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N. Mindu, D. P. Mason, "Derivation of Conservation Laws for the Magma Equation Using the Multiplier Method: Power Law and Exponential Law for Permeability and Viscosity", Abstract and Applied Analysis, vol. 2014, Article ID 585167, 13 pages, 2014. https://doi.org/10.1155/2014/585167
Derivation of Conservation Laws for the Magma Equation Using the Multiplier Method: Power Law and Exponential Law for Permeability and Viscosity
The derivation of conservation laws for the magma equation using the multiplier method for both the power law and exponential law relating the permeability and matrix viscosity to the voidage is considered. It is found that all known conserved vectors for the magma equation and the new conserved vectors for the exponential laws can be derived using multipliers which depend on the voidage and spatial derivatives of the voidage. It is also found that the conserved vectors are associated with the Lie point symmetry of the magma equation which generates travelling wave solutions which may explain by the double reduction theorem for associated Lie point symmetries why many of the known analytical solutions are travelling waves.
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, is the permeability of the medium, and is the viscosity of the matrix phase. The variables , , and and the physical quantities 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. 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 The viscosity is scaled by so that and will be infinite because the matrix viscosity is infinite when the voidage vanishes. In the derivation of (1) it is assumed that the background voidage satisfies .
The partially molten medium consists of a solid matrix and a fluid melt which are modelled as two immiscible fully connected fluids of constant but different densities. The density of the melt is less than the density of the solid matrix and the melt migrates through the compacting medium by the buoyancy force due to the difference in density between the melt and the solid matrix. Changes of phase are not included in the model. It is assumed that the melting has occurred and only migration of the melt under gravity is described by (1) .
In the model proposed by Scott and Stevenson , consider where and . Harris and Clarkson  have investigated this model using Painleve analysis. Mindu and Mason  showed that the magma equation also admits Lie point symmetries other than translations in time and space if the permeability is in the form of an exponential law: Conservation laws for (1) when the permeability and matrix viscosity satisfy the power laws (7) have been obtained using the direct method by Barcilon and Richter  and Harris  and using Lie point symmetry generators by Maluleke and Mason .
In this paper we will derive the conservation laws for the partial differential equation (1) using the multiplier method. We will consider power laws given by (7) and also the exponential laws where and , relating the permeability and matrix viscosity to the voidage. The permeability increases as the voidage increases while the viscosity of the matrix decreases as the voidage increases. The exponential laws are not suitable models when the voidage is small because and . They are suitable for describing rarefaction for which .
An outline of the paper is as follows. In Section 2 we present the formulae and theory that we will use in the paper. In Section 3 conservation laws for the magma equation, with power laws relating the permeability and viscosity to the voidage, are derived using the multiplier method. Further in Section 4 conservation laws for the magma equation, with exponential laws relating the permeability and viscosity to the voidage, are derived using the multiplier method. Finally the conclusions are summarized in Section 5.
2. Formulae and Theory
Consider an th order partial differential equation in the variables , where denotes the collection of th-order partial derivatives of . The equation evaluated on the surface given by (10), where runs from 1 to and is the total derivative defined by is called a conservation law for the differential equation (10). The vector is a conserved vector for the partial differential equation and are its components. Thus, a conserved vector gives rise to a conservation law. A Lie point symmetry generator where runs from 1 to , is said to be associated with the conserved vector for the partial differential equation (10) if [8, 9] The association of a Lie point symmetry with a conserved vector can be used to integrate the partial differential equation twice by the double reduction theorem of Sjöberg .
Conserved vectors for a partial differential equation can be generated from known conserved vectors and Lie point symmetries of the partial differential equation. For where runs from 1 to , is a conserved vector for the partial differential equation although it may be a linear combination of known conserved vectors [8, 9].
We now present the multiplier method for the derivation of conservation laws for partial differential equations. We will outline its application to the partial differential equation (1) in two independent variables.
(1) Multiply the partial differential equation (1) by the multiplier, , to obtain the conservation law where is the partial differential equation (1), and , and The multiplier depends on , , , and the partial derivatives of . The more derivatives included in the multiplier the wider the range of conserved vectors that can be derived.
(3) The determining equation (19) is separated by equating the coefficients of like powers and products of the derivatives of because is an arbitrary function.
