Currently, nanofluid is a hot area of interest for researchers. The nanofluid with bioconvection phenomenon attracted the researchers owing to its numerous applications in the field of nanotechnology, microbiology, nuclear science, heat storage devices, biosensors, biotechnology, hydrogen bomb, engine of motors, cancer treatment, the atomic reactor, cooling of devices, and in many more. This article presents the bioconvection cross-diffusion effects on the magnetohydrodynamic flow of nanofluids on three different geometries (cone, wedge, and plate) with mixed convection. The temperature-dependent thermal conductivity, thermal diffusivity, and Arrhenius activation energy applications are considered on the fluid flow with melting phenomenon. The flow is analyzed under thermal and solutal Robin's conditions. The problem is formulated in the mathematical formulation of partial differential equations (PDEs). The similarity transformations are applied to diminish the governing non-linear coupled boundary value problems into higher-order non-linear ordinary differential equations (ODEs). The resulting expressions/equation numerically tackled utilizing the famous bvp4c package by MATLAB for various interesting parameters. The results were physically and numerically calculated through graphics and tables for the velocity field, energy distribution, nanoparticles concentration, and microorganisms profile for numerous parameters. From the obtained results, we discern that the transfer of heat and mass coefficient is high over a plate and cone in the flow, respectively. The velocity profile is reduced via a larger magnetic parameter. Temperature-dependent thermal conductivity enhances the thermal field. Larger thermophoresis enhanced the concentration of nanoparticles. The microorganisms' Biot number boosts the microorganism’s profile.

Magnetohydrodynamics (MHD) has a vital role in several flow phenomena, including industry. The MHD is applied in various fields, including multidisciplinary technological areas like biochemical manufacturing heat transfer, automotive sector, ceramic technology, aerodynamic performance, metallurgical technology, mental operating techniques, and fluid dynamics.
Magnetohydrodynamic initiation changes the flow field’s preferred direction by
fluctuating boundary layer configuration flow. Electromagnetic fields play an
important role in convection mechanisms like metallic casting, material processing
fields, and nuclear reactor control plants. Zheng et al. [

Several investigators studied heat transfer due to its various applications in
applied technologies, manufacturing, and refrigeration. Thermally conductive fluid
has subsequently become a common research topic for scientists. Nanotechnology
is becoming a subject of significant interest in manufacturing and engineering for
investigators as its requirement grows in industrial sectors. Non-Newtonian fluids
are now a big subject for scientists and physiologists [

Bioconvection occurs because the microorganisms are denser than water;
therefore, a motile gyrotactic microorganism moves upwardly in water. The
production of the base fluid increased as the dynamically self-propelled motile
gyrotactic microorganisms moved in a particular direction. Bioconvection is used
in various areas, including fertilizer, biofuel, and manufacturing systems.
Nanofluid bioconvection is considered for density stratified as well as patron
formation due to nanomaterials apprentice, buoyancy forces, including
microorganisms. Kuznetsov [

Furthermore, affiliation with a related domain that is closely suitable in real
life has a significant impact. Because of their significance in multiple fields of
technology agriculture, including manufacturing applications, the flows generated
through a wedge, cone, and vertical plate have gotten a lot of attention. Vajravelu et
al. [

Whereas mass and heat transport occurs continuously in a moveable fluid.
Compared to the Dufour effect, the Soret effect is negligible in fluids. Numerous
researchers have reported cross-diffusion influences in boundary layer flow in
recent decades. Abd El-Aziz [

MATLAB is an essential software tool for numerically solving and comparing
the outcomes of various models, including physics, technology, and a variety of
others. The variable flowing and thermal characteristics when a decelerating
rotating disc is modified scrutinized by Rafiq et al. [

The study of bioconvection in nanofluid is beneficial in different fields like biofuels cells, biotechnology, bioengineering, oil refineries, biofuels, and agriculture engineering. To the author’s best knowledge, no studies have yet reported analyzing bioconvections for the heat and mass transfer of the MHD flow with cross-diffusion impacts by considering three different geometries by considering the thermal and solutal Robin’s conditions. So, we filled this gap with the assistance of the above-cited studies. The prime motive of this study is to investigate the consequences of the occurrence of motile microorganisms on nanofluid flow with thermal radiation, activation energy, and melting phenomena past the geometries cone, wedge, and the plate. Appropriate similarity transformations are used to convert the PDEs into ODEs, and these equations are solved by numerical technique through shooting methods using bvp4c solver. Outcomes are exposed graphically and conferred numerical values for various physical parameters.

