# Magnetohydrodynamic Thin Film Flow through a Porous Stretching Sheet with the Impact of Thermal Radiation and Viscous Dissipation

Authors: Ziad Khan; Muhammad Jawad; Ebenezer Bonyah; Naveed Khan; Rashid Jan
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

In this paper, the Darcy–Forchheimer laminar thin film flow with MHD and heat transfer on an unsteady horizontal stretched surface is investigated. The impact of thermal radiation and viscous dissipation is also considered for thin film flow. The analysis of heat source with thermal radiation in a boundary layer flow can play a great role in manufacturing engineering procedures such as the production of electric power, solar energy modernization, and astrophysical flow. By using similarity transformation, the system of PDEs is converted to ODEs. Then HAM is applied for the solution of the problem. Moreover, the velocity and temperature profile for various embedded variables are discussed through graphs while the numerical solution of concerned physical quantities such as skin friction and Nusselt number are discussed through tables. The analysis shows that velocity profile is reducing function ofM, K1, S, and F1. The temperature profile is an increasing function S and Ec while a reducing function of Pr and R. The overall behavior of the proposed system is highlighted analytically.


---

## Body

## 1. Introduction

Heat source, fluid flow within liquid film due to an unsteady stretching sheet, plays a very crucial role in many utilization engineering processes and technologies; for example, polymer extrusion, continuous casting, manufacturing plastic film, tinning of copper wires, crystal growth, a polymer sheet, artificial fiber, and wire and fiber coating. In addition to these, many production processes involve heat transmission in various forms such as cooling of a machine tool, pasteurization of food, heat treatment of a part in industrial furnaces, or the temperature control for initiating a chemical reaction. Heat transfer is accomplished in most of these applications using heat transfer instruments such as heat swaps, desiccation, condensers, boilers, and heat sinks. Mathematics plays an important role in different areas of research and has effective outcomes [1–4]. The importance of mathematical tools in the investigation of fluid flow cannot be ignored. Mathematical tools help the researchers to visualize the dynamical behavior of a system and to conceptualize the importance of the input factors of a system. Numerous researchers workout to understand the flow of fluid with different assumptions and in different circumstances. The nanofluid flow over a stretching surface has been originally considered by Wang [5]; they also reduce unstable Navier–Stokes equations using similarity transformation in nonlinear ODEs, which are then calculated by multiple shooting methods. The solutions based on the analysis of a liquid thin film are obtained by Wang [6], who also validated the correctness of the homotopy analysis technique. Sufian et al [7] study the impact of heat source and fluid flow over a stretch cylinder as a function of time. Zhang et al. [8] demonstrated mass diffusion and solutal Marangoni impact on heat transmission and fluid flow in a film toward a stretched sheet. Kalyani et al. [9] investigated the effects of Lorentz force, heat generation, and mass transfer of a Casson incompressible fluid flowing on a surface. They also discussed the impact of thermal radiation on the mass and heat transfer property of flow in a Casson liquid film [10]. The unstable mixed convection flow of the stretch sheet is considered by Devi et al. [11]. Jhankal et al. [12] studied heat source, the forced convective flow of an incompressible fluid past plate inserted in a Darcy–Forchheimer porous medium of the two-dimensional boundary layer MHD flow. The effects of thermal radiation in an upper-convective Maxwell thin film, unsteady stretched sheet inserted in a porous medium are conceptualized by Waheed [13]. Ahmad et al. [14] illustrated a solution for the heat source, MHD flow toward an exponentially stretched surface in the absence of radiation impacts and Darcy’s resistance. On the basis of boundary layer hypothesis and similarity transformation, Sparrow and Gregg [15] were the first to examine the Laminar-film condensing difficulty on a vertical plate. Dandapat and Ray explored the flow of a thin boundary layer across a spinning horizontal disc [16, 17]. Nath and Kumari [18] investigated the time-dependent MHD problem of a fluid dispersed by a spinning incompressible disc. Magnetohydrodynamic and heat flow toward a stretch sheet has been studied by Mohd Nasir et al. [19] in which we consider both the stretching and shrinking cases. Zaidi et al. [20] studied the impacts of thermal radiation, heat creation, and induced magnetic field on hydromagnetic free convection flow of pair stress fluid in an isoflux-isothermal vertical channel. Kishore et al. [21] studied the influence of thermal radiation and viscous dissipation on MHD mass, heat transfer flow past an oscillating vertical plate inserted in a porous medium with unknown surface situations. Elbashbeshy et al. [22] studied the influence of thermal radiation and heat transmission on an unstable extended sheet inserted in a porous medium in the existence of a heat sink. Thermal radioactivity and heat transfer of Sisko fluid on an uneven stretched sheet with a constant magnetic field were investigated by Khan et al. [23]. The authors of [24] discussed mass transfer and unstable boundary layer MHD flow of a UCM fluid in the absence of higher-level chemical reaction. The authors of [25] considered heat transfer, MHD flow in a liquid film over an unsteady stretched sheet. Numerical solution of MHD boundary layer flow of a nanofluid over a moving sheet in the absence of thermal radiation has been carried out by Shateyi and Prakash [26]. The MHD laminar boundary layer flow with heat and mass transport of an electrically conducting water-based nanofluid across a nonlinear stretching sheet with viscous dissipation is studied by Mabood and Khan in [27]. It has been noticed that further investigation is needed to conceptualize the impact of heat transfer, thermal radiation in a fluid film of an unsteady stretching surface inserted in a porous medium in the absence of viscous dissipation and magnetic field impact. Therefore, in the current work, we opt to investigate the impact of heat transfer, thermal radiation in a fluid film of an unsteady stretching surface inserted in a porous medium in the absence of viscous dissipation and magnetic field impact.The aim of the present work is to study Darcy–Forchheimer's laminar thin film flow on an unsteady horizontal stretched surface. The laminar thin-film flow is investigated with the impact of MHD, thermal radiation, and viscous dissipation. The HAM is used to obtain analytic approximations for the solution of nonlinear ODEs. The comparison of HAM and numerical solutions are displayed. The results have been illustrated in graphical or tabular forms followed by a brief discussion and physical interpretations.


