Laminar two-dimensional unsteady mixed-convection boundary-layer flow of a viscous incompressible fluid past asymmetric wedge with variable surface temperature embedded in a porous medium saturated with a nanofluid has been studied. The employed mathematical model for the nanofluid takes into account the effects of Brownian motion and thermophoresis. The velocity in the potential flow is assumed to vary arbitrary with time. The non-Darcy effects including convective, boundary and inertial effects will be included in the analysis. The unsteadiness is due to the time-dependent free stream velocity. The governing boundary layer equations along with the boundary conditions are converted into dimensionless form by a non-similar transformation, and then resulting system of coupled non-linear partial differential equations are solved by perturbation solutions for small dimensionless time until the second order. Numerical solutions of the governing equations are obtained employing the implicit finite-difference scheme in combination with the quasi-linearization technique. To validating the method used, we compared our results with previous results in earlier papers on special cases of the problem and are found to be in agreement. Effects of various parameters on velocity, temperature and nanoparticle volume fraction profiles are graphically presented.
The study of mixed convection flow finds applications in several industrial and technical processes such as nuclear reactors cooled during emergency shutdown, solar central receivers exposed to winds, electronic devices cooled by fans, heat exchanges placed in a low-velocity environment, etc. The mixed convection flows become important when the buoyancy forces due to the temperature difference between the wall and the free stream becomes large.
The study and analysis of heat and mass transfer in porous media has been the subject of many investigations due to their frequent occurrence in industrial and technological applications. Examples of some applications include geothermal reservoirs, drying of porous solids, thermal insulations and many others. Smith [
There is a large body of literature on unsteady, mixed convection, boundary- layer flows past bodies of different geometries. Ibrahim et al. [
The interested reader can find an excellent collection of papers on unsteady convective flow problems over heated bodies embedded in a fluid-saturated porous medium in the book by Pop and Ingham [
Nanofluids with enhanced thermal characteristics have widely been examined to improve the heat transfer performance of many engineering applications [
According to Yacob et al. [
The aim of the present paper is to study the unsteady mixed convection flow along a symmetric wedge embedded in a porous medium saturated with a nanofluid in the presence of first and second orders resistances, which to the best of our knowledge have not been investigated yet. Motivation to study mixed convection in porous media comes from the need to characterize the convective transport processes around deep geological repository for the disposal of high- level nuclear waste, e.g. spent fuel rods from nuclear reactors (see Lai [
The effect of the presence of the buoyancy forces and the isotropic solid matrix on the unsteady mixed convection flow along a symmetric wedge embedded in a porous medium saturated with a nanofluid are considered. In addition, the Brownian motion and the thermophoresis effects are considered. The unsteadiness in the flow field is caused by impulsively creating motion in the free stream and at the same time suddenly raising the surface temperature above its surroundings. The partial differential equations governing the flow and the heat transfer solved numerically using some different numerical methods as the finite difference scheme by Pereyra [
Let us consider an unsteady mixed convection boundary layer flow of an incompressible fluid past along a symmetric wedge with variable surface temperature embedded in a porous medium saturated with a nanofluid as shown in
u ¯ e ( x ¯ ) = U ∞ ( x ¯ L ) m for m ≤ 1 (1)
where L is a characteristic length and m is pressure gradient related in the included angle π β by m = β / ( 2 − β ) . It is clear that for negative values of m the solution becomes singular at x ¯ = 0 . While for m positive the solution can be defined for all values of x ¯ . It is assumed that the variable surface temperature of the wedge is ( T w > T ∞ ) where T ∞ is the ambient temperature of the fluid and
T ¯ is the temperature of the fluid. Under the above-mentioned assumptions, the boundary layer equations governing the flow can be expressed as follows:-
∂ u ¯ ∂ x ¯ + ∂ v ¯ ∂ y ¯ = 0 , (2)
∂ u ¯ ∂ t ¯ + u ¯ ∂ u ¯ ∂ x ¯ + v ¯ ∂ u ¯ ∂ y ¯ = u ¯ e d u ¯ e d x ¯ + υ ∂ 2 u ¯ ∂ y ¯ 2 + υ ε K ( u ¯ e − u ¯ ) + Γ ε 2 K 1 / 2 ( u ¯ e 2 − u ¯ 2 ) + [ ( 1 − φ ∞ ) β T ρ f ( T ¯ − T ∞ ) − ( ρ p − ρ f ) ( φ ¯ − φ ∞ ) / ρ f ] g (3)
∂ T ¯ ∂ t ¯ + u ¯ ∂ T ¯ ∂ x ¯ + v ¯ ∂ T ¯ ∂ y ¯ = α m ∂ 2 T ¯ ∂ y ¯ 2 + τ [ D B ∂ T ¯ ∂ y ¯ ∂ φ ¯ ∂ y ¯ + D T T ∞ ( ∂ T ¯ ∂ y ) 2 ] , (4)
∂ φ ¯ ∂ t ¯ + u ¯ ∂ φ ¯ ∂ x ¯ + v ¯ ∂ φ ¯ ∂ y ¯ = D B ∂ 2 φ ¯ ∂ y ¯ 2 + D T T ∞ ( ∂ 2 T ¯ ∂ y ¯ 2 ) , (5)
where u ¯ and v ¯ are the velocity components in is x ¯ and y ¯ coordinate, respectively. T ¯ is the temperature, φ ¯ is the nanoparticle volume fraction, g is the acceleration due to gravity, ρ f is the density of the base fluid and ν and β T are the viscosity and thermal conductivity of the fluid. While, ρ p is the density of the nanoparticles, ( ρ c ) f is the heat capacity of the fluid and ( ρ c ) p is the effective heat capacity of the nanoparticle material. α m = k m / ( ρ c ) f is the thermal diffusivity, τ = ( ρ c ) p / ( ρ c ) f is the ratio of the effective heat capacity of the nanoparticle material to the heat capacity of the fluid and k m is the effective thermal conductivity of the porous medium. K, ε and Γ are the permeability, the porosity of the porous medium and the empirical constant in the second- order resistance. The coefficients that appear in Equations ((4), (5)) are the Brownian diffusion coefficient D B , and the thermophoretic diffusion coefficient D T .
The boundary conditions for the present problem are
u ¯ = 0 , v ¯ = 0 , T ¯ = T w ( x ¯ ) = T ∞ + ( T r − T ∞ ) ( x ¯ L ) 2 m − 1 , φ ¯ = φ w ( x ¯ ) = φ ∞ + ( φ r − φ ∞ ) ( x ¯ L ) 2 m − 1 , at y ¯ = 0 , u ¯ = u ¯ e ( x ¯ ) , T ¯ → T ∞ , φ ¯ → φ ∞ as y ¯ → ∞ . (6)
We introduce non dimensional dependent and independent variables according to,
x = x ¯ L , y = R e L 1 / 2 y ¯ L , u = u ¯ U ∞ , v = R e L 1 / 2 v ¯ U ∞ , u e = u ¯ e U ∞ , t = U ∞ L t ¯ , T = T ¯ − T ∞ T r − T ∞ , φ = φ ¯ − φ ∞ φ r − φ ∞ . (7)
The velocity over the wedge is now given by u e ( x ) = x m for m ≤ 1 .
