A Leap-Frog Finite Difference Method for Strongly Coupled System from Sweat Transport in Porous Textile Media

In this paper, we present an uncoupled leap-frog finite difference method for the system of equations arising from sweat transport through porous textile media. Based on physical mechanisms, the sweat transport can be viewed as the multicomponent flow that coupled the heat andmoisture transfer, such that the system is nonlinear and strongly coupled.The leap-frogmethod is proposed to solve this system, with the second order accuracy in both spatial and temporal directions. We prove the existence and uniqueness of the solution to the system with optimal error estimates in the discrete L2 norm. Numerical simulations are presented and analyzed, respectively.


Introduction
Single/multicomponent flow in porous textile media attracted considerable attention in the last several decades. See [1][2][3][4] for the single-component models and [5][6][7][8][9] for the multicomponent models. In this paper, we study the multicomponent sweat transport coupled with vapor and heat in porous textile media. In [10], Ye et al. proposed a quasi-steady-state single-component model which consists of a steady-state air equation and dynamic state equations for other components. Under certain conditions, the multicomponent model reduces to a new single-component model, and the physical process can be viewed as sweat transport (vapor and heat flow) governed by the conservation of mass and energy: where is the porosity of the media, is the vapor concentration, is the temperature, is the thermal conductivity, is the latent heat of evaporation/condensation, and is the molecular weight of water. The effective volumetric heat capacity V is defined by where is the molar heat capacity and V is the volumetric heat capacity of fiber.
By Darcy's law, the gas velocity is defined as where is the permeability and is the dynamic viscosity, which usually is density-dependent for the compressible flow.
Here we choose a linear form of fl ] , where ] is a certain constant. By the Hertz-Knudsen equation [11], the phase change rate Γ is defined as Journal of Mathematics where Γ is a positive constant, the saturation pressure sat is determined from experimental measurements [12], and the pressure is given by = , where is the universal gas constant.
Since the right boundary is exposed to environment and the left boundary is connected to the body, we consider commonly used Robin type boundary conditions and the initial conditions Physically, parameters , , , ] , = 1,2, and are nonnegative constants [1,2,6]. We define initial condition parameters 0 ( ) ≥ , 0 ( ) ≥ with and being positive constants.
Due to the strong nonlinearity and the coupling of the system, both theoretical and numerical analyses of the system are difficult. Numerical analysis for some related systems of parabolic/elliptic equations can be found in [13][14][15][16][17][18][19][20]. Existence and uniqueness of a classical solution for a steadystate model was given in [10]. Existence of a weak solution for the corresponding dynamic models was given in [21,22]. Positivity of temperature and nonnegativity of vapor density were also proved here. Recently, a finite difference method second-order in space and first-order in time for the system (6)- (12) was presented in [23], where the backward semiimplicit Euler scheme is applied in the temporal direction and central finite difference approximations are used in the spatial direction. In [23], authors presented optimal error estimates under the assumption that the step size and ℎ are smaller than a positive constant.
In this paper, we propose an uncoupled leap-frog finite difference method for the system (6)- (12) with second-order accuracy in both spatial and temporal directions. We prove the existence and uniqueness of a solution to the finite difference system with optimal error estimates in the discrete 2 norm, under the condition that the mesh size and ℎ are smaller than a positive constant which depends solely upon the physical parameters involved in the equations. Due to the strong nonlinearity and the coupling of equations, the method presented in [23] does not apply to the leap-frog scheme directly. One of the difficulties is to show convergence of the numerical solution without restriction on the grid ratio. In this paper, we assume that the solution ( ( , ), ( , )) to the system (6)- (12) satisfies that min ≤ ( , ) ≤ max , min ≤ ( , ) ≤ max (13) for some positive constants min , max , max , and min .
The manuscript is organized as follows: in Section 2, we present an uncoupled leap-frog finite difference method for the nonlinear sweat transport system. In Section 3, we prove the existence and uniqueness of the solution to the sweat transport system with the optimal error estimate in the discrete 2 norm. Numerical results will be presented in Section 4 to support our theoretical results.

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from which The discrete system is defined by and the discrete initial conditions 0 = 0 ( ) , The computational procedure of the uncoupled leap-frog scheme at each time step is listed below: Step . The vapor concentration +1 can be calculated by solving the tridiagonal linear systems defined in (18)- (20).

