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.

Micropolar fluid is one type of non-Newtonian fluid which consists of non-deformable spherical particles that suspended in viscous medium. In this paper, the problem of two-dimensional boundary layer flow over a permeable shrinking sheet with time dependent velocity in strong concentration micropolar fluid is studied theoretically. The mathematical model is governed by continuity, momentum and microrotation equations. Similarity variables are introduced so that, after performing the similarity transformation on the governing equations, the resulting system of nonlinear ordinary differential equations is then numerically solved using the program bvp4c in Matlab software. The effects of the micropolar material parameter, the unsteadiness parameter, the shrinking parameter and the mass suction parameter to the skin friction coefficient, velocity profiles and microrotation profiles are investigated. It is found that triple solutions exist for some values of the parameters that were considered. Based on the stability analysis that was performed, it showed that only two branches of solutions are categorized as stable, whereas one solution branch is unstable.

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.

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.

A generalization of a boundary value problem treated previously by Magyari et al. (ZAMP 53:782–793, 2002a; Transp Porous Medium 46:91–102, 2002b), Zhang (Math Anal Appl 417:361–375, 2014) and Paullet (Appl Math E Notes 14:123–126, 2014) for a prescribed wall temperature and by Zhang (Appl Math Comput 339:367–373, 2018) for a prescribed wall heat flux is considered. The problem involves a parameter $$\lambda $$ and an exponent p. Numerical solutions are obtained for representative values of $$\lambda $$ and p, a feature of which is the existence of critical values $$\lambda _c$$ of $$\lambda $$ with two solution branches in $$\lambda >\lambda _c$$ , with $$\lambda _c$$ dependent on p. Asymptotic solutions for large $$\lambda $$ and large p are derived for both types of boundary condition. For large $$\lambda $$ , the nature of the solution is essentially different on upper and on the lower branches with similar feature being seen in the behaviour for p large.

This paper extends the recent work by Weidman (2015) on the steady axisymmetric rotational stagnation-point flow impinging on a rotating disk to the case of a permeable stretching/shrinking rotating disk. Similarity variables are used to convert the Navier–Stokes equations to a pair of coupled ordinary differential equations governed by a dimensionless rotation rate α, the velocity of the wall relative to the outer flow λ and the wall transfer rate S. These equations are then solved numerically using the bvp4c function from MATLAB software. The skin friction coefficients or shear stress in both the radial and azimuthal directions are determined. It is found that, for given parameter values, up to three solutions can exist. Asymptotic expressions are derived for large stretching rates λ and for strong fluid withdrawal and injection as well as for rapid rotation rates α, these being compared with the numerically determined values.

The steady mixed convection non-axisymmetric (Homann, Z. Angew. Math. Mech., vol. 16, 1936, pp. 153–164) stagnation-point flow over a vertical flat wall placed in a viscous and incompressible fluid is considered. A similarity solution is derived which involves the dimensionless parameters $\unicode[STIX]{x1D6FE}$ , representing the shear-to-strain-rate ratio, and $\unicode[STIX]{x1D706}$ , a mixed convection parameter. Forced convection, $\unicode[STIX]{x1D706}=0$ , is treated first where solutions additional to those given previously by Weidman (J. Fluid Mech., vol. 702, 2012, pp. 460–469) are found arising from singularities as $\unicode[STIX]{x1D6FE}\rightarrow \pm 1$ . Numerical solutions are obtained for representative values of both $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D706}$ . Critical values $\unicode[STIX]{x1D706}_{c}$ of $\unicode[STIX]{x1D706}$ are seen in opposing flow and these are treated in detail. Asymptotic results for large $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D6FE}$ are derived.

