Imperial College London

DrTobyKirk

Faculty of Natural SciencesDepartment of Mathematics

Chapman Fellow in Mathematics
 
 
 
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toby.kirk12

 
 
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736Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
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22 results found

Tomlinson SD, Mayer MD, Kirk TL, Hodes M, Papageorgiou DTet al., 2024, Thermal resistance of heated superhydrophobic channels with thermocapillary stress, ASME Journal of Heat and Mass Transfer, Vol: 146, ISSN: 0017-9310

A pressure-driven channel flow between a longitudinally ridged superhydrophobic surface (SHS) and solid wall is studied, where a constant heat flux enters the channel from either the SHS or solid wall. First, a model is developed which neglects thermocapillary stresses (TCS) in the transverse direction. The caloric, convective, and total thermal resistance are evaluated, and their dependence on the shape of the liquid–gas interface (meniscus), gas ridge width, texture period, channel height, streamwise TCS, Péclet number, and channel length is established. The caloric resistance is minimized with menisci that protrude into the gas cavity, large slip fractions, small channel heights, and small streamwise TCSs. When heating from the SHS, the convective resistance increases, and therefore, a design compromise exists between caloric and convective resistances. However, when heating from the solid wall, the convective resistance remains the same and SHSs that minimize caloric resistance are optimal. We investigate both water and Galinstan for microchannel applications and find that both configurations can have a lower total thermal resistance than a smooth-walled channel. Heating from the solid wall is shown to always have the lowest total thermal resistance. Numerical simulations are used to analyze the effect of transverse TCSs. Our model captures much of the physics in heated superhydrophobic channels but is computationally inexpensive when compared to the numerical simulations.

Journal article

Hodes M, Kane D, Bazant MZ, Kirk TLet al., 2023, Asymptotic Nusselt numbers for internal flow in the Cassie state, Journal of Fluid Mechanics, Vol: 977, ISSN: 0022-1120

We consider laminar, fully developed, Poiseuille flows of liquid in the Cassie state through diabatic, parallel-plate microchannels symmetrically textured with isoflux ridges. Via matched asymptotic expansions, we develop expressions for (apparent hydrodynamic) slip lengths and Nusselt numbers. Our small parameter is the pitch of the ridges divided by the height of the microchannel. When the ridges are oriented parallel to the flow, we quantify the error in the Nusselt number expressions in the literature and provide a new closed-form result. It is accurate to and valid for any solid (ridge) fraction, whereas previous ones are accurate to and breakdown in the important limit when the solid fraction approaches zero. When the ridges are oriented transverse to the (periodically fully developed) flow, the error associated with neglecting inertial effects in the slip length is shown to be, where is the channel-scale Reynolds number based on its hydraulic diameter. The corresponding Nusselt number expressions' accuracies are shown to depend on the Reynolds number, Péclet number and Prandtl number in addition to. Manipulating the solution to the inner temperature problem encountered in the vicinity of the ridges shows that classic results for the thermal spreading resistance are better expressed in terms of polylogarithm functions.

Journal article

Kirk TL, Lewis-Douglas A, Howey D, Please CP, Chapman SJJet al., 2023, Nonlinear Electrochemical Impedance Spectroscopy for Lithium-Ion Battery Model Parameterization, JOURNAL OF THE ELECTROCHEMICAL SOCIETY, Vol: 170, ISSN: 0013-4651

Journal article

Tomlin RJJ, Roy T, Kirk TLL, Marinescu M, Gillespie Det al., 2022, Impedance Response of Ionic Liquids in Long Slit Pores, JOURNAL OF THE ELECTROCHEMICAL SOCIETY, Vol: 169, ISSN: 0013-4651

Journal article

Couto LD, Drummond R, Zhang D, Kirk T, Howey DAet al., 2022, Identifiability of Lithium-Ion Battery Electrolyte Dynamics, Pages: 1087-1093, ISSN: 0743-1619