The Leap-Frog Scheme and the Optimal Error Estimate
In this section, we will show the existence and uniqueness of the solution to the system (18)-(26) with optimal error estimates in the discrete 2 norm. Let V = {V } =0 and = { } =0 be two mesh functions on Ω ℎ . We define the inner product and norms by Let ( , Θ) be the solution of the system (6)-(12) and = ( , ), Θ = ( , ). The error functions are defined bỹ We state our main result in the theorem below:

the finite difference scheme ( )-( ) is uniquely solvable and
To prove the theorem, we make a stronger assumption that there exists 0 > 0, independent of , ℎ, , such that the inequality, holds for ≤ −1. We prove the assumption and the theorem by induction method. By the initial condition (26), this is true for = 0. In the next subsection we will show that this is also true for = 1. In this part, we let be a generic positive constant, which is associated with the physical parameters , , , min , max , min , max , the parameters involved in initial and boundary conditions and the solution of the system (6)- (12).
is independent of time step , mesh size ℎ, , and constant 0 .

. . e Leap-Frog Scheme and Preliminaries.
For convenience of calculations, we further introduce some notations. Let = ( ) , = ; thus the sweat transport system (6)- (7) can be reduced to with the initial and boundary conditions The discrete leap-frog system (18)-(23) is modified as Let = ( , ) = ( ) ( , ) and = ( , ) = ( , ). We denote by and the corresponding finite difference solution and We get

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and the initial conditions (40) Subtracting the system (36) from the system (38), we get the error equations ̂+ and by (40), we can directly derive the inequalitỹ To prove our main theorem, the following formula will be often used: In the following lemma, we present discrete Sobolev interpolation formulas and the proof can be found in [24].
When ≤ ℎ, with the inverse inequality we havẽ When ℎ ≤ , by taking = 1/2 in Lemma 2, The first part of (54) is obtained and the second part and the inequality (55) can be proved similarly.
In addition, by Lemma 3, there exist constants 3 > 0 and and . . e Existence and Uniqueness. Since the coefficient matrix in the system (18)-(20) is strictly diagonally dominant, thus the system (18)-(20) has a unique solution +1 . Here we will discuss the boundedness of +1 .
. . e Optimal Error Estimate. We have proved the existence and uniqueness of the solution to the system and have derived the estimate (65) for̃+ 1 . In this part, we try to derive an estimate for̃+ 1 .

Numerical Examples
We now numerically evaluate the performance of the proposed leap-frog scheme.
Example . First, we test the accuracy of our algorithm in an artificial example which is taken from [23]. The system is with , , , being constants. We apply the uncoupled leap-frog finite difference method to solve the artificial example. We choose T = 1 and = 1. Since the proposed scheme is of the second order in both spatial and temporal directions, we take = ℎ such that the error bound is proportional to ℎ 2 . We present the 2 -norm errors and the order of convergence ℎ in Table 1 with ℎ = /200, /400, /800 at different time level. We can see clearly from Table 1 that the 2 -norm errors for both components are proportional to ℎ 2 , which confirms our theoretical analysis.
Example . In the second example, we discuss a typical clothing assembly in the textile industry [2,4,25]. The clothing assembly consists of three layers, in the middle is porous fibrous media, and the outside cover is exposed to a cold environment with fixed temperature and relative humidity while the inside cover is exposed to a mixture of air and vapor at higher temperature and relative humidity. In this paper, polyester porous media with laminated or nylon cover materials are tested. To compare with the experimental data in [12], a water equation is added to equations (1)- (2): where is water content, is the density of water, is the porosity with liquid water content, and is the porosity without liquid water content. We have and the effective heat conductivity is defined by where is the thermal conductivity of gas and is the thermal conductivity of the fiber-water mixture [2,6,7], given by The values of these physical parameters for polyester media are presented in Table 2. Other parameters values can be found in [2,6,7].
We apply the uncoupled leap-frog finite difference method for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115). Since only the right side of the water equation includes and , therefore, the water equation is calculated separately. Numerically, at each time step, we first find solution +1 , +1 by procedure (18)-(26), and then +1 can be solved by following nonnormalized discrete formate: Then we evaluate the parameters explicitly in (18)-(26) based on +1 . Here all numerical results are obtained by taking the time step size = 20 and spatial mesh size ℎ = /100. We present numerical results of vapor, temperature, and water content at 8 hours and 24 hours, respectively, for the porous polyester media assembly with laminated cover in Figure 1 and with nylon cover in Figure 2. The comparisons between numerical results of water content and experimental measurements [12] are given in last two subfigures, where the blue lines represent the numerical solution and the red line is given by experimental measurement.

Conclusion
As a subsequent work of [23], we have presented an uncoupled leap-frog finite difference method for the sweat transport system in porous textile media, which is governed by a strongly coupled, nonlinear parabolic system. Optimal 2 error estimates were presented, which imply that the numerical scheme is unconditionally stable. Both theoretical analysis and numerical example indicate that the current scheme is second order accurate in both the temporal and spatial directions. Since the scheme is decoupled for the system, the method can be applied efficiently for problems in higherdimensional space. Under certain time-step restrictions, the analysis can also be extended to the multidimensional problems.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.