Purpose This study aims to investigate the impact of quadratic velocity on MHD flow over a permeable stretching/shrinking sheet using numerical and asymptotic approaches. The velocity of the sheet is given by Uwx=λax+bx2, with a,b>0 and λ as the stretching/shrinking parameter. By incorporating a quadratic velocity term, the study enhances understanding of boundary-layer behavior under varying magnetic field strength and suction/injection rates. Design/methodology/approach Using appropriate similarity transformations, the governing partial differential equations are reduced to a system of ordinary differential equations. Numerical solutions are obtained using MATLAB’s bvp4c solver, while asymptotic approximations are derived for extreme values of the stretching, suction/injection and magnetic field parameters. Both numerical and asymptotic approaches are used for analysis. Findings The results reveal that flow characteristics are significantly influenced by the stretching/shrinking parameter, magnetic field strength and suction/injection rates. The quadratic velocity term affects boundary layer stability and may lead to singularities in the shrinking case, depending on the interaction of suction/injection rate and magnetic field. Asymptotic results match well with numerical results for large parameter values, confirming the method’s reliability. Practical implications The findings are relevant to engineering applications, such as petroleum and chemical engineering, electromagnetic control of fluid flow and polymer processing, including rubber sheet manufacturing and plastic film drawing, offering insights for optimizing flow conditions in industrial applications. Originality/value This study extends classical boundary-layer theory by incorporating a quadratic velocity profile in MHD flow analysis. The strong agreement with existing literature confirms the validity of the approach used. The combined numerical and asymptotic approaches provide a comprehensive understanding of the system behavior, offering new insights into complex fluid interactions under magnetic fields, with the asymptotic solutions being new and original.

The study of the limiting behaviours of a mathematical model is crucial for understanding the stability of the model, gaining insight into asymptotic properties, predicting the long-term behaviour of system as well as validating the model. However, most of the research studies focus on numerical solutions but overlook the asymptotic solutions. In this paper, the limiting behaviours of unsteady mixed convection over a vertical flat plate in a porous medium is studied. The transformed similarity ordinary differential equations are solved both numerically and asymptotically for an unsteadiness parameter and a mixed convection parameter. It is found that the existence of critical value of mixed convection parameter for decelerating flow results in the two branches of possible solutions. For large values of the mixed convection parameter, the results for asymptotic solution agreed well with the numerical solutions especially for accelerating flow.

The purpose of this paper is to describe the stead MHD mixed convection flow over a permeable vertical flat plate embedded in a Darcy–Forchheimer porous medium. Using appropriate similarity variables, the partial differential equations are transformed into ordinary (similar) differential equations, which are numerically solved using the bvp4c function in MATLAB. The numerical results are used to present graphically and in tables, illustrations of the reduced skin friction, reduced Nusselt number, velocity, and temperature profiles. Dual (upper and lower branch) solutions are discovered in this exciting analysis. Although numerous studies on the mixed convection past a vertical plate embedded in a fluid-saturated porous medium exist, none of the researchers have focused on the Darcy–Forchheimer flow with asymptotic solutions. The behavior of the flow and heat transfer has been thoroughly analyzed with the variations in governing parameters, such as Darcy–Forchheimer G, suction/injection S , MHD M, and mixed convection λ parameters.

Purpose The steady laminar wall jet flow over a stretching/shrinking surface in the presence of lateral suction or injection with a convective boundary condition is considered. Design/methodology/approach The partial differential equations for mass, momentum and energy conservation are changed to the system of ordinary differential equations through similarity solution transformations. Solutions, both numerical and asymptotic, to these similarity equations are found in some new ranges of parameters in the governing equations. Findings The equations are solved both asymptotically and numerically for a range of the transpiration parameter S and the flow parameter lambda given in Mahros et al. (2023), thus greatly extending the range of these previous solutions. Asymptotic solutions for both large and small values of the Prandtl number sigma are derived, showing good agreement with additional numerical integrations. It should be noted that in Mahros et al. (2023), only the case when sigma=1 was treated. A solution for large lambda when S=1 is obtained, showing a different asymptotic form to the case when S>0 in Mahros et al. (2023). Multiple solutions were seen by them for S<0 and the nature of the lower solution branch as S -> 0 from below is discussed. The question as to whether the lower branch solutions join as lambda>0 when S<0 is resolved through obtaining an asymptotic solution lambda small. Originality/value The accuracy of the solutions has been checked through a detailed comparison between the solutions obtained numerically and analytically, where excellent agreement has been found. This study is important for scientists working in the area of jet flows to become familiar with the flow properties and behaviour of jets.

