Nomenclature

$\tilde a$amplitude of wavy surface;	Leb bioconvectionLewisnumber;
a,microorganisms concentration difference	Le,Lewis number;
parameter;	Mn,magnetic field parameter;
C,nanoparticle volume fraction;	N,rescaled density of motile microorgan
D,Brownian diffusion coefficient;	isms;
Dt,thermophoretic diffusion coefficient;	Nb,Brownian motion parameter;
Dn,diffusivity of microorganisms;	Nr,buoyancy ratio;
f,dimensionless stream function;	Nt,thermophoresis parameter;
g,acceleration due to gravity;	Nnx,local density number of motile microor
Gr,Grashofnumber;	ganisms;
k,thermal conductivity;	Nux,local Nusselt number;
K,permeability of porous medium;	P,pressure;
${\bar K}$,material parameter;	Peb,bioconvectionPéclet number;
L,characteristic length of wavy surface;	qm,surface mass flux;
qn,surface motile microorganisms flux;	T,temperature;
qw,surface heat flux;	W,constant maximum cell swimming speed;
Ra,Rayleigh number;	(u,v),velocity components of fluid;
Rab,bioconvection Rayleigh number;	(x,y),coordinate axes.
Shx,local Sherwood number;

Greek symbols

α,thermal diffusivity of porous media;	ρf ,density of fluid;
ϕ,dimensionless nanoparticle volume frac	ρp,nanoparticle mass density;
tion;	j,ratio between effective heat capacity of
β,volumetric expansion coefficient;	nanoparticle material and heat capacity
θ,dimensionless temperature;	of fluid;
(ξ,η),non-similarity variables;	μ,dynamic viscosity;
(ρc)p,effective heat capacity of nanoparticle	γ,average volume of microorganisms;
material;	σ,wavysurface;
(ρc)f ,heat capacity of fluid;	ψ,stream function.

Subscripts

w,condition at surface;	∞,condition in free stream.
f,fluid;

1 Introduction

During the past several decades,the natural convection flow and heat transfer have received the attention of researchers owing to their many applications in mechanical,chemical,and civil engineering. These applications include,for example,fibrous insulation,food processing and storage,thermal insulation,geophysical systems,electrochemistry,metallurgy,the underground disposal of nuclear or non-nuclear waste,the cooling system of electronic devices,etc^[1]. The books written by Nield and Bejan^[2],Pop and Ingham^[3],Ingham and Pop^[4],Vafai^{[5, 6]}, and Vadasz^[7] provide a good literature survey for this topic. In addition,the study of heat transfer near irregular surfaces is of fundamental importance because it is often found in many industrial applications. Many researchers are interested in the natural convection flow near the irregular surfaces saturated porous media. Cheng^[8] studied the boundary layer flow and heat transfer near an inclined wavy plate in a bidisperse porous medium with the uniform wall temperature. The author found that the increase in the modified thermal conductivity ratio and the permeability ratio can enhance the natural convection heat transfer of the inclined plate in the bidisperse porous media. The problem of two-dimensional steady-state film condensation on an isothermal finite-size horizontal wavy plate embedded in a porous medium for the case in which the plate faces upward into a region of dry saturated vapor was investigated by Chang^[9]. The results showed that the inclusion of capillary effects in the liquid film analysis has a significant effect on the computed results for the heat transfer coefficient. Elshehawey et al.^[10] used the Brinkmain model to discuss the boundary layer flow of an incompressible viscous fluid moving through a porous medium between two inclined wavy porous plates under the effects of a constant inclined magnetic field that makes an angle with the vertical axis and constant suction/injection. Cheng^[11] studied the effects of spatially wavy surface and fluid inertia on the natural convection heat and mass transfer near a vertical wavy surface embedded in a non-Darcy fluid-saturated porous medium. He observed that the rates of heat and mass transfer tend to decrease as the modified Grashof number increases,while the higher amplitudewavelength ratio decreases the average Nusselt and Sherwood numbers. Mahdy^[12] studied the combined effect of spatially stationary surface waves and the presence of fluid inertia on the free convection along a heated vertical wavy surface embedded in an electrically conducting fluid saturated porous medium subject to the diffusion-thermo (Dufour),thermo-diffusion (Soret),and magnetic field effects. a very good review of convective heat transfer and fluid flow in porous media with nanofluid can be found from Mahdi et al.^[13]. In this review,the porous media characteristics and properties of the nanofluids were mentioned in details. Moreover, studies on convective heat transfer and fluid flow in porous media with nanofluid were reported for the natural,forced,and mixed convection with different geometries containing the wavy case.

