Philip MJ Trevelyan
Welcome to my website.
Turbulent and Potential Flow
- My PhD was concerned with studying theoretically how to minimise the leakages from a fume cupboard with particular emphasis on the effect of a ventilation duct in the room. By assuming that potential flow was valid in the room, the flow pattern was obtained analytically using conformal mappings with complex potentials. However, a physically more realistic approach was to use the k-ε turbulent model for both the fluid flow inside and outside of the fume cupboard.
- After my PhD. I began researching surface tension instabilities of thin films coupled to heat and mass. A single long wave equation was analytically derived, but this method failed for moderate Reynolds numbers. Thus, the weighted integral approach was applied to yield two equations for the fluid flow with additional equations for heat and mass. As multiple travelling wave solitary-like solutions exist, full time dependant calculations were performed to determine which solution occurs.
- In the presence of dispersion we found that the travelling wave solutions were born via non-oscillatory dispersive waves (KdV), unlike the more common oscillatory structures found away from onset. The most common type of solitary-like wave structures consist of a flat film with convex waves on top, however, unusually, concave waves can be the most stable form of solution when the size of the nonlinear term associated with dispersion was sufficiently large and negative.
- One interesting result obtained was that the inclusion of interfacial heat losses can lead to the removal of convex travelling wave solutions when the wall that the thin film is in contact with is heated by a constant flux, however, such travelling wave solutions always exist when the wall is heated to a constant temperature.
Chemically Driven Instabilities
- Later my research was directed towards studying how to generate fluid motion by a chemical reaction via changes in the fluid's physical properties. In particular the way a simple A+B→C chemical reaction can induce buoyancy-driven instabilities was investigated. To determine if, when and where an instability occurs a linear stability analysis was performed.
- Similarly viscous fingering instabilities have been investigated with particular attention to the case when a more viscous liquid is injected into a less viscous liquid. Due to a simple A+B→C chemical reaction, non-monotonic viscosity profiles can induce an instability, even with equal diffusion coefficients. Again, a linear stability analysis was performed to confirm the instability.
Reaction Diffusion Equations
- Large time asymptotic analytical solutions have been obtained for the reaction nA+mB→C in the presence of immiscible liquids. It was found that a reaction front in immiscible liquids could travel in the opposite direction to the same reaction front in two identical miscible liquids. The centre of mass of the product can move in the opposite direction to the centre of mass of the reaction rate.
- Using a higher-order large time asymptotic expansion, a more accurate representation for the reaction front position was found. A class of reaction fronts were found whose position did not scale with the square root of time. This showed that the reaction A+2B→C was the only reaction of the form nA+mB→C with n and m being positive integers in which a reaction front could move to a non-zero finite distance away from its initial position in the large time asymptotic limit.
- Further, the small time asymptotic behaviour of the reaction front formed by A+B→C revealed that the centre of mass of the product concentration distribution is initially located at three quarters of the distance of the centre of mass of the reaction rate from the initial position.
Google Scholar: https://scholar.google.com/citations?hl=en&user=Ath_WSYAAAAJ [says 1492 citations, h-index 22]
Microsoft Academic: https://academic.microsoft.com/profile/e037e334-969j-42eg-8830-j4g6i7hg59hi/PhilipMJTrevelyan/publication/ [says 1397 citations]
Researchgate: https://www.researchgate.net/profile/Philip_Trevelyan [says 1278 citations, h-index 20, RG score 29.4, reads 5066]
Scorpus: https://www.scopus.com/authid/detail.uri?authorId=7801598562 [says 1144 citations, h-index 19]
Publons: https://publons.com/researcher/3518403/philip-trevelyan [says 1101 citations, h-index 19]
Work Email: email@example.com
Faculty of Computing, Engineering and Science
Uiversity of South Wales
CF37 1DL, UK
Lecturer, Mathematics, Glamorgan/South Wales (2010-now).
Post-Doctoral Research Associate positions:
Chemically Driven Instabilities, Nonlinear Physical Chemistry Unit, Brussels (2006-2010).
Reactive Thin Films, Chemical Engineering, Imperial College London (2004-2006).
Heated Thin Films, Chemical Engineering, Leeds (2002-2004).
Thin Films, Chemistry Department, Leeds (1999-2002).
