# br Figs and exhibits the

Figs. 8 and 9 exhibits the dimensionless moving layer thickness 1 (Fo3) in the solid-mush region and 2 ( Fo3) in mush-liquid region re-spectively. As the time Fo3 increases, moving layer thickness 1 (Fo3) and 2 ( Fo3) increases. The effect of Stefan number on moving layer thickness is seen in Figs. 14 and 15. Also, the effect of the Kirchoff number and Biot number is seen in Figs. 16 and 17. Here the tissue is cooled from freezing temperature to the lethal temperature. In parti-cular, the lethal temperature for tissue destruction usually begins around 400C (Zhang, 2009) for diseased tissue like a tumor.

7. Conclusion

In this study, a two-dimensional mathematical bio-heat transfer model of lung tumor tissue during the freezing process in cryosurgery has been developed and then used the Modified Legendre wavelets Galerkin method to obtain the results. Firstly, our problem is trans-formed into non-dimensional form and then applying finite difference method in our problem to convert it Tigecycline into initial boundary value pro-blem of ordinary matrix differential equation in stage 1 and moving boundary value problem in stage 2 and 3 by imposing on it at a constant temperature(I kind), a constant heat flux(II kind) and a constant heat transfer coefficient(III kind) in each stage. After this, we obtained the system of Sylvester equations by using Legendre wavelet Galerkin

Fig. 13. Moving layer thickness 2 in stage2 with different Stefan number (a) I kind (b) II kind (c) III kind.

Fig. 14. Moving layer thickness 1 in stage3 with different Stefan number (a) I kind (b) II kind (c) III kind.

Fig. 15. Moving layer thickness 2 in stage3 with different Stefan number (a) I kind (b) II kind (c) III kind.

method which are solved by generalized inverse technique (Bartels-Stewart algorithm). In each stage, we have applied this technique to carry out the temperature distribution. From our calculations, we see that the effect of relaxation time is negligible in the mushy and frozen region and present only in unfrozen region. We observe from the figures of moving layer thickness and temperature distribution that there is slight variation not too much with the effect of different boundary condition. In Stage1, the tissue is cooled up to the liquidus temperature Tl ( 10 C) where unfrozen region is formed and then in Stage 2, the tissue is cooled up to the freezing temperature Ts ( 8 0C) where the mushy
**r>
region is formed and in Stage3, the tissue is cooled up to the lethal temperature where the frozen region is formed. The effect of Stefan number on moving layer thickness is seen in stage 2 and 3. Also, the effect of Kirchoff number on moving layer thickness has been seen in Stage 3 of boundary condition II kind and the effect of Biot number on moving layer thickness has been seen in stage3 of boundary condition
III kind. Moving layer thickness increases as the Kirchoff number in-creases and moving layer thickness decreases as the Biot number in-creases. Although a two-dimensional mathematical bio-heat transfer model is considered in our investigation, genital herpes can further be extended for
the three-dimensional mathematical bio-heat transfer model to attain more physically realistic results. Lastly, this model is beneficial for the experimental analysis of the freezing process of cryosurgical treatment.
Acknowledgments
The research of the first author is supported by University Grant Commission New Delhi, India, grant no. 19/06/2016(i)EU-V. The au-thors express their sincere thanks to DST-Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University Varanasi, India, for alloting the required facilities. The authors are grateful to the Reviewer for their valuable comments.
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