The macroscopic convection flow of fluid caused by the density gradient created by collective swimming of motile microorganisms is known as the bioconvection flow. These self-propelled motile microorganisms increase the density of the base fluid by swimming in a particular direction, thus causing bioconvection. Khan et al.^[14] used the Oberbeck-Boussinesq approximation and similarity transformation to study the magnetohydrodynamic (MHD) boundary layer flow, heat and mass transfer of a water-based nanofluid containing gyrotactic microorganisms over a vertical plate in the presence of the Navier slip effect. They found that the magnetic,buoyancy, nanofluid,and bioconvection parameters reduce the Nusselt,Sherwood,and density numbers of microorganisms. Xu and Pop^[15] discussed analytically the mixed convection flow and heat transfer over a stretching surface with uniform free stream in the presence of both nanoparticles and gyrotactic microorganisms. Their results showed that the local Nusselt number reduces almost linearly as the thermophoresis parameter increases. The onset of bioconvection in a horizontal layer filled with a nanofluid that also contains gyrotactic microorganisms was presented by Kuznetsov^[16]. a linear instability analysis and a Galerkin method were used to obtain an analytical solution for the critical Rayleigh number for the non-oscillatory situation. Xu and Pop^[17] used the model proposed by Kuznetsov and Nield^[18] to discuss the mixed bioconvection flow in a horizontal channel filled with a nanofluid that contains both nanoparticles and gyrotactic microorganisms. They observed that the increase of the Lewis number and the thermophoresis parameter helps to enhance the temperature distributions for the positive Reynolds numbers,but it results in the decrease of temperature profiles for the negative Reynolds number. Mahdy^[19] studied the same problem for a vertical cone in porous media,where bioconvection takes place in a peristaltic flow in an asymmetric channel filled by nanofluid containing gyrotactic microorganism. The bio-nano-engineering model was reported by Akbar^[20]. Moreover, Raees et al.^[21] presented the case of mixed convection in gravity-driven nano-liquid film containing both nanoparticles and gyrotactic microorganisms. An excellent review for the current problem can be found in Refs. ^{[22, 23, 24, 25]},in which the cases of the stretching sheet,the square porous cavity,the non-Newtonian nanofluids in porous media,and a permeable vertical plate, were presented,respectively. More studies can be found in Refs. ^{[26, 27, 28, 29]}.

Mahdy and Ahmed^[30] performed a numerical analysis to examine laminar free convective of a nanofluid along a vertical wavy surface saturated porous medium. They found that the heat and mass transfer rates decrease by increasing either the buoyancy ratio number or the thermophoresis parameter. However,they neglected the effects of inertia as well as the presence of both nanoparticles and gyrotactic microorganisms. Therefore,the main objective of the present study is to investigate the MHD boundary layer flow and heat transfer over a vertical wavy surface saturated isotropic porous medium in the presence of both nanoparticles and gyrotactic microorganisms. The partial differential equations governing the problem are,firstly, transformed into a vertical flat plate,and the non-similar solutions are used to solve the resulting system. The obtained numerical results are presented in terms of the local Nusselt number,the local Sherwood number,the local density number of the motile microorganisms,the velocity profiles,the temperature distributions,and the rescaled density of the motile microorganisms.

2 Problem analysis

Let us consider a two-dimensional steady free convection boundary layer flow over a vertical wavy plate placed in a non-Darcian nanofluid saturated porous medium containing gyrotactic microorganisms subject to a magnetic field. The coordinate system is chosen such that the ${\tilde x}$-axis is along the wavy surface whereas the ${\tilde y}$-axis is normal outward,as shown in Fig. 1. The wavy surface profile is chosen to be