PurposeThe purpose of this study is to obtain both the numerical and asymptotic solutions of the unsteady mixed convection flow and heat transfer over an expanding or contracting cylinder which is placed vertically.Design/methodology/approachSolutions of the governing ordinary (similarity) differential equations for the fluid flow and temperature field are obtained using the function bvp4c from MATLAB. The problem involves the Prandtl numberσ, the mixed convection parameterλand unsteadiness parameterSthat characterize an expanding or contracting cylinder. This solution approach is capable of producing multiple solutions once the necessary assumptions are provided.FindingsIt is found that solutions exist for all negative values ofS, expanding cylinder, and only for small positive values ofS, contracting cylinder. Further, more than one solution is observed; numerical computation shows that the critical point ofSbecomes singular asλapproaching zero. For the case of expanding cylinder, the mixed convection parameter has a significant effect on both the flow and heat transfer characteristics. Asymptotic analysis also shows that whenσis large, dual solutions exist for some values ofSandλ.Originality/valueThe present results are new and original for the study of the unsteady mixed convection flow and heat transfer over an expanding/contracting cylinder where numerical solutions are obtained for representative values of the involved parameters. Asymptotic solutions for largeλand largeσare derived.

PurposeThis paper aims to study the magnetohydrodynamic (MHD) wall jet of a hybrid nanofluid flow over a moving surface with a thermally convective surface, wall moving with suction/injection. Design/methodology/approachOn using appropriate similarity transformations, the governing equations that describe the model are converted into a system of nonlinear ordinary differential equations. These equations are solved both analytically and numerically using standard two-point boundary-value problem solvers and Chebyshev pseudospectral differentiation matrix method, respectively. FindingsThese results show that the HNF is heating/cooling with growth of the positive/negative values of the parameter measuring the velocity of the moving surface. The temperature distributions increase, where the thermal boundary layer gets thicker, as the magnetic field strengthens and with an increase in the absolute value of the Biot number. Originality/valueThe current findings for the HNFs are new and original. They generalize successfully the problems investigated previously by different researchers for the cases of fluids and also nanofluids.

This chapter discusses viscous fluids. The unsteady mixed convection flow at a three-dimensional stagnation point has been discussed with governing equations. Mixed convection boundary layer flow near the stagnation point on a vertical surface with slip is described further. This chapter finally concludes with mixed convection nonaxisymmetric Homann stagnation-point flow.

The first section of this chapter discusses the problem of steady two-dimensional mixed convection flow of a micropolar fluid near a stagnation-point flow over a stretching plate. The next section discusses the oblique stagnation–slip flow of a micropolar fluid toward a stretching/shrinking surface. The problem of forced convection flow over a wedge in a micropolar fluid is also discussed.

This chapter discusses the mixed convection flow in porous medium with problem formulation. The next section deals with Steady mixed convection flow over a permeable vertical thin cylinder in a porous medium with basic equations and similarity transformation. Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids and mixed convection boundary layer flow along a vertical cylinder embedded in a porous medium filled by a nanofluid have been discussed in this chapter.

The model proposed by Postelnicu et al. [17, 18] for the natural convection boundary-layer flow on a vertical surface in a porous medium driven by spatially-dependent localised internal heating is discussed further. Their results for a prescribed wall temperature characterised by the parameter $$\lambda $$ are extended to a consideration of the singularity seen in the solution as $$\lambda \rightarrow -\frac{1}{2}$$ and to the asymptotic limit $$\lambda \rightarrow \infty $$ , where convection resulting from the wall temperature becomes more important near the wall and showing a region of reversed flow/temperatures below ambient away from the wall. The case, not treated by Postelnicu et al., when there is a prescribed surface heat flux is also treated, finding that the solution became singular as $$\lambda \rightarrow -\frac{1}{3}$$ , the nature of which is discussed. The large $$\lambda $$ limit is also obtained again finding that the dominant effect is the convection resulting from the wall heat transfer though here the temperature within the boundary layer remains above ambient.

This chapter discusses convective flows with internal heat generation. Flows with local temperature dependent heat generation in porous media have been discussed. The internal heating that take a modified form of Arrhenius kinetics is also considered. This chapter also discusses the flows with spatially dependent heat generation.

This chapter discusses stretching and shrinking sheets in nanofluids. This chapter begins with the discussion on flow and heat transfer over an unsteady shrinking sheet with suction in a nanofluid using Buongiorno's model. The next section considers the uniform flow of a nanofluid over a biaxial stretching surface. The next section discusses the stretching/shrinking of a curved surface in a hybrid nanofluid. This chapter concludes with magnetohydrodynamic flow and heat transfer over a permeable stretching/shrinking sheet in a hybrid nanofluid with a convective boundary condition.

The forced convection flow near a stagnation point on a stretching/shrinking cylinder derived originally by Wang [1] and later extended by Lok and Pop [2] is further considered through obtaining the asymptotic behaviour for the relevant dimensionless parameters, namely the Prandtl number σ, the wall velocity λ, the wall transpiration rate S and the Reynolds number R. Additional numerical results are also obtained to confirm our asymptotic predictions. Critical values are seen in the numerical results for negative values of λ with an asymptotic solution valid for large negative λ gives an asymptotic expression for these critical values. It is found that multiple solutions are possible for the limiting range of shrinking and suction parameters. Besides, the asymptotic behaviour is similar for the case of large suction and large Reynolds number. Also, the asymptotic analysis shows that for the solution to continue to large shrinking parameter, the suction parameter should be positive and large.