Modules Involved With
Current Modules (2020-21):
AM0S01 - Foundations of Mathematics
MS1S461 - Mathematics and Statistics for Computing
MS2S04 - Further Calculus
MS3S20 - Partial Differential Equations
MS3S29 - Final Year Project
MS4S03 - Numerical Methods for Geophysical Flows
MS4D01 - MMath Project
AM0H01 - Foundation Quantitative Methods
AM0H02 - Foundation Mathematics
AM1S50 - Mathematics for Mechanical and Aeronautical Engineers
AM2H41 - Engineering Mathematics 2
MS0D02 - Mathematical Applications and Investigations
MS0T03 - Mathematical Applications and Investigations
MS1S13 - Method and Techniques of Mathematics
MS2S16 - Algorithms
53. P.M.J. Trevelyan, A. De Wit and J. Kent, Rayleigh-Taylor Instabilities of Classical Diffusive Density Profiles for Miscible Fluids in Porous Media: A Linear Stability Analysis, accepted to J. Eng. Math. in (2021).
52. M.J.A. Choudhury, P.M.J. Trevelyan and G.P. Boswell, Mathematical modelling of fungi-initiated siderophore-iron interactions, Mathematical Medicine & Biology, 37, 515-550, (2020).
51. S. Cowell, J. Kent and P.M.J. Trevelyan, Rayleigh-Taylor Instabilities in Miscible Fluids with Initially Piecewise Linear Density Profiles, J. Eng. Math., 121, 57-83, (2020).
https://doi.org/10.1007/s10665-020-10039-6 [1 Citation]
50. P.M.J. Trevelyan and A.J. Walker, Asymptotic Properties of Radial A+B→C reaction fronts, Phys. Rev. E. 98, 032118, (2018).
https://doi.org/10.1103/PhysRevE.98.032118 [5 Citations]
49. M.J.A. Choudhury, P.M.J. Trevelyan and G.P. Boswell, A mathematical model of nutrient influence on fungal competition, Journal of Theoretical Biology, 438, 9-20, (2018).
https://doi.org/10.1016/j.jtbi.2017.11.006 [9 Citations]
48. M.J.A. Choudhury, P.M.J. Trevelyan and G.P. Boswell, Determining the kinematic properties of an advancing front using a decomposition method, IAENG, 46, 578-584, (2016).
http://www.iaeng.org/IJAM/issues_v46/issue_4/IJAM_46_4_23.pdf. [1 Citation]
47. V. Loodts, P.M.J. Trevelyan, L. Rongy, A. De Wit, Density profiles around A+B→C reaction-diffusion fronts in partially miscible systems: A general classification, Phys. Rev. E. 94, 043115, (2016).
https://doi.org/10.1103/PhysRevE.94.043115 [23 Citations]
46. P.M.J. Trevelyan, C. Almarcha and A. De Wit, Buoyancy-driven instabilities around miscible A+B→C reaction fronts: A general classification, Phys. Rev. E. 91, 023001, (2015).
https://doi.org/10.1103/PhysRevE.91.023001 [53 Citations]
45. J. Gandhi and P.M.J. Trevelyan, Onset conditions for a Rayleigh-Taylor instability with step function density profiles, J. Engng. Math. 86, 31-48, (2014).
https://doi.org/10.1007/s10665-013-9649-2 [9 Citations]
44. C. Almarcha, P. M. J. Trevelyan, P. Grosﬁls and A. De Wit, Thermal effects on the diffusive layer convection instability of an exothermic acid-base reaction front, Phys. Rev. E. 88, 033009, (2013).
https://doi.org/10.1103/PhysRevE.88.033009 [23 Citations]
43. J. Carballido-Landeira, P.M.J. Trevelyan, C. Almarcha and A. De Wit, Mixed-mode instability of a miscible interface due to coupling between Rayleigh-Taylor and double-diffusive convection modes, Phys. Fluids. 25, 024107, (2013).
https://doi.org/10.1063/1.4790192 [46 Citations]
42. P.M.J. Trevelyan, Approximating the large time asymptotic reaction zone solution for fractional order kinetics AnBm, Discrete and Continuous Dynamical Systems – Series S, 5, 219-234, (2012).