where ${\tilde a}$ is the amplitude of the wavy surface,and L is the characteristic length of the wavy surface. At this boundary (i.e.,${\tilde y}$ = ${\tilde σ}$),the fluid temperature ${\tilde T}$ ,the nanoparticle fraction ${\tilde C}$,and the density of motile microorganisms ${\tilde N}$ have a fixed value and are denoted by ${\tilde T}$_w,${\tilde C}$_w,and ${\tilde N}$_w,respectively. The ambient values,attained as ${\tilde y}$ tends to infinity,of ${\tilde T}$,${\tilde C}$,and ${\tilde N}$ are denoted by${\tilde T}$_∞,${\tilde C}$_∞,and ${\tilde N}$_∞,respectively. Homogeneity and the local thermal equilibrium in the porous medium are assumed. In addition,the nanoparticle suspension is assumed to be stable (there is no nanoparticle agglomeration). The presence of nanoparticles is assumed to have no effect on the direction in which microorganisms swim and their swimming velocities. This is a reasonable assumption if the nanoparticle suspension is dilute (nanoparticle concentration is less than 1%). Bioconvection induced flow only takes place in a dilute suspension of nanoparticles. Otherwise, a large concentration of nanoparticles results in a large suspension viscosity,which suppresses bioconvection. Moreover,the Oberbeck-Boussinesq approximation is valid,and the nanoparticle concentration is dilute. Invoking these assumptions,the governing partial differential equations can be written in the usual dimensional forms of

In the previous equations,${\tilde u}$ and ${\tilde v}$ are the velocity components along the ${\tilde x}$- and ${\tilde y}$-axes,μ is the dynamic viscosity of the fluid,α and β are the thermal diffusivity and the volumetric expansion coefficient,respectively,ρ_f is the density of the base fluid,ρ_p is the density of nanoparticles, ρ_m∞ is the microorganism density,and γ is the average volume of microorganisms. W is the constant maximum cell swimming speed,D,D_t,and D_n are the Brownian,thermophoretic diffusion,and diffusivity of microorganisms coefficients,respectively,and j = ∈(ρc)_p/(ρc)_f is the ratio of the effective heat capacity of the nanoparticle material to the heat capacity of the fluid. It is clear that the case of Darcy law is recovered when ${\bar K}$ = 0. K is the permeability of the porous medium,and ${\bar K}$ is a material parameter which may be thought of as a measure of the inertial impedance of the matrix. ${\tilde V}$ = (${\tilde u}$,${\tilde v}$) is the velocity flux vector. In addition,both the permeability K and the material parameter ${\bar K}$ can be determined from the widely-known correlations proposed by Ergun’s relations^[31],

where d indicates the particle diameter,and ∈ represents the porosity. It is well known that the Darcy flow model breaks down when the inertia resistance becomes comparable with the viscous resistance. Thus,the term (${\bar K}$/ν)$\sqrt {{{\tilde u}^2} + {{\tilde v}^2}} $ must be added.

Now,the dimensional boundary conditions for the problem are

In order to obtain the non-dimensional form of the previous equations,we define the following transformations: The stream function ${\tilde ψ}$ is defined as usual (${\tilde u}$ =${{\partial \tilde \varphi } \over {\partial \tilde y}}$and ${\tilde v}$ =$ - {{\partial \tilde \varphi } \over {\partial \tilde x}}$). Using the above transformation, Eqs. (3)-(7) and Eq. (10) turn into the following forms: Here,in Eqs. (11)-(14),${\hat V}$ = (${\left( {{{\partial \hat \varphi } \over {\partial \hat x}}} \right)^2} + {\left( {{{\partial \hat \varphi } \over {\partial \hat y}}} \right)^2}$)^1/2 is the non-dimensional velocity,Gr is the Grashof number,M_n is the magnetic field parameter,Ra is the thermal Rayleigh number,Ra_b is the bioconvection Rayleigh number,N_r is the buoyancy ratio parameter,N_t is a modified diffusivity ratio parameter (somewhat similar to the Soret parameter that arises in the crossdiffusion phenomena in solutions),N_b is the Brownian motion parameter,the parameter Le is the traditional Lewis number (the ratio of the Schmidt number to the Prandtl number Pr), Le_b is the bioconvection Lewis number,Pe_b is the bioconvection Péclet number,and a is the bioconvection constant. Furthermore,these dimensionless parameters are defined as The effect of the wavy surface can be transferred from the boundary conditions into the governing equations by means of the coordinate transformation given by Applying the previous transformations into Eqs. (11)-(14) with the consideration that the Darcy-Rayleigh number is very large (Ra → ∞) in the boundary-layer regime,we obtain the governing equations as follows:

