The effects of heat generation on the Magnetohydrodynamics (MHD) natural convection flow along a vertical flat plate with non-uniform surface temperature are analyzed. The governing differential equations are transformed by introducing appropriate non-similarity variables to render them dimensionless, and then numerically resolved. Results for the details of velocity profiles, temperature profiles, local skin frictions, and rate of heat transfer are displayed graphically, and numerical tabulated data for skin friction and rate of local heat transfer are entered in tables with heat generation parameters and Joule Heating parameters.
Natural convection plays a vital part in processes like those involving strong gravitational fields on a global scale or geological processes, and the effects of heat conduction play a significant role in these situations. Both viscous dissipation and pressure work were initially incorporated into the energy equation by Ackroyd, (1962). He demonstrated that the effect of pressure work dominates over that of viscous dissi-pation. Magnetohydrodynamics (MHD) free convec-tion flow with conduction and Joule heating along a vertical flat plate has been studied by Parveen and Alim, (2011) who have looked at the effects of viscous dissipation and temperature-dependent thermal con-ductivity. In the context of heat generation Molla et al. (2004) investigated the issue of natural convection flow along a vertical wavy surface with a constant surface temperature. Stress analysis of the natural convection flow in a vertical flat plate subjected to joule heating and heat conduction (Alam et al., 2007; Islam et al., 2020).
According to Prabhakar and Prabhakararao, (2013) a vertical conical annular porous medium is affected by a temperature variation described by a Power law. Free convection was conducted by Alam et al. (2007) using a vertically permeable circular cone, pressure work, and a temperature gradient across the cones surface. The subject of the effect of a magnetic field on heat and fluid flow over a wavy surface has been explored by Tashtoush and Al-Odat, (2004). Some natural convection flows are affected by pressure stress work and viscous dissipation, as mentioned by Joshi and Gebhart, (1981). MHD-free convection from a vertical plate with power-law change in surface temperature, as studied by Abo-Eldahab and El Aziz, (2005) is subject to the effects of viscous dissipation and Joule heating. Mahajan and Gebhart, (1989) have investigated the role of viscous dissipation in buoyancy-induced flows. Sparrow and Cess, (1961) investigate how magnetic fields influence heat transport via free convection. Shariful and Mohammadein, (2004) investigated the impact of thermal-diffusion and diffusion thermo on heat and mass transport in magnetohydrodynamics. Poots et al. (1961) perform magneto hydrodynamics with laminar natural convection flow. The presence of heat generation prompted Molla et al. (2006) to explore magnetohydrodynamic natural convection flow on a spherical surface with a homogeneous heat flux. With a thin vertical plate and non-uniform internal heat generation Mendez and Trevino, (2000) study conjugates conduction-natural convection heat transfer. Magneto-hydrodynamic free convection in a strong cross flow field was described by Kuiken, (1970). Free convection from a vertical permeable circular cone with a non-uniform surface heat flux is shown by Hossain and Paul, (2001).
We have investigated the skin friction and local heat transfer coefficient as functions of Prandtls number Pr, magnetic parameter M, Joule heating parameter J, and heat generation parameter Q, as well as the effects of velocity and temperature fields.
Formulation of the Problem
Take into account laminar free convection flow down a vertical plate placed in a stable environment, where u and v represent the velocity components in the directions, respectively, and where is vertically up and is the coordinate perpendicular to x.
For steady, two-dimensional flow of the boundary layer continuity equation, momentum equation and energy equation including heat generation term and Joule heating term are given below.
Where, is the thermal diffusivity.
Where, for heated upward flows, x is taken vertically up from the active leading edge, and for cooled downward flows, x is taken vertically down. The temperature of quiescent ambient fluid at large values of y is taken to be constant.
Where is the temperature at the wall, is the fluid temperature, is the kinematics velocity, is the fluid thermal expansion coefficient, is the magnetic field strength, is the heat generation coefficient, is the specific heat at the constant pressure, is the fluid density and is the pressure. The terms for heat generation and Joule heating are the final two terms in the energy equation, respectively.