https://doi.org/10.3934/dcdss.2012.5.219 [2 Citations]
41. L.A. Riolfo, Y. Nagatsu, S. Iwata, R. Maes, P.M.J. Trevelyan and A. De Wit, Experimental evidence of reaction-driven miscible viscous fingering, Phys. Rev. E., 85, 015304, (2012).
https://doi.org/10.1103/PhysRevE.85.015304 [57 Citations]
40. P.M.J. Trevelyan, A. Pereira and S. Kalliadasis, Dynamics of a reactive thin film, Math. Model. Nat. Phenom., 7, 99-145, (2012).
https://doi.org/10.1051/mmnp/20127408 [5 Citations]
39. C. Almarcha, Y. R’Honi, Y. De Decker, P. M. J. Trevelyan, K. Eckert, and A. De Wit, Convective mixing induced by acid-base reactions, J. Phys. Chem. B., 115, 9739-9744, (2011).
https://doi.org/10.1021/jp202201e [54 Citations]
38. P.M.J. Trevelyan, C. Almarcha and A. De Wit, Buoyancy-driven instabilities of miscible two-layer stratifications in porous media and Hele-Shaw cells, J. Fluid Mech., 670, 38-65, (2011).
https://doi.org/10.1017/S0022112010005008 [113 Citations]
37. S. Kuster, L.A. Riolfo, A. Zalts, C. El Hasi, C. Almarcha, P.M.J. Trevelyan, A. De Wit and A. D’Onofrio, Differential diffusion effects on buoyancy-driven instabilities of acid-base fronts: the case of a color indicator, J. Phys. Chem. Chem. Phys., 13, 17295-17303 (2011).
https://doi.org/10.1039/C1CP21185D [39 Citations]
36. M. Mishra, P.M.J. Trevelyan, C. Almarcha and A. De Wit, Influence of double diffusive effects on miscible viscous fingering, Phys. Rev. Lett., 105, 204501 (2010).
https://doi.org/10.1103/PhysRevLett.105.204501 [83 Citations]
35. S.H. Hejazi, P.M.J. Trevelyan, J. Azaiez and A. De Wit, Viscous fingering of a miscible reactive A+B→C interface: A linear stability analysis, J. Fluid Mech., 652, 501-528 (2010).
https://doi.org/10.1017/S0022112010000327 [95 Citations]
34. L. Rongy, P.M.J. Trevelyan and A. De Wit, Influence of buoyancy-driven convection on the dynamics of A+B→C reaction fronts in horizontal solution layers, Chem. Eng. Sci., 65, 2382-2391 (2010).
https://doi.org/10.1016/j.ces.2009.09.022 [28 Citations]
33. C. Almarcha, P.M.J. Trevelyan, L.A. Riolfo, A. Zalts, C. El Hasi, A. D’Onofrio and A. De Wit, Active role of a color indicator in buoyancy-driven instabilities of chemical fronts, J. Phys. Chem. Lett., 1, 752-757 (2010).
https://doi.org/10.1021/jz900418d [62 Citations]
32. C. Almarcha, P.M.J. Trevelyan, P. Grosfils and A. De Wit, Chemically Driven Hydrodynamic Instabilities, Phys. Rev. Lett., 104, 044501 (2010).
https://doi.org/10.1103/PhysRevLett.104.044501 [146 Citations]
31. P.M.J. Trevelyan, Analytical small time asymptotic properties of A+B→C fronts, Phys. Rev. E, 80, 046118 (2009).
https://doi.org/10.1103/PhysRevE.80.046118 [11 Citations]
30. P.M.J. Trevelyan, Higher-order large-time asymptotics for a reaction of the form nA+mB→C, Phys. Rev. E, 79, 016105 (2009).
https://doi.org/10.1103/PhysRevE.79.016105 [8 Citations]
29. C. Ruyer-Quil, P.M.J. Trevelyan, F. Giorgiutti-Dauphine, C. Duprat and S. Kalliadasis, Film flows down a fiber: Modeling and influence of streamwise viscous diffusion, Eur. Phys. J. Special Topics, 166, 89-92 (2009).
