Onate et al 2010a - Revision history

Cinmemj at 09:39, 4 January 2019

2019-01-04T09:39:07Z

Cinmemj at 09:32, 4 January 2019

2019-01-04T09:32:17Z

Cinmemj at 09:31, 4 January 2019

2019-01-04T09:31:26Z

Cinmemj at 13:32, 3 January 2019

2019-01-03T13:32:01Z

Cinmemj at 13:31, 3 January 2019

2019-01-03T13:31:21Z

Cinmemj at 13:12, 3 January 2019

2019-01-03T13:12:23Z

Cinmemj at 13:10, 3 January 2019

2019-01-03T13:10:26Z

Cinmemj at 13:08, 3 January 2019

2019-01-03T13:08:59Z

Cinmemj at 13:04, 3 January 2019

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Cinmemj at 13:03, 3 January 2019

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@@ Line 766: / Line 766: @@
 The authors thank Dr. T. Ohlemiller at NIST for providing experimental results for the 2D rectangular slab problem <span id='citeF-22'></span>[[#cite-22|[22]]]. Special thanks are given to Mr. P. Ryzhakov and to Drs. J. Marti and F. del Pin from CIMNE for their advice during this research. The support of Dr. P. Dadvand in the development of the PFIRE code using the Kratos software platform <span id='citeF-23'></span>[[#cite-23|[23]]] is acknowledged.
-===REFERENCES===
+==REFERENCES==
 <div id="cite-1"></div>

@@ Line 618: / Line 618: @@
 ===4.3 Inclusion of gasification effects===
-The same problem was solved next ''including gasification effects''. The parameters in the gasification heat flux model chosen for the computations (Eqs.([[#eq-13|13]]) &#8211; ([[#eq-15|15]]) are given in Table [[#table-1|I]].
+The same problem was solved next ''including gasification effects''. The parameters in the gasification heat flux model chosen for the computations (Eqs.([[#eq-13|13]]) &#8211; ([[#eq-15|15]])) are given in Table [[#table-1|I]].
 The general trend of the numerical results including gasification is not very different from the results shown in Figures [[#img-6|6]] and [[#img-7|7]]. Gasification effects however have an influence on the mass loss rate.

← Older revision		Revision as of 09:31, 4 January 2019
Line 579:		Line 579:

	<div class="center" style="font-size: 75%;">		<div class="center" style="font-size: 75%;">
−	'''Box II'''. Definition of the viscosity-temperature dependence for the polymer melt experiment of Figure 5</div>	+	'''Box II'''. Definition of the viscosity-temperature dependence for the polymer melt experiment of Figure [[#img-5\|5]]</div>

@@ Line 292: / Line 292: @@
 The effect of gasification can be introduced by adding a (nonlinear) energy loss term in the energy equation. This term represents the energy that migrates from the condensed phase object to the gas due to the gasification of a part of the material during the heating process. The volumetric gasification heat flux has the following form
-{| class="floating_imageSCP" style="text-align: left; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 70%;font-size: 85%;"
+{| class="floating_imageSCP" style="text-align: left; border: 1px solid #BBB; margin: 1em auto; width: 50%;max-width: 50%;font-size: 85%;"
 |-
 |style="padding-left:10px;"|1. LOOP OVER TIME STEPS, <math>t=1</math>, NTIME
@@ Line 308: / Line 308: @@
 |style="padding-left:10px;"|<math>\bullet </math> Compute  nodal pressures from Eq.[[#eq-10|(10)]]
 |-
-|<math>{\boldsymbol L} {~}^{t+1}\bar {\boldsymbol p}^{i+1}=-{\boldsymbol G}^T {~}^{t+1}\bar {\boldsymbol v}^{i+1}- {\boldsymbol  Q}{~}^{t+1}\bar{\boldsymbol \pi }^i</math>
+|style="padding-left:10px;"|<math>{\boldsymbol L} {~}^{t+1}\bar {\boldsymbol p}^{i+1}=-{\boldsymbol G}^T {~}^{t+1}\bar {\boldsymbol v}^{i+1}- {\boldsymbol  Q}{~}^{t+1}\bar{\boldsymbol \pi }^i</math>
 |-
 |style="padding-left:10px;"|<math>\bullet </math> Compute nodal pressure gradient projections from Eq. [[#eq-11|(11)]]