The growing need for improved battery fast charging algorithms and management systems is pushing forward the development of high-fidelity electrochemical models of cells. Critical to the accuracy of these models is their parameterisation, however this challenge remains unresolved, both in terms of theoretical analysis and practical implementation. This paper develops a framework to analyse from impedance measurements the identifiability of electrolyte dynamics-a subcomponent of a general Li-ion model that is key to enabling accurate fast charging simulations. By assuming that the electrolyte volume fractions in the electrode and separator regions are equal, an analytic expression for the impedance function of the electrolyte dynamics is obtained, and this can be tested for structural identifiability. It is shown that the only parameters of the electrolyte model that may be identified are the diffusion time scale and a geometric coupling parameter. Simulations highlight the identifiability issues of electrolyte dynamics (relating to symmetric cells) and explain how the electrolyte parameters might be identified.

Conference paper

Kirk TL, Evans J, Please CP, Chapman SJet al., 2022, MODELING ELECTRODE HETEROGENEITY IN LITHIUM-ION BATTERIES: UNIMODAL AND BIMODAL PARTICLE-SIZE DISTRIBUTIONS, SIAM Journal on Applied Mathematics, Vol: 82, Pages: 625-653, ISSN: 0036-1399

In mathematical models of lithium-ion batteries, the highly heterogeneous porous electrodes are frequently approximated as comprising spherical particles of uniform size, leading to the commonly used single-particle model (SPM) when transport in the electrolyte is assumed to be fast. Here electrode heterogeneity is modeled by extending this to a distribution of particle sizes. Unimodal and bimodal particle-size distributions (PSD) are considered. For a unimodal PSD, the effect of the spread of the distribution on the cell dynamics is investigated, and choice of effective particle radius when approximating by an SPM assessed. Asymptotic techniques are used to derive a correction to the SPM valid for narrow, but realistic, PSDs. In addition, it is shown that the heterogeneous internal states of all particles (relevant when modeling degradation, for example) can be efficiently computed after the fact. For a bimodal PSD, the results are well approximated by a double-particle model (DPM), with one size representing each mode. Results for lithium iron phosphate with a bimodal PSD show that the DPM captures an experimentally observed double plateau in the discharge curve, suggesting it is entirely due to bimodality.

Journal article

Yariv E, Kirk TL, 2021, Longitudinal thermocapillary slip about a dilute periodic mattress of protruding bubbles, IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), Vol: 86, Pages: 490-501, ISSN: 0272-4960

A common realization of superhydrophobic surfaces comprises of a periodic array of cylindrical bubbles which are trapped in a periodically grooved solid substrate. We consider the thermocapillary animation of liquid motion by a macroscopic temperature gradient which is longitudinally applied over such a bubble mattress. Assuming a linear variation of the interfacial tension with the temperature, at slope \sigma T, we seek the effective velocity slip attained by the liquid at large distances away from the mattress. We focus upon the dilute limit, where the groove width 2c is small compared with the array period 2l. The requisite velocity slip in the applied-gradient direction, determined by a local analysis about a single bubble, is provided by the approximation \begin{align} \pi \frac{G\sigmaT c^2}{\mu l} I(\alpha), \end{align∗}wherein G is the applied-gradient magnitude, \mu is the liquid viscosity and I(\alpha) , a non-monotonic function of the protrusion angle \alpha , is provided by the quadrature, \begin{align} I(\alpha) = \frac{2}{\sin\alpha} \int 0^\infty\frac{\sinh s\alpha}{ \cosh s(\pi-\alpha) \sinh s \pi} \, \textrm{d} s. \end{align}

Journal article

Kirk TL, Please CP, Jon Chapman S, 2021, Physical Modelling of the Slow Voltage Relaxation Phenomenon in Lithium-Ion Batteries, JOURNAL OF THE ELECTROCHEMICAL SOCIETY, Vol: 168, ISSN: 0013-4651

Journal article

Alexander JP, Kirk TL, Papageorgiou DT, 2020, Stability of falling liquid films on flexible substrates, Journal of Fluid Mechanics, Vol: 900, Pages: A40-1-A40-33, ISSN: 0022-1120