https://doi.org/10.1140/epjst/e2009-00884-0 [14 Citations]
28. A. Pereira, P.M.J. Trevelyan, U. Thiele and S. Kalliadasis, Interfacial instabilities driven by chemical reactions, Eur. Phys. J. Special Topics, 166, 121-125 (2009).
https://doi.org/10.1140/epjst/e2009-00891-1 [6 Citations]
27. P.M.J. Trevelyan, D.E. Strier and A. De Wit, Analytical asymptotic solutions of nA+mB→C reaction-diffusion equations in two-layer systems: A general study, Phys. Rev. E, 78, 026122 (2008).
https://doi.org/10.1103/PhysRevE.78.026122 [12 Citations]
26. L. Rongy, P.M.J. Trevelyan and A. De Wit, Dynamics of A+B→C reaction fronts in the presence of buoyancy-driven convection, Phys. Rev. Lett., 101, 084503 (2008).
https://doi.org/10.1103/PhysRevLett.101.084503 [67 Citations]
25. C. Ruyer-Quil, P.M.J. Trevelyan, F. Giorgiutti-Dauphine, C. Duprat and S. Kalliadasis, Modelling film flows down a fibre, J. Fluid Mech., 603, 431-462 (2008).
https://doi.org/10.1017/S0022112008001225 [116 Citations]
24. P.M.J. Trevelyan, B. Scheid, C. Ruyer-Quil and S. Kalliadasis, Heated falling films, J. Fluid Mech., 592, 295-334 (2007).
https://doi.org/10.1017/S0022112007008476 [81 Citations]
23. A. Pereira, P.M.J. Trevelyan, U. Thiele and S. Kalliadasis, Dynamics of a horizontal thin liquid film in the presence of reactive surfactants, Phys. Fluids, 19, 112102 (2007).
https://doi.org/10.1063/1.2775938 [51 Citations]
22. A. Pereira, P.M.J. Trevelyan, U. Thiele and S. Kalliadasis, Interfacial hydrodynamic waves driven by chemical reactions, J. Engng. Math., 59, 207-220 (2007).
https://doi.org/10.1007/s10665-007-9143-9 [14 Citations]
21. S. Saprykin, P.M.J. Trevelyan, R. Koopmans and S. Kalliadasis, Free-surface thin-film flows over uniformly heated topography, Phys. Rev. E, 75, 026306 (2007).
https://doi.org/10.1103/PhysRevE.75.026306 [66 Citations]
20. A. Pereira, P.M.J. Trevelyan, U. Thiele and S. Kalliadasis, Thin Films in the Presence of Chemical Reactions, Fluid Dyn. Mater. Process., 3, 303-316 (2007).
https://doi.org/10.3970/fdmp.2007.003.303 [5 Citations]
19. P.M.J. Trevelyan and S. Kalliadasis, Wave dynamics on a thin liquid film falling down a heated wall, J. Engng. Math., 50, 177-208 (2004).
https://doi.org/10.1007/s10665-004-1016-x [52 Citations]
18. P.M.J. Trevelyan and S. Kalliadasis, Dynamics of a reactive falling film at large Peclet numbers. I Long-wave approximation, Phys. Fluids, 16, 3191-3208 (2004).
https://doi.org/10/1063/1.1767834 [38 Citations]
17. P.M.J. Trevelyan and S. Kalliadasis, Dynamics of a reactive falling film at large Peclet numbers. II Nonlinear waves far from criticality: Integral-boundary-layer approximation, Phys. Fluids, 16, 3209-3226 (2004).
https://doi.org/10.1063/1.1767835 [27 Citations]
16. P.M.J. Trevelyan, S. Kalliadasis, J.H. Merkin and S.K. Scott, Dynamics of a vertically falling film in the presence of a first-order chemical reaction, Phys. Fluids, 14, 2402-2421 (2002).
https://doi.org/10.1063/1.1485761 [27 Citations]
15. P.M.J. Trevelyan, S. Kalliadasis, J.H. Merkin and S.K. Scott, Mass transport enhancement in regions bounded by rigid walls, J. Engng. Math., 42, 45-64 (2002).
https://doi.org/10.1023/A:1014369607387 [10 Citations]
14. P.M.J. Trevelyan, S. Kalliadasis, J.H. Merkin and S.K. Scott, Circulation and reaction enhancement of mass transport in a cavity, Chem. Eng. Sci., 56, 5177-5188 (2001).
