The basic idea of dispersing particles into liquids can be traced back to the published theoretical work of Maxwell [1]. The particles that had dimensions in the micrometer and millimeter range were considered in the study. However, particles in the order of millimeters or even micrometers cause problems of rapid settling, clogging, and abrasion [2]. The rapid development of nanotechnology makes the generation of nanometer–sized particles possible and eliminates problems arising from micrometer–sized particles. Nanofluids, first proposed by Choi [3], are a new class of heat transfer fluids that are superior to conventional micrometer–sized particles based fluids in terms of thermo–physical properties. The nanoparticles used in creating nanofluids include metals (gold, silver, copper), oxides (alumina, zirconia, silica, titania), and some other compounds (carbon nanotubes, silicon carbide, graphene). Buongiorno [4] developed an analytical model to explain the convective heat transfer in nanofluids. He considered seven slip mechanisms and concluded that Brownian diffusion and thermophoresis are the two most important slip mechanisms in nanofluids. Aly et al. [5] studied the effects of thermal radiation and suction/injection on the heat transfer flow of water based nanofluid past a vertical plate, and reported that a constant temperature was maintained on the plate. Moshizi et al. [6] considered mixed convection of Al₂O₃–water nanofluid inside a vertical micro–annulus with two different types of heat fluxes imposed at the walls. They noted that both temperature and concentration dependent buoyancy forces affect flow fields and nanoparticle migration. Sheikholeslami et al. [7] numerically investigated the magnetohydrodynamic (MHD) flow of CuO–water nanofluid flow in a porous semi annulus with constant heat flux. Sheikholeslami [8] numerically investigated the CuO–water nanofluid inside a porous channel in the presence of a magnetic field. They reported a significant enhancement in heat transfer rate with a corresponding rise in the Hartmann number. Kanimozhi et al. [9] evaluated the impact of coupled buoyancy and thermocapillary driven convection in a cylindrical porous annulus saturated with water based Ag–MgO hybrid nanofluid along with viscous dissipation effects. At varying levels of random motion of nanoparticles, a higher volume percentage resulted in a decreased heat transmission rate. Several researchers have extended heat transfer flow problems from Newtonian fluids to nanofluids using different nanoparticles such as Cu–water [10]; CNTs–water [11]; Al₂O₃–water/ethylene glycol [12]; Fe₃O₄–water [13]; TiO₂–water [14]; MWCNTs–water [15].

Literature review revealed that most articles on nanofluids mainly focus on the impacts of nanoparticle dimensions, types, dispersibility of nanoparticles, and the base fluids. Some recent work suggests that the shapes of nanoparticles have a significant impact on the thermo–physical properties of nanofluids. Elias et al. [16] examined the effects of five types of nanoparticles shape (such as cylindrical, bricks, blades, spherical, and platelets) based nanofluids to improve the performance of a shell and tube heat exchanger. Boehmite alumina (γ–AlOOH) nanoparticles of different shapes were dispersed in ethylene glycol or water based nanofluids. They determined that cylindrically shaped nanoparticles showed a better heat transfer rate and higher entropy generation. Jeong et al. [17] investigated the viscosity and thermal conductivity of ZnO nanofluids with nanoparticle shapes of nearly rectangular and a sphere, under various volume concentrations of the nanoparticles, ranging from 0.05 to 5.0 vol.%. They reported that the viscosity and conductivity of ZnO nanofluid with rectangular shape nanoparticles was better than with sphere shape nanoparticles. Vanaki et al. [18] numerically studied the effects of different nanofluids on the thermal and flow fields through transversely wavy wall channels with different phase shifts between the upper and lower wavy walls. They found that the SiO₂–EG nanofluid with platelets shape nanoparticles gave the highest heat transfer enhancement. The shape effects of nano–size particles in Cu–H₂O nanofluid on entropy generation was analyzed by Ellahi et al. [19]. Their study concluded that platelets shape nanoparticles gained maximum temperature than brick and cylinder shape nanoparticles.

