Research

The following paragraphs serve as a brief introduction to some interesting, hopefully novel, research problems in convection and pattern formation that I am either currently investivating or interested in studying more in the future. For any queries, comments or interest in collaboration do not hesitate to email grp39@cam.ac.uk.

Porous Rayleigh-Bénard convection with a horizontally-heterogeneous permeability

After non-dimensionalisation, the problem of studying Rayleigh-Bénard convection in a hetereogenous porous rectangle can be formulated as the initial boundary value problem

\[\begin{split} \begin{align*} &\text{Find} \\ &c(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}, \\ &\textbf{u}(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}^2, \\ &p(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R} \\ &\text{where}~\Omega=[0,L]\times[0, 1] \\ &\text{and}~\textbf{x}=(x,y)\\ &\text{such that} \\ &\mathbb{IBVP} \begin{cases} \frac{\partial c}{\partial t} + \textbf{u}\cdot\nabla c = \frac{1}{Ra}\nabla^2c & \\ \nabla\cdot\textbf{u} = 0 & \\ \textbf{u}=-\mathsf{K}(\textbf{x})\cdot(\nabla p + c\,\textbf{e}_y) & \forall(\textbf{x}, t)\in\Omega\times[0,\infty) \\ c=1-y + \mathcal{N}(\textbf{x}) & \forall(\textbf{x}, t)\in\Omega\times\{0\} \\ c=1 & \forall(\textbf{x}, t)\in\{(x,y)\in\partial\Omega: y=0\} \times [0,\infty] \\ c=0 & \forall(\textbf{x}, t)\in\{(x,y)\in\partial\Omega: y=1\} \times [0,\infty] \\ \frac{\partial c}{\partial x} = 0 & \forall(\textbf{x}, t)\in\{(x,y)\in\partial\Omega: x=0, x=L\} \times [0,\infty] \\ \textbf{n}\cdot\textbf{u} = 0 & \forall(\textbf{x}, t)\in\partial\Omega \times [0,\infty] \\ \end{cases}~. \end{align*} \end{split}\]

The fluid is assumed to be isoviscous and hence, in non-dimensional variables, has a temperature-independent unit viscosity, whilst its density has been taken has linearly decreasing in temperature such that \(\rho(c)=-c\) combines with the effect of gravity acting vertically in the negative \(y\)-direction to give the buoyancy term \(c\,\textbf{e}_y\) in Darcy’s law. As a simple of model of heterogeneity, we have assumed a decoupling of porosity from permeability such that the porosity is constant throughout the domain and a prescribed permeability \(\mathsf{K}(\textbf{x})\) is imposed, rather than prescribing a porosity \(\phi(\textbf{x})\) and a constitutive relation \(\mathsf{K}(\phi)\). With this approach only the permeability \(\mathsf{K}(\textbf{x})\) appears in the governing equations because the constant porosity is taken into the Rayleigh number

\[Ra=\frac{HK_{\text{ref}}\,g\Delta\rho}{\mu\phi D_{\text{mol}}}\]

formed as the ratio of convective to diffusive speeds. We wish to study the effect of an isotropic but horizontally-heterogeneous permeability

\[\begin{split} \begin{align*} \mathsf{K}(\textbf{x})&=K(x)\mathsf{I} \\ K(x)&=1 + \kappa\sin\left(\frac{j\pi x}{L}\right) \end{align*} \end{split}\]

where \(0\leq\kappa<1\) and \(j\in\{0, 1, 2, \dots\}\) are parameters characterising the amplitude and frequency of the horizontal permeability perturbation away from unity.

• How is the onset of convection affected by \(\kappa, j\)?

• How is the vertical flux scaling as a function of \(Ra\) affected by \(\kappa, j\)?

• How is the plume structure affected by \(\kappa, j\)?

Porous convection from an internal point source

\[\begin{split} \begin{align*} &\text{Find} \\ &c(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}, \\ &\textbf{u}(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}^2, \\ &p(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R} \\ &\text{where}~\Omega=[0,1]\times[0, 1] \\ &\text{and}~\textbf{x}=(x,y)\\ &\text{such that} \\ &\mathbb{IBVP} \begin{cases} \frac{\partial c}{\partial t} + \textbf{u}\cdot\nabla c = \frac{1}{Ra}\nabla^2c + S(\textbf{x})& \\ \nabla\cdot\textbf{u} = 0 & \\ \textbf{u}=-\mathsf{K}(\textbf{x})\cdot(\nabla p + c\,\textbf{e}_y) & \forall(\textbf{x}, t)\in\Omega\times[0,\infty) \\ c=\mathcal{N}(\textbf{x}) & \forall(\textbf{x}, t)\in\Omega\times\{0\} \\ c=0 & \forall(\textbf{x}, t)\in\{(x,y)\in\partial\Omega: y=0, y=1\} \times [0,\infty] \\ \frac{\partial c}{\partial x} = 0 & \forall(\textbf{x}, t)\in\{(x,y)\in\partial\Omega: x=0, x=L\} \times [0,\infty] \\ \textbf{n}\cdot\textbf{u} = 0 & \forall(\textbf{x}, t)\in\partial\Omega \times [0,\infty] \\ \end{cases}~. \end{align*} \end{split}\]

\[S(\textbf{x})=s\exp\left(-\frac{(x-\tfrac{1}{2})^2+(y-\tfrac{1}{2})^2}{\ell}\right)\]

• How is the onset of convection affected by \(s, \ell\)?

• How can we model the circulation pattern around the point source?