progress

1 files changed, 19 insertions, 4 deletions

diff --git a/typst/main.typ b/typst/main.typ
index 7d9c539..e6b5570 100644
--- a/typst/main.typ
+++ b/typst/main.typ
@@ -94,17 +94,32 @@ than the possibly more correct value by Proudman and Pearson@proudman_pearson_19
 //$ F = F_S ( 1 + (3/8)"Re" + (9/40)("Re")^2 * (log "Re" + gamma + (5/3) log 2 - (323/360)) + (27/80)*"Re"^3 log "Re" ) $
 //)
 
-Using spherical coordinates $(r,theta,phi.alt)$ and no external force we receive the velocity components
+//Using spherical coordinates $(r,theta,phi.alt)$ and no external force we receive the velocity components
+
+//#math.equation(
+//block: true,
+//$ v_r = U cos theta (1- (3r)/(2a) + r^3 / (2a^3)) $
+//$ v_theta = - U cos theta (1- (3r)/(4a) + r^3 / (4a^3)) $
+//)
+
+== Lubrication Forces for Near Contact
+
+When spheres or a sphere near a straight wall approach each other with a small separation $h<<r$ the flow in the gap is dominated by
+lubrication forces. For a sphere approaching a straight wall with velocity $U$ normal to the wall we
+have 
 
 #math.equation(
 block: true,
-$ v_r = U cos theta (1- (3r)/(2a) + r^3 / (2a^3)) $
+$ F_(L) tilde.eq (6 pi mu r^2 U) / h (1-(9h)/(16r) + ...) $
 )
 
+With two spheres of equal radius r and otherwise the same conditions approaching
 
-Do the analitical derivation from sphere and stokes here TODO
+#math.equation(
+block: true,
+$ F_(L) tilde.eq (6 pi mu r^2 U) / h (1-(h)/(5r) + ...) $
+)
 
-== Lubrication forces between solid particles
 
 == Clogging and Bridging