Monfared et al. [20] investigated the turbulent convection flow of boehmite–alumina incorporated with various shapes of nanoparticles inside a heat exchanger under entropy generation effects. They revealed that platelet shape nanoparticles led to a stronger frictional entropy generation rate. Dogonchi et al. [21] studied nanoparticle shape effects on water based CuO–nanofluid using CVFEM. The study concluded that platelet shape nanoparticles exhibit a strong heat transfer rate. Shakiba et al. [22] studied the mixed convection of MHD flow of nanofluid inside two vertical annulus under a radial magnetic field with respect to the suction or injection using the analytical solution method. The impact of Brownian motion and shape factor on the thermal state of CuO–water nanofluid was also considered. The highest overall rate of heat transfer was achieved in the case of platelet shape nanoparticles. Sheikholeslami et al. [23] performed a simulation using water based Fe₃O₄–nanofluids. Their results showed that platelet shape nanoparticles should be used to obtain the highest Nusselt number. Hosseinzadeh et al. [24] examined micropolar MHD fluid flow over a vertical plate where three different base fluids including water, ethylene glycol, and ethylene glycol/water (50%–50%) were used. Their findings indicated that for water-based fluids, the temperature profile of lamina–shaped nanoparticles was 38.09% higher than brick–shaped nanoparticles. MHD mixed convection water based MoS₂–GO hybrid nanofluid flow through an upright cylinder with different shapes of nanoparticles was investigated by Chu et al. [25]. Their results showed that the blade–shaped nanoparticles of MoS₂–GO hybrid nanofluid had the maximum temperature. Khashi’ie et al. [26] analyzed the effects of spherical, brick, and blade shape nanoparticles on Cu–Al₂O₃ water based nanofluid flow in respect of the two important of parameters of EMHD and thermal radiation. Their results indicated that the maximum heat transfer enhancement occured in the case of blade shape nanoparticles. Anwar et al. [27] examined the heat transfer rates of different shape nanoparticles in sodium alginate based MoS₂–Co hybrid nanofluid. They used spherical, cylindrical, blade, platelet, and brick shape nanoparticles and found that maximum heat transfer rate occured in the case of blade shape nanoparticles. Shahsavar et al. [28] performed entropy generation analysis using boehmite–alumina nanofluid containing five different nanoparticles shapes (spherical, bricks, blades, cylindrical, and platelet). They reported that spherical shape nanoparticles had the minimum impact on entropy generation. Ganesh et al. [29] performed a numerical experiment through FEM on the convection of heat generating γ Al₂O₃–H₂O nanofluid filled in a square cavity with multiple circular-, square-, and triangular-shaped obstacles. They found a higher heat transfer rate in the cavity with triangular obstacles. Cao et al. [30] remarked that the accompanying influence on the thermophoretic force and the type of the random motion of spherical carbon nanotubes, cylindrical graphene, and platelet alumina nanoparticles affects heat transfer and mass transfer rates. Xiu et al. [31] noted that because there are numerous platelet aluminium, cylindrical magnesium oxide, and titanium dioxide nanoparticles in the flow of water-based ternary hybrid nanofluids, heat transfer is significant even at the start of the fluid flow. Saleem et al. [32] discovered that the friction between the layers of a water–based ternary–hybrid nanofluid and the wall can be reduced by increasing the density of spherical nanoparticles.

In the above work, Newtonian fluids were taken as a base fluid. However, few researchers have considered non–Newtonian based nanofluids in their research. It is well recognized that non–Newtonian fluids are encountered in numerous transport processes such as central heating systems, molten polymers, and nuclear reactors [33]. Sodium alginate natural convection based Fe₂O₃–nanofluid with accretion/ablation was illustrated by Hussanan et al. [34]. The mixed convection ethylene glycol based nanofluids flow, together with the functional shape effects of nanoparticles was investigated by Memon et al. [35]. Hussanan et al. [36] highlighted the impacts of magnetic and non–magnetic nanoparticles on viscoplastic Casson-based mixture hybrid nanofluid. Kalaivanan et al. [37] investigated the influence of Arrhenius activation energy in heat and mass transfer of second–grade nanofluid flow and noted that the thickness of the boundary layer increased with non–dimensional activation energy. Influence of Arrhenius activation energy on the momentum, energy, and mass transport of a second-grade magneto nanofluid flow with elastic deformation effects was discovered by Kalaivanan et al. [38]. Hammachukiattikul et al. [39] discussed the heat transfer flow of viscoplastic Casson-based hybrid nanofluid with additional factors. These factors include radiation, heat source/sinks, and inclined Lorentz force. Ganesh et al. [40] investigated the thermal and hydraulic characteristics of a Casson-based MWCNT nanofluid inside a wavy square enclosure containing a circular–shaped obstacle. Ganesh et al. [41] explored the incompressible, two–dimensional, buoyancy driven convection flow of Casson-based MWCNT nanofluid in a square enclosure with a circular barrier and found that by increasing the MWCNT volume fraction along the wavy bottom wall, Casson parameter, Rayleigh number, and the heat transfer rate increased. To our knowledge, nanoparticle shape effects on non–Newtonian viscoplastic sodium alginate (SA) based Cu–nanofluid has not been explored yet. The purpose of this article is three–fold. Firstly, to formulate the SA based Cu–nanofluid flow over a stretching/shrinking sheet. Secondly, to analyze magnetic field and viscous dissipation effects. Finally, to solve the governing non–linear system numerically as well as analytically by using Runge–Kutta–Fehlberg (RKF) and MAPLE built–in differential equation solver, Dsolve, methods. Thermo-physical properties of Cu and SA are given in Table 1.

