kill(all)$

fluid : Cauchy momentum equation,version 1.0
peter.vlasschaert@gmail.com,11/06/2019

load packages
----------------------

load(lrats)$

see also : see website (  = cylindrical coordinates)
===================
----------------------------------------------------------------------------------------------------------------------------------
=> fluid : How to find differentation unit vector(cylindrical coordinates),version 1.0
=> fluid  : divergence of tensor cylindrical coordinates( = tensor),version 1.0.(why)
=> fluid : Why , div van tensor (cylindrical coordinates),version 1.0
=> fluid :Material Derivative,'cylindrical' coordinates,version 1.0,corrected (11/06/2019)
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Cauchy momentum equation .  ( = cylindrical coordinates)
=================================================

Du/dt  = 1/ρ*div(σ) + g

u =ur*er+uθ*eθ + uz*ez  
g = body forces per unit mass
σ =  second order ( = stress tensor)
ρ = density ( @ point ,continuum)
u = flow velocity vector field

rem :  ur = ur(r,θ,z,t) = dr/dt
            uθ= uθ(r,θ,z,t) = r*dθ/dt = r*ω
            uz = uz(r,θ,z,t) = dz/dt
rem : {er,eθ,ez} unit vectors cylindrical coordinates

tensor :  'second order ': σ.
-----------------------------------------

p1:σ=expand(eθ^2*σθθ+eθ*(ez*σθz+er*σθr+ez*σzθ+er*σrθ)+ez^2*σzz+er*ez*σzr+er*ez*σrz+er^2*σrr);

index notation :σ

p1a:lhs(p1)=expand(eθ^2*σ[θ,θ]+eθ*(ez*σ[θ,z]+er*σ[θ,r]+ez*σ[z,θ]+er*σ[r,θ])+ez^2*σ[z,z]+er*ez*σ[z,r]+er*ez*σ[r,z]+er^2*σ[r,r]);

rem : er^2 = er*er
           eθ^2 = eθ*eθ
           ez^2 = ez*ez

relation between : cartesian coordinates , cylindrical coordinates (see website)
-----------------------------------------------------------------------------------------------------------------------

cylindrical coordinates: 
1:r, 2:θ,3:z
cartesian coordinates :
1:x,2:y,3:z

differentation :::  unit vectors: see 'above'
----------------------------------------------------------------

p2:['diff(er,θ) = eθ,'diff(eθ,θ) = -er];

cylindrical coordinates:gradient  ( = gradf)
--------------------------------------------

gradf : cylindrical coordinates
=========================

p3:gradf(m) := diff(m,r)*er+1/r*diff(m,θ)*eθ+diff(m,z)*ez;

Material Derivative  ( = Du/dt)
========================
result  => vector : 'first order tensor'

er: components
------------------------

p4a:ar=-uθ^2/r+(('diff(ur,θ,1))*uθ)/r+('diff(ur,z,1))*uz+'diff(ur,t,1)+ur*('diff(ur,r,1));

index notation :
=============

p4a1:a[r]=-u[θ]^2/r+(('diff(u[r],θ,1))*u[θ])/r+('diff(u[r],z,1))*u[z]+'diff(u[r],t,1)+u[r]*('diff(u[r],r,1));

eθ: components
-------------------------

p4b:aθ=(uθ*('diff(uθ,θ,1)))/r+uz*('diff(uθ,z,1))+'diff(uθ,t,1)+ur*('diff(uθ,r,1))+(ur*uθ)/r;

index notation :
=============

p4b1:a[θ]=(u[θ]*('diff(u[θ],θ,1)))/r+u[z]*('diff(u[θ],z,1))+'diff(u[θ],t,1)+u[r]*('diff(u[θ],r,1))+(u[r]*u[θ])/r;

ez: components
------------------------

p4c:az=(('diff(uz,θ,1))*uθ)/r+uz*('diff(uz,z,1))+'diff(uz,t,1)+ur*('diff(uz,r,1));

index notation :
=============

p4c1:a[z]=(('diff(u[z],θ,1))*u[θ])/r+u[z]*('diff(u[z],z,1))+'diff(u[z],t,1)+u[r]*('diff(u[z],r,1));

rem : {ar,aθ,az} =acceleration

div(σ)
=====
result  => vector : 'first order tensor'.
-------------------------------------------------------
use : index notation.

er: components
------------------------

p5a:'diff(σ[θ,r],θ,1)/r-σ[θ,θ]/r+'diff(σ[z,r],z,1)+'diff(σ[r,r],r,1)+σ[r,r]/r;

eθ: components
-------------------------

p5b:'diff(σ[θ,θ],θ,1)/r+σ[θ,r]/r+'diff(σ[z,θ],z,1)+'diff(σ[r,θ],r,1)+σ[r,θ]/r;

ez: components
-------------------------

p5c:'diff(σ[θ,z],θ,1)/r+'diff(σ[z,z],z,1)+'diff(σ[r,z],r,1)+σ[r,z]/r;