## Fluid Dynamics and Mechanobiology

Today I want to discuss some of the mathematics I’ve been doing, and mention where I hope my research goes. My doctoral research hasn’t exactly narrowed down on a particular question yet, as we are asking around to find experimental collaborators, but I have been learning quite a lot in the way of solid and fluid mechanics, mechanobiology, and a host of related ideas.

First I want to discuss a classical problem in fluid mechanics (flow through a cylindrical pipe), and then a slight modification of it that makes it a bit more interesting, and possibly related to biological capillary flow. The notes are in part for my own review, but any comments or questions would be very welcome. After this I will briefly touch on the kinds of problems my research interests are becoming, although everything at the moment is so interesting that this might change.

1. Brief Background on Fluid Dynamics

Let ${\boldsymbol{u}= (u,v,w)}$ be the velocity field of the fluid we are considering. The Navier-Stokes equations can then be written as

$\displaystyle \rho\left(\frac{\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u} \cdot \nabla)\boldsymbol{u}\right) = -\nabla p + \boldsymbol{F} + \nu\Delta\boldsymbol{u} \ \ \ \ \ (1)$

where ${\rho}$ is the density of the liquid, ${p}$ is the pressure, ${\nu}$ is the viscosity and ${\boldsymbol{F}}$ are any body forces. The left hand side of this equation can be written using the material derivative, which accounts for convection of the flow. Overall these equations account for conservation of momentum of the fluid. Conservation of mass is also useful, but here I will make the assumption that the fluid is incompressible, and that the density is constant. This means all flows are divergenceless, e.g.

$\displaystyle \nabla \cdot \boldsymbol{u} = 0. \ \ \ \ \ (2)$

We let ${L,U}$, and ${p^*}$ be characteristic length, velocity, and pressure scales respectively. We recall the definition of the Reynold’s number as

$\displaystyle Re = \frac{\rho U L}{\nu}. \ \ \ \ \ (3)$

If we scale equations (1) using these, ${\frac{L}{U}}$ as a typical time scale, and scaling the body force ${\boldsymbol{F}}$ with ${{p^*}{L}}$ we find

$\displaystyle \frac{\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u} \cdot \nabla)\boldsymbol{u} =\frac{p*}{\rho U^2}( -\nabla p + \boldsymbol{F}) + \frac{1}{Re}\Delta\boldsymbol{u}. \ \ \ \ \ (4)$

Now we haven’t specified the pressure scale ${p^*}$ in terms of dimensional quantities yet. It may be easy to see from (4) how one could choose this scale to focus on viscous terms when ${Re \ll 1}$, or inertial terms when ${Re \gg 1}$. We will concentrate on the former case, and take ${p^* = \frac{U}{L}}$ so that (4) can be well approximated by

$\displaystyle \Delta\boldsymbol{u} + \boldsymbol{F} = \frac{1}{\nu}\nabla p. \ \ \ \ \ (5)$

Equations (5) are known as the equations of Stokes Flow. Note that this is a linearisation that required assumptions on the Reynold’s number, and hence on characteristic length and velocity scales. For certain problems these assumptions break down, but they do allow us to do quite a bit more analysis than the general Navier-Stokes equations.

In Hagen-Poiseuille flow, we are interested in axisymmetric laminar flow in an (infinitely long) pipe driven by a constant pressure gradient (${\boldsymbol{F} = 0}$). We write the velocity ${\boldsymbol{u} = (u_r,u_\theta,u_z)= (u,v,w)}$ in cylindrical coordinates where ${r}$ is the radius from the center of the pipe, ${\theta}$ the angle from some prescribed plane, and ${z}$ the location down the length of the pipe, again from some initial position. I will direct you to the solution of that here, rather than repeating the procedure. Note particularly that the fluid velocity is assumed to only have a nonzero component in the ${z}$ direction, and that it is a function only of the radius ${r}$. The boundary conditions in this case reduce to ${w = 0}$ on ${r=R}$ where we have scaled the radial coordinate using ${L=R}$ as the radius of the pipe, and that the velocity is finite at ${r=0}$. The solution profile given by a pressure gradient ${\frac{\partial p}{\partial z}}$ is

$\displaystyle w = -\frac{1}{4}\frac{\partial p}{\partial z}(1-r^2). \ \ \ \ \ (6)$

Note that this is using all of the scalings above. Usually we take the pressure gradient to be negative in the positive ${z}$ direction so that the fluid flows in this direction, with this parabolic velocity profile.