(4) When is a solution of the partial differential equation, , (16) becomes a conservation law. The condition is imposed on (16). The product of the multiplier and the partial differential equation is then written in conserved form by elementary manipulations. This yields the conserved vectors by setting all the constants equal to zero except one in turn.
3. Conservation Laws for the Magma Equation with Power Law Permeability and Viscosity by the Multiplier Method
When the permeability and viscosity are related to the voidage by the power laws (7) the magma equation becomes
3.1. Lower Order Conservation Laws
In order to derive conservation laws for (20) consider first a multiplier of the form A multiplier for the partial differential equation has the property where The determining equation for the multiplier is where is defined by (18). Separating (24) with respect to products and powers of the partial derivatives of we obtain the following system of equations: Equation (26) is the same as (25). It is readily verified that every solution of (25) is a solution of (27). We therefore need to consider only (25). The general solution of (25) is
There are several cases to consider depending on the values of and . The special cases are illustrated as lines and points in the plane in Figure 1.
(i), , . From (22) and (28), Equation (30) is satisfied for arbitrary functions . When is a solution of the partial differential equation (20), then Hence, any conserved vector of the partial differential equation (20) with and satisfying the conditions of this case and with multiplier of the form is a linear combination of the two conserved vectors The conserved vector (32) is the elementary conserved vector.
3.2. The Search for Higher Order Conservation Laws
We now consider a multiplier of the form As before the determining equation for the multiplier is where is given by (23). By equating the coefficient of the highest order derivative term, , to zero in (48) we have and therefore Hence, (47) does not give a new multiplier or a new conserved vector.
Consider next the multiplier The determining equation for the multiplier is where is given by (23). By Equating the coefficients of , , and to zero in (52), we obtain the following system of equations: From (55) it follows that Substituting (56) into (53) we find that and therefore Thus, (51) becomes Now substitute (59) into (54) which gives The solution to (60) is where is a constant. Equation (59) becomes
Now substitute (62) into (48) and then equate the coefficient of in (48) to zero. This gives and integrating (63) we have Thus, (62) becomes Lastly, substitute (65) into (48) and equate the coefficient of to zero, which gives It follows from (66) that there are two cases.
Case 1 (). If , then from (66) we have . Therefore, Thus, does not give a new multiplier and therefore new conservation laws will not be derived.
Case 2 (). If , then from (66), we have . Thus, (65) becomes Now substitute (68) into (48) and equate the coefficient of in (48) to zero, which gives Solving (69) we have provided that and . Since , these two special cases correspond to the points and on the plane in Figure 1.
Consider first the general case excluding the points and . Equation (68) becomes Substituting (71) into (48) we find that . Hence, Since the multiplier (72) contains two constants, and , it leads to two conserved vectors. The conserved vector corresponding to , is the elementary conserved vector (32). The constants , lead to the conserved vector
Consider with and . The differential equation (69) becomes The general solution of (75) is When the multiplier (68) becomes On substituting (77) into the determining equation (48) we find that and the multiplier reduces to The multiplier (78) again contains two arbitrary constants, and . Setting the constants , gives the elementary conserved vector (32). Setting , leads to the conserved vector
Consider next multipliers of the form The determining equation for the multiplier is where is given by (23). Equating the coefficient of in (82) to zero, we find that and therefore which has already been considered.
Harris  proved that, except possibly for the two special cases with and with , there are no more independent conserved vectors. She proved this result using the direct method for conservation laws.
The multipliers and the corresponding conserved vectors for the partial differential equation (20) are listed in Table 1. This table was presented by Maluleke and Mason  without the multipliers. These conserved vectors agree with the results obtained by Barcilon and Richter  and Harris .
3.3. Association of Lie Point Symmetries with Conserved Vectors
The Lie point symmetries for the partial differential equation (20) are listed in Table 2. These Lie point symmetries were derived by Maluleke and Mason [7, 12]. Using (14) we will investigate which of the Lie point symmetries are associated with the conserved vectors for the Magma equation (20).
(i), . Consider first the Lie symmetry generator and the elementary conserved vector (32). Applying (14) we find that (85) is associated with the conserved vector (32) provided that , that is, provided that
(ii), , . Consider next the Lie point symmetry generator (85), with the conserved vector (33). Applying (14) we find that (85) is associated with (33) provided that , that is, provided is given by (86).
(vi), . Consider next