@@ Line 294: / Line 294: @@
 {| class="floating_imageSCP" style="text-align: left; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 70%;font-size: 85%;"
 |-
-|1. LOOP OVER TIME STEPS, <math>t=1</math>, NTIME
+|style="padding-left:10px;"|1. LOOP OVER TIME STEPS, <math>t=1</math>, NTIME
 |-
-|Known values
+|style="padding-left:10px;"|Known values
 |-
-|<math>{~}^t\bar {\boldsymbol x},{~}^t\bar {\boldsymbol v},{~}^t\bar {\boldsymbol p},{~}^t\bar{\boldsymbol \pi }, {~}^t\bar {\boldsymbol T},{~}^t\mu , {~}^t{\boldsymbol f}, {~}^t{\boldsymbol q}, {~}^tC, {~}^tV, {~}^tM</math>
+|style="padding-left:10px;"|<math>{~}^t\bar {\boldsymbol x},{~}^t\bar {\boldsymbol v},{~}^t\bar {\boldsymbol p},{~}^t\bar{\boldsymbol \pi }, {~}^t\bar {\boldsymbol T},{~}^t\mu , {~}^t{\boldsymbol f}, {~}^t{\boldsymbol q}, {~}^tC, {~}^tV, {~}^tM</math>
 |-
-|2. LOOP OVER NUMBER OF ITERATIONS, <math>i=1</math>, NITER
+|style="padding-left:10px;"|2. LOOP OVER NUMBER OF ITERATIONS, <math>i=1</math>, NITER
 |-
-<math>\bullet </math> Compute the nodal velocities by solving Eq. [[#eq-9|(9)]]
+|style="padding-left:10px;"|<math>\bullet </math> Compute the nodal velocities by solving Eq. [[#eq-9|(9)]]
 |-
-|<math>\displaystyle \left[\frac{1}{\Delta t}{\boldsymbol M} +{\boldsymbol K}\right]{~}^{t+1}\bar  {\boldsymbol v}^{i+1}= {~}^{t+1}{\boldsymbol f}+{\boldsymbol G} {~}^{t+1}\bar {\boldsymbol p}^i +\frac{1}{\Delta  t}{\boldsymbol M} {~}^t\bar {\boldsymbol v}</math>
+|style="padding-left:10px;"|<math>\displaystyle \left[\frac{1}{\Delta t}{\boldsymbol M} +{\boldsymbol K}\right]{~}^{t+1}\bar  {\boldsymbol v}^{i+1}= {~}^{t+1}{\boldsymbol f}+{\boldsymbol G} {~}^{t+1}\bar {\boldsymbol p}^i +\frac{1}{\Delta  t}{\boldsymbol M} {~}^t\bar {\boldsymbol v}</math>
 |-
-|<math>\bullet </math> Compute  nodal pressures from Eq.[[#eq-10|(10)]]
+|style="padding-left:10px;"|<math>\bullet </math> Compute  nodal pressures from Eq.[[#eq-10|(10)]]
 |-
 |<math>{\boldsymbol L} {~}^{t+1}\bar {\boldsymbol p}^{i+1}=-{\boldsymbol G}^T {~}^{t+1}\bar {\boldsymbol v}^{i+1}- {\boldsymbol  Q}{~}^{t+1}\bar{\boldsymbol \pi }^i</math>
 |-
-|<math>\bullet </math> Compute nodal pressure gradient projections from Eq. [[#eq-11|(11)]]
+|style="padding-left:10px;"|<math>\bullet </math> Compute nodal pressure gradient projections from Eq. [[#eq-11|(11)]]
 |-
-|<math>{~}^{t+1}\bar {\boldsymbol \pi }^{i+1}= - \hat {\boldsymbol M}_D [{\boldsymbol Q}^T] {~}^{t+1}\bar {\boldsymbol  p}^{i+1}\quad ,\quad \hat {\boldsymbol M}_D=diag[\hat {\boldsymbol M}]</math>