2. A Perturbed Boundary Problem

We consider Hagen-Poiseuille flow, except with the boundary being sinusoidally perturbed. Everything is still assumed to be axisymmetric, so it seems appropriate to let our velocity profile look like ${\boldsymbol{u} = (u, 0, w)}$, and depend on ${r}$ and ${z}$. That is, we consider the Stokes’ and Continuity equations and the following boundary conditions,

$\displaystyle \begin{array}{rcl} \frac{\rho}{\nu} \left(\frac{\partial u^*}{\partial t^*} + u^* \frac{\partial u^*}{\partial r^*} + w^* \frac{\partial u^*}{\partial z^*}\right) = -\frac{1}{\nu}\frac{\partial p^*}{\partial r^*} + \frac{1}{r^*}\frac{\partial}{\partial r^*}(r^*\frac{\partial u^*}{\partial r^*}) + \frac{\partial^2 u^*}{\partial z^{*2}} - \frac{u^*}{r^{*2}} \\ \frac{\rho}{\nu} \left(\frac{\partial w^*}{\partial t^*} + u^* \frac{\partial w^*}{\partial r^*} + w^* \frac{\partial w^*}{\partial z^*}\right) = -\frac{1}{\nu}\frac{\partial p^*}{\partial z^*} + \frac{1}{r}\frac{\partial}{\partial r^*}(r^*\frac{\partial w^*}{\partial r}) + \frac{\partial^2 w^*}{\partial z^{*2}} \\ \frac{1}{r^*}\frac{\partial}{\partial r^*}(r^* u^*) + \frac{\partial w^*}{\partial z^*} = 0\\ u^*(R+\epsilon R \sin(\frac{z^*}{\lambda}), z^*) = 0\\ w^*(R+\epsilon R \sin(\frac{z^*}{\lambda}), z^*) = 0. \end{array}$

If we introduce new axial and radial coordinates we can create a further simplification of the problem. Let ${z = \frac{z^*}{\lambda}}$ and ${r = \frac{r^*}{R}}$, where ${R}$ is the mean radius of the pipe. We also scale the pressure by ${p = \frac{R^3}{Q \nu \lambda}p^*}$, and the fluid velocities by ${u = \frac{\lambda}{Q}u^*}$ and ${w = \frac{R}{Q} w^*}$, where ${Q}$ is a flux rate per unit length (e.g. ${Q = \frac{1}{\pi R}\int_0^{2 \pi}\int_0^Ru^* r^* dr^* d\theta}$). Let ${t = \frac{Q}{\lambda R}t^*}$. Scaling and rearranging, our equations now look like:

$\displaystyle \begin{array}{rcl} \delta^4 Re \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z}\right) = - \frac{\partial p}{\partial r} + \delta^2\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial u}{\partial r}) + \delta^4\frac{\partial^2 u}{\partial z^2} - \delta^2\frac{u}{r^2}\\ \delta^2 Re \left(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial r} + w \frac{\partial w}{\partial z}\right) =-\frac{\partial p}{\partial z} + \frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial w}{\partial r}) + \delta^2\frac{\partial^2 w}{\partial z^2}\\ \frac{1}{r}\frac{\partial}{\partial r}(r u) + \frac{\partial w}{\partial z} = 0\\ u^*(1+\epsilon \sin(z), z) = 0\\ w^*(1+\epsilon \sin(z), z) = 0 \end{array}$

where ${\delta = \frac{R}{\lambda}}$ is the ratio of the mean pipe radius to the wavelength of the sinusoidal boundary, and ${Re = \frac{\rho \lambda Q}{\nu R}}$ is the Reynold’s number. There are some details about when the terms involving ${Re}$ are negligible, but we will assume that the flow rates are small and viscosity large enough that ${Re \ll 1}$ for the rest of this discussion.a bit of the biology that I have been learning

We approach this problem by expanding our solutions in the small parameters ${\epsilon}$ and ${\delta^2}$,

$\displaystyle \begin{array}{rcl} u = u_0 + \epsilon u_1 + \delta^2 u_2 + \dots\\ w = w_0 + \epsilon w_1 + \delta^2 w_2 + \dots\\ p = p_0 + \epsilon p_1 + \delta^2 p_2 + \dots \end{array}$

The boundary conditions can also be written as

$\displaystyle \begin{array}{rcl} u(1+\epsilon \sin(z), z) = u(1,z) + \epsilon \sin(z) \frac{\partial u}{\partial r}(1,z) + \dots = 0\\ w(1+\epsilon \sin(z), z) = w(1,z) + \epsilon \sin(z) \frac{\partial w}{\partial r}(1,z) + \dots = 0, \end{array}$

where we have expanded them around the linear boundary at ${r=1}$. Substituting our expansions above into these boundary equations, and matching powers of ${\epsilon}$ and ${\delta^2}$, we can determine boundary conditions for our expanded solution.

