SS laminar blood flow through a spherical aneurysm

[HappMatt]

Homework Statement

What are the equations that govern the steady laminar flow of blood, modeled as an isothermal,
incompressible, Newtonian fluid, through an aneurysm in the aorta, which is idealized as a “spherical
bulge” located in a straight tube?

Homework Equations

??
Assumptions: Newtonian, SS, incompresible, isothermal ,laminar


beyond this I'm confused on how to model this. there are lots of models that show flow around a sphere but not flow through a hollow sphere. I'm thinking that I going to have to use Bernoulli and possible a mass and momentum balance but what's confusing me is how to go from the cylindrical flow of the capillary to the spherical flow of the aneurysm. I'm not sure if there is some approximation I am supposed to make or what. I guess I am looking for suggestions on how to approach this.
thanks

The Attempt at a Solution

 

[Valkarie]

The equations that govern the steady laminar flow of blood, modeled as an isothermal, incompressible, Newtonian fluid, through an aneurysm in the aorta, which is idealized as a “spherical bulge” located in a straight tube, can be derived using the Navier-Stokes equations. The Navier-Stokes equations are a set of partial differential equations that describe the motion of an incompressible, viscous fluid. In this case, the equations are given by:Conservation of Mass: ∇⋅u = 0 Conservation of Momentum: ∂u/∂t + (u⋅∇)u = -1/ρ∇p + ν∇²u Where u is the velocity vector, p is the pressure, ρ is the density, and ν is the kinematic viscosity.For a spherical bulge, these equations can be simplified by assuming axisymmetric flow, where the velocity vector only has a radial component. Additionally, we can assume that the pressure gradient is negligible, and that the flow is steady and laminar. Under these assumptions, the above equations reduce to: Conservation of Mass: du/dr = 0 Conservation of Momentum: dP/dr + μ(d²u/dr²) = 0 Where P is the pressure, μ is the dynamic viscosity, and u is the radial velocity.These equations can then be solved numerically to obtain the velocity profile and pressure distribution in the aneurysm, given appropriate boundary conditions.

 

[piercas]

I would approach this problem by first identifying the key parameters and assumptions that need to be considered. These include the properties of the fluid (Newtonian, incompressible, isothermal), the geometry of the aneurysm (spherical bulge in a straight tube), and the type of flow (steady laminar).

Next, I would review the relevant equations that govern the flow of a Newtonian fluid, such as the Navier-Stokes equations and the continuity equation. These equations can be used to derive the velocity and pressure profiles for the fluid flow through the aneurysm.

To simplify the problem, I would consider the aneurysm as a hollow sphere with a constant radius and assume that the fluid flow is axisymmetric. This allows us to use the cylindrical coordinates and apply the equations for flow through a pipe to the spherical geometry.

To model the flow, I would use computational fluid dynamics (CFD) software to solve the governing equations numerically. This would provide a visual representation of the flow patterns and allow for the calculation of important parameters such as the velocity and pressure distribution, as well as the shear stress on the aneurysm wall.

In terms of validation, I would compare the results from the CFD simulation to experimental data or other validated models to ensure the accuracy of the solution. If needed, I may also make simplifying assumptions or use empirical correlations to approximate the flow through the aneurysm.

Overall, the approach to modeling the steady laminar flow of blood through a spherical aneurysm would involve identifying key parameters, using relevant equations, and utilizing numerical methods such as CFD to solve the problem.