## 2. Mathematical Expressions of the Problem

### 2.1. Basic Equations and Boundary Constraints

Assume a thin liquid film extending on a stretched surface with a laminar unsteady flow which arises from a narrow slit where x-coordinates are parallel to the slit and y-coordinates are perpendicular to the sheet as exhibited in Figure1. Along x-axis, the stretching sheet is stretched by two equal and opposite forces. Further, the latent heat effect is neglected due to evaporation by assuming the liquid to be nonvolatile. Moreover, due to a relatively thin film, the buoyancy force is also neglected.Figure 1
Geometric representation of thin film to illustrate its structure.The continuous sheet velocity is given byUx,t=cx/1−γt [25], where c and γ are constants having per-time dimension. The term cx2/ν1−γt is localized Reynolds number on the sheet velocity U, and the stretched rate c/1−γt reduces with time as 0≤γ<1. The nomenclature of this research work has been presented.The stretched sheet’s surface temperatureTs is determined as [25](1)Tsx,t=T0−Trefcx22ν1−γt−3/2,where T0 is the slit temperature and Tref is the temperature with reference, which might be a constant temperature with a reference or a constant temperature difference. In this work, Tref will be interpreted as 0≤Tref≤T0. Normally, a transverse magnetic field is applied to the extended surface, which is believed to be changeable in nature and chosen as(2)Bx,t=B01−γt−1/2.The temperature and velocity influence of a thin film obeys the following governing time-dependent equations with the boundary conditions [25].(3)∂u∂x+∂v∂y=0,(4)∂u∂t+u∂u∂x+v∂u∂y=ν∂2u∂y2−σB02ρu−νε2ku−Fu2,(5)∂T∂t+u∂T∂x+v∂T∂y=kρCp∂2T∂y2+μρCp∂u∂y2+1ρCp∂qr∂y,(6)wheny=0:u=U,v=0,T=Tsandwheny=h:∂u∂y=∂T∂y=0,v=dhdt,where u and v denote the velocity terms in the directions of x-axis and y-axis, respectively, μ is the dynamic viscosity, B0 is the magnetic field, ρ is the density, T is the temperature, k is the thermal diffusivity, ν is the kinematic viscosity, t is the time, σ is the electrical conductivity, and h is the uniform thickness of the film sheet. In the x- and y-directions, the two terms of equation (3) reflect the net outflow (outflow minus inflow) of mass, for an unsteady flow the sum of these terms must be zero. The net rate at which x-momentum leaves due to fluid motion across its limits is signified by the 2nd and 3rd components on the left side of equation (4) while the net force owing to viscous shear stresses is shown off by the 1st component on the right side. The net rate at which thermal energy escapes due to fluid motion is represented by the 2nd and 3rd terms on the left side of equation (5). The 1st component on the right side of equation (5) shows the net input of thermal energy owing to y-direction, while the 2nd term represents the viscous dissipation that remains.In this case, when the influence of surface tension is insignificant at a distance ofy=h from the surface, shear stresses operating in planes parallel to the fluid velocity are defined as(7)τ=μ∂u∂y,where μ is a fluid property and is known as the dynamic viscosity and ∂u/∂y is the rate of shear stress.The flow of heat aty=h is defined by(8)q=−k∂T∂y.In the above equation, the constant k is called thermal conductivity. Equation (8) tells us that if temperature decreases with y then q will be positive and it will flow in the y-direction, if temperature increases with y then q will be negative and it will flow opposite in the y-direction.


