We introduce a mathematical model of unsteady thermoelectric MHD flow and heat transfer of two immiscible fractional secondgrade fluids, with thermal fractional parameters
In recent years, the requirement of modern technology has stimulated interest in flow and heat transfer studies which involve interaction of several phenomena such as heat exchangers, transport of heat or cooled fluids, chemical processing equipment, and microelectronic cooling [
Many problems relating to plasma physics, aeronautics, and geophysics and in petroleum industry and so forth involve multilayered fluid flow situations [
Thermoelectric magnetohydrodynamics (TEMHD) theory was originally developed by Shercliff with direct application to a fusion environment [
Recently, the fractional derivatives have been found to be quite flexible in describing the behaviors of viscoelastic fluids and are studied by many mathematicians considering various motions of such fluids [
ElShahed [
In this work, we introduce a mathematical model of unsteady flow and heat transfer of two immiscible thermoelectric fluids with thermomechanical fractional parameters
Consider unsteady laminar stratified two immiscible fluid flows through nonconducting half space
The regions
A constant magnetic field of strength
The following assumptions are required.
The stress tensor which is different from zero for generalized secondgrade viscoelastic fluid is given by [
where
where
The modified Ohm and Fourier laws defined by Shercliff [
where
A mathematical model of fractional heat conduction equation by using the Taylor series expansion of time fractional order developed by Jumarie in the context of thermoelectric MHD [
where
The magnetic induction has one constant nonvanishing component:
The Lorentz force
Under these assumptions the governing equations for such flow will take the following form.
The energy equation:
The thermal boundary and interface condition for the two fluids are written as
The hydrodynamic boundary and interface condition for the two fluids are considered as
Therefore, (
The thermal boundary and interface condition in nondimensional form become
By the same manner, we get
From (
Two viscoelastic fluids were chosen for the purposes of numerical evaluations, namely, high density polyethylene (HDPE) and polypropylene (PP) [
The constants of the problem.
















The computations were carried out for the case of two immiscible fluids (HDPE/PP) and the case of a single fluid (HDPE) on the whole region.
The temperature, the velocity, and the stress were calculated by using the numerical method of the inversion of the Laplace transform outlined above. The FORTRAN programming language was used on a personal computer. The accuracy maintained was 6 digits for the numerical program.
Figure
Temperature, velocity, and stress distribution for
Temperature, velocity, and stress distribution for
Temperature, velocity, and stress distribution for
Temperature, velocity, and stress distribution for
Temperature, velocity, and stress distribution for
Temperature, velocity, and stress distribution for HDPE/PP,
In Figure
Figures
Figures
Figure
Tables
Temperature values of HDPE and HDPE/PP fluids for different






HDPE  HDPE/PP  HDPE  HDPE/PP  HDPE  HDPE/PP  


0.0  1.000001  1.000001  1.000001  1.000001  1.000001  1.000001 
0.4  0.087479  0.087478  0.0000007  0.00000008  0.0000007  0.00000008 
0.8  0.000621  0.000611  0.0000005  0.00000006  0.0000005  0.00000005 
1.2  0.0000007  0.000000  0.0000003  0.000000  0.0000003  0.000000 
1.6  0.0000005  0.000000  0.0000002  0.000000  0.0000002  0.000000 
2.0  0.0000003  0.000000  0.000000  0.000000  0.000000  0.000000 




0.0  1.000001  1.000001  1.000001  1.000001  1.000001  1.000001 
0.4  0.420681  0.421193  0.579084  0.578967  0.582487  0.582363 
0.8  0.106871  0.112543  0.234755  0.249912  0.250953  0.279957 
1.2  0.015363  0.002334  0.030723  0.000204  0.054408  0.0000208 
1.6  0.001059  0.0000008  0.0000311  0.0000009  0.000000  0.000000 
2.0  0.0000085  0.00000005  0.000000  0.000000  0.000000  0.000000 
Temperature values of HDPE and HDPE/PP fluids for different






HDPE  HDPE/PP  HDPE  HDPE/PP  HDPE  HDPE/PP  


0.0  1.000001  1.000001  1.000001  1.000001  1.000001  1.000001 
0.4  0.000031  0.000030  0.000001  0.000001  0.000001  0.000001 
0.8  0.000001  0.000001  0.0000008  0.0000007  0.0000008  0.0000007 
1.2  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000 
1.6  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000 
2.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000 




0.0  1.000001  1.000001  1.000001  1.000001  1.000001  1.000001 
0.4  0.581747  0.581597  0.582525  0.582342  0.582487  0.582363 
0.8  0.246487  0.274262  0.250745  0.279630  0.250953  0.279957 
1.2  0.051544  0.0000035  0.054211  0.0000240  0.054408  0.0000208 
1.6  0.0000203  0.000000  0.0000426  0.0000010  0.0000332  0.000000 
2.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000 
Velocity values of HDPE and HDPE/PP fluids for different






HDPE  HDPE/PP  HDPE  HDPE/PP  HDPE  HDPE/PP  


0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000 
0.6  0.002939  0.003013  0.002620  0.002688  0.002616  0.002684 
1.2  0.001571  0.001570  0.001397  0.001398  0.001395  0.001396 
1.8  0.000838  0.000868  0.000745  0.000769  0.000744  0.000768 
2.4  0.000447  0.000479  0.000397  0.000423  0.000397  0.000422 
3.0  0.000238  0.000265  0.000212  0.000233  0.000211  0.000232 




