@Rouben Rostamian  

Thanks for observing carefully. 

I am looking for an asymptoic expansion in terms of the parameter and  the initial guess as  (r^2 - h^2)/4. I am seeing this equations and try to find using maple

@Rouben Rostamian  

Sorry you are correct it should be w(r) only.

@Rouben Rostamian  

Thanks a lot

@Rouben Rostamian  

Thanks for your suggestions but still error in plot the graphs; 

Please see it where i am commiting mistake

Download Error.mw

@Rouben Rostamian  

Like this graph already same type of graph got it 

@Rouben Rostamian  

My h(z) is defined as: where delta=0.2, d=0.2

@Rouben Rostamian  

Thanks  for wonderful explanation  but my h is not constant it's vary in different range ie piecewise function  then for this function we have to proceed in a same way or not  

(OR)

for  h range from 0 to 0.2 it is 1

for h=0.2 to 0.4 it is 1-(delta/(2))*(1 + cos((2*(Pi)/L)*(z - d1 - L/2)))

for h=0.4 to 1 again 1.

how to combine all those in one graphs for different range

L:=0.4:delta:=0.1:
d1:=0.3: d2:=0.3: z:=1:

h:=z->piecewise( z<=d1,   1,
                 z<=d1+L,   1-(delta/(2))*(1 + cos((2*(Pi)/L)*(z - d1 - L/2))), 
                 z<=d1+L+d2,  1 ):

@Rouben Rostamian   

These are the equations and boundary conditons with the graphs; please help me to resolve the error:

Thanks in advance

@Rouben Rostamian  

here maple file please see it

Download PDES.mw

@Rouben Rostamian  

The domain of w(r,z) and theta(r,z) is

0<=r<=h(z)

0<=z<=infinity 

@Rouben Rostamian  

Domain is 0 to 1

@Rouben Rostamian  

Thanks for pointing errors. Still after making corrections as you mention  i am getting errors "  Error, final value in for loop must be numeric or character   Error, unable to execute "

Please see it 

restart:
with(PDEtools):with(plots):
L:=0.4:
d1:=0.3: d2:=0.3: delta:=0.25:

h:=z->piecewise( z<=d1,   1,
                 z<=d1+L,   1-(delta/(2))*(1 + cos((2*(Pi)/L)*(z - d1 - L/2))), 
                 z<=d1+L+d2,  1 ):

a:=x4*S*Gr/(4*x1*x5)*sin(alpha):
b:=1/Da+(x3*M/(x1*(1+m^2))):
c:=(1/x1)*DP:
x1:=1/((1-phi1)^2.5*(1-phi2)^2.5):
x5:=khnf/kf:
x3:=shnf/sf:
x4:=(1-phi2)*((1-phi1)+phi1*(RBs1)/(RBf))+phi2*RBs2/RBf:
shnf:=sbf*((ss2+2*sbf-2*phi2*(sbf-ss2))/(ss2+2*sbf+phi2*(sbf-ss2))):
sbf:=sf*((ss1+2*sf-2*phi1*(sf-ss1))/(ss1+2*sf+phi1*(sf-ss1))):
ss2:=2.7*10^(-8):
ss1:=59.6*10^(6):
sf:=6.67*10^(-1):
khnf:=kbf*((ks2+2*kbf-2*phi2*(kbf-ks2))/(ks2+2*kbf+phi2*(kbf-ks2))):
kbf:=kf*((ks1+2*kf-2*phi1*(kf-ks1))/(ks1+2*kf+phi1*(kf-ks1))):
ks1:=401:
ks2:=76.5:
kf:=0.4972:
RBs1:=8933*16.7*10^(6):
RBs2:=6320*18*10^(6):
RBf:=1063*1.8*10^(6):
DP:=(1/192)*(18432*F-77*b^2*h(z)^8+308*b*h(z)^8*a+768*h(z)^6*a+12*b*h(z)^8)/(h(z)^4*(b*h(z)^2-6)):
conds:= w(h(z),0)=0, theta(h(z),0)=0, D[1](w)(r,0)=0, D[1](theta)(r,0)=0:
pdes:= diff(w(r, z), r, r)+(1/r)*(diff(w(r, z), r))-(1/Da+(x3/x1)*M)*w(r, z)-(1/x1)*(DP+theta(r,z)*x4*sin(alpha))=0,
         diff(theta(r, z), r,r)+(1/r)*(diff(theta(r, z), r))+(S/x5)=0:
pars:= { F=1.5,Da=0.1,phi1=0.01,phi2=0.02,alpha=Pi/4,S=0.1,Gr=2,m=0.5}:
         
  PrVals:=[2, 5, 7,9]:
  colors:=[red, green, blue, black]:
  for j from 1 to numelems(PrVals) do
      pars1:=`union`( pars, {M=PrVals[j]}):
      pdSol:= pdsolve( eval([pdes], pars1),
                       eval([conds], pars1),
                       numeric
                     );
      plt[j]:=pdSol:-plot( w(r,z), z=1, r=0..1, numpoints=200, color=colors[j]);
  od:
  plots:-display( [seq(plt[j], j=1..numelems(PrVals))]);
Error, final value in for loop must be numeric or character
Error, unable to execute seq
 

@Carl Love 

By refering the article by applying HPM  i am trying to plot the graphs 

ARTICLE:"https://www.sciencedirect.com/science/article/abs/pii/S0026286221000625" author S Das et. al

@mmcdara 

If we run the codes   as corected by Carl Love 26822  we should get the graph like that

@Carl Love 

Here solid lines  are hybrid nano fluid and dashes are nano fluid,

why reverse graph is coming sir