The linear stability of a liquid film falling down an inclined flexible plane under the influence of gravity is investigated using analytical and computational techniques. A general model for the flexible substrate is used leading to a modified Orr–Sommerfeld problem addressed numerically using a Chebyshev tau decomposition. Asymptotic limits of long waves and small Reynolds numbers are addressed analytically and linked to the computations. For long waves, the flexibility has a destabilising effect, where the critical Reynolds number decreases with decreasing stiffness, even destabilising Stokes flow for sufficiently small stiffness. To pursue this further, a Stokes flow approximation was considered, which confirmed the long-wave results, but also revealed a short wave instability not captured by the long-wave expansions. Increasing the surface tension has little effect on these instabilities and so they were characterised as wall modes. Wider exploration revealed mode switching in the dispersion relation, with the wall and surface mode swapping characteristics for higher wavenumbers. The zero-Reynolds-number results demonstrate that the long-wave limit is not sufficient to determine instabilities so the numerical solution for arbitrary wavenumbers was sought. A Chebyshev tau spectral method was implemented and verified against analytical solutions. Short wave wall instabilities persist at larger Reynolds numbers and destabilisation of all Reynolds numbers is achievable by increasing the wall flexibility, however increasing the stiffness reverts back to the rigid wall limit. An energy decomposition analysis is presented and used to identify the salient instability mechanisms and link them to their physical origin.

Journal article

Kirk T, Karamanis G, Crowdy D, Hodes Met al., 2020, Thermocapillary stress and meniscus curvature effects on slip lengths in ridged microchannels, Journal of Fluid Mechanics, Vol: 894, ISSN: 0022-1120

Pressure-driven flow in the presence of heat transfer through a microchannel patterned with parallel ridges is considered. The coupled effects of curvature and thermocapillary stress along the menisci are captured. Streamwise and transverse thermocapillary stresses along menisci cause the flow to be three-dimensional, but when the Reynolds number based on the transverse flow is small the streamwise and transverse flows decouple. In this limit, we solve the streamwise flow problem, i.e. that in the direction parallel to the ridges, using a suite of asymptotic limits and techniques – each previously shown to have wide ranges of validity thereby extending results by Hodes et al. (J. Fluid Mech., vol. 814, 2017, pp. 301–324) for a flat meniscus. First, we take the small-ridge-period limit, and then we account for the curvature of the menisci with two further complementary limits: (i) small meniscus curvature using boundary perturbation; (ii) arbitrary meniscus curvature but for small slip (or cavity) fractions using conformal mapping and the Poisson integral formula. Heating and cooling the liquid always degrade and enhance (apparent) slip, respectively, but their effect is greatest for large meniscus protrusions, with positive protrusion (into the liquid) being the most sensitive. For strong enough heating the solutions become complex, suggesting instability, with large positive protrusions transitioning first.

Journal article

Mayer M, Hodes M, Kirk T, Crowdy Det al., 2019, Effect of surface curvature on contact resistance between cylinders, Journal of Heat Transfer, Vol: 141, ISSN: 0022-1481

Due to the microscopic roughness of contacting materials, an additional thermal resistance arises from the constriction and spreading of heat near contact spots. Predictive models for contact resistance typically consider abutting semi-infinite cylinders subjected to an adiabatic boundary condition along their outer radius. At the nominal plane of contact, an isothermal and circular contact spot is surrounded by an adiabatic annulus and the far-field boundary condition is one of constant heat flux. However, cylinders with flat bases do not mimic the geometry of contacts. To remedy this, we perturb the geometry of the problem such that, in cross section, the circular contact is surrounded by an adiabatic arc. When the curvature of this arc is small, we employ a series solution for the leading-order (flat base) problem. Then, Green's second identity is used to compute the increase in spreading resistance in a single cylinder, and thus the contact resistance for abutting ones, without fully resolving the temperature field. Complementary numerical results for contact resistance span the full range of contact fraction and protrusion angle of the arc. The results suggest as much as a 10–15% increase in contact resistance for realistic contact fraction and asperity slopes. When the protrusion angle is negative, the decrease in spreading resistance for a single cylinder is also provided.