Now,Eqs. (16)-(19) may be reduced to a form more convenient for the numerical solution by considering the following transformations:

Therefore,we get the governing boundary layer equations as follows: subject to the dimensionless boundary conditions The primes denote partial derivative with respect to η. The results of practical interest in many applications are the local Nusselt number Nu_x,the local Sherwood number Sh_x,and the local density number of the motile microorganisms Nn_x which are defined as where ${\tilde q}$_w,${\tilde q}$_m,and ${\tilde q}$_n are the wall heat,the wall mass,and the wall motile microorganisms fluxes,respectively,and are defined as and n = (-$\dot \sigma /\sqrt {1 + {{\dot \sigma }^2}} ,1/\sqrt {1 + {{\dot \sigma }^2}} $) is the unit vector normal to the wavy plate. Using the previous transformations,we get the local Nusselt and Sherwood numbers and the density of the motile microorganisms from the following expressions:

3 Numericalsection

In this section,the governing equations (21)-(24) together with the boundary conditions (25) are solved numerically using the fourth-order Runge-Kutta method. In this technique, the solutions are divided to steps,namely,the first level of truncation and the second level truncation. For the first level of truncation,the derivatives with respect to ξ in Eqs. (21)-(24) are neglected. Thus,Eqs. (21)-(24) can be written as

subject to the following boundary conditions:

In the second step,all the neglected terms in the first truncation level are restored,and the following new variables are introduced:

Thus,the governing equations are

subject to the boundary conditions Here,the new unknown variables F₁,Θ,Φ,and χ₁ need four additional equations with the appropriate boundary conditions to be evaluated. These can be obtained by differentiating Eqs. (37)-(40) with respect to ξ and neglecting the terms ${{\partial {F_1}} \over {\partial \xi }}$,${{\partial {\chi _1}} \over {\partial \xi }}$,${{\partial {F_1}} \over {\partial \xi }}$,and ${{\partial {\Phi _{}}} \over {\partial \xi }}$. The resulting equation together with Eqs. (31)-(34) and Eqs. (37)-(40) is solved numerically using the Runge-Kutta method with a shooting technique. The step size Δη = 0.05 is used to obtain the numerical solution with five-decimal place accuracy as the criterion of convergence. Additional details for the present method can be found in Ref. ^[32]. Besides,Table 1 shows a comparison of the maximum relative error defined by ε = $\left[{{{N{u_{est}} - Nu} \over {Nu}}} \right]$applicable for each in [0,0.5] for a vertical flat plate at ξ = 0,M_n = 0,Gr = 0,and Ra_b = 0,where Nu_estRa^-1/2 = 0.444 + C_rN_r +C_bN_b +C_tN_t,in which C_r,C_b,and C_t are the coefficients in the linear regression. It is observed that there is excellent agreement found between the present results and those obtained by Nield and Kuznetsov^[33].

4 Results and discussion

The obtained results are presented in terms of the local Nusselt number,the local Sherwood number,and the local density number of the motile microorganisms for wide ranges of the governing parameter. In the presented study,the Grashof number Gr is considered to vary from 0 to 10,the magnetic field parameterM_nvaries from 0 to 5,the thermophoresis parameter N_t varies from 0.1 to 1.2,the Brownian motion parameter N_b varies from 0.5 to 1.0,the buoyancy ratio varies from 0.1 to 0.4,and the bioconvection Rayleigh number varies from 0 to 0.4.

Figures 2-4 show the effect of the Grashof number Gr on the local Nusselt Nu_x,the local Sherwood number Sh_x,and the local density number of the motile microorganisms Nn_x, respectively,for different values of the bioconvection Rayleigh number Ra_b. It is observed that the increase in the Grashof number Gr causes a reduction in the profiles of Nu_x,Sh_x, and Nn_x. This is due to the fact that the increase in Gr reduces the dimensionless velocity and increases the dimensionless temperature,the dimensionless concentration,and the rescaled density of motile microorganisms within the boundary layer and consequently decreases Nu_x, Sh_x,and Nn_x. In addition,an increase in the bioconvection Rayleigh number Ra_b results in a clear reduction in the thermal boundary layer thickness,the concentration boundary layer thickness,and the motile microorganisms boundary layer thickness,and as a result,Nu_x,Sh_x, and Nn_x decrease as Ra_b increases. This is due to the fact that Ra_b reduces the dimensionless velocity and increases the dimensionless temperature,the dimensionless concentration,and the dimensionless rescaled density of motile microorganisms,and as a result,Nu_x,Sh_x,and Nn_x decrease with an increase in the bioconvection Rayleigh number.

