Abstract
The migration of melt through the mantle of the Earth is governed by a third-order nonlinear partial differential equation for the voidage or volume fraction of melt. The partial differential equation depends on the permeability of the medium which is assumed to be a function of the voidage. It is shown that the partial differential equation admits, as well as translations in time and space, other Lie point symmetries provided the permeability is either a power law or an exponential law of the voidage or is a constant. A rarefactive solitary wave solution of the partial differential equation is derived in the form of a quadrature for the exponential law for the permeability.
1. Introduction
The one-dimensional migration of melt upwards through the mantle of the Earth is governed by the third-order nonlinear partial differential equation where is the voidage or volume fraction of melt, is time, is the vertical spatial coordinate, and is the permeability of the medium. The special case in which is a power law, has been studied extensively, and solitary wave solutions have been derived [1–12]. In this paper will initially not be specified. For arbitrary forms of , (1) does not depend explicitly on and , and therefore it admits the Lie point symmetries We will determine the forms of for (1) to admit other Lie point symmetries besides the Lie point symmetries (3). This would be a significant property for (1) to posses because invariant solutions could then be constructed. One of the forms obtained for is an exponential law relating the permeability to the voidage. We will derive a new rarefactive solitary wave solution of (1) with the exponential law for the permeability.
The variables , , , and in (1) are dimensionless. The voidage is scaled by the background voidage . The background state is therefore defined by . The characteristic length in the -direction, which is vertically upwards, is the compaction length defined by where is the coefficient of shear viscosity of the melt and and are the bulk and shear viscosity of the solid matrix, respectively. We will assume that and are constants as did Barcilon and Richter [3]. Scott and Stevenson [1] assume that and are power laws of the voidage . The characteristic time is defined by where is the acceleration due to gravity and is the difference between the density of the solid matrix and the density of the melt. The permeability is scaled by and therefore When the voidage is zero, the permeability must also be zero and therefore
In the derivation of (1), it is assumed that the background voidage satisfies . An outline of the derivation of (1) when satisfies the power law (2) is given by Nakayama and Mason [5]. The derivation is readily extended to the general case in which .
An outline of the paper is as follows. In Section 2 the Lie point symmetries of (1) are investigated and the forms of are determined for (1) to admit, as well as the Lie point symmetries (3), other Lie point symmetries. In Section 3 a new rarefactive solitary wave solution of (1) is obtained when depends on through an exponential law. Finally, the conclusions are summarized in Section 4.
2. Lie Point Symmetries
In this section will not be specified initially. We will investigate the Lie point symmetries of (1) and the forms of for these symmetries to exist.
Equation (1) is as follows: where a subscript denotes partial differentiation. We look for Lie point symmetries of the form The coefficients and in the Lie point symmetry (9) should not be confused with the bulk and shear viscosities in the characteristic quantities (4) and (5). The invariance criterion is where the third prolongation of is of the form The remaining terms in are not required in (10) and with summation over the repeated index, , from 1 to 2. The total derivatives and are defined by
We replace in the determining equation (10) using the partial differential equation (8) and then separate the determining equation according to the following partial derivatives of : It follows directly from (14) that where , , , and still have to be determined. Using (15), the invariance criterion separates further into the following system of equations:
For arbitrary forms of , (16) to (21) are satisfied by where and are constants and (1) admits the Lie point symmetries (3). We will seek possible forms of for which (1) admits, as well as (3), other Lie point symmetries. From (18) we see that there are two general cases depending on whether the permeability, , depends on or is constant.
2.1. Permeability Depends on Voidage
Consider first the case in which depends on the voidage, ; that is, the permeability is not constant, so that Then, from (18), , where is a constant. By differentiating (20) with respect to we find that , and from (21) it follows that , where is a constant. Hence, from (15), From (17), where and are constants. Also from (16) and (19), and therefore where is a constant. Equations (16) to (21) reduce to It is readily verified that if satisfies (29), then satisfies (28) identically. Equation (28) therefore does not need to be considered further.
When (23) is satisfied Lie point symmetries of (1) exist provided that satisfies (29) and are given by (24), (25), and (27). This case separates into two subcases depending on whether or .
2.1.1. The Case
Consider first . The general solution of the ordinary differential equation (29) for is where is a constant. But since , it follows that , and since we obtain Since is not a constant, . Equations (27), (25), and (24) become The three Lie point symmetries of (1) with the power law (30) for are presented in Table 1. The results agree with those derived by Maluleke and Mason [7, 9] for the generalized magma equation with , where is the exponent in the power law relating the bulk and shear viscosities of the solid matrix to the voidage.
2.1.2. The Case ,
When but , the general solution of (29) is where is a constant. Since is not constant, , and because the permeability increases as the voidage increases, . Also, and therefore When , then If is large, is small. However, the exponential law (34) for does not satisfy the condition . It is not a suitable model for physical phenomena with small values of the voidage. For instance, it would not be suitable to describe compressive solitary waves which contain [8, 10–12]. It is suitable for describing rarefactive solitary waves which satisfy and this will be considered in Section 3. Equations (27), (25), and (24) become The three Lie point symmetries of (1) with the exponential law (34) for are presented in Table 1.
2.2. Constant Permeability
Finally, consider constant permeability, . Since , it follows that . The model does not satisfy and it cannot be used to describe physical phenomena in which the permeability depends on voidage.
When , (16) to (21) are reduced to Integrating (37) and (38) once with respect to gives where and are constants. By eliminating , we obtain and therefore where and are constants. From (40), Thus, from (15) where is arbitrary and satisfies (39) which is the partial differential equation (1) with . The Lie point symmetries of (1) with constant permeability are presented in Table 1. The results agree with those derived by Maluleke and Mason [7, 9] for and .