### 2.2. Similarity Transformations

Here, we use the stream functionψx,y,t to specify the velocity terms:(9)u=∂ψ∂y=bx1−γtf′η,(10)v=−∂ψ∂x=−νb1−γt12fη,satisfying continuity equation (3). New dependent and independent variables, f, θ, and η are, respectively, introduced as(11)fη=ψx,y,tνb/1−γt1/2x,θη=T0−Tx,y,tTrefbx2/2ν1−γt−3/2,η=bν1−γt1/2y.Substitute the similarity transformation equations (9)–(11) into equations (4) and (5) to obtain a set of nonlinear ODEs in the form of(12)f‴−M+K1f′+ff″−Sf′+η2f″−1+F1f′2=0,(13)1+43Rθ″+EcPrf″2−PrS23θ−ηθ′+2θf′−θ′f=0.Following are the corresponding boundary constraints of the differential equations:(14)f0=0,f′0=1,θ0=1,fβ=Sβ2,θ′β=0,f″β=0,where the prime denotes derivative in terms of η, S=γ/c is the unsteadiness parameter, M=σB2/ρc1−γt is the magnetic parameter, K1=νε2/kc1−γt is the porosity parameter, Ec=U2/CpTs−T0 is the Eckert number, Pr=μCp/k is the Prandtl number, and R is thermal radiation. Despite the fact that the dimensionless film thickness β is constant and only relies on S, the current film thickness h is only depending on time t. So the free surface β denotes the value of similarity variable by η such that(15)β=cν1−γt1/2h.By differentiating with respect tot, we have(16)dhdt=−γβ2cν−1/21−γt−1/2.


### 2.3. Physical Interest of the Problem

In this subsection, we calculated the skin friction coefficientCf and local Nusselt number Nux as(17)Cf=−2τsρU2.And(18)Nux=xkTrefqs−qr.From equations (7) and (8), at the surface y = 0, the surface shear stress and surface heat flux are defined as(19)τs=μ∂u∂yy=0,(20)qs=−k∂T∂yy=0.And the Rosseland estimationqr is given by [23](21)qr=−16σ∗T033k∗∂T∂yy=0,where σ∗ is the Stefan–Boltzman constant and has the numerical value(22)σ∗=5.670×10−8Wm−2K4.Inserting equations (19)–(21) into equations (17) and (18) to obtain Cf and Nux in terms of Rex,(23)Cf=−2Rex−1/2f″0,Nux=1+43RRex3/221−γt−1/2θ′0,where Rex=cx2/ν1−γt is the local Reynolds number and R=4σ∗T03/kk∗ is the radiation parameter.