The steady 2D flow of SA based Cu–nanofluids with five shapes of nanoparticles is considered over a stretching sheet having a sheet velocity assumed as uw(x)=bx , where b is a constant, as shown in Fig. 1. The x and y are taken along and normal to the sheet, respectively. It is also assumed that the temperature of the sheet is Tw(x)=T∞+Ax2 [42].

The rheological behavior of Cu–nanofluid followed by [36,39] can be described in the form of continuity, momentum, and energy equations

(1) ∂u∂x+∂v∂y=0, 

(2) u∂u∂x+v∂u∂y=νNF(1+1γ)∂2u∂y2−σNFρNFB02u, 

(3) u∂T∂x+v∂T∂y=KNF(ρCp)NF∂2T∂y2−μNF(ρCp)NF(1+1γ)(∂u∂y)2, 

where u and v are the components of velocity, γ is the viscoplastic parameter and B0 is the magnetic field. Other symbols and quantities are defined in the nomenclature. The boundary conditions are

(4) u=αuw(x), v=0aty=0; u→0asy→∞, 

(5) T=Tw aty=0; T→T∞ asy→∞.

The viscosity of SA based Cu–nanofluid with different shapes of nanoparticle, except spherical, is calculated from the following correlation:(6) μNF=(1+δ1ϕ+δ2ϕ2)μBF,  where the constants δ1 and δ2 are known as viscosity coefficients based on nanoparticle shapes as given by Timofeeva et al. [43], see Table 2.

In the case of spherical nanoparticle shape, viscosity of SA based Cu–nanofluid is(7) μNF=[1(1−ϕ)2.5]μBF.

Pak et al. [44] introduced a correlation to predict the density of nanofluid, which is defined as(8) ρNF=[(1−ϕ)+ϕρNPρBF]ρBF.

The electrical conductivity of SA based Cu–nanofluid given by Sheikholeslami [45] which is of the form(9) σNF=[1+3(σNPσBF−1)ϕ(σNPσBF+2)−(σNPσBF−1)ϕ]σBF.

Many relationships proposed for the thermal conductivity of a two–phase mixture, in which Maxwell [1] developed a model for the thermal conductivity of a two–phase mixture based on thermal conductivity of base fluids, spherical particles and particle volume fraction. Hamilton et al. [46] expanded Maxwell model for non–spherical particle shapes. Their model allows calculating the effective thermal conductivity of a two–phase mixture under the empirical shape factor. This model is stated as(10) KNF=[KNP+(n−1)KBF+(n−1)(KNP−KBF)ϕKNP+(n−1)KBF−(KNP−KBF)ϕ]KBF,  where n= shape factor (for sphere n=3 , for cylinder n=6 ). Moreover, n=3ψ is empirical shape factor as given by Timofeeva et al. [43], see Table 3.

To simplify the Eqs. (2) and (3), following variables [11,39] are defined(11) η=ybνBF, u=bxFη(η), v=−bνBFF(η), θ(η)=T−T∞Tw−T∞.

Using Eq. (11), governing Eqs. (2) and (3) are reduced as follows:

(12) μNFμBFρBFρNF(1+1γ)Fηηη(η)+F(η)Fηη(η)−Fη2(η)−MρBFρNFσNFσBFFη(η)=0, 

(13) 1PrKNFKBF(ρCp)BF(ρCp)NFθηη (η)+F(η)θη(η)−2Fη(η)θ(η)+μNFμBF(ρCp)BF(ρCp)NF(1+1γ)EcFηη2(η)=0, 

where Fη(η) represents an ordinary derivative with respect to η , M2=σBFB02bρBF is the magnetic parameter, Pr=(ρCp)BFνBFKBF is the Prandtl number and Ec=b2A(Cp)BF=uw2(Tw−T∞)(Cp)BF is the Eckert number. The corresponding boundary conditions in the non–dimensional form are

(14) F(η)=0, Fη(η)=α, θ(η)=1atη=0, 

(15) Fη(η)→0, θ(η)→0 asη→∞.

