A Third-Order -Laplacian Boundary Value Problem Solved by an SL Lie-Group Shooting Method
The boundary layer problem for power-law fluid can be recast to a third-order -Laplacian boundary value problem (BVP). In this paper, we transform the third-order -Laplacian into a new system which exhibits a Lie-symmetry SL. Then, the closure property of the Lie-group is used to derive a linear transformation between the boundary values at two ends of a spatial interval. Hence, we can iteratively solve the missing left boundary conditions, which are determined by matching the right boundary conditions through a finer tuning of . The present SL Lie-group shooting method is easily implemented and is efficient to tackle the multiple solutions of the third-order -Laplacian. When the missing left boundary values can be determined accurately, we can apply the fourth-order Runge-Kutta (RK4) method to obtain a quite accurate numerical solution of the -Laplacian.
The power-law fluids have been called the Ostwald-de Waele fluids which have been well examined in the past several decades, because the constitutive equation of such a fluid not only gives a good expression for a large portion of the non-Newtonian fluids but also encompasses a Newtonian fluid as well. The theoretical boundary layer theory for power-law fluids was first investigated by Schowalter , and then Acrivos et al.  obtained a similarity solution. The experimental results that a significant drag reduction can be achieved by injecting fluid into the boundary layer motivated the investigations of the non-Newtonian boundary layer flows with injection or suction at the surface. Flows with suction or injection through a porous wall are of practical interest for cooling, delaying transition to turbulence, and the prevention of separation in an adverse pressure gradient.
The drag force inside the shear layer is a consequence of pressure distribution on the surface. Realizing the nature of this force by a mathematical modelling to predict the drag force and the associated behavior of fluid flow has been the focus of considerable research. The reason for the interest in the analysis of the boundary layer flows along solid surfaces is the possibility of applying the theory to the efficient design of supersonic and hypersonic flights. Besides, the mathematical model considered in the present research has importance in studying many problems of engineering, meteorology, and oceanography, for example, Howell et al. , Nachman and Callegari , Ozisik , Schlichting , Shu and Wilks , and Zheng and Zhang .
We assume that the moving flat plate is semi-infinite with a porous surface and that the plate is moving at a constant speed in the direction parallel to an oncoming flow with a constant speed . By the assumption of incompressibility and the conservation of momentum, the laminar flow satisfies In the above, and are the coordinates attached to the plate in the horizontal and perpendicular directions, and and are, respectively, the velocity components of the flow in the and directions. The fluid density is assumed to be a constant.
The shear stress is governed by a power law where is a constant and the power reflects the discrepancy to the Newtonian fluids (with ). The case with is the power law of pseudoplastic fluids, and is the dilatant fluids. The corresponding boundary conditions are given by
After introducing a similarity variable and a stream function with we can obtain which is subjected to the following boundary conditions: In the above, is the velocity ratio. When , we have a reverse flow attached near the boundary. When , the speed of the oncoming fluid is larger than that of the plate. When , the speed of the moving plate is faster than the speed of the oncoming fluid. The term is a constant related to the situation of suction if it is negative or injection if it is positive.
Previously, the author and his coworkers have developed the Lie-group shooting method based on the Lorentz-group [9, 10] to solve the boundary layer equations [11–14]. Liu et al.  have found the multiple solutions of boundary layer problem by an enhanced fictitious time integration method for the nonlinear algebraic equations discretized from the governing equation. In this paper, we propose a more simple and powerful Lie-group shooting method directly based on the three-dimensional special linear group to solve the -Laplacian boundary layer equations with multiple solutions. Liu  used the Lie-group shooting method to solve the eigenvalue problem of the second-order Sturm-Liouville equation. Recently, Liu  has successfully applied the Lie-group shooting method to effectively solve the Falkner-Skan boundary layer problem. The present study is an extension of these researches.
The operator in (6) is one sort of the third-order -Laplacian . Besides the non-Newtonian fluid [19–21], the -Laplacian operator arises in some different physical and mathematical modelling of the combustion theory , population dynamics [23, 24], and also the Monge-Kantorovich partial differential equation . There had been many papers for studying the second-order -Laplacian about its existence of positive solutions. Only a few papers are devoted to the numerical solution of the third-order -Laplacian.