## 2.1. Basic Equations and Boundary Constraints

Assume a thin liquid film extending on a stretched surface with a laminar unsteady flow which arises from a narrow slit where x-coordinates are parallel to the slit and y-coordinates are perpendicular to the sheet as exhibited in Figure1. Along x-axis, the stretching sheet is stretched by two equal and opposite forces. Further, the latent heat effect is neglected due to evaporation by assuming the liquid to be nonvolatile. Moreover, due to a relatively thin film, the buoyancy force is also neglected.Figure 1
Geometric representation of thin film to illustrate its structure.The continuous sheet velocity is given byUx,t=cx/1−γt [25], where c and γ are constants having per-time dimension. The term cx2/ν1−γt is localized Reynolds number on the sheet velocity U, and the stretched rate c/1−γt reduces with time as 0≤γ<1. The nomenclature of this research work has been presented.The stretched sheet’s surface temperatureTs is determined as [25](1)Tsx,t=T0−Trefcx22ν1−γt−3/2,where T0 is the slit temperature and Tref is the temperature with reference, which might be a constant temperature with a reference or a constant temperature difference. In this work, Tref will be interpreted as 0≤Tref≤T0. Normally, a transverse magnetic field is applied to the extended surface, which is believed to be changeable in nature and chosen as(2)Bx,t=B01−γt−1/2.The temperature and velocity influence of a thin film obeys the following governing time-dependent equations with the boundary conditions [25].(3)∂u∂x+∂v∂y=0,(4)∂u∂t+u∂u∂x+v∂u∂y=ν∂2u∂y2−σB02ρu−νε2ku−Fu2,(5)∂T∂t+u∂T∂x+v∂T∂y=kρCp∂2T∂y2+μρCp∂u∂y2+1ρCp∂qr∂y,(6)wheny=0:u=U,v=0,T=Tsandwheny=h:∂u∂y=∂T∂y=0,v=dhdt,where u and v denote the velocity terms in the directions of x-axis and y-axis, respectively, μ is the dynamic viscosity, B0 is the magnetic field, ρ is the density, T is the temperature, k is the thermal diffusivity, ν is the kinematic viscosity, t is the time, σ is the electrical conductivity, and h is the uniform thickness of the film sheet. In the x- and y-directions, the two terms of equation (3) reflect the net outflow (outflow minus inflow) of mass, for an unsteady flow the sum of these terms must be zero. The net rate at which x-momentum leaves due to fluid motion across its limits is signified by the 2nd and 3rd components on the left side of equation (4) while the net force owing to viscous shear stresses is shown off by the 1st component on the right side. The net rate at which thermal energy escapes due to fluid motion is represented by the 2nd and 3rd terms on the left side of equation (5). The 1st component on the right side of equation (5) shows the net input of thermal energy owing to y-direction, while the 2nd term represents the viscous dissipation that remains.In this case, when the influence of surface tension is insignificant at a distance ofy=h from the surface, shear stresses operating in planes parallel to the fluid velocity are defined as(7)τ=μ∂u∂y,where μ is a fluid property and is known as the dynamic viscosity and ∂u/∂y is the rate of shear stress.The flow of heat aty=h is defined by(8)q=−k∂T∂y.In the above equation, the constant k is called thermal conductivity. Equation (8) tells us that if temperature decreases with y then q will be positive and it will flow in the y-direction, if temperature increases with y then q will be negative and it will flow opposite in the y-direction.


## 2.2. Similarity Transformations

Here, we use the stream functionψx,y,t to specify the velocity terms:(9)u=∂ψ∂y=bx1−γtf′η,(10)v=−∂ψ∂x=−νb1−γt12fη,satisfying continuity equation (3). New dependent and independent variables, f, θ, and η are, respectively, introduced as(11)fη=ψx,y,tνb/1−γt1/2x,θη=T0−Tx,y,tTrefbx2/2ν1−γt−3/2,η=bν1−γt1/2y.Substitute the similarity transformation equations (9)–(11) into equations (4) and (5) to obtain a set of nonlinear ODEs in the form of(12)f‴−M+K1f′+ff″−Sf′+η2f″−1+F1f′2=0,(13)1+43Rθ″+EcPrf″2−PrS23θ−ηθ′+2θf′−θ′f=0.Following are the corresponding boundary constraints of the differential equations:(14)f0=0,f′0=1,θ0=1,fβ=Sβ2,θ′β=0,f″β=0,where the prime denotes derivative in terms of η, S=γ/c is the unsteadiness parameter, M=σB2/ρc1−γt is the magnetic parameter, K1=νε2/kc1−γt is the porosity parameter, Ec=U2/CpTs−T0 is the Eckert number, Pr=μCp/k is the Prandtl number, and R is thermal radiation. Despite the fact that the dimensionless film thickness β is constant and only relies on S, the current film thickness h is only depending on time t. So the free surface β denotes the value of similarity variable by η such that(15)β=cν1−γt1/2h.By differentiating with respect tot, we have(16)dhdt=−γβ2cν−1/21−γt−1/2.