The governing Equations (2)-(5) can be written as
∂ u ∂ x + ∂ v ∂ y = 0 , (8)
∂ u ∂ t + u ∂ u ∂ x + v ∂ v ∂ y = u e d u e d x + ∂ 2 u ∂ y 2 + ε D a L R e L ( u e − u ) + Γ ε 2 D a L 1 / 2 ( u e 2 − u 2 ) + λ ( T − N r φ ) , (9)
∂ T ∂ t + u ∂ T ∂ x + v ∂ T ∂ y = 1 P r ∂ 2 T ∂ y 2 + N o B ∂ T ∂ y ∂ φ ∂ y + N o T ( ∂ T ∂ y ) 2 , (10)
∂ φ ∂ t + u ∂ φ ∂ x + v ∂ φ ∂ y = 1 S c ∂ 2 φ ∂ y 2 + 1 S c N o T N o B ∂ 2 T ∂ y 2 , (11)
where D a L = K / L 2 is the Darcy number, λ = G r L / R e L 2 is the mixed convection parameter, R e L = U ∞ L / ν is the Reynolds number, G r L = g β ( T r − T ∞ ) L 3 / ν 2 is the Grashof number, N r = ( ρ p − ρ f ) ( φ r − φ ∞ ) ρ f ( 1 − φ ∞ ) ( T r − T ∞ ) is the nanofluid buoyancy ratio parameter, N o T = τ ν ( D T T ∞ ) ( T r − T ∞ ) is the thermophores is parameter, N o B = τ ν D B ( φ r − φ ∞ ) is the Brownian motion parameter, P r = α m / ν is the Pran- dtl number and S c = ν / B B is the Schmidt number.
For t ≥ 0 and m ≤ 1 the boundary conditions then may be written as:
u = 0 , v = 0 , T = x 2 m − 1 , φ = x 2 m − 1 at y = 0 , u = u e ( x ) = x m , T → 0 , φ → 0 as y → ∞ (12)
The number of independent variables in the governing Equations (8)-(11) can be reduced from three to two by applying the following transformations as
η = x ( m − 1 ) / 2 ( 2 ξ ) − 1 / 2 y , ψ = x ( m + 1 ) / 2 ( 2 ξ ) 1 / 2 f ( η , ξ ) , T = x 2 m − 1 θ ( η , ξ ) , φ = x 2 m − 1 ϕ ( η , ξ ) , ξ = 1 − e − τ , τ = x m − 1 t , (13)
where η is a non-dimensional similarity variable and ψ is the stream function, which is defined in the usual way, namely u = ∂ ψ / ∂ y and v = − ∂ ψ / ∂ y .
Substituting the transformations (13) into Equations (9)-(11), we obtained the following transformed equations for the momentum and thermal boundary layer equations:
f ‴ + η ( 1 − ξ ) f ″ + [ ( m + 1 ) ξ − ( m − 1 ) ( 1 − ξ ) ln ( 1 − ξ ) ] f f ″ + 2 ξ [ γ x ( 1 − f ′ ) + ( m + Δ x ) ( 1 − f ′ 2 ) ] = 2 ( 1 − m ) ξ ( 1 − ξ ) ln ( 1 − ξ ) ( f ′ ∂ f ′ ∂ ξ − f ″ ∂ f ∂ ξ ) + 2 ξ ( 1 − ξ ) ∂ f ′ ∂ ξ − 2 λ ξ ( θ − N r ϕ ) , (14)
1 P r θ ″ + η ( 1 − ξ ) θ ′ + [ ( m + 1 ) ξ − ( m − 1 ) ( 1 − ξ ) ln ( 1 − ξ ) ] f θ ′ − 2 ( 2 m − 1 ) ξ θ f ′ = 2 ( 1 − m ) ξ ( 1 − ξ ) ln ( 1 − ξ ) ( f ′ ∂ θ ∂ ξ − θ ′ ∂ f ∂ ξ ) + 2 ξ ( 1 − ξ ) ∂ θ ∂ ξ − N B θ ′ ϕ ′ − N T θ ′ 2 (15)
1 S c ϕ ″ + η ( 1 − ξ ) ϕ ′ + [ ( m + 1 ) ξ − ( m − 1 ) ( 1 − ξ ) ln ( 1 − ξ ) ] f ϕ ′ − 2 ( 2 m − 1 ) ξ ϕ f ′ = 2 ( 1 − m ) ξ ( 1 − ξ ) ln ( 1 − ξ ) ( f ′ ∂ ϕ ∂ ξ − ϕ ′ ∂ f ∂ ξ ) + 2 ξ ( 1 − ξ ) ∂ ϕ ∂ ξ − N T S c N B θ ″ (16)