+|style="padding-left:10px;"|<math>{~}^{t+1}\bar {\boldsymbol \pi }^{i+1}= - \hat {\boldsymbol M}_D [{\boldsymbol Q}^T] {~}^{t+1}\bar {\boldsymbol  p}^{i+1}\quad ,\quad \hat {\boldsymbol M}_D=diag[\hat {\boldsymbol M}]</math>
 |-
-|<math>\bullet </math> Compute nodal temperatures from Eq.[[#eq-12|(12)]]
+|style="padding-left:10px;"|<math>\bullet </math> Compute nodal temperatures from Eq.[[#eq-12|(12)]]
 |-
-|<math>\displaystyle \left[\frac{1}{\Delta t}{\boldsymbol C} +{\boldsymbol H}\right]{~}^{t+1}\bar  {\boldsymbol T}^{i+1}={~}^{t+1}{\boldsymbol q}-\frac{1}{\Delta t}{\boldsymbol C} {~}^t\bar {\boldsymbol T}</math>
+|style="padding-left:10px;"|<math>\displaystyle \left[\frac{1}{\Delta t}{\boldsymbol C} +{\boldsymbol H}\right]{~}^{t+1}\bar  {\boldsymbol T}^{i+1}={~}^{t+1}{\boldsymbol q}-\frac{1}{\Delta t}{\boldsymbol C} {~}^t\bar {\boldsymbol T}</math>
 |-
-|<math>\bullet </math> Update position of analysis domain nodes:
+|style="padding-left:10px;"|<math>\bullet </math> Update position of analysis domain nodes:
 |-
-|<math>{~}^{t+1}\bar {\boldsymbol x}^{i+1}= {~}^t{\boldsymbol x}^i+ {~}^{t+1}{\boldsymbol  v}^{i+1}\Delta t</math>
+|style="padding-left:10px;"|<math>{~}^{t+1}\bar {\boldsymbol x}^{i+1}= {~}^t{\boldsymbol x}^i+ {~}^{t+1}{\boldsymbol  v}^{i+1}\Delta t</math>
 |-
-|Define new “cloud” of nodes <math>^{t+1} {C}^{i+1}</math>
+|style="padding-left:10px;"|Define new “cloud” of nodes <math>^{t+1} {C}^{i+1}</math>
 |-
-|<math>\bullet </math> Update viscosity values in terms of temperature
+|style="padding-left:10px;"|<math>\bullet </math> Update viscosity values in terms of temperature
 |-
-|<math>{~}^{t+1}\mu =\mu (^{t+1}\bar {\boldsymbol T}^{i+1})</math>
+|style="padding-left:10px;"|<math>{~}^{t+1}\mu =\mu (^{t+1}\bar {\boldsymbol T}^{i+1})</math>
 |-
-|Check convergence  for all variables <math>\to </math> NO <math>\to </math>  Next iteration <math>i\to i+1</math>
+|style="padding-left:10px;"|Check convergence  for all variables <math>\to </math> NO <math>\to </math>  Next iteration <math>i\to i+1</math>
 |-
-|<math>~~~~~~~~~~\downarrow </math> YES
+|style="padding-left:10px;"|<math>~~~~~~~~~~\downarrow </math> YES
 |-
-|Next time step <math>t\to t+1</math>
+|style="padding-left:10px;"|Next time step <math>t\to t+1</math>
 |-
-|<math>\bullet </math> Identify new analysis domain boundary: <math>^{t+1}V</math>
+|style="padding-left:10px;"|<math>\bullet </math> Identify new analysis domain boundary: <math>^{t+1}V</math>
 |-
-|<math>\bullet </math> Generate mesh: <math>{~}^{t+1}M</math>
+|style="padding-left:10px;"|<math>\bullet </math> Generate mesh: <math>{~}^{t+1}M</math>
 |-
-|Go to 1
+|style="padding-left:10px;"|Go to 1
 |}

