Non Newtonian Blood Pulse Flow in a Vein
The present issue deals with the non-newtonian blood pulse flow in a vessel.
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Blood Flow Project Description
The present issue deals with the blood flow in a vessel. The fluid used in this simulation is blood, which is a non-Newtonian fluid. Non-Newtonian fluids are fluids whose viscosity varies with their reaction rate, meaning they lack a specific viscosity. In this type of fluid, the relationship between the shear stress variations and the applied stress rates is nonlinear, and therefore, there is no constant coefficient of fit for viscosity. We do this transient simulation for 0.5 s. By the way, we apply a User-Defined-Function (UDF) to model the pulse of the blood flow.
We use several assumptions for the present simulation:
The solver is Pressure-Based, the simulation is Unsteady, and we also consider the gravity effect.
Geometry & Mesh of the Vein
We design the 3-D geometry of the present model by Gambit software and then linked to the Fluent software. The present model consists of a cylindrical vessel and two smaller vessels with a smaller size and diameter, such that one branch has a 90-degree angle and the other, a branch with a 45-degree curvature. In fact, the present model consists of one input section and two output sections.
We do the meshing of the present model by Gambit software. The mesh was unstructured and non-conformal and the element number is equal to 397388.
The table presents summaries of the problem definition and problem-solving steps:
|Boundary conditions (Blood Flow in a Vein)|
|Velocity inlet||Inlet type|
|Pressure outlet||Outlet type|
|0 Pa||gauge pressure|
|0 Pa||gauge pressure|
|Solution Methods (Blood Flow in a Vein)|
|second-order upwind||pressure||Spatial discretization|
Use the CARREAU model to define viscosity (Blood Flow in a Vein):
Since the fluid used in this simulation is blood at a density of 1050 kg.m-3, and blood is a Non-Newtonian fluid type, the Carreau equation with certain parameters has been used. In general, Newtonian fluids have constant viscosity during force application, but non-Newtonian fluids have variable viscosity during force application, which has different types. Time-dependent non-Newtonian fluids have two categories: Reopectic, such as printer ink and cream, which increase their viscosity over time, and Thixotropic, such as honey, whose viscosity They decrease when the force is applied.
Time-independent non-Newtonian fluids are also divided into three groups: Dilatants such as starch and clay whose viscosity depends only on the amount of applied force, and Pseudoplastic, such as greases, paints, soaps, and ketchup, which are inversely correlated with the amount of force applied. There are also other groups called Bingham, such as toothpaste and silica nanocomposites, which have a stress threshold or a certain amount of force and tension to flow. In the current simulation, blood is used as a Pseudoplastic non-Newtonian fluid defined by the Carreau model. This model attempts to create a wide range of fluids by establishing a curve commensurate with both the Newtonian and non-Newtonian fluid functions of the type of subtractive stress (Pseudoplastic). The relationship between viscosity and strain rate is as follows:
In the above relation, n represents the degree of deviation from Newtonian fluids, η0 and ηꝏ, respectively, representing high and low viscosity values, λ the time constant value, the function H (T) indicating the temperature dependence as Arrhenius law, Ƴ Symbol of strain rate and η symbol of viscosity. All of the above values are entered in the Carreau model section. 𝛂 also indicates the ratio of activation energy to thermodynamic constant and T𝛂 denotes the reference temperature.
UDF code to define the velocity (Pulse Blood Flow)
Since the flow of blood is not constant at a constant velocity and pulsed, the velocity function must be used periodically using the User-Defined Function code, then the UDF code is read in the Fluent software. Use the following function to define the pulse rate:
In the above relation, “t” denotes the local time in any time period derived from the following relation:
In the above relation, [ ] represents the integer component, or the largest integer smaller than the integer, and the current time represents the moment or time passing or progressing in the simulation.
All files, including Geometry, Mesh, Case & Data, are available in Simulation File. By the way, Training File presents how to solve the problem and extract all desired results.
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