where γ x = D a − 1 R e L − 1 x 1 − m is the local permeability parameter, Δ x = F x is the inertia coefficient parameter (see Chamkha [
N T = N o T x 2 m − 1 = τ ν ( D T T ∞ ) ( T r − T ∞ ) x 2 m − 1 = τ ν ( D T T ∞ ) ( T w − T ∞ ) , N B = N o B x 2 m − 1 = τ ν D B ( φ r − φ ∞ ) x 2 m − 1 = τ ν D B ( φ w − φ ∞ ) (17)
The boundary conditions to be satisfied by the Equations (14)-(16) are
f ( 0 , ξ ) = f ′ ( 0 , ξ ) = 0 , θ ( 0 , ξ ) = 1 , ϕ ( 0 , ξ ) = 1 , f ′ ( ∞ , ξ ) = 1 , θ ( ∞ , ξ ) = 0 , ϕ ( ∞ , ξ ) = 0 , (18)
In the above equations, prime denotes differentiation of the functions with respect to η only.
Now to find the numerical solutions we can get the easiest form by using the transformations ξ = 1 − e − τ from Equations (14)-(16) applicable for 0 ≤ τ ≤ ∞ .
f ‴ + η e − τ f ″ + [ ( m + 1 ) ( 1 − e − τ ) + ( m − 1 ) τ e − τ ] f f ″ + 2 ( 1 − e − τ ) [ γ x ( 1 − f ′ ) + ( m + Δ x ) ( 1 − f ′ 2 ) ] = 2 ( m − 1 ) τ ( 1 − e − τ ) ( f ′ ∂ f ′ ∂ τ − f ″ ∂ f ∂ τ ) + 2 ( 1 − e − τ ) ( ∂ f ′ ∂ τ ) − 2 λ ( 1 − e − τ ) ( θ − N r ϕ ) (19)
1 P r θ ″ + η e − τ θ ′ + [ ( m + 1 ) ( 1 − e − τ ) + ( m − 1 ) τ e − τ ] f θ ′ − 2 ( 2 m − 1 ) ( 1 − e − τ ) θ f ′ = 2 ( m − 1 ) τ ( 1 − e − τ ) ( f ′ ∂ θ ∂ τ − θ ′ ∂ f ∂ τ ) + 2 ( 1 − e − τ ) ∂ θ ∂ τ − N B θ ′ ϕ ′ − N T θ ′ 2 (20)
1 S c ϕ ″ + η e − τ ϕ ′ + [ ( m + 1 ) ( 1 − e − τ ) + ( m − 1 ) τ e − τ ] f ϕ ′ − 2 ( 2 m − 1 ) ( 1 − e − τ ) ϕ f ′ = 2 ( m − 1 ) τ ( 1 − e − τ ) ( f ′ ∂ ϕ ∂ τ − ϕ ′ ∂ f ∂ τ ) + 2 ( 1 − e − τ ) ∂ ϕ ∂ τ − N T S c N B θ ″ (21)
The boundary conditions to be satisfied by the above equations are
f ( 0 , τ ) = f ′ ( 0 , τ ) = 0 , θ ( 0 , τ ) = 1 , ϕ ( 0 , τ ) = 1 , f ′ ( ∞ , τ ) = 1 , θ ( ∞ , τ ) = 0 , θ ( ∞ , τ ) = 0. (22)
In practical applications, two quantities of physical interest are to be determined, such as, surface shear stress and the rate of heat and mass transfer at the surface. These may obtained in terms of the skin friction coefficient (wall shear stress) C f , local Nusselt number N u x and the local Sherwood number S h x , which are defined by:
C f = μ ρ f U e 2 ( x ¯ ) ( ∂ u ¯ ∂ y ¯ ) y = 0 , N u x = x ¯ T w − T ∞ ( ∂ T ¯ ∂ y ¯ ) y = 0 , S h x = x ¯ φ w − φ ∞ ( ∂ φ ¯ ∂ y ¯ ) y = 0 (23)