This paper is organized as follows. In Section 2, we consider (6) as a special case of the third-order -Laplacian, and then by a translation of the variable to a new variable with a positive value, we can transform the -Laplacian into a new system, which exhibits a Lie-symmetry of . We introduce some mathematical requirements of the Lie-group formulation of the resulting ODEs and construct a Lie-group shooting method based on the Lie-group . The boundary conditions that the present Lie-group shooting method can be applied to are discussed. In Section 3, we test the performance of the newly developed Lie-group shooting method for several examples, and in Section 4, the multiple solutions of the boundary layer problem of power-law fluids are investigated. Finally, we draw conclusions in Section 5.
2. An Lie-Group Shooting Method
Equation (6) is a special case of the following -Laplacian: of which the boundary conditions have many types to be discussed below. In the above, , where .
In order to develop an Lie-group shooting method, we suppose that , such that there exists a constant rendering Then, (8) becomes
2.1. A Group-Preserving Scheme
The differential equations system (15) is highly nonlinear due to the appearance of and in the coefficient matrix; however, it allows a Lie-symmetry : because of . Here, for saving notations, we use and . The above is a fundamental matrix of (15).
Accordingly, we can develop a group-preserving scheme (GPS) to solve (15) where denotes the numerical value of at the discrete space and . This Lie-symmetry is known as the three-dimensional real-valued special linear group, denoted by .
2.2. A Generalized Midpoint Rule
Now, we prove the following closure property of the Lie-group : By the assumptions of and , we have and . Then, by using the following result: it is straightforward to verify that , which means that . Thus, we have proven (21).
Because each in (20) is an element of the Lie-group , and by the above closure property of the Lie-group is also a Lie-group element of , denoted by . Hence, we have This is a one-step Lie-group transformation from to , acting by .
However, it is very hard to obtain an exact solution of because the differential equations system is highly nonlinear. Before the derivation of a suitable form for , let us recall the mean value theorem for a continuous function , which is defined in an interval of . The mean value theorem asserts that there exists at least one , such that the following equality holds where the value of depends on the function . In terms of the weighting factor , we can write . Therefore, it means that there exists at least one , such that (24) is satisfied. The above theorem enables us to evaluate the value of the integral in (24) by an area of a rectangle with a width times a height , where is calculated by a mid-point rule with a suitable .
Because is a solution of (16), we can formally write it by an exponential mapping When is not a constant matrix, in general we do not have a closed-form solution of . However, motivated by the above mean value theorem and to be a reasonable approximation, we can calculate in (23) by a generalized mid-point rule, which is obtained from an exponential mapping of by taking the values of the variables in at a suitable mid-point: , where is an unknown constant to be determined by the shooting method. So we can compute this by which is corresponding to a constant matrix where and are supposed to be constant.
This Lie-group element generated from such a constant matrix has a closed-form solution. If and , then we have where denotes the length of the interval .
If and , then we have In the above, we have taken
For the special case of , we can derive
2.3. Specification of Boundary Conditions
For the third-order ODEs, there are several different type boundary conditions. In this section, we study this problem that under what type boundary conditions the present Lie-group shooting method is applicable.
Let denote, respectively, the left-end and right-end boundary values of . For linear type boundary conditions (separable or nonseparable), we can describe the boundary conditions by the following equation: where both and are matrices and is a constant vector, which might be zero.
In order to demonstrate the above idea about the specification of the boundary conditions, of which the present Lie-group shooting method is applicable, let us take and and (6) under the following boundary conditions: where we have replaced by a finite number . In terms of (33), we have Then, by (28) or (29), we can obtain Thus, the above matrix is not invertible, because of . So we can conclude that for the power-law fluid under the boundary conditions (36), the present method is not applicable. In Section 4, we will give another type approach.
Physically, we can specify that at a large distance from the boundary layer the shear stress is quite small. Then, instead of (36), we can specify where is a small number. In terms of (33), we have Then, by (28) or (29), we can obtain which is invertible, due to . Then, the present method is applicable. Below, we discuss the Lie-group shooting solution for the boundary layer problem of the power-law fluid under the boundary conditions (39).
2.4. An Lie-Group Shooting Method
In order to demonstrate the application of the Lie-group shooting method to find the missing left boundary conditions, as a representative case, let us take and , and then we can recover to (6) which is subjected to the following boundary conditions: where and are given constants, and we use a large value , say , to replace the last boundary condition in (6), and also is changed to as just mentioned above.