## 2.3. Physical Interest of the Problem

In this subsection, we calculated the skin friction coefficientCf and local Nusselt number Nux as(17)Cf=−2τsρU2.And(18)Nux=xkTrefqs−qr.From equations (7) and (8), at the surface y = 0, the surface shear stress and surface heat flux are defined as(19)τs=μ∂u∂yy=0,(20)qs=−k∂T∂yy=0.And the Rosseland estimationqr is given by [23](21)qr=−16σ∗T033k∗∂T∂yy=0,where σ∗ is the Stefan–Boltzman constant and has the numerical value(22)σ∗=5.670×10−8Wm−2K4.Inserting equations (19)–(21) into equations (17) and (18) to obtain Cf and Nux in terms of Rex,(23)Cf=−2Rex−1/2f″0,Nux=1+43RRex3/221−γt−1/2θ′0,where Rex=cx2/ν1−γt is the local Reynolds number and R=4σ∗T03/kk∗ is the radiation parameter.


## 3. Solution by HAM

HAM is applied for the solutions of equations (12) and (13) with boundary constraints (12). Mathematica is used for this purpose. The modeled equations are solved by HAM, the basic derivation is identified as(24)Lθ^θ^=θ^″−θ^,Lf^f^=f^‴−f^′.Linear operatorsLf^, and Lθ^ are specified by(25)Lf^γ1+γ2e−η+γ3eη=0,Lθ^γ4e−η+γ5eη=0.The nonlinear operatorsNf^ and Nθ^ are defined as follows:(26)Nf^f^η;ζ=f^ηηη−S12ηf^ηη+f^η−M+K1f^η−1F1f^η2+f^f^ηηNθ^θ^η;ζ,f^η;ζ=1+43Rθ^ηη+EcPrf^ηη2+PrS2ηθ^η+3θ^+2θ^f^η−f^θ^η.The0thorder system for equations (12) and (13) is presented as(27)1−ζLf^f^η;ζ−f^0η=pℏf^Nf^f^η;ζ,1−ζLθ^θ^η;ζ−θ^0η=pℏθ^Nθ^θ^η;ζ,f^η;ζ,while BCs are as follows(28)f^η;ζ|η=0=0,∂f^η;ζ∂η|η=0=1,∂2f^η;ζ∂η2|η=β=0,∂θ^η;ζ∂η|η=β=0f^η;ζ|η=β=Sβ2,θ^η;ζ|η=0=1.ℏθ^ and ℏf^ are utilized to standard for the solution confluence whereas the embedding constraint is ζ∈0,1; when ζ=0 and ζ=1, we get the following:(29)θ^η;1=θ^η,f^η;1=f^η.Through Taylor’s series,f^η;ζ and θ^η;ζ are expanded for ζ=0:(30)θ^η;ζ=θ^0η+∑n=1∞θ^nηζn,f^η;ζ=f^0η+∑n=1∞f^nηζnf^nη=1n!∂f^η;ζ∂η|p=0,θ^nη=1n!∂θ^η;ζ∂η|p=0,while BCs are(31)f^′0=1,f^0=0,θ^0=1,f^″β=0,θ^β=0,f^β=Sβ2.Now(32)ℜnf^η=f^n−1′′′−Sf^n−1′+12ηf^n−1′′−M+K1f^n−1′−1−F1f^n−1′2+∑j=0w−1f^w−1−jf^j′′ℜnθ^η=1+43Rθ^n−1″+PrEcf^n−1′′2+S2ηθ^n−1′+3θ^n−1−2∑j=0w−1θ^w−1−jf^j′−∑j=0w−1θ^w−1−j′f^j,while(33)χn=0,ifn≤11,ifn>1.


