Fluid Mechanics

Abstract

This chapter provides an overview of fluid mechanics, emphasizing applications in models of blood flow. It begins with the physics of fluid motion, including the concepts of viscosity and idealized Newtonian fluids, before proceeding to the Navier Stokes equations for incompressible fluids. The concepts of laminar and non-laminar flow are also introduced, including Reynolds number and turbulent flow. Finally, the chapter describes techniques for modelling blood flow, including the use of Windkessel models (hydraulic circuit equivalents) that can be incorporated as boundary conditions in finite element models of blood flow, as well as non-Newtonian aspects of blood flow, which may be of relevance at low blood shear rates or small vessel diameters. Detailed examples of models solved in COMSOL include laminar flow in a circular tube, a multiphysics model of drug delivery in a coronary stent, aortic blood flow, as well as model of axial streaming of a blood cell using COMSOL’s moving mesh interface. The chapter ends with a small set of theoretical and computational COMSOL problems, with fully-worked answers provided in the solution section of the text.

Problems

9.1

Consider a circular tube of diameter D and length L containing an incompressible Newtonian fluid of density $\rho$ and viscosity $\mu$. Initially, the upstream and downstream pressures at each end of the tube are 0 and the fluid is at rest. Suddenly, the upstream pressure is instantaneously stepped to a value of P. By approximating this system as a concentric set of circular tubes of fluid sliding past each other, determine the governing 1D PDE and associated boundary conditions for the axial fluid velocity v in terms of radial position r and time t. Use COMSOL to solve this PDE for the following parameters: $D=10\,\mathrm {mm}$, $L=100\,\mathrm {mm}$, $\rho =1000\,\mathrm {kg}\,\mathrm {m}^{-3}$, $\mu =2\,\mathrm {mPa}\,\mathrm {s}$, and $P=10\,\mathrm {mmHg}$. Plot the axial velocity as a function of radial position for times 0, 0.5, 1, and 1.5 s.

9.2

In the example of Sect. 9.1.1, a simple axisymmetric COMSOL model was implemented to determine the steady-state fluid velocity profile in a cylindrical tube of diameter $2\,\mathrm {cm}$, length $15\,\mathrm {cm}$, fluid viscosity $3.5\,\mathrm {mPa}\,\mathrm {s}$, and pressure differential of $100\,\mathrm {Pa}$ between the ends of the tube. Use COMSOL to perform a mesh convergence analysis on this model, solving for a free-triangular mesh automatically generated at maximum element sizes of 0.1–1 mm in steps of 1 mm. Plot the axial velocity at the inlet against maximum element size. What maximum element size is required to achieve an axial velocity error of less than 5%?

9.3

The descending thoracic and abdominal aorta may be modelled as a cylindrical tube of diameter 20 mm and length 30 cm, terminated at its distal end by a simple RC hydraulic-circuit, similar to the example of Sect. 9.4.2. In this case, however, the inlet pressure P(t) is a pressure pulse whose values are shown below as a function of time:

Time (ms)	Pressure (mmHg)	Time (ms)	Pressure (mmHg)	Time (ms)	Pressure (mmHg)
0	72	360	104	720	86
40	78	400	93	760	86
80	90	440	81	800	83
120	112	480	77	840	79
160	132	520	77	880	77
200	148	560	78	920	75
240	149	600	81	960	73
280	135	640	85	1000	72
320	122	680	86

As in Sect. 9.4.2, assume the material parameters of blood are $\mu = 3.5\,\mathrm {mPa}\,\mathrm {s}$ and $\rho = 1100\,\mathrm {kg}\,\mathrm {m}^{-3}$. Also assume the blood is initially at rest: for this to be the case, the initial values of pressure at both ends of the tube must be the same (72 mmHg). Simulate this model using COMSOL and plot aortic blood flow as a function of time over 1 s.

HINT: Use COMSOL’s interpolation function feature to specify the inlet pressure waveform as a table .