Consider the boundary layer flow past a stretchable wedge. The wedge is assumed to be moving with a velocity u
w
(x) in a water-based nanofluid saturated by gyrotactic microorganisms. The free stream velocity is taken to be u
e
(x). Further, there is no nanoparticle agglomeration, the effect of nanoparticles on the swimming direction of microorganisms and on the velocity of swimming of microorganisms. The assumption to be valid, we assume that the suspension of nanoparticles is dilute. To formulate the flow phenomena, we have considered a Cartesian coordinate system. The coordinates along the surface and normal to it, are denoted by x and y, respectively (see Fig. 1). A uniform magnetic field is applied parallel to the y-axis. The induced magnetic field is assumed to be negligible. The viscous dissipation and joule heating effects are also taken into consideration while modeling the energy equation. At the surface of the wedge, a constant suction or injection is imposed. Under the aforesaid assumptions, and using the scale analysis of Buongiorno (2006) and Kuznetsov (2010), the boundary layer equations governing the flow can be written as follows:
$$ \frac{{\partial\check{u}}}{{\partial\check{x}}} + \frac{{\partial\check{v}}}{{\partial\check{y}}} = 0, $$
(1)
$$ \check{u} \frac{{\partial \check{u} }}{{\partial \check{x} }} + \check{v} \frac{{\partial \check{u} }}{{\partial \check{y} }} = u_{e} \frac{{du_{e} }}{{d\check{x} }} + \upsilon \frac{{\partial^{2} \check{u} }}{{\partial \check{y}^{2} }} - \frac{{\sigma B_{0}^{2} }}{\rho }\left( {\check{u} - u_{e} } \right), $$
(2)
$$ \check{u} \frac{{\partial \check{T} }}{{\partial \check{x} }} + v\frac{{\partial \check{T} }}{{\partial \check{y} }} = \alpha \frac{{\partial^{2} \check{T} }}{{\partial \check{y}^{2} }} + \tau \left[ {D_{B} \frac{{\partial \check{C} }}{{\partial \check{y} }}\frac{{\partial \check{T} }}{{\partial \check{y} }} + \left( {\frac{{D_{T} }}{{T_{\infty } }}} \right)\left( {\frac{{\partial \check{T} }}{{\partial \check{y} }}} \right)^{2} } \right] + \frac{\mu }{{\left( {\rho C_{p} } \right)_{f} }}\left( {\frac{{\partial \check{u} }}{{\partial \check{y} }}} \right)^{2} + \frac{{\sigma B_{0}^{2} }}{{\left( {\rho C_{p} } \right)_{f} }}\left( {u - u_{e} } \right) ^{2} , $$
(3)
$$ \check{u} \frac{{\partial \check{C} }}{{\partial \check{x} }} + \check{v} \frac{{\partial \check{C} }}{{\partial \check{y} }} = D_{B} \frac{{\partial^{2} \check{C} }}{{\partial \check{y}^{2} }} + \left( {\frac{{D_{T} }}{{T_{\infty } }}} \right)\frac{{\partial^{2} \check{T} }}{{\partial \check{y}^{2} }}, $$
(4)
$$ \check{u} \frac{{\partial \check{n} }}{{\partial \check{x} }} + \check{v} \frac{{\partial \check{n} }}{{\partial \check{y} }}+ \frac{{\partial (\check{n}\overbrace{v})}} {{\partial \check{y} }} = D_{n} \frac{{\partial^{2} \check{n} }}{{\partial \check{y}^{2} }}.$$
(5)
In above equations, \( \check{u} \) and \( \check{v} \) are the components of velocity in \( \check{x} \) and \( \check{y} \) directions, respectively. \( \check{T} \), is the temperature of the fluid, ρis the density of nanofluid, \( \mu \) is the viscosity of nanofluid and \( \alpha \) is the thermal diffusivity of nanofluid. Moreover, \( \tau = \frac{{\left( {\rho C} \right)_{p} }}{{\left( {\rho C} \right)_{f} }} ; \) where C is the volumetric expansion coefficient and \( \rho_{p} \) the density of the particles. Furthermore, \( \check{n} \) is the density of the motile microorganisms, \( \overbrace{v}=\left( {\frac{(bWc)}{\Delta C}}\right)\nabla C \), is the velocity vector representing the cell swimming in nanofluids, Dn is the diffusivity of microorganisms, b is the chemotaxis constant [m] and Wc is the maximum cell swimming speed [m/s].
The boundary conditions for the problem are:
$$ \check{u} = u_{w} \left( x \right),\quad\check{v} = v_{0} ,\quad\check{T} = T_{w} , \quad D_{B} \frac{{d\check{C} }}{{d\check{y} }} + \frac{{D_{T} }}{{T_{\infty } }} \frac{{d\check{T} }}{{d\check{y} }} = 0,\quad \check{n} = n_{w},\quad {\text{at}}\;\check{y} = 0, $$
$$ \check{u} = u_{e} \left( x \right),\quad \check{T} = T_{\infty },\quad C = C_{\infty },\quad \check{n} = n_{\infty } \quad{\text{as}}\;y \to \infty , $$
(6)
For a mathematical analysis of the problem, we assume that u
w
(x) and u
e
(x) have the following form:
$$ u_{w} \left( x \right) = ax^{m} ,\quad u_{e} \left( x \right) = cx^{m} , $$
where, a and c are positive constants; besides, \( m = \frac{\beta }{{\left( {2 - \beta } \right)}}\left( {0 \le m \le 1} \right) \cdot \beta \) here is Hartee pressure gradient parameter which corresponds to β = Ω/2 for a total wedge angle Ω.