Journal article

Game S, Hodes M, Kirk T, Papageorgiou DTet al., 2018, Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges for Arbitrary Meniscus Curvature, JOURNAL OF HEAT TRANSFER-TRANSACTIONS OF THE ASME, Vol: 140, ISSN: 0022-1481

Journal article

Crowdy D, Hodes M, Kirk T, 2018, Spreading and contact resistance formulae capturing boundary curvature and contact distribution effects, Journal of Heat Transfer, Vol: 140, ISSN: 0022-1481

There is a substantial and growing body of literature which solves Laplace's equation governing the velocity field for a linear-shear flow of liquid in the unwetted (Cassie) state over a superhydrophobic surface. Usually, no-slip and shear-free boundary conditions are applied at liquid–solid interfaces and liquid–gas ones (menisci), respectively. When the menisci are curved, the liquid is said to flow over a “bubble mattress.” We show that the dimensionless apparent hydrodynamic slip length available from studies of such surfaces is equivalent to (i) the dimensionless spreading resistance for a flat, isothermal heat source flanked by arc-shaped adiabatic boundaries and (ii) the dimensionless thermal contact resistance between symmetric mating surfaces with flat contacts flanked by arc-shaped adiabatic boundaries. This is important because real surfaces are rough rather than smooth. Furthermore, we demonstrate that this observation provides a significant source of new and explicit results on spreading and contact resistances. Significantly, the results presented accommodate arbitrary solid-to-solid contact fraction and arc geometry in the contact resistance problem for the first time. We also provide formulae for the case when each period window includes a finite number of no-slip (or isothermal) and shear free (or adiabatic) regions and extend them to the case when the latter are weakly curved. Finally, we discuss other areas of mathematical physics to which our results are directly relevant.

Journal article

Karamanis G, Hodes M, Kirk T, Papageorgiou DTet al., 2018, Solution of the extended Graetz-Nusselt problem for liquid flow over isothermal parallel ridges, Journal of Heat Transfer: Transactions of the ASME, Vol: 140, Pages: 1-15, ISSN: 0022-1481

We consider convective heat transfer for laminar flow of liquid between parallel plates. The configurations analyzed are both plates textured with symmetrically aligned isothermal ridges oriented parallel to the flow, and one plate textured as such and the other one smooth and adiabatic. The liquid is assumed to be in the Cassie state on the textured surface(s) to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). We solve for the developing three-dimensional temperature profile resulting from a step change of the ridge temperature in the streamwise direction assuming a hydrodynamically developed flow. Axial conduction is accounted for, i.e., we consider the extended Graetz–Nusselt problem; therefore, the domain is of infinite length. The effects of viscous dissipation and (uniform) volumetric heat generation are also captured. Using the method of separation of variables, the homogeneous part of the thermal problem is reduced to a nonlinear eigenvalue problem in the transverse coordinates which is solved numerically. Expressions derived for the local and the fully developed Nusselt number along the ridge and that averaged over the composite interface in terms of the eigenvalues, eigenfunctions, Brinkman number, and dimensionless volumetric heat generation rate. Estimates are provided for the streamwise location where viscous dissipation effects become important.

Journal article

Kirk TL, 2018, Asymptotic formulae for flow in superhydrophobic channels with longitudinal ridges and protruding menisci, Journal of Fluid Mechanics, Vol: 839, Pages: R31-R312, ISSN: 0022-1120

This paper presents new asymptotic formulae for flow in a channel with one or both walls patterned with a longitudinal array of ridges and arbitrarily protruding menisci. Derived from a matched asymptotic expansion, they extend results by Crowdy (J. Fluid Mech., vol. 791, 2016, R7) for shear flow, and thus make no restriction on the protrusion into or out of the liquid. The slip length formula is compared against full numerical solutions and, despite the assumption of small ridge period in its derivation, is found to have a very large range of validity; relative errors are small even for periods large enough for the protruding menisci to degrade the flow and touch the opposing wall.