Equations are too solved subject to the boundary conditions.
Where is the free stream velocity.
The following generalizations are introduced to obtain the equations in terms of generalized stream and temperature functions and . Now letting.
Where A stands for the downstream tem-perature differential down the x-axis that would have occurred in the absence of the joule heating component. The real-world Grash of the number has an association with .
ψ_yy=c(ξ) b^2 (ξ)f(η),ψ_yyy=c(ξ)b^3 (ξ)f(η),
ψ_xy=c_ξ (ξ)b(ξ)f(η)+c(ξ)b_ξ (ξ)f(n)+c(ξ)b(ξ)f(η)∂/∂ξ(η)
ψ_x=c_ξ (ξ)f(η)+c(ξ)f(η)η/(b(ξ)) b_ξ (ξ)
So the Joule heating term
Temperature profiles
φ_y=φ(η)b(ξ)
φ_yy=φ(η)b^2 (ξ)
φ_x=η/(b(ξ)) b_ξ (ξ).φ(η)
Therefore, the momentum equation
⇒(f^2 (η)-f(η)f(η))/(b(ξ)) c_ξ (ξ)+(c(ξ)b_ξ (ξ))/(b^2 (ξ)) f^2 (η)=f(η)+gβφΔT/(c(ξ)b^3 (ξ)ν^2 )-(σβ_0^2)/(μb^2 ) f^
Therefore, the momentum equation is
(7)
Also, the energy equation
█(@⇒c(ξ)(b_ξ (ξ))/(b^2 (ξ)) f(η)φ(η)+φ" " c(ξ)(f(η)〖(ΔT)〗_ξ)/(b(ξ)ΔT)-(c_ξ (ξ)f(η)φ(η))/(b(ξ))-η (c(ξ)b_ξ (ξ))/(b^2 (ξ)) f(η)φ(η))
=1/p_r φ(η)+Q_o/(b^2 (ξ)νρC_p ) φ+(σβ_0^2)/ρΔT ν〖 c〗^2 (ξ)f^2 (η)
Divide
(8)
The heat generation term and the Joule heating effect are the final two terms in the energy equation (8). It is widely understood that when these concepts are ignored,, similarity solutions exist for a power law surface (at =0) temperature distributions.
∴c_x=d/dx[4(1/4 Gr_x )^(1⁄4)]=d/dx[4(1/4 (gβx^3 nx^n)/ν^2 )^(1⁄4)]=4.(1/4 gβn/ν^2 )^(1⁄4) d/dx(x^((n+3)/4))
b_x=d/dx [1/x[1/4 (gβx^3 nx^n)/ν^2 ]^(1⁄4) ]=[1/4 gβn/ν^2 ]^(1⁄4) (n-1)/4 x^((n+3)/4-2 )=(n-1)/(4x^2 )(1/4 Gr_x )^(1⁄4)
Now
Then for the momentum equation
f(η)+gβφΔT/(c(ξ)b^3 (ξ)ν^2 )+(c_ξ (ξ))/(b(ξ)) f(η)f(η)-[(c_ξ (ξ))/(b(ξ))+(c(ξ)b_ξ (ξ))/(b^2 (ξ))]f^2 (η)+(σβ_0^2)/(μb^2 ) f^=0 (9)
(10)
Where, , Hartman number related to MHD.
And energy equation
1/p_r φ(η)+(n+3)f(η)φ(η)-4nf(η)φ(η)-Q_o/(b^2 (ξ)νρC_p ) φ+(16σβ_0^2)/ρΔT ν
(1/4 Gr_ξ )^(1/2) f^2 (η)" "("11")
Therefore Joule heating term are included, ( ) and are functions of and for the power law case. To retain Joule heating terms to the first order are chosen
So, the equation (11) becomes
(12)
Where, Q=(Q_0 " " ξ^2)/(ν" " ρ" " C_p " " 〖(1/4 Gr_ξ)〗^(1/2) " " )
Q is the dimensionless heat generation parameter.