We seek a similarity solution for the Eqs. (1)–(4) of the form,
$$ \psi = \left( {\frac{{2U_{e} x\nu }}{1 + m}} \right)^{{\frac{1}{2}}} F\left( \eta \right),\quad \theta \left( \eta \right) = \frac{{T - T_{\infty } }}{{T_{w} - T_{\infty } }},\quad \phi \left( \eta \right) = \frac{{\check{C} - C_{\infty } }}{{C_{\infty } }},\quad \eta = \left( {\frac{{\left( {1 + m} \right)U_{e} }}{2x\nu }} \right)^{{\frac{1}{2}}} y. $$
(7)
\( \psi \) in Eq. (7) is the stream function that can be defined in a usual way. Besides, \( u = \frac{\partial \psi }{\partial x} \) and \( v = - \frac{\partial \psi }{\partial y} \). Using Eq. (5) and the stream function into Eqs. (1)–(4), we get the following system of nonlinear differential equations,
$$ F^{{{\prime \prime \prime }}} + FF^{{\prime \prime }} + \left( {\frac{2m}{m + 1}} \right)\left( {1 - \left( {F^{{\prime }} } \right)^{2} } \right) + M^{2} \left( {1 - \left( {F^{{\prime }} } \right)} \right) = 0, $$
(8)
$$ \theta^{{\prime \prime }} + \Pr f\theta^{{\prime }} + \Pr Nb\phi^{{\prime }} \theta^{{\prime }} + \Pr Nt\theta^{{{\prime }2}} + \Pr Ecf^{{{\prime \prime }2}} + \Pr EcM^{2} \left( {f^{{\prime }} - 1} \right)^{2} = 0, $$
(9)
$$ \phi^{{\prime \prime }} + LePrf\phi^{{\prime }} + \frac{Nt}{Nb}\theta^{{\prime \prime }} = 0. $$
(10)
$$ \chi^{{\prime \prime }} + PrLb\left( {f\chi^{{\prime }} } \right) - Pe\left( {\phi^{{\prime }} \chi^{{\prime }} + \phi^{{\prime \prime }} \left( {\sigma + \chi } \right)} \right) = 0. $$
(11)
The boundary conditions also get transformed to
$$ F\left( 0 \right) = \frac{2}{m + 1}S,\quad F^{{\prime }} \left( 0 \right) = 0,\quad \theta \left( 0 \right) = 1,\quad Nb\phi^{{\prime }} \left( 0 \right) + Nt\theta^{{\prime }} \left( 0 \right) = 0,\quad \chi \left( 0 \right) = 1 $$
(12)
$$ F^{{\prime }} \left( \infty \right) = 1,\quad \theta \left( \infty \right) = 0,\quad \phi \left( \infty \right) = 0, \quad \chi \left( \infty \right) = 0. $$
(13)
In the above equations, m is the pressure gradient parameter and S is the suction/injection parameter. S > 0 shows that there is injection at the wall while S < 0 corresponds to the cases involving suction at the wall. \( M = \frac{{2\sigma B_{0}^{2} }}{{\rho \left( {1 + m} \right)U_{e} \left( x \right)}} \), \( Ec = \frac{{u_{w}^{2} }}{{\left( C \right)_{f} \left( {T_{w} - T_{\infty } } \right)}}, \)
\( Pr = \frac{\nu }{\alpha } \), \( Le = \frac{\alpha }{{D_{B} }} \), \( Nb = \frac{{\left( {\rho C} \right)_{p} D_{B} C_{\infty } }}{{\left( {\rho C} \right)_{f} \nu }} \), \( Nt = \frac{{\left( {\rho C} \right)_{p} D_{T} \left( {T_{w} - T_{\infty } } \right)}}{{\left( {\rho C} \right)_{f} T_{\infty } \nu }} \), \( Lb = \frac{\alpha }{Dn}, \quad Pe = \frac{bWc}{Dm} \) and \( \sigma = \frac{{n_{\infty } }}{{\Delta n_{w} }}. \) represent magnetic number, Prandtl number, Lewis number, Brownian motion parameter, thermophoresis parameter, bioconvection Lewis number, the bioconvection Pecket number and the bioconvection constant, respectively.
Some physical quantities of interest are the skin friction coefficient and Nusselt number. They are defined respectively as:
$$ Re_{x}^{{\frac{1}{2}}} C_{f} = \sqrt {\frac{m + 1}{2}} F^{{\prime \prime }} \left( 0 \right), $$
and
$$ Re_{x}^{ - 1/2} Nu = \sqrt {\frac{m + 1}{2}} \theta^{{\prime }} \left( 0 \right). $$
It is pertinent to mention here that the Sherwood number for the passive control model becomes identically zero, i.e.
$$ Re_{x}^{ - 1/2} Sh = \sqrt {\frac{m + 1}{2}} \phi^{{\prime }} \left( 0 \right) = 0, $$
The microorganism density number in dimensionless form will reduce to:
$$ Re_{x}^{ - 1/2} Nn = \sqrt {\frac{m + 1}{2}} \chi^{{\prime }} \left( 0 \right). $$
Here, \( Re_{x} = \frac{{u_{e} x}}{\nu } \) is the local Reynold number.