@@ Line 769: / Line 769: @@
 <div id="cite-1"></div>
-'''[[#citeF-1|[1]]]'''  Zhang J, Shields TJ, Silcock GWH.  Effect of melting behaviour on flame spread of thermoplastics. ''Fire & Materials'' 1997; '''21'''(1):1&#8211;6.
+[[#citeF-1|[1]]]  Zhang J, Shields TJ, Silcock GWH.  Effect of melting behaviour on flame spread of thermoplastics. ''Fire & Materials'' 1997; 21(1):1&#8211;6.
 <div id="cite-2"></div>
-'''[2]'''  Fleischmann CM, Hill GR. Burning behaviour of upholstered furniture. Interflam 2004; 907&#8211;916.
+[2]  Fleischmann CM, Hill GR. Burning behaviour of upholstered furniture. Interflam 2004; 907&#8211;916.
 <div id="cite-3"></div>
-'''[[#citeF-3|[3]]]'''  Sherratt J, Drysdale D. The effect of the melt-flow process on the fire behaviour of thermoplastics. ''Interflam'' 2001; 149&#8211;159.
+[[#citeF-3|[3]]]  Sherratt J, Drysdale D. The effect of the melt-flow process on the fire behaviour of thermoplastics. ''Interflam'' 2001; 149&#8211;159.
 <div id="cite-4"></div>
-'''[[#citeF-4|[4]]]'''  Butler KM, Ohlemiller TJ, Linteris GT. A progress report on numerical modeling of experimental polymer melt flow behavior. ''Interflam'' 2004; pp. 937&#8211;948.
+[[#citeF-4|[4]]]  Butler KM, Ohlemiller TJ, Linteris GT. A progress report on numerical modeling of experimental polymer melt flow behavior. ''Interflam'' 2004; pp. 937&#8211;948.
 <div id="cite-5"></div>
-'''[[#citeF-5|[5]]]'''  Idelsohn SR, Oñate E, Del Pin F.  A lagrangian meshless finite element method applied to fluid-structure interaction problems. ''Computer and Structures'' 2003b; '''81''':655&#8211;671.
+[[#citeF-5|[5]]]  Idelsohn SR, Oñate E, Del Pin F.  A lagrangian meshless finite element method applied to fluid-structure interaction problems. ''Computer and Structures'' 2003b; 81:655&#8211;671.
 <div id="cite-6"></div>
-'''[[#citeF-6|[6]]]'''  Idelsohn SR, Oñate E, Del Pin F.   The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. ''International Journal for Numerical  Methods in Engineering'' 2004; '''61''':964&#8211;989.
+[[#citeF-6|[6]]]  Idelsohn SR, Oñate E, Del Pin F.   The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. ''International Journal for Numerical  Methods in Engineering'' 2004; 61:964&#8211;989.
 <div id="cite-7"></div>
-'''[7]'''  Idelsohn SR, Oñate E, Calvo N, Del Pin F.   The meshless finite element method.  ''International Journal for Numerical  Methods in Engineering'' 2003a; '''58'''(6): 893&#8211;912.
+[7]  Idelsohn SR, Oñate E, Calvo N, Del Pin F.   The meshless finite element method.  ''International Journal for Numerical  Methods in Engineering'' 2003a; 58(6): 893&#8211;912.
 <div id="cite-8"></div>
-'''[[#citeF-8|[8]]]'''  Oñate E, Idelsohn SR, Del Pin F, Aubry R.  The particle finite element method. An overview. ''International Journal of Computational Methods'' 2004; '''1'''(2):267&#8211;307.
+[[#citeF-8|[8]]]  Oñate E, Idelsohn SR, Del Pin F, Aubry R.  The particle finite element method. An overview. ''International Journal of Computational Methods'' 2004; 1(2):267&#8211;307.
 <div id="cite-9"></div>
-'''[[#citeF-9|[9]]]'''  Aubry R, Idelsohn SR, Oñate E.  Particle finite element method in fluid-mechanics including thermal convection-diffusion. ''Computers and Structures'' 2005; '''83'''(17-18):1459&#8211;1475.
+[[#citeF-9|[9]]]  Aubry R, Idelsohn SR, Oñate E.  Particle finite element method in fluid-mechanics including thermal convection-diffusion. ''Computers and Structures'' 2005; 83(17-18):1459&#8211;1475.
 <div id="cite-10"></div>
-'''[[#citeF-10|[10]]]'''  Aubry R, Idelsohn SR, Oñate E.   Fractional step like schemes for free surface problems with thermal coupling using the Lagrangian PFEM. ''Computational Mechanics'' 2006; '''38'''(4-5):294&#8211;309.
+[[#citeF-10|[10]]]  Aubry R, Idelsohn SR, Oñate E.   Fractional step like schemes for free surface problems with thermal coupling using the Lagrangian PFEM. ''Computational Mechanics'' 2006; 38(4-5):294&#8211;309.
 <div id="cite-11"></div>
-'''[[#citeF-11|[11]]]'''    Oñate E, Idelsohn SR, Celigueta MA, Rossi R.  Advances in the particle finite element method for the analysis of fluid-multibody interaction and bed erosion in free surface flows. ''Computer Methods in Applied Mechanics and Engineering'' 2008; '''197'''(19-20):1777&#8211;1800.
+[[#citeF-11|[11]]]    Oñate E, Idelsohn SR, Celigueta MA, Rossi R.  Advances in the particle finite element method for the analysis of fluid-multibody interaction and bed erosion in free surface flows. ''Computer Methods in Applied Mechanics and Engineering'' 2008; 197(19-20):1777&#8211;1800.
 <div id="cite-12"></div>
-'''[[#citeF-12|[12]]]'''  Butler KM, Oñate E, Idelsohn SR, Rossi R. Modeling polymer melt flow using the particle finite element method. 11th International Interflam Conference (''Interflam'07''), September 3&#8211;5 2007, London, England, Volume 2,  pp. 929&#8211;940.