Equations (1.10) and (1.12) are numerically integrated for the vertical surface case, with the following boundary conditions
(13)
Skin Friction Coefficient and Heat Transfer Coefficient
For the flat surface the heat flux is given by,
We know,
∂T/∂y=bΔTφ(η)
The proper and give the following heat flux for the power law case.
Where, ΔT=d(x)=Nx^n,b(x)=1/x[1/4 Gr_x ]^(1⁄4),Gr_x=(gβx^3 Nx^n)/ν^2
Heat transfer coefficient (Nusselt Number)
=
Local Nusselt Number, Nu_ξ=" "-〖(1/4 " " Gr_ξ)〗^(1/4) " " φ^ (0)
The viscous stress and the Skin friction coefficient defined on a convection velocity, are know from
∴τ(ξ)=μν" " c(ξ)f(η) 〖{b(ξ)}〗^2
When
Therefore, Skin friction coefficient
C_fξ=(2" " μ" " ν" " c(ξ)b^2 (ξ)f^ (0))/(ρν^2 " " c^2 (ξ)b^2 (ξ)〖f^〗_max^2 (0))∝(2" " μ" " ν" " c(ξ)b^2 (ξ)f^ (0))/(ρν^2 " " c^2 (ξ)b^2 (ξ))
C_fξ=(f^ (0))/(2〖(1/4 " " Gr_ξ)〗^(1/4) " " )
The Effects of heat generation on MHD Natural Convection Flow along a Vertical Flat Plate with Non-Uniform Surface Temperature have been analyzed in this present work. The settings of the parameter controlling heat generation in this simulation are Q = 1.50, 1.20, 0.90, 0.50, magnetic parameter M = 0.50, 0.40, 0.30, 0.20, 0.10, joule heating parameter J = 0.50, 0.40, 0.30, 0.20, 0.10, and Prandlts number Pr = 7.00, 3.00, 1.00, 0.70. If we know the functions f(ξ, η), (ξ, η) and their derivatives for the various heat generating parameter values Q, magnetic parameter M, Prandtls number Pr, and the Joule heating parameter J. Fig. 2–5 show the velocity and tem-perature curves obtained from the solution . Also, the local skin friction (ξ, 0) and local heat transfer (ξ, 0) obtained from figure 6 to 9. Fig. 2a and 2b display results for the effect of the heat generation parameter Q = 1.50, 1.20, 0.90, 0.50 for various values of the controlling parameter M = 0.50, Pr = 0.70, J = 0.50, n = 1.00 on the velocity profile (ξ, η) increa-ses with the increase of heat generation parameter Q which indicates that the heat generation parameter increases the fluid motion. In Fig. 2b it is shown that the temperature profile (ξ, η) increase for increasing values of Q with another controlling parameter. The effects of magnetic parameter or Hartmann number M = 0.50, 0.40, 0.30, 0.20, 0.10 for Q = 1.50, Pr = 0.70, J = 0.50 and n = 1.00 on the velocity profiles and temperature profiles are shown in Fig. 3a and 3b. Fig. 3a and 3b show the impacts of magnetic parameter M on the velocity and temperature curves, respectively. The velocity profiles are found to decrease as the value of M increases, whereas the temperature profiles increase as the value of M increases. In Fig. 4a and 4b illustrate the effect of Joule heating parameter J = 0.50, 0.40, 0.30, 0.20, 0.10 for M = 0.50, Pr=0.70, Q = 1.50 and n = 1.00 on the velocity profiles and temperature profiles are shown in Fig. 4a and 4b. Fig. 4a and 4b show the impact of the Joule heating parameter J on the velocity and temperature curves, respectively. The velocity profiles in this image grow as the value of J increases, and the temperature profiles also increase with the value of the Joule heating parameter J increases.