## 4. Result and Discussion

The present work analyzes the Darcy–Forchheimer laminar thin film flow with MHD and heat transfer on an unsteady horizontal stretched surface in the absence of viscous dissipation, thermal radiation, and magnetic field of impact. The numerical, as well as analytical solutions, is given. Using similarity transformations, the governing partial differential equations of heat transfer flow are transformed into a nonlinear ordinary differential equation. Geometric representations of the problem are plotted in Figure1 and the impact of physical parametric quantity on f′η and θη curves are depicted in Figures 2–5. Figures 2 and 3 show the impact of M, K1, S, and F1 on f′η. The velocity profile reduces when (M) increases, as shown in Figure 2(a). This is due to the reducing nature of the Lorentz forces which reduces the motion of thin-film fluid in the boundary layer and raises its temperature. Figures 2(b) and 3(b) convey that the fluid velocity drop by rising values of (K1) and (F1). This is because the increasing hydromagnetic boundary layer increases the strength of porosity. Figure 3(a) illustrates the distribution of velocity gradient in the film decreases by rising values of unsteadiness parameter. As the value of (S) in the system rises, the thickness of the boundary layer declines, preventing the establishment of a transition from laminar to turbulent flow. The effect of embedded parameters Pr, R, S, and Ec on temperature profiles is exhibited in Figures 4 and 5. Figure 4(a) displays the variation of temperature reduces, as the Prandtl number increases. Being the ratio of molecular diffusivity of momentum to the molecular diffusivity of heat, it demonstrates the relative thickness of the momentum boundary layer to the thermal boundary layer. The rise in Prandtl number (Pr) decreases the temperature of the fluid. Increasing the Prandtl number, the thermal conductivity of the fluid decreases and the rate of heat transfer through stretch surface reduces; consequently, the temperature and thermal boundary layer reduce. Figure 4(b) demonstrates the temperature gradient for different values of (R). As the values of (R) upsurge, the temperature of thin film declines. Physically, when the radiation parameter increases, the Rosseland radiative absorption coefficient k∗ declines as defined by the expression for R=4σ∗T03/kk∗. Therefore, the heat flux diminishes. That is why the thin film temperature reduces. Figure 5(a) discusses the impact of unsteadiness parameter (S) on temperature profiles. As the values of (S) rise, the thermal boundary layer grows as a result, and the temperature field becomes increment for the nanofluid. Figure 5(b) demonstrates that increases in the value of (Ec) produce more temperature in the thin film. Eckert number is the ratio among kinetic energy and enthalpy, which is detected in the dissipation function of the energy equation, also known as the dissipation parameter. Ec  = 0 means there is no viscous dissipation. Figure 5(b) displays that the behavior of (Ec) is directly proportional to the temperature, as (Ec) upsurge viscous dissipation rises, i.e., kinetic energy transformed into heat energy which in turn raises the temperature.Figure 2
(a) Influence ofM on f′η. (b) Influence of K1 on f′η.
(a)(b)Figure 3
(a) Impact ofS on f′η. (b) Impact of F1 on f′η.
(a)(b)Figure 4
(a) Influence ofPr on θη. (b) Influence of R on θη.
(a)(b)Figure 5
(a) Influence ofS on θη. (b) Influence of Ec on θη.
(a)(b)The physical significance of embedded parameters are displayed in Table1 and the numerical values of skin friction and local Nusselt number are displayed in Tables 2 and 3. Table 2 demonstrates that the skin friction coefficient increases by rising various parameters, namely, S, F1, M, and K1. As these parameters increase, the unsteady parameter, the skin friction enhances due to the exertion of a drag force in the fluid thin film. The Lorentz force is a rising magnetic field that increases the resistive force on fluid movement. That is why more values of (M) enhance Cf. Table 3 lists the numerical values of Nux for a few physical