+[[#citeF-12|[12]]]  Butler KM, Oñate E, Idelsohn SR, Rossi R. Modeling polymer melt flow using the particle finite element method. 11th International Interflam Conference (''Interflam'07''), September 3&#8211;5 2007, London, England, Volume 2,  pp. 929&#8211;940.
 <div id="cite-13"></div>
-'''[[#citeF-13|[13]]]'''  Rossi R, Butler KM, Oñate E,  Idelsohn SR. Modeling polymer melt flow using the Particle Finite Element Method. ''Advanced Research Workshop on Fire Computer Modeling'', Gijón, Spain, October 18-20, 2007.
+[[#citeF-13|[13]]]  Rossi R, Butler KM, Oñate E,  Idelsohn SR. Modeling polymer melt flow using the Particle Finite Element Method. ''Advanced Research Workshop on Fire Computer Modeling'', Gijón, Spain, October 18-20, 2007.
 <div id="cite-14"></div>
-'''[[#citeF-14|[14]]]'''  Edelsbrunner H, Mucke EP.  Three dimensional alpha shapes. ''ACM Trans. Graphics'' 1999; '''13''':43&#8211;72.
+[[#citeF-14|[14]]]  Edelsbrunner H, Mucke EP.  Three dimensional alpha shapes. ''ACM Trans. Graphics'' 1999; 13:43&#8211;72.
 <div id="cite-15"></div>
-'''[[#citeF-15|[15]]]'''  Zienkiewicz OC, Taylor RL, Nithiarasu N. ''The Finite Element Method. Vol. 3 Fluid Mechanics'', Elsevier, 2005.
+[[#citeF-15|[15]]]  Zienkiewicz OC, Taylor RL, Nithiarasu N. ''The Finite Element Method. Vol. 3 Fluid Mechanics'', Elsevier, 2005.
 <div id="cite-16"></div>
-'''[[#citeF-16|[16]]]'''  Oñate E. Derivation of stabilized equations for numerical solution of advective-diffusive transport and fluid flow problems. ''Computer Methods in Applied Mechanics and Engineering'' 1998; '''151''':233-265.
+[[#citeF-16|[16]]]  Oñate E. Derivation of stabilized equations for numerical solution of advective-diffusive transport and fluid flow problems. ''Computer Methods in Applied Mechanics and Engineering'' 1998; 151:233-265.
 <div id="cite-17"></div>
-'''[[#citeF-17|[17]]]'''  Oñate E. A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation. ''Computer Methods in Applied Mechanics and Engineering'' 2000; '''182'''(3-4):355-370.
+[[#citeF-17|[17]]]  Oñate E. A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation. ''Computer Methods in Applied Mechanics and Engineering'' 2000; 182(3-4):355-370.
 <div id="cite-18"></div>
-'''[[#citeF-18|[18]]]'''  Oñate E. Possibilities of finite calculus in computational mechanics. ''International Journal for Numerical Methods in Engineering'' 2004; '''60'''(1):255&#8211;281.
+[[#citeF-18|[18]]]  Oñate E. Possibilities of finite calculus in computational mechanics. ''International Journal for Numerical Methods in Engineering'' 2004; 60(1):255&#8211;281.
 <div id="cite-19"></div>
-'''[[#citeF-19|[19]]]'''  Zienkiewicz OC, Jain P C, Oñate E.  Flow of solids during forming and extrusion: Some aspects of numerical solutions. ''International Journal of Solids and Structures'' 1978; '''14''':15&#8211;38.
+[[#citeF-19|[19]]]  Zienkiewicz OC, Jain P C, Oñate E.  Flow of solids during forming and extrusion: Some aspects of numerical solutions. ''International Journal of Solids and Structures'' 1978; 14:15&#8211;38.
 <div id="cite-20"></div>
-'''[[#citeF-20|[20]]]'''  Zienkiewicz OC, Oñate E, Heinrich JC.  A general formulation for the coupled thermal flow of metals using finite elements. ''International Journal for Numerical Methods in Engineering'' 1981; '''17''':1497&#8211;1514.
+[[#citeF-20|[20]]]  Zienkiewicz OC, Oñate E, Heinrich JC.  A general formulation for the coupled thermal flow of metals using finite elements. ''International Journal for Numerical Methods in Engineering'' 1981; 17:1497&#8211;1514.
 <div id="cite-21"></div>
-'''[[#citeF-21|[21]]]'''  Butler KM.   A model of melting and dripping thermoplastic objects in fire. Fire and Materials 2009, San Francisco, USA, January 26&#8211;28, 2009, pp. 341&#8211;352.
+[[#citeF-21|[21]]]  Butler KM.   A model of melting and dripping thermoplastic objects in fire. Fire and Materials 2009, San Francisco, USA, January 26&#8211;28, 2009, pp. 341&#8211;352.
 <div id="cite-22"></div>
-'''[[#citeF-22|[22]]]'''  Ohlemiller TJ, Shields JR.  Aspects of the Fire Behavior of Thermoplastic Materials.  ''NIST Technical Note 1493'' 2008.
+[[#citeF-22|[22]]]  Ohlemiller TJ, Shields JR.  Aspects of the Fire Behavior of Thermoplastic Materials.  ''NIST Technical Note 1493'' 2008.
 <div id="cite-23"></div>
-'''[[#citeF-23|[23]]]'''  Davdand P, Rossi R, Oñate E. An object-oriented environment for developing finite element codes for multi-disciplinary applications. ''Archives of Computational Methods in Engineering'' 2009, accepted for publication.
+[[#citeF-23|[23]]]  Davdand P, Rossi R, Oñate E. An object-oriented environment for developing finite element codes for multi-disciplinary applications. ''Archives of Computational Methods in Engineering'' 2009, accepted for publication.
 ==APPENDIX==