Fig. 5a depicts the velocity profiles for different values of prandtls number, Pr = 7.00, 3.00, 1.00, 0.70 while the other controlling parameter M = 1.50, Q = 1.00, J = 0.50, n = 1.00. Corresponding distribution of the temperature profiles (ξ, η) in the fluids is shown in Fig. 5b. Fig. 5a shows that increasing the prandtls number decreases the fluid velocity. Fig. 5b, on the other hand, shows that when the prandtls number Pr increases, the temperature profiles within the boundary layer drop. Numerical values of the skin friction (ξ, 0) and the local heat transfer coefficient (ξ, 0) are depicted graphically in Fig. 6a and 6b respectively against ξ for different values of the heat generation parameter Q = 1.50, 1.20, 0.90, 0.50 for M = 0.50, Pr = 0.70, J = 0.50, n = 1.00. It is seen from fig 6a that the skin friction (ξ, 0) increases when the heat genera-tion parameter Q increases. It is also observed in fig 6b, local heat transfer coefficient (ξ, 0) increases while the heat generation parameter Q increases. The effects of magnetic parameter or Hartmann number M = 0.50, 0.40, 0.30, 0.20, 0.10 for Q = 1.50, Pr = 0.70, J = 0.50 and n = 1.00 on the local skin friction coefficient (ξ, 0) and local heat transfer coefficient (ξ, 0) are shown in Fig. 7a and 7b. Fig. 7a and Fig. 7b it is evident that the increasing value of M leads to increase the skin friction coefficient (ξ, 0) and decrease local heat transfer coefficient (ξ, 0). The variation of skin friction and heat transfer for different values of Joule heating parameter J = 0.50, 0.40, 0.30, 0.20, 0.10 against ξ for M = 0.50, Pr=0.70, Q = 1.50 and n = 1.00 are shown in figure 2.8a and 2.8b. Figure 2.8a and figure 2.8b it is found that the increasing values of Joule heating parameter J leads to increase both the skin friction coefficient (ξ, 0) and local heat transfer coefficient (ξ, 0). From Fig. 2a, it is observed that increase of the value of prandtls number Pr = 7.00, 3.00, 1.00, 0.70 leads to increase of the value of skin friction Coefficient (ξ, 0) and local heat transfer coefficient (ξ, 0) also increases shown in figure 2.9b for the same values of Prandlts number Pr against ξ M = 1.50, Q = 1.00, J = 0.50, n = 1.00.
From the present investigation, the following conclusions may be drawn. Increased values of the heat generation parameter Q leads to increase the velocity profiles, the temperature profiles, the local skin friction coefficient (ξ, 0) and the local heat transfer coefficient while M = 0.50, Pr = 0.70, J = 0.50, Vd = 0.60, n = 1.00 . The effect of magnetic parameter or Hartmann Number M is to increase the temperature profiles, the local skin friction coefficient (ξ, 0) but the local heat transfer coefficient, the velocity profiles decrease with the increasing value of M. Increased values of the Joule heating parameter J led to increase the velocity profiles, the skin friction coefficient, the temperature profile and the local heat transfer coefficient. It has been observed that the skin friction and the local heat transfer coefficient increases for increasing value of Pr but the temperature profile, the velocity profile decreases with the increasing value of Pr.
Thanks to the Mathematics Department of Dhaka University of Engineering and Technology (DUET), Gazipur, Bangladesh for giving us opportunity to complete the research work.
The authors declare that there are no conflics of interest regarding the publication of this research article.
Academic Editor
Dr. Toansakul Tony Santiboon, Professor, Curtin University of Technology, Bentley, Australia.
PhD Researcher, Department of Mathematics, Dhaka University of Engineering & Technology (DUET), Gazipur, Bangladesh.
Akter M, Sarker SPK, and Alam MM. (2023). Magnetohydrodynamics (MHD) effects on heat generation and joule heating with non-uniform surface temperature and natural convection flow over a vertical flat plate, Int. J. Mat. Math. Sci., 5(2), 09-18. https://doi.org/10.34104/ijmms.023.09018