The stepping technique developed for solving the initial value problem (IVP) requires both the initial conditions of , and for the third-order ODEs. Starting from the initial values of , and , we can numerically integrate the following IVP step by step from to : where some unknown initial values are to be found by the Lie-group shooting method.
In (43), and are given, but is an unknown constant to be determined such that we can satisfy the target equation of . Starting from an initial guess of , and , we can solve the unknown initial value by the following iterative processes: which are obtained from (23). Inserting the initially guessed values of , and and the given values of , and into (28) or (29) with a specified , we can evaluate , and then by (48), we can generate the new values of , and , until they are convergent. If the new values of , and converge to satisfy the following convergence criterion: then the iterations stop. Here, and denote, respectively, the th and the th iteration values of . They are defined similarly for and .
For a trial , we can calculate from the above equations by a few iterations and then numerically integrate (44) by the fourth-order Runge-Kutta method (RK4) from to and compare the end value of with the exact one , which is a target equation to be matched. Indeed, we need to find the root of the equation , where is a numerically integrated result, depending on . It can be done in practice by adjusting the value of to a point such that the curve of mismatching error is intersected with the zero line at that point.
3. Numerical Examples
Example 1. First, we consider the following -Laplacian:
We assume that the closed-form solution is . Hence,
We can use the following equations to iteratively solve the unknown initial value of :
We take for and for and use the Lie-group shooting method developed in Section 2.4 by adjusting the value of . If the target equation is satisfied, then we obtain the numerical solution.
The convergence criterion is . Although under this stringent convergence criterion the iteration process to find is convergent very fast as shown in Figure 1(a), where for the iteration numbers are between 16 and 23 for the case . In Figure 1(a), we also plot the mismatching errors with respect to in a range for , while in a range for . Both have an intersection point with the zero line. Then, through a finer tuning of the value to for the case and to for the case , we can match the right-end boundary condition very precisely with an error in the order . The numerical solutions of and are, respectively, plotted in Figures 1(b) and 1(c), which are almost coincident with the closed-form solutions. Therefore, we plot the numerical errors, which are the absolute differences between exact solutions and numerical solutions, in Figure 2 for and . It can be seen that the numerical results are quite accurate.
Example 2. Then, we consider a different case of (50) by
Similarly, the closed-form solution is . Hence,
We can use the following equations to iteratively solve the unknown initial value of :
We take and . If the target equation is satisfied, then we obtain the numerical solution. In Figure 3(a), we plot the mismatching error and the number of iterations with respect to in a range . Then through a finer tuning of the value to , we can match the right-end boundary condition very precisely with an error in the order . Upon comparing the numerical solutions with the closed-form solutions, the numerical errors of and are, respectively, plotted in Figures 3(b) and 3(c). It can be seen that for the accuracy is in the order of , while that for the accuracy is in the order of .
Example 3. We consider the same equation (53) but under the following boundary conditions:
When the closed-form solution is given by
the term is given by
For the above boundary conditions, we can use the following equations to iteratively solve the unknown initial value of :
We take and . If the target equation is satisfied, then we obtain the numerical solution. When we plot the mismatching error with respect to in a range in Figure 4, we find that there exist two intersection points at and , which means that there exist two solutions. In (57), we only give one exact solution, but we do not have another solution as given in a closed-form.
Then, through a finer tuning of the value of , we can match the right-end boundary condition very precisely with the error in the order , and the first numerical solution is obtained with , while the second numerical solution is obtained with . Upon comparing the first numerical solution with the closed-form solution in (57), the numerical errors of , and are, respectively, plotted in Figures 5(a), 5(b), and 5(c). It can be seen that all the accuracies are in the order of . In Figure 6, we compare the first numerical solution and the second numerical solution with the exact one. It can be seen that when the first numerical solution is almost coincident with the exact solution, the second numerical solution is obviously different from the first numerical solution. We can also observe that the second numerical solution satisfies the boundary conditions in (56) very precisely.
Example 4. Then, we consider a more general boundary conditions for the following -Laplacian:
Similarly, we consider a translation with , such that we have
In terms of (33), we can write
Thus, from (62) and (23), we can solve The above five equations can be used to iteratively solve the five unknowns of , and . We note that by the last boundary condition in (61).