By introducing the non-dimensional variables (7) and the transformation (13), the skin friction coefficient, C f , the local Nusselt number, N u x and the local Sherwood number, S h x can now defined by:
C f R e x 1 / 2 = [ 2 ( 1 − e − τ ) ] − 1 / 2 f ″ ( 0 , τ ) , N u x R e x − 1 / 2 = − [ 2 ( 1 − e − τ ) ] − 1 / 2 θ ′ ( 0 , τ ) , S h x R e x − 1 / 2 = − [ 2 ( 1 − e − τ ) ] − 1 / 2 ϕ ′ ( 0 , τ ) . (24)
Perturbation solutions method for small time τ ≪ 1
For small τ Equations (19)-(21) become
f ‴ + η f ″ + 2 m τ f f ″ + 2 τ [ γ x ( 1 − f ′ ) + ( m + Δ x ) ( 1 − f ′ 2 ) + λ ( θ − N r ϕ ) ] = 2 ( m − 1 ) τ 2 [ f ′ ∂ f ′ ∂ τ − f ″ ∂ f ∂ τ ] + 2 τ [ ∂ f ′ ∂ τ ] (25)
1 P r θ ″ + η θ ′ + 2 m τ f θ ′ − 2 ( 2 m − 1 ) τ θ f ′ + N B θ ′ ϕ ′ + N T θ ′ 2 = 2 ( 1 − m ) τ 2 [ f ′ ∂ θ ∂ τ − θ ′ ∂ f ∂ τ ] + 2 τ ∂ θ ∂ τ (26)
1 S c ϕ ″ + η ϕ ′ + 2 m τ f ϕ ′ − 2 ( 2 m − 1 ) τ ϕ f ′ + N T S c N B θ ″ = 2 ( 1 − m ) τ 2 [ f ′ ∂ ϕ ∂ τ − ϕ ′ ∂ f ∂ τ ] + 2 τ ∂ ϕ ∂ τ (27)
and the corresponding boundary conditions (22) become
f ( 0 , τ ) = f ′ ( 0 , τ ) = 0 , θ ( 0 , τ ) = 1 , ϕ ( 0 , τ ) = 1 , f ′ ( ∞ , τ ) = 1 , θ ( ∞ , τ ) = 0 , ϕ ( ∞ , τ ) = 0 (28)
The resulting system of Equations (25)-(27), along with boundary conditions (28) is solved using the perturbation technique. Now, the non-dimensional stream function and the temperature functions can be written as:
f ( η , τ ) = f 0 ( η ) + τ f 1 ( η ) + τ 2 f 2 ( η ) + ⋯ (29)
θ ( η , τ ) = θ 0 ( η ) + τ θ 1 ( η ) + τ 2 θ 2 ( η ) + ⋯ (30)
ϕ ( η , τ ) = ϕ 0 ( η ) + τ ϕ 1 ( η ) + τ 2 ϕ 2 ( η ) + ⋯ (31)
Substituting Equations (29)-(31) into Equations (25)-(27) and equating the various coefficients of power of τ to zero (here we collect terms up to the second power of O ( τ 2 ) , we can get the following sets of ordinary differential equations:
Zero order:
f ‴ 0 + η f ″ 0 = 0 (32)
1 P r θ ″ 0 + η θ ′ 0 + N B θ ′ 0 ϕ ′ 0 + N T θ ′ 0 2 = 0 , (33)
1 S c ϕ ″ 0 + η ϕ ′ 0 + N T S c N B θ ″ 0 = 0 , (34)
with the corresponding boundary conditions:
f 0 ( 0 ) = f ′ 0 ( 0 ) = 0 , θ 0 ( 0 ) = ϕ 0 ( 0 ) = 1 , f ′ 0 ( ∞ ) = 1 , θ 0 ( ∞ ) = ϕ 0 ( ∞ ) = 0 (35)
First order:
f ‴ 1 + η f ″ 1 − 2 f ′ 1 = − 2 m ( 1 − f ′ 0 2 + f 0 f ″ 0 ) + 2 ( γ x f ′ 0 + Δ x f ″ 0 2 ) − 2 λ ( θ 0 − N r ϕ 0 ) , (36)
1 P r θ ″ 1 + η θ ′ 1 − 2 θ 1 + N B θ ′ 1 ϕ ′ 1 + N T θ ′ 1 2 = − 2 m f 0 θ ′ 0 + 2 ( 2 m − 1 ) θ 0 f ′ 0 , (37)
1 S c ϕ ″ 1 + η ϕ ′ 1 − 2 ϕ 1 + N T S c N B θ ″ 1 = − 2 m f 0 ϕ ′ 0 + 2 ( 2 m − 1 ) ϕ 0 f ′ 0 , (38)
with the corresponding boundary conditions:
f 1 ( 0 ) = f ′ 1 ( 0 ) = 0 , θ 1 ( 0 ) = ϕ 1 ( 0 ) = 0 , f ′ 1 ( ∞ ) = 0 , θ 1 ( ∞ ) = ϕ 1 ( ∞ ) = 0 (39)
Second order:
f ‴ 2 + η f ″ 2 − 4 f ′ 2 = 2 [ ( 1 − 2 m ) f 1 f ″ 0 + ( 3 m − 1 ) f ′ 1 f ′ 0 − m f 0 f ″ 1 − λ ( θ 1 − N r ϕ 1 ) ] + 2 γ x f ′ 1 + 4 Δ x f ′ 0 f ′ 1 (40)
1 P r θ ″ 2 + η θ ′ 2 − 4 θ 2 + N B θ ′ 2 ϕ ′ 2 + N T θ ′ 2 2 = 2 [ ( 1 − 2 m ) f 1 θ ′ 0 + ( 3 m − 2 ) θ 1 f ′ 0 − m θ ′ 1 f 0 + ( 2 m − 1 ) θ 0 f ′ 1 ] (41)
1 S c ϕ ″ 2 + η ϕ ′ 2 − 4 ϕ 2 + N T S c N B θ ″ 2 = 2 [ ( 1 − 2 m ) f 1 ϕ ′ 0 + ( 3 m − 2 ) ϕ 1 f ′ 0 − m ϕ ′ 1 f 0 + ( 2 m − 1 ) ϕ 0 f ′ 1 ] (42)
With the corresponding boundary conditions:
f 2 ( 0 ) = f ′ 2 ( 0 ) = 0 , θ 2 ( 0 ) = 0 , θ 2 ( 0 ) = 0 , f ′ 2 ( ∞ ) = 0 , θ 2 ( ∞ ) = 0 , ϕ 2 ( ∞ ) = 0 (43)
where primes denote differentiation with respect to η . Knowing the values of f ″ 0 ( 0 ) , f ″ 1 ( 0 ) , f ″ 2 ( 0 ) , θ ′ 0 ( 0 ) , θ ′ 1 ( 0 ) , θ ′ 2 ( 0 ) , ϕ ′ 0 ( 0 ) , ϕ ′ 1 ( 0 ) and ϕ ′ 2 ( 0 ) from the solutions of Equations (32)-(43), we get the values of the skin friction coefficient C f , local Nusselt number N u x and the local Sherwood number S h x , from the following expressions:
C f R e x 1 / 2 = ( 2 τ ) − 1 / 2 ( f ″ 0 ( 0 ) + τ f ″ 1 ( 0 ) + τ 2 f ″ 2 ( 0 ) ) , N u x R e x − 1 / 2 = − ( 2 τ ) − 1 / 2 ( θ ′ 0 ( 0 ) + τ θ ′ 1 ( 0 ) + τ 2 θ ′ 2 ( 0 ) ) , S h x R e x − 1 / 2 = ( 2 τ ) − 1 / 2 ( ϕ ′ 0 ( 0 ) + τ ϕ ′ 1 ( 0 ) + τ 2 ϕ ′ 2 ( 0 ) ) . (44)
The sets of ordinary differential Equations (32)-(34) are solved successively by giving appropriate initial guess values for f ″ i ( 0 ) , θ ′ i ( 0 ) , ϕ ′ i ( 0 ) , i = 0 , 1 , 2 to match the values with the corresponding boundary conditions at f ′ i ( ∞ ) , θ i ( ∞ ) , ϕ i ( ∞ ) , i = 0 , 1 , 2 . The numerical values of the coefficients of skin friction, C f R e x 1 / 2 and the Nusselt number N u x R e x − 1 / 2 for smaller values of τ ∈ [ 0 , 1 ] obtained by perturbation method while P r = 1.0 and m = 0.2 are shown in
Numerical results presented for some representative values of the governing parameters govern this problem. In order to see the physical insight, the numerical values of velocity f ′ ( η ) , temperature θ ( η ) , and nanoparticle volume fraction ϕ ( η ) within the boundary layer computed for different parameters as unsteadiness parameter τ , mixed convection parameter λ , nanofluid buoyancy ratio parameter Nr, thermophoresis