https://doi.org/10.1016/S0009-2509(01)00179-8 [10 Citations]
13. L.M. Conroy, P.M.J. Trevelyan and D.B. Ingham, An analytical, numerical, and experimental comparison of the fluid velocity in the vicinity of an open tank with one and two lateral exhaust slot hoods and a uniform crossdraft, Ann. Occup. Hyg., 44, 407-419 (2000).
https://doi.org/10.1093/annhyg/44.6.407 [15 Citations]
12. P.M.J. Trevelyan, L. Elliott and D.B. Ingham, Potential flow in a semi-infinite channel with multiple sub-channels using the Schwarz-Christoffel transformation, Comput. Meth. Appl. Mech. Eng., 189, 341-359 (2000).
https://doi.org/10.1016/S0045-7825(99)00299-6 [4 Citations]
11. P.M.J. Trevelyan, L. Elliott and D.B. Ingham, A numerical method for Schwarz-Christoffel conformal transformation with application to potential flow in channels with oblique sub-channels, CMES-Comp. Model. Eng., 1, 117-122 (2000).
Conference Proceedings / Extended Abstracts
10. L.A. Riolfo, Y. Nagatsu, P.M.J. Trevelyan, A. De Wit, Chemically-driven miscible viscous fingering: how can a reaction destabilize typically stable displacements?, European Conference on Complex Systems, Springer Proceedings in Complexity, (2013).
https://doi.org/10.1007/978-3-319-00395-5_2 [2 Citations]
9. P.M.J. Trevelyan, D.E. Strier and A. De Wit, Asymptotic reaction-diffusion profiles in two-layer systems, Mathematics in Chemical Kinetics and Engineering. (Ghent, Belgium, February 2009).
8. A. Pereira, P.M.J. Trevelyan and S. Kalliadasis, Hydrodynamics of reactive thin films, Proceedings of the 22nd International Congress on Theretical Applied Mechanics, paper 1132. (Adelaide, Australia, August 2008).
7. S. Kalliadasis and P.M.J. Trevelyan, Dynamics of a reactive falling film at large Peclet numbers, Proceedings of the 21st International Congress on Theoretical Applied Mechanics, paper FM14_L10220 (Warsaw, Poland, August 2004).
6. Toth, Boissonade, Scott, Westerhoff, Jonnalagadda, Gaspar, Trevelyan, Showalter, Snita, Marek, Mayama, Dewel, Simon, Sorensen, Epstein, Satnoianu, Harrison, Merkin, Hemming, Hantz, Noszticzius, Miller, Hauser, Sielewiesiuk, General discussion, Faraday Discussions, 2002, 120, 407-419. (Manchester, UK, September 2001).
5. P.M.J. Trevelyan, D.B. Ingham and L. Elliott, The effects of ventilation and sash handles on the flow in fume cupboards, Progress in Modern Ventilation, Proceedings of Ventilation 2000, 2, Proceedings of the 6th International Symposium on Ventilation for Contaminant Control, 2000, 2, 80-83. (Helsinki, Finland, June 2000).
4. P.M.J. Trevelyan, L. Elliott and D.B. Ingham, Boundary integral approach to determine the potential fluid flow in a channel with multiple sub-channels, in Proceedings of 2nd UK Conference on Boundary Integral Methods, (L. Wrobel, S.N. Chandler-Wilde, Eds.), Brunel University Press, 1999, 291-302. ISBN 1-902316-00-2. (London, UK, September 1999).
3. P.M.J. Trevelyan, L. Elliott and D.B. Ingham, Boundary integrals applied to potential flow in channels/oblique sub-channels, Proceedings of the 1st International Conference on Boundary Element Techniques, (M.H. Aliabadi, Ed.), Queen Mary and Westfield College, University of London, 341-348. ISBN 0 904 188 531 (London, UK, July 1999).
2. P.M.J. Trevelyan, L. Elliott and D.B. Ingham, Effects of a ventilation duct on the performance of a fume cupboard, RoomVent98, 1998, 1, 385-391. (Stockholm, Sweden, June 1998).
1. P.M.J. Trevelyan, L. Elliott and D.B. Ingham, Ventilation near a fume cupboard, Proceedings of the Institution of Chemical Engineers, 1998, paper 140. (Newcastle, UK, April 1998).
