• Learning Objectives

•  Determine the criterion for grid convergence for a laminar one-dimensional channel flow
•  Solve nonlinear one-dimensional channel flow for non-Newtonian fluid

Problem 1

Apply the finite volume technique to discretize and solve one-dimensional fully developed laminar flow between two horizontal parallel plates governed by

𝑢𝑢 is the velocity, 𝑃𝑃𝑥𝑥 is the pressure gradient and µ is the viscosity. In conservative form, this can be written as

In your solution show:
• A sketch for the cells, clearly marking faces and nodes for internal and boundary cells.
• Apply the linear approximation and use Dirichlet (velocity specified) boundary conditions to determine the approximate equations for internal and boundary cells.
• Compute the velocity distribution and compare your result with the exact solution, by adapting one of the uploaded codes. The number of grid cells is left up to you to determine. The solution must be grid converged.
For the numerical solution, let 𝑃𝑃𝑥𝑥 = 2µ, ℎ = 0.1, 𝑢𝑢(0) = 𝑢𝑢1 = 0.01, & 𝑢𝑢(ℎ) = 𝑢𝑢2 = 0. For grid convergence, you may define an error norm , and require that the error is less than 0.01𝑢 = 0.01 × 0.01. The exact solution is given by

Problem 2

Apply the finite volume technique to discretize and solve one-dimensional fully developed laminar non-Newtonian flow between two horizontal parallel plates governed by,

𝑢𝑢 is the velocity, 𝑃𝑃𝑥𝑥 is the pressure gradient. For non-Newtonian fluids, the viscosity µ𝑒𝑒 depends on the flow strain rate, which for one-dimensional fully developed flow is approximated by,

where 𝜇𝜇𝑜𝑜, 𝜅𝜅 are constants. Non-Newtonian fluids exist in several important applications, particularly in fluids using in printing, molten plastics used in 3D printers and most important for us, in blood and rheological flows. For more insight, you can check https://www.rheosense.com/applications/viscosity/newtonian-non-newtonian to learn more about
the shear thinning and thickening effects. As this is a non-linear problem, it is highly recommended
you follow the suggested algorithm: