From 0f4d9cfbb2d87431afc6a7566bd0ec8d240263c3 Mon Sep 17 00:00:00 2001 From: "Claudius \"keldu\" Holeksa" Date: Tue, 11 Nov 2025 19:42:18 +0100 Subject: progress --- typst/main.typ | 14 ++++++++++---- 1 file changed, 10 insertions(+), 4 deletions(-) (limited to 'typst/main.typ') diff --git a/typst/main.typ b/typst/main.typ index 546dbad..47724fc 100644 --- a/typst/main.typ +++ b/typst/main.typ @@ -40,7 +40,8 @@ Explores the effect of it in porous media and examines the more recent particle == Stokes Flow // Write about Navier -The incompressible Navier Stokes equations are a set of equations +The incompressible Navier Stokes equations are a set of equations defined with the density $rho$, the dynamic viscosity $mu$ and the +velocity field $u(x,t)$ #math.equation( block: true, @@ -50,6 +51,9 @@ $ ρ((∂)/(∂t)u + (u · ∇)u) = −∇p + μ∇²u + f,\ While we are often interested in Navier-Stokes flows, a generalized analytical solution does not exist, most analytical approaches rely on a set of restrictions or assumptions. +With the Reynolds number defined by +$ "Re" = (rho U L ) / mu +$ Luckily we decided to work on a fluid which tends $"Re" arrow 0$ or at least $"Re" << 1$ where due to the viscous forces the inertial term is negligable and is assumed to be zero. Which is why we arrive at this term @@ -66,7 +70,7 @@ a far-field criterion == Solid-Fluid interaction with Stokes flow -With the previous assumptions a solid sphere of radius $r$ moving with velocity $v$ in an unbounded creeping flow, +With the previous assumptions a solid sphere of radius $r$ moving with a relative velocity $v$ in an unbounded creeping flow, we receive the drag force provided by Stokes #math.equation( @@ -74,8 +78,10 @@ block: true, $ F = 6 pi mu r v $ ) -where according to Proudman and Pearson@proudman_and_pearson the predicted drag is two percent lower than possibly more correct value -by Proudman and Pearson@proudman_and_pearson. +where according to Proudman and Pearson@proudman_pearson_1957 at a Reynolds number of 0.05 the predicted drag is two percent lower +than the possibly more correct value by Proudman and Pearson@proudman_pearson_1957. + + // Insert analytical solution -- cgit v1.2.3