variables, that is, Ec, S, Pr, and R. The Eckert number is normally made up of the nanofluid thermal conductivity components, which is used to raise the temperature field, which has the reverse effect on cooling operations, and hence the Nusselt number reduced for high values of (Ec). With an increase in (S), the boundary layer temperature of fluid raises, increasing the heat source of the thin film fluid; as a result, Nux boosts. In this table, we may observe that the Nusselt number upsurges as (R) grows, but Nux drops as (Pr) grows. Table 4 displays excellent agreement between the HAM and ND solve solution through computer-based package Mathematica 11.1.0.Table 1
Dimensionless groups.
GroupSymbolDefinitionPhysical interpretationLocal Nusselt numberNux1+4/3RRex3/2/21−γt−1/2θ′0Basic dimensionless convective heat transfer coefficient (ratio of convective heat transfer to conduction in a fluid slab of thickness h)Local Reynolds numberRexρUx/μThe ratio of inertia to viscous forcesEckert numberEcU2/CpTs−T0The kinetic energy of the flow relative to the boundary layer's enthalpy differencePrandtl numberPrμCp/kThe ratio of kinematic viscosity (ν) to thermal conductivity (k)Skin frictionCf−2τs/ρU2Dimensionless surface shear stressTable 2
Impact of different physical parameters onCf  =  −2Rex−1/2f″0 .
SF1MK1−2Rex−1/2f″00.30.30.40.32.861 450 120.52.714 365 430.72.015 367 980.21.835 540 230.31.735 701 410.41.491 365 230.41.074 130 360.90.786 521 511.00.540 378 130.30.731 560 270.40.472 765 310.60.190 438 16Table 3
Influence of various physical parameters overNux  =  δy/δx1+4/3RRex3/2/21−γt−1/2θ′0.
EcSPrR1+4/3RRex3/2/21−γt−1/2θ′00.30.24.50.11.142 021 630.51.075 243 010.70.853 456 250.21.431 804 290.31.674 597 360.71.851 286 254.50.562 621 075.50.351 293 286.50.168 435 680.11.437 510 200.51.637 920 811.52.024 412 56Table 4
Comparison of HAM and ND solve solution.S  = 0.25, K1  = 0.5, β  = 0.2, Pr  = 6.4, M  = 0.5, Rex  = 3.5, Ec  = 0.8.
ηfηf′ηθηHAMND solveHAMND solveHAMND solve0000.045 721 90.045 718 80.740.740.50.052 695 870.052 596 440.064 868 650.064 868 470.606 868 860.606 867 991.00.071 730 280.071 729 840.134 258 920.134 257 890.686 235 350.686 235 121.50.162 572 450.162 571 960.134 575 430.134 554 840.742 782 080.742 781 932.00.337 234 520.337 134 070.132 731 740.132 721 720.776 867 070.776 856 892.50.278 375 420.278 374 780.126 932 250.126 932 181.035 200 891.035 200 563.00.351 556 170.351 454 080.118 621 510.118 621 371.044 847 061.044 846 823.50.360 926 140.360 723 020.054 304 930.054 304 661.051 488 791.051 478 654.00.517 202 490.517 062 340.055 871 030.055 870 941.058 526 061.058 525 884.50.524 530 420.524 432 050.057 471 840.057 471 721.061 174 991.061 174 585.00.534 362 760.534 260 340.063 165 890.063 165 811.062 544 471.062 544 565.50.548 134 330.548 035 120.061 417 890.061 417 491.046 048 421.045 066 836.00.512 423 520.512 424 090.059 671 220.059 680 991.006 788 691.006 487 926.50.543 545 750.543 545 140.056 754 230.056 754 010.958 425 870.958 424 847.00.555 015 960.555 014 850.052 641 630.052 641 450.962 045 720.962 045 667.50.551 920 460.551 920 130.051 719 470.051 718 240.927 408 130.927 407 99


## 5. Concluding Remarks

The boundary layer two-dimensional Darcy–Forchheimer laminar thin film flow over a horizontal stretched sheet with the impact of thermal radiation and viscous dissipation are considered. The cause of heat source and thermal radiation over an unsteady stretched surface inserted in a porous medium in the absence of a magnetic field has been solved by HAM. The important key findings of this research work are given below:(i)
Velocity profile reduces for raise in values ofM and K1(ii)
When the values ofS and F1 increase, the velocity profile reduces(iii)
Temperature field increases with an increase in the values ofS and Ec(iv)
With increasing values ofPr and R, the temperature field reduces(v)
Skin friction coefficient rises as upsurges the values of physical parameters that isS, F1, M, and K1(vi)
By increasing the values ofS and R, the local Nusselt number upsurge while it declines for more values of Ec and Pr



---

*Source: 1086847-2022-01-19.xml*