parameter NT, Brownian motion parameter NB, first resistant parameter γ , second resistant parameter Δ , Prandtl number Pr and Schmidt number Sc. In addition, numerical results obtained to discuss the effects of the governing parameters on the skin friction coefficient, local Nussel number and local Shrewood number and displayed in tabular and graphical forms.
Figures 2-4 displayed the effects of nanonfluid parameters, thermophoresis parameter NT, Brownian motion parameter NB and buoyancy ratio parameter Nr. From these figures, it is observed that the velocity and the temperature profiles
τ | Harris et al. [ | Hossian et al. [ | Present result | |||
---|---|---|---|---|---|---|
C f R e x 1 / 2 | N u x R e x − 1 / 2 | C f R e x 1 / 2 | N u x R e x − 1 / 2 | C f R e x 1 / 2 | N u x R e x − 1 / 2 | |
0.01 | 5.65797 | 5.64360 | 5.66886 | 5.64525 | 5.65769 | 5.63468 |
0.10 | 1.83491 | 1.78946 | 1.83693 | 1.76140 | 1.83312 | 1.75951 |
0.20 | 1.33334 | 1.26902 | 1.33189 | 1.22357 | 1.33090 | 1.22543 |
0.40 | 0.99341 | 0.90234 | 0.99548 | 0.83937 | 0.99038 | 0.83739 |
1.0 | 0.72370 | 0.57925 | 0.72665 | 0.46370 | 0.72116 | 0.46104 |
increase with an increase in both NT and NB. However, for the nanoparticle volume fraction profiles there is a crossing over point where the volume fraction profile decreases before that point and slightly increases after that (
The effects of the mixed convection parameter λ on the non-dimensional velocity, temperature and nanoparticle volume fraction are illustrated in
From this figure, it is observed that the velocity increases as λ increases, however; both the temperature and nanoparticle fraction profiles slightly decrease with increasing values of λ .
and nanoparticle volume fraction. An increase in γ or Δ leads to an increase in fluid velocity and a decrease on both temperature and nanoparticle volume fraction. This is due the fact that an increase in γ or Δ implies that there is a decrease in the resistance of the porous medium which tends to accelerate the fluid velocity in the boundary layer region.
to decrease velocity, temperature and species concentration profiles. This is consistent with the fact that, increase in Sc means decrease of molecular diffusivity those results in decease of concentration boundary layer. Hence species concentration is higher for small values of Sc and lower for large value of Sc.