@@ Line 579: / Line 579: @@
 <div class="center" style="font-size: 75%;">
-'''Box II'''. De¯nition of the viscosity-temperature dependence for the polymer melt experiment of
+'''Box II'''. Definition of the viscosity-temperature dependence for the polymer melt experiment of Figure 5</div>
 Figure 5</div>

@@ Line 548: / Line 548: @@
 |}
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;max-width: 50%;"
 |-
 <pre>
@@ Line 556: / Line 555: @@
 def f1(Tc):
-return 10**(14.48 - 0.13858*Tc + 5.5960e-4*Tc*Tc - 7.8665e-7*Tc*Tc*Tc)
+    return 10**(14.48 - 0.13858*Tc + 5.5960e-4*Tc*Tc - 7.8665e-7*Tc*Tc*Tc)
 def f2(Tc):
-return 10**(53.19 - 0.2542*Tc + 2.9879e-4*Tc*Tc)
+    return 10**(53.19 - 0.2542*Tc + 2.9879e-4*Tc*Tc)
 def AuxFunction(T):
-Tc = T - 273
+    Tc = T - 273
-if( Tc <= 25):
+    if( Tc <= 25):
-mu = 1e6
+        mu = 1e6
-elif (Tc > 25 and  Tc <= 200):
+    elif (Tc > 25 and  Tc <= 200):
-mu = 1e6*(200-Tc)/175 + f1(200)
+          mu = 1e6*(200-Tc)/175 + f1(200)
-elif(Tc > 200 and Tc <= 350):
+    elif(Tc > 200 and Tc <= 350):
-mu = f1(Tc)
+         mu = f1(Tc)
-elif(Tc > 350 and Tc < 425 ):
+    elif(Tc > 350 and Tc < 425 ):
-mu = f2(Tc)
+         mu = f2(Tc)
-else:
+    else:
-mu = f2(425)
+         mu = f2(425)
-return mu
+    return mu
 </pre>
 Figure [[#img-6.a|6a]] shows four snapshots of the time evolution of the melt flow into the catch pan using the finer grid and a heat flux level of 20 kW/m<math display="inline">^{2}</math>.