Journal article

Kadoko J, Karamanis G, Kirk T, Hodes Met al., 2017, One-Dimensional Analysis of Gas Diffusion-Induced Cassie to Wenzel State Transition, JOURNAL OF HEAT TRANSFER-TRANSACTIONS OF THE ASME, Vol: 139, ISSN: 0022-1481

Journal article

Karamanis G, Hodes M, Kirk T, Papageorgiou DTet al., 2017, Solution of the Graetz-Nusselt problem for liquid flow over isothermal parallel ridges, Journal of Heat Transfer: Transactions of the ASME, Vol: 139, Pages: 1-12, ISSN: 0022-1481

We consider convective heat transfer for laminar flow of liquid between parallel plates that are textured with isothermal ridges oriented parallel to the flow. Three different flow configurations are analyzed: one plate textured and the other one smooth; both plates textured and the ridges aligned; and both plates textured, but the ridges staggered by half a pitch. The liquid is assumed to be in the Cassie state on the textured surface(s), to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. Heat is exchanged with the liquid either through the ridges of one plate with the other plate adiabatic, or through the ridges of both plates. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). Axial conduction is neglected and the inlet temperature profile is arbitrary. We solve for the three-dimensional developing temperature profile assuming a hydrodynamically developed flow, i.e., we consider the Graetz–Nusselt problem. Using the method of separation of variables, the thermal problem is essentially reduced to a two-dimensional eigenvalue problem in the transverse coordinates, which is solved numerically. Expressions for the local Nusselt number and those averaged over the period of the ridges in the developing and fully developed regions are provided. Nusselt numbers averaged over the period and length of the domain are also provided. Our approach enables the aforementioned quantities to be computed in a small fraction of the time required by a general computational fluid dynamics (CFD) solver.

Journal article

Hodes M, Kirk TL, Karamanis G, MacLachlan Set al., 2017, Effect of thermocapillary stress on slip length for a channel textured with parallel ridges, JOURNAL OF FLUID MECHANICS, Vol: 814, Pages: 301-324, ISSN: 0022-1120

Journal article

Kirk TL, Hodes M, Papageorgiou DT, 2016, Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature, Journal of Fluid Mechanics, Vol: 811, Pages: 315-349, ISSN: 1469-7645

We investigate forced convection in a parallel-plate-geometry microchannel with superhydrophobic walls consisting of a periodic array of ridges aligned parallel to the direction of a Poiseuille flow. In the dewetted (Cassie) state, the liquid contacts the channel walls only at the tips of the ridges, where we apply a constant-heat-flux boundary condition. The subsequent hydrodynamic and thermal problems within the liquid are then analysed accounting for curvature of the liquid–gas interface (meniscus) using boundary perturbation, assuming a small deflection from flat. The effects of this surface deformation on both the effective hydrodynamic slip length and the Nusselt number are computed analytically in the form of eigenfunction expansions, reducing the problem to a set of dual series equations for the expansion coefficients which must, in general, be solved numerically. The Nusselt number quantifies the convective heat transfer, the results for which are completely captured in a single figure, presented as a function of channel geometry at each order in the perturbation. Asymptotic solutions for channel heights large compared with the ridge period are compared with numerical solutions of the dual series equations. The asymptotic slip length expressions are shown to consist of only two terms, with all other terms exponentially small. As a result, these expressions are accurate even for heights as low as half the ridge period, and hence are useful for engineering applications.

Journal article

Lam LS, Hodes M, Karamanis G, Kirk T, MacLachlan Set al., 2016, Effect of Meniscus Curvature on Apparent Thermal Slip, JOURNAL OF HEAT TRANSFER-TRANSACTIONS OF THE ASME, Vol: 138, ISSN: 0022-1481

Journal article

Kadoko J, Karamanis G, Kirk T, Hodes Met al., 2016, ANALYSIS OF GAS DIFFUSION-INDUCED CASSIE TO WENZEL STATE TRANSITION ON A STRUCTURED SURFACE, ASME Summer Heat Transfer Conference, Publisher: AMER SOC MECHANICAL ENGINEERS

Conference paper

Karamanis G, Hodes M, Kirk T, Papageorgiou DTet al., 2016, Nusselt Numbers for Fully-Developed Flow Between Parallel Plates with One Plate Textured with Isothermal Parallel Ridges, ASME Summer Heat Transfer Conference, Publisher: AMER SOC MECHANICAL ENGINEERS

Conference paper

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