Setting ${\epsilon=\delta=0}$ gives the zero order problem as

$\displaystyle \begin{array}{rcl} 0 = \frac{\partial p_0}{\partial r}\\ \frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial w_0}{\partial r}) = \frac{\partial p_0}{\partial z}\\ \frac{1}{r}\frac{\partial}{\partial r}(r u_0) + \frac{\partial w_0}{\partial z} = 0\\ u_0(1,z) = 0\\ w_0(1,z) = 0. \end{array}$

This problem for ${u_0,w_0}$ will give us the Hagen-Poiseuille solution, as symmetry arguments will collapse the solution into simple parabolic unidirectional flow. Matching terms of order ${\epsilon}$ we have

$\displaystyle \begin{array}{rcl} 0 = \frac{\partial p_1}{\partial r}\\ \frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial w_1}{\partial r}) = \frac{\partial p_1}{\partial z}\\ \frac{1}{r}\frac{\partial}{\partial r}(r u_1) + \frac{\partial w_1}{\partial z} = 0\\ u_1(1,z) = -\sin(z) \frac{\partial u_0}{\partial r}(1,z)\\ w_1(1,z) = -\sin(z) \frac{\partial w_0}{\partial r}(1,z) \end{array}$

and for ${O(\delta^2)}$ we have,

$\displaystyle \begin{array}{rcl} \frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial u_0}{\partial r}) - \frac{u_0}{r^2} = \frac{\partial p_2}{\partial r}\\ \frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial w_2}{\partial r}) + \frac{\partial^2 w_0}{\partial z^2} = \frac{\partial p_2}{\partial z}\\ \frac{1}{r}\frac{\partial}{\partial r}(r u_2) + \frac{\partial w_2}{\partial z} = 0\\ u_2(1,z) = 0)\\ w_2(1,z) = 0. \end{array}$

We know that for flow in a pipe, the (dimensional) pressure drop can be computed as ${\Delta P = \frac{8 L Q}{\nu \pi R^2}}$. Because of our pressure scaling, this quantity will appear exactly in the ${p_0}$ term. Therefore, the (nondimensional) solution for problem (6) is

$\displaystyle u_0 = 0,\quad w_0 = \frac{2}{\pi}(1-r^2),\quad p_0 = -\frac{8}{\pi} z \ \ \ \ \ (7)$

The solution for problem (6) is

$\displaystyle u_1 = 2\frac{r-r^3}{\pi}\cos(z),\quad w_1 = \frac{4}{\pi}\sin(z)(2r^2-1),\quad p_1 = -\frac{32}{\pi}\cos(z) \ \ \ \ \ (8)$

If we assume that problem (6) is small (That is, ${\delta^2}$ is negligible compared with ${\epsilon}$), our approximate velocities look like

$\displaystyle \begin{array}{rcl} u(r,z) = 2 \epsilon\frac{r-r^3}{\pi}\cos(z)\\ w(r,z) = \frac{2}{\pi}(1-r^2) + \epsilon(\frac{4}{\pi}\sin(z)(2r^2-1)). \end{array}$

Below are two plots of these solutions as velocity fields, with some characteristic streamlines, for different values of ${\epsilon}$.

Finally, I should note that this problem was posed to me by my supervisor, and is related to research she did some years back studying the glycocalyx coating of cells in capillaries and other small vessels. In that model, the flow is allowed to leak out of the boundary through poroelastic effects, and the perturbation approach is more focused on the thin-film approximation (e.g. on $\delta$).

3. Insights into Tissue Mechanics

There is quite a lot of research in tissue-level biology at the moment, being done by very interdisciplinary groups. I mentioned some of this in a previous blog post, but now that I am in a very active mathematical biology group, I’ve seen much of the open problems first hand. Some of the things that immediately jump out at me are the difficulties in modelling Soft Matter, which is very applicable in biology. Much of our theories of solid mechanics rely on small deformations to have nice linear models, and these just don’t accurately model soft tissue. Likewise, physiological fluids are often composed of many different components which can be modelled in a variety of ways, for example as mixtures. I am increasingly becoming interested in working in these areas, especially where they overlap such as in development, or in the functioning of complex physiologies (e.g. the heart).

I will write a bit more in a later post on angiogenesis and vasularization, particularly with respect to the applications in cancer biology, wound healing, and tissue engineering. This is the general area which I hope to do my doctoral work in, although as I mentioned at the beginning this will depend very much on the experimental collaborations that occur.

As always, comments and questions are very much appreciated.