There are therefore three forms of for which (1) has Lie point symmetries in addition to (3), namely, the power law (31), the exponential law (34), and constant permeability.The Lie point symmetries of (1) with the three forms of are presented in Table 1. Equation (1) with the power law has been studied in detail. In Section 3 we will consider the exponential law and investigate rarefactive solitary wave solutions of (1) with given by (34).
3. Rarefactive Solitary Wave
When the permeability satisfies the exponential law (34), the partial differential equation (1) becomes We now derive a rarefactive solitary wave solution, with , of the partial differential equation (46).
The solution is an invariant solution of (46) provided that where is a linear combination of the Lie point symmetries of (46). The Lie point symmetries of (46) are given in Table 1. Consider the invariant solution generated by the linear combination (47) becomes where and are constants. The general solution of (49) is readily derived, and since we obtain where . The group invariant solution (50) is a travelling wave solution and the constant is the dimensionless speed of the wave.
Substitute (50) into (46). This gives the third-order ordinary differential equation and integrating (51) once with respect to we obtain where is a constant of integration. Since the right hand side of (52) depends only on , we integrate (52) with respect to . Now and (52) becomes but and integrating (54) once with respect to , we obtain where is a constant.
In order to obtain the three arbitrary constants, , , and , we impose three boundary conditions suitable for a rarefactive solitary wave. The background state is . Three boundary conditions for a rarefactive solitary wave are
The amplitude of the solitary wave is and has a local maximum when . Using (52) and (56), the three boundary conditions give Hence, and, expressed in terms of , Equation (56) becomes where Now from (53) and the boundary conditions (57), and since the left hand side of (61) is nonnegative, for a solitary wave to exist it is necessary that for .
Before proceeding further with the solution, we first investigate the properties of . In order to do that we define From (59), it follows that Equation (62) can be written in terms of as follows Now, and since for , it follows that for and , Hence, is an increasing function of . Also, Consider first . Since is an increasing function of , it follows that and for . Also, it can be verified that for . Hence from (66), for and a rarefactive solitary wave solution exists. Consider next . Then from (64), as , and since and is an increasing function of , it follows that . We still have by (68) and (70) is still satisfied for . Hence from (66), for and a rarefactive solitary wave solution does not exist.
The difference between for and is illustrated in Figure 1 for , and 2. For the permeability decreases as the voidage, , increases, which is generally not observed physically. When the permeability satisfies the power law (30), a rarefactive solitary wave solution exists for and does not exist for [5].
Now from (61) and (62), the rarefactive solitary wave solution is given by the following: where the wave speed is given by (59). In Figure 2, the solitary wave solution (71) is plotted against for a range of values of . In order to compare the solutions for different values of , the length is scaled in all cases with the same characteristic length, namely, calculated for . We see that the width of the solitary wave increases as increases. The increase in the permeability has the effect of spreading the solitary wave.
Consider now the dependence of the wave velocity, , on the amplitude of the solitary wave, . Since , it follows from (68) that The velocity increases as the amplitude of the solitary wave increases, and therefore larger amplitude waves travel faster. This property also holds for the solitary waves described by the power law (31) for [5]. Equation (72) is a special case of the general result that is an increasing function of for which was central to the proof of the existence of solitary wave solutions for . There is therefore a close connection between the property that larger amplitude solitary waves travel faster and the existence of solitary wave solutions. We also determine from (69) the limiting value This limiting value also holds for the dimensionless velocity of rarefactive solitary waves for the power law (31) with .
In Figure 3, the dimensionless wave velocity, , of the solitary wave, given by (59), is plotted against the amplitude for and 2. In order to compare for different values of , the wave velocity is scaled for each value of by the same characteristic velocity calculated for . We see that increases steadily with for both cases in agreement with (72). We also see that as increases the velocity increases. The increase in the permeability as increases allows the melt to propagate at a greater speed through the solid matrix.
4. Conclusions
One of the functional forms which the permeability must satisfy for the partial differential equation (1) to possess Lie point symmetries besides translation in time and space is the power law. This relation between the permeability and the voidage has been studied extensively to model the migration of melt through the mantle of the Earth. Mainly, travelling wave solutions have been considered, but more general group invariant solutions which include the third Lie point symmetry have been investigated by Harris and Clarkson [8].
Unlike the power law, the exponential law relating the permeability to the voidage is not valid at because, when the voidage vanishes, the permeability should also vanish. It can be used when the migration of melt through the Earth’s mantle is modelled as a rarefactive solitary wave with . For both the power law and exponential law, larger amplitude waves travel faster. There is a close connection between this property and the existence of solitary wave solutions.
The conservation laws for the partial differential equation (1) with the power law for have been investigated by several authors [3, 6, 9]. The number of conservation laws is finite, except possibly for which is not physical. This indicates that the solitary waves are not solitons. The conservation laws for the partial differential equation (46) with the exponential law for still need to be investigated. If it is found that the number of conservation laws is finite it would indicate that the rarefactive solitary waves with the exponential law are also not solitons.
We have assumed that the bulk viscosity, , and shear viscosity, , of the solid matrix are constant. A large amount of research has been performed on models in which is related to the voidage by a power law with exponent [1, 4–12]. The Lie group analysis for the permeability considered here could be extended to include the viscosity of the solid matrix.
Acknowledgments
N. Mindu thanks Shanduka and Tomorrow Trust for financial support. D. P. Mason thanks the National Research Foundation, Pretoria, South Africa, for financial support. The authors thank a referee for going through the paper thoroughly and suggesting improvements to the text.