Assume an incompressible, convective, time independent, laminar MHD flow over geometries of three different shapes like cone, wedge and plate. The melting mechanism is measured. In the plane

For the above considered problem the basic coupled nonlinear dimensional equations are given below [

The applied boundaries conditions are [

If

If

If

Introduced the suitable similarities variables

Transformation equations are as follows:

The dimension-less conditions are as follows:

In above equations

The friction factor

In this portion the nonlinear dimensionless ordinary differential equations with appropriate boundary conditions are solved numerically through the solver bvp4c package in computational MATLAB. The higher order ODEs are converted into linear one by introducing the following variables such as

With

In this section all the results of graphs are embellished and created by using the computational software MATLAB–16a (bvp4c) solver. For the understanding of figures, we must have to know about the color of curves demonstrated. The black curve demonstrates for wedge, red for cone and blue for sheet. The iterations of effective goods parameters increased or decreased with their individual behaviors. In this segment the graphic outcome interoperate for the different values of different parameters, like

The influence of buoyancy ratio parameter

Heat and mass transformation of the nanofluid bioconvection flow under the melting mechanisms is scrutinized in the existence of cross-diffusion impacts. Mathematical modeling is fixed to scrutinize the flowing performance on vertically
three different geometries (cone, wedge, and plate). Some significant remarks are synopsized below:

Fluid exposure, the fluid velocity is highest for the flow over the plate as compared with the other two geometries (cone and wedge).

The transformation rate of mass and heat is much better for the flow via cone than flow over the wedge.

It is investigated that both the Brownian motion diffusion parameter and thermophoretic diffusion parameter are directly proportional to the temperature of the fluid. At the same time, concentration is directly proportional to the thermophoretic parameter but inversely proportional to the Brownian diffusion parameter.

The temperature and concentration profile performance increased for the highest value in the thermophoretic parameter.

The microorganisms’ density is inversely proportional to the Peclet number and Lewis number.

This study analyzed the flow characteristics of non-Newtonian fluids extended for the present mathematical model.

Reducing the amount of melting enhanced the heat transfer rate and velocity of the fluid.

Concentration of Nanomaterials, mol L^{–1}

Peclet Number

Half Angle of Cone, Radian

Full Angle of Wedge, Radian

Brownian Diffusion Coefficient, m^{2} s^{–1}

Real Constants

Thermophoresis Diffusion Coefficient, m^{2} s^{–1}

Grashof Number

Cell Swimming Speed, m s^{–1}

Thermal Biot Number

Stream Velocity Function,

Energy, Concentration, and Bioconvection Parameters

Concentration of Microorganisms, m^{–3}

Volumetric Thermal Expansion Coefficients, 1/Kelvin

Volumetric Concentration Expansion Coefficients 1/mol L^{–1}

Volumetric Microorganisms Expansion Coefficients, 1/m^{3}

Microorganisms Coefficient, m^{2} s^{–1}

Activation Energy, J

Magnetic Parameter

Solutal Biot Number

Mixed Convection Parameter

Microorganism Biot Number

Temperature Dependent Thermal Conductivity

Temperature of Nanoliquid, K

Radiation Parameter

Velocity Component, m s^{–1}

Prandtl Number

Coordinates System, m

Bioconvection Rayleigh Number

_{f}

Convective Heat Transfer Coefficient, Wm^{2} K^{–1}

Thermophoresis Parameter

Concentration Diffusivity

Melting Parameter

Brownian Motion Parameter

Variable Thermal Conductivity

Lewis Number

Variable Mass Diffusivity

Temperature Difference Parameter

Microorganisms Difference Parameter

Chemical Reaction Parameter

Skin Friction Coefficient

Melting Parameter

Local Density Number of Microorganisms

Local Nusselt Number

_{h}

Local Sherwood Number

Gravitation Force,

Characteristic Length

Radius of Cone, m

Heat Capacitance, Jm^{–3} K^{–1}

Density of Nanoliquid, kg m^{–3}

Wall

Fluid

Ambient

Free–Stream