FEATool Multiphysics  v1.16.1 Finite Element Analysis Toolbox
ex_nonnewtonian1.m File Reference

## Description

EX_NONNEWTONIAN1 2D Example for non-Newtonian flow in a channel.

[ FEA, OUT ] = EX_NONNEWTONIAN1( VARARGIN ) Sets up and solves stationary non-Newtonian flow in a rectangular channel. The fluid is governed by a power-law model such that the viscosity can be expressed as mu_eff = mu0*K*D^(n-1). A constant pressure drop is prescribed resulting in an analytical solution for the velocity u(y) = n/(n+1)*(1/(mu0*K)*dp)^(1/n)*((h/2)^((n+1)/n)-abs(h/2-y)^((n+1)/n)). Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
rho         scalar {1}             Density
mu0         scalar {1.29684}       Newtonian viscosity
dp          scalar {1}             Pressure drop
n           scalar {0.25}          Power-law exponent
K           scalar {1}             Power-law constant
h           scalar {1}             Channel height
l           scalar {0.2}           Channel length
igrid       scalar 1/{0}           Cell type (0=quadrilaterals, 1=triangles)
hmax        scalar {0.1}           Max grid cell size
sf_u        string {sflag1}        Shape function for velocity
sf_p        string {sflag1}        Shape function for pressure
iplot       scalar 0/{1}           Plot solution and error (=1)
.
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct


# Code listing

 cOptDef = { ...
'rho',      1;
'mu0',      1.29684;
'K',        1;
'n' ,       0.25;
'dp' ,      1;
'h',        1;
'l',        2;
'igrid',    1;
'hmax',     0.05;
'sf_u',     'sflag2';
'sf_p',     'sflag1';
'iplot',    1;
'fid',      1 };
[got,opt] = parseopt(cOptDef,varargin{:});
fid       = opt.fid;

% Geometry and grid parameters.
h         = opt.h;       % Height of rectangular domain.
l         = opt.l;       % Length of rectangular domain.
% Discretization parameters.
sf_u      = opt.sf_u;    % FEM shape function type for velocity.
sf_p      = opt.sf_p;    % FEM shape function type for pressure.

% Geometry definition.
gobj = gobj_rectangle( 0, l, 0, h );
fea.geom.objects = { gobj };
fea.sdim = { 'x' 'y' };   % Coordinate names.

% Grid generation.
if ( opt.igrid==1 )
fea.grid = gridgen(fea,'hmax',opt.hmax,'fid',fid);
else
fea.grid = rectgrid(round(l/opt.hmax),round(h/opt.hmax),[0 l;0 h]);
if( opt.igrid<0 )
end
end
n_bdr = max(fea.grid.b(3,:));           % Number of boundaries.

% Boundary conditions.
dtol      = opt.hmax;
i_inflow  = findbdr( fea, ['x<',num2str(dtol)] );     % Inflow boundary number.
i_outflow = findbdr( fea, ['x>',num2str(l-dtol)] );   % Outflow boundary number.

% Problem definition.
fea.phys.nn.eqn.coef{1,end} = { opt.rho };
fea.phys.nn.eqn.coef{2,end} = { 1 };   % Select power-law viscosity model.
fea.phys.nn.eqn.coef{4,end} = { opt.mu0 };
fea.phys.nn.eqn.coef{6,end} = { opt.K };
fea.phys.nn.eqn.coef{7,end} = { opt.n };
fea.phys.nn.eqn.coef{13,end}{1} = [num2str(opt.dp),'*(1-x/',num2str(l),')'];   % Initial pressure.
fea.phys.nn.sfun = { sf_u sf_u sf_p };           % Set shape functions.
fea.phys.nn.bdr.sel([i_inflow,i_outflow]) = 4;
fea.phys.nn.bdr.coef{4,end}{3,i_inflow}   = l/opt.dp;   % Set inflow pressure.

% Parse and solve problem.
fea = parsephys(fea);   % Check and parse physics mode.
fea = parseprob(fea);   % Check and parse problem struct.

[fea.bdr.n{1}{[i_inflow,i_outflow]}] = deal( '-p*nx' );   % Offset natural Neumann BC.
[fea.bdr.d{2}{[i_inflow,i_outflow]}] = deal(0);           % Set v=0 at inlet and outlet.

fea.sol.u = solvestat( fea, 'relchg', false, 'fid', fid, 'maxnit', 50 );   % Call to stationary solver.

% Postprocessing.
s_velm = 'sqrt(u^2+v^2)';
s_refsol  = [num2str(opt.n/(opt.n+1)*(1/(opt.mu0*opt.K)*opt.dp)^(1/opt.n)),'*(',num2str((h/2)^((opt.n+1)/opt.n)),'-abs(',num2str(h/2),'-y)^',num2str((opt.n+1)/opt.n),')'];
p = [ l/2*ones(1,25); linspace(0,h,25) ];
u = evalexpr( s_velm, p, fea );
u_ref = evalexpr( s_refsol, p, fea );
s_velm = 'sqrt(u^2+v^2)';
if ( opt.iplot>0 )
figure
subplot(3,1,1)
postplot(fea,'surfexpr',s_velm,'evaltype','exact')
title('Velocity field')

subplot(3,1,2)
postplot(fea,'surfexpr','miu_nn')
title('Effective viscosity')

subplot(3,1,3)
plot( u, p(2,:) )
hold on
plot( u_ref, p(2,:), 'k.' )
legend( 'Computed solution', 'Reference solution','Location','West')
title( 'Velocity profile at x=l/2' )
ylabel( 'y' )
end

% Error checking.
err = sqrt(sum((u-u_ref).^2)/sum(u_ref.^2));
if( ~isempty(fid) )
fprintf(fid,'\nL2 Error: %f\n',err)
fprintf(fid,'\n\n')
end

out.err  = err;
out.pass = err<0.05;
if ( nargout==0 )
clear fea out
end