As a demonstrative case, we take , that is, , and to be the exact solution, where can be computed from (60) by inserting the above . The boundary conditions are given by
We take . If the target equation is satisfied, then we obtain the numerical solution. When we plot the mismatching error with respect to in a finer range in Figure 7, we find that there exists one intersection point at . We can match the right-end boundary condition very precisely with an error being . Because there are many equations to be solved iteratively, the number of iterations as shown in Figure 7(a) is between 45 and 48, which is higher than the previous three examples. In Figure 7(b), we compare the numerical solution of with the exact solution , whose numerical error as shown in Figure 7(c) is quite accurate in the order of .
Alternatively, we consider a nonlinear perturbation of the above example under the same boundary conditions but with where we also fix . For this problem, we do not have a closed-form solution. However, we take , and by taking and , we can obtain two numerical solutions as shown in Figure 8. The numerical solution as shown by the dashed line is quite unstable. For the purpose of comparison, we also plot the numerical solutions obtained in the last example in (64) by the dashed-dotted lines in Figure 8. It can be seen that the solid lines are somewhat perturbed from the ones of the dashed-dotted lines, but the unstable ones are quite different from the above two solutions.
4. Power-Law Fluids
In this section, we consider the boundary layer problems of power-law fluid in (6). We use the Lie-group shooting method developed in Section 2.3 by adjusting the value of . However, before that we need to treat the difficulty mentioned in Section 2.3 if we use the boundary conditions in (36).
First, we need to point out that for this boundary layer problem we only consider that the function is convex, that is, . So the term defined in (30) is positive. Then, from (28), (23), and (32), we have where .
Let . From (67), we can derive a scalar equation to solve : Because , and are given from the boundary conditions, and is selected, we can apply the Newton method to solve the above equation, whose solution is denoted by . Hence, we have by the definition of . Furthermore, by (30), we have From (68) and (71), we can solve where . When is solved, is determined by (66); hence, (72) can be used iteratively to solve the two unknowns of and . We can satisfy the target equation by selecting the best value of .
Example 5. We fix , and , and the convergence criterion is . Although under this stringent convergence criterion the iteration process to find is convergent very fast as shown in Figure 9(a), where for the iteration numbers are all to be four. In Figure 9(a), we plot the mismatching error with respect to in the same range. It can be seen that the mismatching error curve is intersected with the zero line at a point near to 0.68. Then, through a finer tuning of the value to , we can match the right-end boundary condition very precisely with an error in the order of . The unknown initial value of (or ) is obtained. The numerical results of , and are, respectively, plotted in Figures 9(b)–9(d).
Example 6. We fix , and , and the convergence criterion is . As shown in Figure 10(a), where for the iteration numbers are all to be four, the mismatching error curve is intersected with the zero line at two points a and b. Then, through a finer tuning of the values to and , we can match the right-end boundary condition very precisely with the errors in the order of , and thus we obtain two numerical solutions as compared in Figures 10(b)–10(d). For the first solution, the unknown initial value of (or ) is obtained, while for the second solution the unknown initial value of (or ) is obtained. The numerical results of and are very close to that obtained by Liu .
Example 7. We fix , and . As shown in Figure 11(a), the mismatching error curve is intersected with the zero line at three points a, b, and c. Then, through a finer tuning of the values to , and , we can obtain three corresponding numerical solutions as compared in Figures 11(b)–11(d). For the first solution, the unknown initial value of is obtained, and for the second solution the unknown initial value is , while that for the third solution the unknown initial value is . For the last solution, grows rapidly after .
In the present paper, we have offered a rather accurate and simple method with only a few iterations to find the unknown left boundary conditions by applying the Lie-group shooting method to the third-order -Laplacian boundary value problems. Also, as an application, we have solved the boundary layer problems of power-law fluids by the present method. The Lie-group shooting method allows us to express the missing left-end boundary conditions by the closed-form functions of , where the best is determined iteratively by matching the right-end boundary conditions. Because the iterations to find the missing left-end boundary conditions are convergent very fast, the Lie-group shooting method based on is quite computationally efficient. The new method was effective to find the multiple solutions, although for the highly nonlinear case with multiple unknown left boundary conditions.
The paper was supported by the Project NSC-99-2221-E-002-074-MY3, and the 2011 Outstanding Research Award from Taiwan’s National Science Council and the 2011 Taiwan Research Front Award from Thomson Reuters, granted to the author, are highly appreciated.
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