Figures 10-17 are presented to illustrate the variations of the local rate of shear stress, the local rate of heat transfer and the local rate of mass transfer for different values of the governing parameters.
increases in the velocity and temperature profiles while its volume fraction profile decreases. This yields reductions in both the local Nusselt number and the local Sherwood number and enhancement in the local skin-friction coefficient.
The effects of the buoyancy ratio parameter Nr are illustrated in
The effects of mixed convection parameter λ on skin friction coefficient, local Nusselt number and local Sherwood number are are shown in
The effects of both the first resistance parameter γ and the second resistance parameter Δ on the local skin-friction coefficient, the local Nusselt number and the local Sherwood number are illustrated in
The effects of Prandtl number Pr and Schmidt number Scon the behaviors of the local skin-friction coefficient, local Nusselt number and the local Sherwood number are illustrated on
Finally, from Figures 10-17, it is observed that as the unsteadiness parameter τ increases the local rate of shear stress increases while, both of the local rate of heat transfer and the local rate of mass transfer decreases.
In the present work, the problem of unsteady mixed convection flow along a sym- metric wedge embedded in a porous medium saturated with a nanofluidis studied theoretically. The resulting system of nonlinear partial differential equations is treated using the Sparrow-Quack-Boerner local non-similarity numerical method. The obtained system is solved numerically using an efficient numerical shooting technique with a fourth-fifth-order Runge-Kutta method scheme (MATLAB pac- kage). The solutions for the flow and the heat and mass transfer characteristics are evaluated numerically for various values of the governing parameters, namely the unsteadiness parameter τ , mixed convection parameter λ , nanofluid buo- yancy ratio parameter Nr, thermophoresis parameter NT, Brownian motion parameter NB, first resistance parameter γ , second resistance parameter Δ , Pran- dtl number Pr and the Schmidt number Sc. The following are brief summary conclusions drawn from the analysis:
1) The thickness of the momentum boundary layer slightly decreases with an increase in the nanofluid buoyancy ratio parameterNr. However, it increases with the increase of all other parameters.
2) The thickness of the thermal boundary layer increases with an increase in both of NT and NB parameters while it slightly increases with an increase inNr parameter.
3) The nanoparticle volume fraction boundary-layer thickness decreases obviously with an increase in both NT andNB parameters and slightly increases with an increase in theNr parameter.
4) The buoyancy parameter λ enhances obviously the momentum boundary- layer thickness and slightly reduce both of the thermal and the nanoparticle volume fraction boundary-layer thicknesses.
5) Both the first and the second resistance parameters γ and Δ enhance the momentum boundary-layer thickness and reduce both of the thermal and the nanoparticle volume fraction boundary-layer thicknesses.
6) The magnitude of the skin-friction coefficient f ″ ( 0 ) decreases with increasing values of the nanofluid buoyancy ratio parameter Nr and increases with the increase of all other parameters.
7) The local Nusselt number decreases with all the nanofluid parametersNT,NB andNr. While, it decreases with increases in the Prandtl number Pr and the Schmidt numberSc.
8) The local Sherwood number increases as theNT parameter increases. However, it decreases as either ofNB,Nr,Pr or Sc increases.
9) The unsteadiness parameter τ and the resistance parameters γ and Δ enhance the local skin-friction coefficient, local Nusselt number and the local Sher- wood number.
The authors would like to thank Institute of Scientific Research and Revival of Islamic Heritage at the Umm Al-Qura University (Project ID No. 43405010) for the financial support.
The authors declare no conflicts of interest regarding the publication of this paper.
Abualnaja, K.M., Elgendy, M.S. and Ibrahim, F.S. (2021) Unsteady Mixed Convection Flow along Sym- metric Wedge with Variable Surface Temperature Embedded in a Porous Medium Saturated with a Nanofluid. Journal of Applied Mathematics and Physics, 9, 101-126. https://doi.org/10.4236/jamp.2021.91008