The steady MHD oblique stagnation-point flow towards a stretching/shrinking surface in a viscous and electrically conducting fluid and in the presence of a uniform magnetic field is studied. The external magnetic field is parallel to the dividing streamline of the oblique stagnation-pint flow. The governing Navier–Stokes equations are reduced to a system of two ordinary differential equations. Solutions of these equations are evaluated numerically for various values of the governing parameters, namely the magnetic parameter M, the stretching/shrinking parameter \(\lambda \) and the two constants \(\alpha \) and \(\beta \) arising in the model. It is found that dual (upper and lower branch) solutions exist and that there is a critical value \(\lambda _c\) of \(\lambda \), dependent on M, with solutions only in \(\lambda \ge \lambda _c\) with the lower solution branch terminating as \(\lambda \rightarrow -(1+M)\). It is shown that the values of \(\lambda _c\) increase as M is increased thus extending the range of similarity solutions in the opposing flow regime. It is also found that the stagnation line is displaced due to the effect of MHD with a region of reversed flow being observed near to the wall for a shrinking surface.

A detailed study of the problem of the boundary-layer flow on a shrinking permeable surface near a forward stagnation point with an outer flow u∞∝ xm, a tangential wall velocity uw ∝ xm, and velocity slip on the surface considered previously by Fauzi et al. [1] for m=1 (stagnation point flow) is presented. Further numerical results are obtained and the asymptotic behaviour of the flow under various conditions of the governing parameters is described. Four cases of the problem are considered, namely an impermeable fixed wall, an impermeable moving wall, a permeable fixed wall and a permeable moving wall. For the case of an impermeable fixed wall, it is found that there is a critical value βc of β=2m/(m+1) dependent on the velocity slip parameter A, and that this critical value approaches a finite limit as A increases. For the case of impermeable moving wall, the critical value is negative, decreasing as A is increased. Asymptotic solutions for both strong suction and strong blowing are obtained for the permeable fixed wall. For the case of permeable moving wall, the critical values λc of the parameter λ, the ratio of the wall velocity to the outer flow, found by Fauzi et al. [1] are completed and plotted against the governing suction parameter S. It is seen that λc becomes large as suction is increased.

A detailed study of the problem of unsteady separated stagnation-point flow toward a stretching or shrinking sheet is presented. By use of a similarity transformation, the partial differential equations are reduced to a system of ordinary differential equations which are then solved numerically by use of the collocation method. The bvp4c Matlab program is used to perform the computation. This method is capable of finding multiple solutions for the flow reattachment case. The streamlines of the separation flow are presented for the cases when the origins of the stagnation flow and the stretching sheet are aligned and non-aligned. Comparison with results from the open literature reveals excellent agreement.

The mixed convection boundary-layer flow on one face of a semi-infinite vertical surface embedded in a fluid-saturated porous medium is considered when the other face is taken to be in contact with a hot or cooled fluid maintaining that surface at a constant temperature \(T_\mathrm{{f}}\). The governing system of partial differential equations is transformed into a system of ordinary differential equations through an appropriate similarity transformation. These equations are solved numerically in terms of a dimensionless mixed convection parameter \(\epsilon \) and a surface heat transfer parameter \(\gamma \). The results indicate that dual solutions exist for opposing flow, \(\epsilon <0\), with the dependence of the critical values \(\epsilon _\mathrm{{c}}\) on \(\gamma \) being determined, whereas for the assisting flow \(\epsilon >0\), the solution is unique. Limiting asymptotic forms for both \(\gamma \) small and large and \(\epsilon \) large are also discussed.

The steady mixed convection flow near an axisymmetric stagnation point on a stretching or shrinking vertical cylinder is considered. The equations for the fluid flow and temperature fields reduce to similarity form that involves a Reynolds number R, a mixed convection parameter λ, a parameter γ representing the motion of the cylinder and the Prandtl number σ. Numerical solutions of the similarity equations are obtained for representative values of these parameters, which show the existence of critical values λc = λc (R,γ,σ) with the existence of dual solutions in the opposing (λ < 0) case. The variation of the skin friction coefficient f″(1) and heat flux θ′(1) with λ and R when σ = 1.0 are shown graphically. Also variations of λc with R, and γ are determined again in the case when σ = 1.0. It is found that in the aiding flow (λ > 0) case solutions are possible for all λ and the asymptotic solution in the limit λ → ∞ is obtained. The nature of the solution in the asymptotic limit of large values of R is also treated in the cases when λ is of O (1) and when λ is of O (R).