The non–linear differential Eqs. (12) and (13) with the boundary conditions (14) and (15) have been solved numerically by the Runge–Kutta–Fehlberg (RKF) method using MAPLE software (Version 13) and analytically with the help of MAPLE built–in differential equation solver Dsolve. The system of differential equations consists of third–order velocity and second–order temperature equations are reduced in simultaneous ordinary differential equations. The step size Δy=0.001 and a convergence criterion of 10−6 are considered in the numerical computations. The asymptotic boundary conditions (15) are substituted by using a value of 12 for the similarity variable ηmax as Fη(12)=0, θ(12)=0.

The effect of different physical parameters, namely, the viscoplastic parameter γ , magnetic parameter M, Eckert number Ec , Prandtl number Pr on velocity and temperature fields are shown in graphs. Figs. 2–5 describe the temperature field θ(η) variation at different values of viscoplastic parameter γ along a stretching sheet for SA based nanofluid when M>0 (in the presence of magnetic field) and α>0 (in case of stretching sheet), for different shapes of Cu nanoparticles such as bricks, cylinders, platelets, and blades respectively. It was noticed that temperature increased with increasing values of γ for bricks, cylinders, platelets, and blades shapes of Cu nanoparticles, respectively. To this end, increases in the values of γ led to decreases in yield stress which improved the temperature field. Figs. 6–9 show the influence of different values of Prandtl number Pr on the temperature field θ(η) for bricks, cylinders, platelets, and blades shapes of Cu nanoparticles when M>0 and α>0 . It was found that increases in Pr decreased the temperature of bricks, cylinders, platelets, and blades shapes of Cu nanoparticles. Physically, the increase in Pr number means that momentum diffusivity dominated over thermal diffusivity which caused the decrease in temperature field. Moreover, these figures show the comparison of temperature field θ(η) for four different types of base fluids such as water (Pr=6.2) , sodium alginate (Pr=6.45) , kerosene (Pr=21) and engine oil (Pr=6450) containing bricks, cylinders, platelets, and blades shapes of Cu nanoparticles. The temperature of water and sodium alginate-based Cu–nanofluids was noticed to be higher than kerosene and engine oil based Cu–nanofluids. The reason for this is that water and sodium alginate have better thermal conductivity than kerosene and engine oil-based fluids. Therefore, water and sodium alginate-based fluids raised the temperature θ(η) of the Cu–nanofluid.

Figs. 10–13 show the temperature profile θ(η) for different values of Eckert number Ec for brick, cylinder, platelet, and blade shapes of Cu–nanoparticles respectively when M>0 and α>0 . It was observed that temperature increased for all shapes of nanoparticles respectively with increasing values of Ec . This was due to increases in Ec resulting in enhancement of the kinetic energy, as it is a well–known fact that temperature is defined as average kinetic energy. Moreover, the dissipative frictional forces between layers were enhanced with the increases in Ec and consequently, the temperature of the Cu–nanofluid rose. Velocity fields Fη(η) for different shapes of nanoparticles (bricks, cylinders, platelets, and blades) respectively for different values of viscoplastic parameter γ when M>0 and α>0 are shown in Figs. 14–17. It was observed that velocity field decreased for all nanoparticles with increasing values of γ . To this end, increases in values of γ led to reductions in yield stress, therefore the velocity field Fη(η) reduced. Figs. 18–21 present the variations in velocity profile Fη(η) with the variations in stretching sheet parameters α when M>0 for different shapes of nanoparticles. It was observed that the increases in the α increased the velocity field Fη(η) . Figs. 22–25 show the impact of different values of magnetic parameter M when α>0 on the velocity field Fη(η) for different shapes of nanoparticles (bricks, cylinders, platelets and blades), respectively. It was observed that increases in magnetic parameter M decreased the velocity field Fη(η) . Resultant resistive type forces due to enhancement of magnetic field strength, also called Lorentz forces with respect to the aligned magnetic field play a key factor in slowing the fluids motion. Hence, increasing values of M immediately cause an increase in drag force which decreases the velocity profile. To test the present numerical results, the results of velocity and temperature fields for Cu–nanofluid are compared with those obtained by Saleem et al. [48] in Figs. 26 and 27, for blade and platelet shapes nanoparticles when M=0 (n the absence of magnetic field) and γ→∞ (in case of Newtonian fluid), respectively. Both of these results for velocity and temperature fields of Cu–nanofluid are found in an excellent agreement.

The present article examined nanoparticles shape effects on sodium alginate-based Cu–nanofluid over a stretching/shrinking sheet. The vital findings are

Velocity field decreases for all shapes of Cu nanoparticles for magnetic parameter and viscoplastic parameter.