0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000 
0.6  0.025095  0.023630  0.029555  0.027560  0.029917  0.027718 
1.2  0.015306  0.013589  0.018988  0.016542  0.019510  0.016752 
1.8  0.008268  0.008287  0.010202  0.010043  0.010472  0.010187 
2.4  0.004441  0.005030  0.005475  0.006068  0.005620  0.006164 
3.0  0.002385  0.003040  0.002938  0.003651  0.003016  0.003712 
Velocity values of HDPE and HDPE/PP fluids for different






HDPE  HDPE/PP  HDPE  HDPE/PP  HDPE  HDPE/PP  


0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000 
0.6  0.002372  0.002243  0.003562  0.003525  0.002616  0.002684 
1.2  0.000000  0.000001  0.001608  0.001491  0.001395  0.001396 
1.8  0.000000  0.000000  0.000725  0.000820  0.000744  0.000768 
2.4  0.000000  0.000000  0.000327  0.000451  0.000397  0.000422 
3.0  0.000000  0.000000  0.000147  0.000248  0.000211  0.000232 




0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000 
0.6  0.103453  0.096869  0.036271  0.032740  0.029917  0.027718 
1.2  0.029575  0.014052  0.022237  0.017948  0.019510  0.016752 
1.8  0.000684  0.008051  0.010770  0.010925  0.010472  0.010187 
2.4  0.000007  0.004603  0.005210  0.006615  0.005620  0.006164 
3.0  0.000000  0.002627  0.002518  0.000000  0.003016  0.000000 
Stress values of HDPE and HDPE/PP fluids for different






HDPE  HDPE/PP  HDPE  HDPE/PP  HDPE  HDPE/PP  


0.0  1.662574  1.648661  1.631704  1.617246  1.623545  1.608846 
0.6  −0.19194  −0.201456  −0.225428  −0.234513  −0.22988  −0.23904 
1.2  −0.11365  −0.123767  −0.120332  −0.129629  −0.12270  −0.13202 
1.8  −0.06069  −0.069907  −0.064232  −0.072580  −0.06549  −0.07385 
2.4  −0.032418  −0.039437  −0.034286  −0.040603  −0.034957  −0.041275 
3.0  −0.017314  −0.022221  −0.018302  −0.022696  −0.018659  −0.023051 




0.0  1.392190  1.323119  1.241577  1.100666  1.231832  1.070555 
0.6  −0.00054  −0.079583  0.183463  0.013853  0.190674  −0.003849 
1.2  −0.22762  −0.277199  −0.29069  −0.315591  −0.25326  −0.303997 
1.8  −0.13970  −0.180148  −0.19048  −0.207686  −0.19573  −0.202145 
2.4  −0.07573  −0.115562  −0.10268  −0.134565  −0.10550  −0.132031 
3.0  −0.04088  −0.073419  −0.05534  −0.086082  −0.05687  −0.084999 
Stress values of HDPE and HDPE/PP fluids for different






HDPE  HDPE/PP  HDPE  HDPE/PP  HDPE  HDPE/PP  


0.0  0.938168  0.931532  1.553433  1.530803  1.623545  1.608846 
0.6  −0.051838  −0.048786  −0.246749  −0.264793  −0.229881  −0.239039 
1.2  −0.000012  −0.000402  −0.114039  −0.141048  −0.122702  −0.132024 
1.8  0.000000  −0.000216  −0.052630  −0.078897  −0.065493  −0.073849 
2.4  0.000000  −0.000116  −0.024256  −0.044096  −0.034957  −0.041275 
3.0  0.000000  −0.000062  −0.011164  −0.024627  −0.018659  −0.023051 




0.0  0.903874  0.900399  1.182936  1.043413  1.231832  1.070555 
0.6  −0.12195  −0.195861  0.155454  −0.024652  0.190674  −0.003849 
1.2  −0.21473  −0.435217  −0.25726  −0.318253  −0.25326  −0.303997 
1.8  −0.00902  −0.264756  −0.18232  −0.212666  −0.19573  −0.202145 
2.4  −0.00013  −0.159753  −0.09039  −0.139431  −0.10551  −0.132031 
3.0  −0.00001  −0.095721  −0.04474  −0.090033  −0.05687  −0.084999 
Consider
Magnetic induction
Specific heat at constant pressure of region
Caputo fractional time derivative operator of order
Electric density
Lorentz force
Constant magnetic field
Heaviside unit step function
Conduction current density vector
Thermal conductivity of region
Seebeck coefficient of region
Hartmann number
Prandtl number
Heat conduction vector of region
Time
Temperature of the fluid in region
Temperature of the fluid away from the plate
Reference temperature
Dimensionless figureofmerit
Thermal fractional parameters of region
Material modulus of region
Mechanical fractional parameters of region
Viscosity of region
Density of region
Electrical conductivity of region
Dimensionless viscoelastic parameter of region
Kinematic viscosity in Region1
Peltier coefficient of region
Thermal relaxation time of region
Stress tensor
Fluid in Region1
Fluid in Region2.
The authors declare that there is no conflict of interests regarding the publication of this paper.