hoc dimensional considerations.In fact, , ) where r is defined by equation ( 5) of reference [2].In what follows, it proves convenient to transform the system (3.10)into the form obtained by Bird [2] by making the transformation to get (the dash denotes differentiation with respect to 'new'  ) ; 0

Reduction to an initial value problem: further group analysis
We wish to solve the system (3.11) to find the dimensionless velocity, , F from which we can get the x-component of the velocity, , u from .UF That is, we wish to solve the ordinary differential boundary value problem ; Bird solved (4.1) using a semi-analytic method.[2].In what follows, we solve the system (4.1) for , F following Phan-Thien [8], by turning (4.1) into an initial value problem.To do this we introduce, first, the change of variable , 1 F G   when the system (4.1)becomes (the variable G is not to be confused with the transformation group) ; 0 The boundary value problem (4.2) may be transformed into an initial value problem now by a one-parameter stretching group transformation (see, for example, the books by Na [7] or Bluman and Coles [3] or Ames [1]).If we define  Transforming the boundary condition at infinity, we find that With everything now in place, the solution procedure is as follows.First, for any , n we solve the initial value problem ; 0 (4.12)

Results and conclusions
The initial value problems (4.11) and (4.12) were solved using a standard fourthorder Runge-Kutta routine and the results of this solution process are presented in Table 1 and Figure 1, for the range of values of n considered by Bird [2].
Table 1.Numerical values for calculating velocity profiles.The approach here is restricted to a basic similarity analysis through the medium of stretching groups.Phan-Thien [8] has developed a Lie-group solution process for problems slightly more general than (1.1)/(1.2),from which similarity solutions may be extracted.The method developed here is, hopefully, simpler.In conclusion, an example of a similarity approach to the group analysis of partial differential equations has been presented, the partial differential equation, describing unsteady flow in a pseudo-plastic fluid, being reduced to an ordinary differential equation.The complete transformation, including boundary conditions, has led to an ordinary differential boundary value problem which in turn was transformed to an initial value problem and solved in a standard manner.

2 )
The demand that equation (1.1) and the boundary conditions (1.2) be invariant under the 5-parameter group transformation (2.1), leads to relations that determine both the order r of the transformation group (2.2) and the values of the group indices

3 S
we seek an invariant, , independent linear homogeneous partial differential equations, in four variables, that is, by partial differentiation of (3.4) with respect to substituting that the relations (3.6) into equations (3.3) and (3.5): this leads to two sets of simultaneous equations for

2 
we require another equation, and we obtain this by forcing a condition on the unknown initial condition

solutions to unsteady pseudoplastic flow 1467
[1,3,7]ity solutions, which, in turn, allows a similarity transformation of the partial differential boundary value problem (1.1)/(1.2) into an ordinary differential boundary value problem.The final part of the reduction process, presented in section 4, uses another[1,3,7]Similarity 1.1)/(1.2) into an ordinary differential boundary value problem.Critical to this analysis is the determination of the group invariants of [4,8]h that the implicit solution is written as[4,8] Robinvelocity has fallen-off to % 1 of the velocity of the moving wall (the xz-plane).The numbers in brackets in Table1refer directly to Bird's results.The comparison with Bird's results in Table1help to validate the solution method.
being the values of the reduced variable  for which the fluid 1474 W.