Drying viscoelastic materials: a non-Fickian approach

Abstract

In this paper, we study a system of partial differential equations defined in a moving domain. This system is defined by a heat equation and a diffusion equation for a concentration of non-Fickian type whose diffusion coefficient depends on the temperature, completed with suitable initial and boundary conditions. The non-Fickian mass flux is established considering the viscoelastic properties of the medium where the strain depends on the temperature and on the concentration. The initial boundary value problem (IBVP) analyzed can be used to describe the drying of viscoelastic materials where the internal structure offers a resistance to the movement of the moisture molecules and a consequent delay in the moisture removal. Due to heat transference into the materials and moisture removal, shrinkage of the medium occurs. The stability of the IBVP defined in a moving domain is analyzed and its qualitative behavior is numerically studied.

Appendix: A moving discrete model

The IBVP (43)–(50) is solved numerically using a finite difference discretization in the uniform meshes in [0, 1] and $[0,\tau _f]$ defined by

$$\begin{aligned}&\{\xi _i,~~ i=0,1,\dots ,N,~~ \xi _0=0,~~ \xi _N=1,~~ \xi _i-\xi _{i-1}=\varDelta \xi ,~~ i=1,2,\dots ,N\},\\&\{\tau _j,~~ j=0,1,\dots ,M_\tau , ~~\tau _0=0, ~~\tau _{M_\tau }=\tau _f,~~ \tau _j-\tau _{j-1}=\varDelta \tau ,~~ j=1,2,\dots ,M_\tau \}. \end{aligned}$$

By $g_i^j,f_i^j,\ell _j$ we represent the finite difference approximations for $g(\xi _i,\tau _j), f(\xi _i,\tau _j)$ and $\ell (\tau _j)$, respectively, defined by the IMEX (implicit–explicit) method presented in what follows. The finite difference approximation for (46) is the one proposed by Furzeland in Crank (1984)

$$\begin{aligned} \frac{\ell _{j+1}-\ell _j}{\varDelta \tau }={\overline{D}}(f_N^j)\frac{M_0}{\ell _j} \frac{3g^j_N-4g^j_{N-1} +g^j_{N-2}}{2\varDelta \xi }, \end{aligned}$$

(53)

for $j=0,\dots ,M_\tau -1$. Equations (43)–(45) are discretized by

$$\begin{aligned} \frac{ g_i^{j+1}-g_i^j}{\varDelta \tau }= & {} \frac{1}{\ell _j^2}\bigg ({\overline{D}}(f_{i+1}^j)\frac{g_{i+1}^{j+1} -g_i^{j+1}}{(\varDelta \xi )^2}-{\overline{D}}(f_{i}^j)\frac{g_{i}^{j+1} -g_{i-1}^{j+1}}{(\varDelta \xi )^2}\bigg )\nonumber \\&+\frac{D_{{\text {vnon}}}}{\ell _j^2}\frac{\sigma _{n,i-1}^{j}-2\sigma _{n,i}^{j} +\sigma _{n,i+1}^{j}}{(\varDelta \xi )^2}\nonumber \\&+\frac{\xi _i}{\ell _j}\frac{\ell _{j+1}-\ell _j}{\varDelta \tau }\frac{g^{j+1}_{i+1} -g^{j+1}_{i-1}}{2\varDelta \xi }, \end{aligned}$$

(54)

$$\begin{aligned} \frac{ f_i^{j+1}-f_i^j}{\varDelta \tau }= & {} \frac{Le}{\ell _j^2}\frac{f_{i-1}^{j+1}-2f_i^{j+1} +f_{i+1}^{j+1}}{(\varDelta \xi )^2}\nonumber \\&+ \frac{\xi _i}{\ell _j} \frac{\ell _{j+1}-\ell _j}{\varDelta \tau } \frac{f^{j+1}_{i+1} -f^{j+1}_{i-1}}{2\varDelta \xi }, \end{aligned}$$

(55)

$$\begin{aligned} \frac{\sigma _{n,i}^{j+1}-\sigma _{n,i}^{j}}{\varDelta \tau }= & {} -E_{{\text {non}}}\sigma _{n,i}^{j}\nonumber \\&+ \frac{\xi _i}{\ell _j} \frac{\ell _{j+1} -\ell _j}{\varDelta \tau } \bigg (\mu _{{\text {non}}}\alpha _T(T_{{\text {air}}}-T_0) \frac{f^{j+1}_{i+1}-f^{j+1}_{i-1}}{2\varDelta \xi }\nonumber \\&+ \mu _{{\text {non}}}\alpha _M M_0 \frac{g^{j+1}_{i+1}-g^{j+1}_{i-1}}{2\varDelta \xi } +\frac{\sigma _{n,i+1}^j-\sigma _{n,i-1}^j}{2\varDelta \xi }\bigg )\nonumber \\&- \mu _{{\text {non}}}\alpha _T(T_{{\text {air}}}-T_0) \frac{f_{i}^{j+1}-f_i^j}{\varDelta \tau }-\mu _{{\text {non}}}\alpha _MM_0 \frac{g_{i}^{j+1}-g_i^j}{\varDelta \tau }, \end{aligned}$$

(56)

respectively, for $i=1,\dots ,N-1$ and $j=0,\dots ,M_\tau -1$, where

$$\begin{aligned} {\overline{D}}(v_{i}^n)=\frac{{\overline{D}}(v_{i}^n)+{\overline{D}}(v_{i-1}^n)}{2}. \end{aligned}$$

The initial condition (47) becomes

$$\begin{aligned} f_i^0=0,\,\, g_i^0=1,\,\, \sigma _{n,i}^0=1,\,\, \ell _0=1,\,\, i=1,\dots ,N-1. \end{aligned}$$

(57)

The boundary condition (48)–(50) are discretized using forward and backward finite differences for space derivatives, that is, by

$$\begin{aligned}&\displaystyle f_1^{j+1}=f_{0}^{j+1} \,\, , \,\, g_1^{j+1}=g_{0}^{j+1},&\end{aligned}$$

(58)

$$\begin{aligned}&\displaystyle -\frac{f_{N}^{j+1}-f_{N-1}^{j+1}}{\varDelta \xi }+\lambda \frac{g_{N}^{j+1}-g_{N-1}^{j+1}}{\varDelta \xi }=\ell _jBi(f_{N}^{j+1}-1),&\end{aligned}$$

(59)

$$\begin{aligned}&\displaystyle -{\overline{D}}(f_{N}^j)\frac{g_{N}^{j+1}-g_{N-1}^{j+1}}{\varDelta \xi }-D_{{\text {vnon}}}\frac{\sigma _{n,N}^{j}-\sigma _{n,N-1}^{j}}{\varDelta \xi }=\ell _jBi_{\text {m}}\left( g_{N}^{j+1}-\frac{M_{\text {e}}}{M_0}\right) ,&\end{aligned}$$

(60)

respectively, for $j=0,\dots ,M_\tau -1$.