This chapter discusses the study of the dynamics of fluid flow against a surface known as stagnation-point flow. The unsteady separated stagnation-point flow has been described and solved numerically using the bvp4c program in Matlab that is efficient in finding multiple solutions for the boundary value problem. Axisymmetric rotational stagnation-point flow over a permeable stretching/shrinking rotating disk is further described in this chapter. Both numerical and asymptotic solutions are discussed in detail. Finally, this chapter concludes with magnetohydrodynamic oblique stagnation-point flow toward a stretching/shrinking surface.

Basic equations and mathematical methods are discussed in this chapter. Thermophysical properties of fluids and nanoparticles is studied. Some numerical methods such as Matlab program bvp4c, Keller-Box method, and Runge–Kutta method have been described in this chapter. This chapter concludes with analytical solution methods.

Purpose The purpose of this paper is to numerically analyze the stagnation point flow of Cu-Al2O3/water hybrid nanofluid with mixed convection past a flat plate and circular cylinder. Design/methodology/approach The similarity equations that reduced from the boundary layer and energy equations are solved using the bvp4c solver. The duality of solutions is observed within the specific range of the control parameters, namely, mixed convection parameter λ, curvature parameter γ and nanoparticles volumetric concentration ϕ1 for alumina, while for copper ϕ2. The stability analysis is also designed to justify the particular solutions’ stability. Additionally, the idea to obtain the solution for large value of λ and γ is also presented in this paper. Findings Two solutions exist in opposing and assisting flows up to a critical value λc where λc lies in the opposing region. An upsurge of the curvature parameter tends to extend the critical value (delay the separation process), whilst increase the heat transfer performance of the working fluid. Meanwhile, the application of hybrid Cu-Al2O3/water nanofluid also can decelerate the separation of laminar boundary layer flow and produce higher heat transfer rate than the Cu–water nanofluid and pure water. Originality/value The results are new and original. This study benefits to the other researchers, specifically in the observation of the fluid flow characteristics and heat transfer rate of the hybrid nanofluid. Also, this paper features with the mathematical formulation for the solution with large values of λ and γ.

In this note, we extend the problem treated in (Lok, Math Modelling Anal 24:617–634 (2019)) to the case of permeable surface which is shrinking in mutually orthogonal directions. Both numerical and asymptotic solutions are obtained for two important governing parameters, $$\gamma $$ the shrinking rate and S characterizing the fluid transfer through the boundary. In this problem, a restriction on S is required for a solution to exist. This contrasts with the problem in (Lok, Math Modelling Anal 24:617–634 (2019)) where no restriction on S is needed. Numerical solutions show that for a fixed value of S, two critical points $$\gamma _c$$ are observed for $$S > 2$$ . Conversely, two critical points $$S_c$$ are found for a given value of $$\gamma $$ when $$S > 2$$ . A discussion on the nonexistence of solution for $$S = 2$$ is given and asymptotic solutions for S large and $$(S-2)$$ small are also presented.

The Falkner–Skan equation, defined by the parameter \(\beta \), is considered subject to a free streamline (zero wall shear) boundary condition. Solutions are found only in \(\beta <0\), the solution becoming singular as \(\beta \rightarrow 0\). Several sets of solutions are seen in \(\beta <0\), each emerging from the trivial solution \(f \equiv \eta \) at \(\beta =-\frac{1}{2} -k, \,k=0,1,2,\ldots \). The first of these sets of solutions has \(f'(0)\) monotone with \(\beta \), the solution terminating as \(\beta \rightarrow 0\) and becoming singular as \(\beta \rightarrow -1\). The other sets of solutions each have a saddle-node bifurcation giving two solution branches, becoming singular and terminating as \(\beta \rightarrow -1\). The asymptotic limits of \(\beta \rightarrow 0\) and \(\beta \rightarrow -1\) are discussed.

In a recent paper, Al-Housseiny and Stone (J Fluid Mech 706:597–606, 2012) considered the dynamics of a stretching surface and how this interacts with the boundary-layer flow it generates. These authors discussed the cases $$c= -3$$ for an elastic sheet and $$c= -1$$ for the viscous fluid, c being representative for the stretching velocity of the sheet. The aim of the present paper is to extend the analysis of Al-Housseiny and Stone (2012) to the general values of c, to allow for both a stretching and a shrinking sheet and for the surface to be permeable through the parameter S, where $$S>0$$ for the fluid withdrawal and $$S < 0$$ for fluid injection. Both the cases $$S=0$$ (impermeable surface) and $$S \ne 0$$ (permeable surface) are considered for both stretching surfaces and shrinking surfaces. In all these cases, asymptotic solutions are presented for large values c and S (both withdrawal and injection).