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In this paper a model is developed to describe the three dimensional contact melting process of a cuboid on a heated surface. The mathematical description involves two heat equations (one in the solid and one in the melt), the Navier-Stokes equations for the flow in the melt, a Stefan condition at the phase change interface and a force balance between the weight of the solid and the countering pressure in the melt. In the solid an optimised heat balance integral method is used to approximate the temperature. In the liquid the small aspect ratio allows the Navier-Stokes and heat equations to be simplified considerably so that the liquid pressure may be determined using an igenfunction expansion and finally the problem is reduced to solving three first order ordinary differential equations. Results are presented showing the evolution of the melting process. Further reductions to the system are made to provide simple guidelines concerning the process. Comparison of the solutions with experimental data on the melting of n-octadecane shows excellent agreement.

The work in this paper deals with the development of momentum and thermal boundary layers when a power law fluid flows over a flat plate. At the plate we impose either constant temperature, constant flux or a Newton cooling condition. The problem is analysed using similarity solutions, integral momentum and energy equations and an approximation technique which is a form of the Heat Balance Integral Method. The fluid properties are assumed to be independent of temperature, hence the momentum equation uncouples from the thermal problem. We first derive the similarity equations for the velocity and present exact solutions for the case where the power law index n = 2. The similarity solutions are used to validate the new approximation method. This new technique is then applied to the thermal boundary layer, where a similarity solution can only be obtained for the case n = 1.

Hypothesis: The standard model for diffusion and surface kinetics driven growth of a single spherical particle in solution is applied incorrectly throughout the literature. This leads to inaccurate values for parameter values, such as the diffusion and surface kinetics coefficients. The model cannot even distinguish between diffusion or surface kinetics driven growth. Findings: It is shown that crystal growth occurs in two distinct stages. The standard model only holds during the late time. Fitting to experimental data, including the early time, leads to incorrect values for the coefficients. It is shown that diffusion and surface kinetics are interchangeable in the model and so indistinguishable. The growth is controlled by a single non-dimensional group. Previous studies, where more independent parameters are calculated have redundancy. The Gibbs-Thomson relation plays an important role but, in the cases studied here, this is only noticeable during the first growth stage where the model does not hold. For the first time an explicit relation for the variation of the radius with time is given. Excellent agreement with experimental data on CdSe growth is shown. © 2018 Elsevier Inc.

In this paper, we study the melting of a spherical nanoparticle. The model differs from previous ones in that a number of features have been incorporated to match experimental observations. These include the size dependence of the latent heat and a cooling condition at the boundary (as opposed to the fixed temperature condition used in previous studies). Melt temperature variation and density change are also included. The density variation drives the flow of the outer fluid layer. The latent heat variation is modelled by a new relation, which matches experimental data better than previous models. A novel form of Stefan condition is used to determine the position of the melt front. This condition takes into account the latent heat variation, the energy required to create new surface and the kinetic energy of the displaced fluid layer. Results show that melting times can be significantly faster than predicted by previous theoretical models; for smaller particles, this can be around a factor 3. This is primarily due to the latent heat variation. The previously used fixed temperature boundary condition had two opposing effects on melt times: the implied infinite heat transfer led to faster melting but also artificially magnified the effect of kinetic energy, which slowed down the process. We conclude that any future models of nanoparticle melting must be based on the new Stefan condition and account for latent heat variation. © 2016, Springer-Verlag Berlin Heidelberg.

A mathematical model is developed to describe the melting of nanowires. The first section of the paper deals with a standard theoretical situation, where the wire melts due to a fixed boundary temperature. This analysis allows us to compare with existing results for the phase change of nanospheres. The equivalent solidification problem is also examined. This shows that solidification is a faster process than melting; this is because the energy transfer occurs primarily through the solid rather than the liquid which is a poorer conductor of heat. This effect competes with the energy required to create new solid surface which acts to slow down the process, but overall conduction dominates. In the second section, we consider a more physically realistic boundary condition, where the phase change occurs due to a heat flux from surrounding material. This removes the singularity in initial melt velocity predicted in previous models of nanoparticle melting. It is shown that even with the highest possible flux the melting time is significantly slower than with a fixed boundary temperature condition. © 2016, Springer Science+Business Media Dordrecht.

In this paper we discuss three examples where continuum theory may be applied to describe nanoscale phenomena: \begin{enumerate} \item[1.] Enhanced flow in carbon nanotubes (CNTs) -- This model shows that the experimentally observed enhancement can be explained using standard flow equations but with a depletion layer between the liquid and solid interfaces. \item[2.] Nanoparticle melting -- Nanoparticles often exhibit a sharp increase in melting rate as the size decreases. A mathematical model will be presented which predicts this phenomena. \item[3.] Nanofluids -- Experimental results concerning the remarkable heat transfer characteristics of nanofluids are at times contradictory. We develop a model for the thermal conductivity of a nanofluid, which provides much higher predictions than the standard Maxwell model and a better match to data. \end{enumerate}

In this paper we analyse the time-dependent heat equations over a finite domain to determine expressions for the thermal diffusivity and conductivity of a nanofluid. Due to the complexity of the standard mathematical analysis of this problem we employ a well-known approximate solution technique known as the Heat Balance Integral Method. This allows us to derive simple analytical expressions for the thermal properties, which appear to depend primarily on the volume fraction and liquid properties. The model is shown to compare well with experimental data taken from the literature even up to relatively high concentrations and predicts significantly higher values than the Maxwell model for volume fractions approximately greater than 1

We develop a mathematical model to simulate the solidification process of a non-Newtonian power-law fluid flowing through a circular cross-section microchannel. The initial system consists of three partial differential equations, describing the fluid flow and temperature in the liquid and solid, which are solved over a domain specified by the Stefan condition. This is reduced to solving a partially coupled system consisting of a single partial differential equation and the Stefan condition. Results show qualitative differences, depending on the power law index and imposed flow conditions, between Newtonian and non-Newtonian solidification. The model behaviour is illustrated using power law models for blood and polyethylene oxide.