Conditions for a Quadratic Equation with Complex Coefficients to have both Roots inside the Unit Circle
Any quadratic equation with complex coefficients can be written in the form:
z2 + (x + iy) z + C = 0
where x, y and C are real numbers.
If x2/(1+C)2 + y2/(1-C)2 < 1 and |C| < 1 then |z| < 1.
(i) If y = 0, then |x| < 1+C < 2 implies |z| < 1 (This is equivalent to the Jury condition for a quadratic equation with real coefficients).
(ii) If x = 0, then |y| < 1-C < 2 implies |z| < 1.
(iii) If C = 1, then y = 0 and |x| < 2 implies |z| < 1.
(iv) If C = -1, then x = 0 and |y| < 2 implies |z| < 1.
Fun Stuff: Fractal Patterns
Consider the discrete mapping zn+1 = f(zn , k). The points that lie inside the Mandelbrot set are the values of k (complex) which lead to |z| remaining finite as the integer n tends to infinity, starting from z0=0. The boundary of the Mandelbrot set is a fractal. Below some fractal patterns are sketched in the complex k plane (k=x+iy where x & y are real with i2 = -1) for some simple mapping functions.
Quadratic: zn+1 = zn2 + k
Notice that the fractal is symmetric about the real axis, i.e. θ=0 where θ is the angle from the real axis measured in the anti-clockwise direction. The boundary of the largest area is known to be the cardioid 2r=1-cos(θ) shifted along the real axis by ¼. In Cartesian coordinates this is given by 16(x2+y2) = 3 ± 2(3-8x)½. Also an infinite number of circles are attached to the cardioid. It is important to note that the Mandelbrot set is connected, however, if an insufficient resolution is used then some of the peninsulas can appear like islands.
Cubic: zn+1 = zn3 + k
We notice that this has 2 lines of symmetry, the real and imaginary axis, i.e. θ=0 & π/2. Further, there are an infinite number of cardioids attached to the main boundary.
Quartic: zn+1 = zn4 + k
Now there are 3 lines of symmetry: θ=0, π/3 & 2π/3.
Quintic: zn+1 = zn5 + k
The 4 lines of symmetry are: θ=0, π/4, π/2 & 3π/4.
Sextic: zn+1 = zn6 + k
This has 5 lines of symmetry: θ=0, π/5, 2π/5, 3π/5 & 4π/5.
These fractals were plotted using the output from a fortran 77 program like this:
DO 10, I=1,N
DO 20, J=1,N
C Define first point z(n)=U+iV and k=X+iY
DO 30, ITERATION=1,M
C Calculate z(n+1) = z(n)**2 + k where z(n+1)=P+iQ
C If |z|>2 stop iterating
If (U**2+V**2.GT.4.D0) GOTO 100
100 WRITE(99,25) X,Y,ITERATION
This program calculates the number of iterations required until the magnitude of z is greater than 2. This yields a measure of the rate of divergence for the points outside the Mandelbrot set.
Mandelbrot sets are not the only type of fractals. Another type are Julia sets which are similar to the Mandelbrot sets, as they also come from a discrete mapping zn+1= f (zn, k). The difference between them is that the Mandelbrot set is the set of values of k which lead to |z| remaining finite (starting from z0=0) whilst the Julia set is the set of values of z0 that lead to |z| remaining finite for a given value of k. Thus for a given mapping, an infinite number of Julia sets can be obtained by choosing different values of k.
Below the Julia set for the quadratic mapping zn+1 = zn2 + k with k = -0.9 is illustrated where z=x+iy. The contours represent the number of iterations required until the magnitude of z is greater than 2, to measure the rate of divergence for the points outside the Julia set.
Another example of a Julia set, using the same quadratic mapping zn+1 = zn2 + k, but now for k=0.36+0.36i yields a very different picture:
In this case the Julia set is almost empty.
The Julia set for the cubic mapping zn+1 = zn3 + k with k=-0.5+0.1i is:
However, if we change k to -0.516+0.1i the Julia set becomes:
Finally, part of the Julia set is illustrated for the mapping
zn+1 = zn3 +1.75zn2 -0.495 +0.2i