The effects of the magnetic field parameterM_non the local Nusselt number Nu_x,the local Sherwood number Sh_x,and the local density number of the motile microorganisms Nn_x are displayed in Figs. 5-7,respectively. The results show that asM_nincreases,Nu_x,Sh_x,and Nn_x increase. The reason for these behaviors is due to an additional resistance induced by the magnetic field that increases the thermal boundary layer thickness and consequently increases Nu_x,Sh_x,and Nn_x.

Figure 8 shows the effect of the thermophoresis parameter on the profiles of the local Nusselt number for different values of the bioconvection Rayleigh number. The figure shows a clear reduction in the local Nusselt number obtained by increasing Nt. This can be attributed to the temperature difference in the boundary layer region which increases by increasing N_t that leads to increasing the temperature distributions and decreasing the thermal boundary layer thickness. On the contrary,the increase in N_t results in an increase in the local Sherwood number and the local density number of the motile microorganisms. This behavior can be noted from Figs. 9 and 10 which display the effects of N_t on Sh_x and Nn_x.

Figure 11 depicts the effect of the Brownian motion parameter on the profiles of the local Nusselt number for different values of the bioconvection Rayleigh number. It is clear that Nu_x takes the higher values at small values of N_b. The reason for that is due to the viscosity of the nanofluid that increases as N_b increases which in turn decreases the velocity profiles and decreases the thermal boundary layer thickness. Figure 12 andFigure 13 show the opposite behavior for the local Sherwood number Sh_x and the local density number of the motile microorganisms Nn_x. As it can be seen from these figures,the increase in N_b results in an increase in the concentration boundary layer thickness and the rescaled density of motile microorganisms boundary layer thickness that increases Sh_x and Nn_x.

Figures 14-16 display the profiles of the local Nusselt number Nu_x,the local Sherwood number Sh_x,and the local density number of the motile microorganisms Nn_x under the effect of the buoyancy ratio N_r. Like the effect of Gr,the increase in N_r causes a significantly reduction in Nu_x,Sh_x,and Nn_x. In fact,the increase in N_r declares the fluid motion which increases the temperature distributions,the concentration distribution,and the distributions of the rescaled density of motile microorganisms,and as a result,the gradients of temperature, concentration,and rescaled density of motile microorganisms decrease. Consequently,Nu_x, Sh_x,and Nn_x decrease. All these behaviors are presented with the referenced case of M_n = 1, a = 0.2,Le = 6,N_r = 0.3,N_t = N_b = 0.6,Le_b = 5,Pe_b = 0.4,Ra_b = 0.3,and a = 0.1.

5 Conclusions

The problem of laminar MHD non-Darcian free convection from a vertical wavy surface in porous media saturated by nanofluid containing gyrotactic microorganisms is investigated in the current paper. a non-similar solution and the fourth-order Runge-Kutta method are used to solve the partial differential equations governing the problem. It is found that the increase in the Grashof number reduces the local Nusselt number,the local Sherwood number, and the local density number of the motile microorganisms. In addition,as the magnetic field parameter increases,the local Nusselt number,the local Sherwood number,and the local density number of the motile microorganisms decrease. The increase in the bioconvection parameter leads to decreasing the dimensionless velocity,the local Nusselt number,the local Sherwood number,and the local density number of the motile microorganisms. a clear reduction in the local Nusselt number is obtained by increasing the thermophoresis parameter or the Brownian motion parameter,while the local Sherwood number and the local density number of the motile microorganisms increase by increasing the thermophoresis parameter or the Brownian motion parameter. As the buoyancy ratio increases,the local Nusselt number,the local Sherwood number,and the local density number of the motile microorganisms decrease.