Experiments were carried out in order to evaluate the particle removal efficiency from a wafer surface by means of a buffing disk in a process for flow rates of chemical media ranging from 3.33×10−6m3/s to 6.67×10−6m3/s, buffing disc pressures of 6.894kPa to 20.684kPa and three relative speeds were used in the experiments. A mathematical model which considered the toppling of a particle as a result of forces due to friction, hydrodynamic drag, adhesion and capillary phenomena was developed in order to correlate the particle removal efficiency with the flow rate, buffing disc pressure and relative rotational speed. Results of simulations of the model showed a good agreement with the experimental data with satisfactory correlation coefficients.

In this paper, the steady axisymmetric stagnation point flow of a viscous and incompressible fluid over a shrinking circular cylinder with mass transfer (suction) is studied. The flow is induced by a cylinder shrinking with a linear velocity distribution from the stagnation line. The fluid flow solution is an exact solution of the Navier-Stokes equation, which is reduced to a nonlinear ordinary differential equation. This equation is solved numerically for some values of the governing parameters that involves a Reynolds number R, a shrinking parameter λ, and a suction parameter γ. The effects of these governing parameters on the velocity and temperature profiles, skin friction coefficient, Nusselt number, as well as the distributions of the streamlines are investigated. The obtained results for the case of fixed cylinder are compared with those from the open literature and it is shown that they are in excellent agreement. The solutions are non-unique for some values of λ (<0). The streamlines show that the flow structures become complicated due to the shrinking and suction effects.

PurposeThe purpose of this paper is to theoretically investigate the steady two‐dimensional magnetohydrodynamic (MHD) boundary layer flow over a shrinking sheet. The effects of stretching and shrinking parameter as well as magnetic field parameter near the stagnation point are studied.Design/methodology/approachA similarity transformation is used to reduce the governing partial differential equations to a set of nonlinear ordinary differential equations which are then solved numerically using Keller‐box method.FindingsThe solution is unique for stretching case; however, multiple (dual) solutions exist for small values of magnetic field parameter for shrinking case. The streamlines are non‐aligned and a reverse flow is formed near the surface due to shrinking effect.Practical implicationsThe flow due to a stretching or shrinking sheet is relevant to several practical applications in the field of metallurgy, chemical engineering, etc. For example, in manufacturing industry, polymer sheets and filaments are manufactured by continuous extrusion of the polymer from a die to a windup roller, which is located at a finite distance away. In these cases, the properties of the final product depend to a great extent on the rate of cooling which is governed by the structure of the boundary layer near the stretching surface.Originality/valueThe present results are original and new for the MHD flow near the stagnation‐point on a shrinking sheet. For shrinking case, the velocity on the boundary is towards a fixed point which would cause a velocity away from the sheet. Therefore, this paper is important for scientists and engineers in order to become familiar with the flow behaviour and properties of such MHD flow and the way to predict the properties of this flow for the process equipments.

The unsteady boundary layer flow of a micropolar fluid induced by a twodimensional body, which is started impulsively from rest, is studied in this paper. The variation with time t of the external stream V (t) is assumed to be of the form V (t) = 1 - exp (-alpha t(m)), where alpha >= 0 means a coefficient of acceleration and m is an arbitrary integral value. The problem is formulated for the flow at the rear stagnation point on an infinite plane wall. Numerical solutions of the unsteady boundary layer equations are obtained using an implicit finite- difference scheme known as the Keller's box method. Results are given for the velocity and microrotation profiles, as well as for the dimensionless time elapsed before the boundary layer begins to separate from the wall. It is found that the dimensionless time elapsed before separation takes place is lower for a micropolar fluid (K not equal 0) than for a Newtonian fluid (K = 0), where K denotes the micropolar or material parameter.

This paper considers the problem of steady two-dimensional boundary layer flow of a micropolar fluid near an oblique stagnation point on a fixed surface with Navier’s slip condition. It is shown that the governing nonlinear partial differential equations admit similarity solutions. The resulting nonlinear ordinary differential equations are solved numerically using the Keller box method for some values of the governing parameters. It is found that the flow characteristics depend strongly on the micropolar and slip parameters.

Drying of temperature sensitive materials such as food and agricultural products, particularly at temperatures sensitive enough to cells, lead to probable cell damage and gelatinization. Such effects lead to surface hardened or case hardened layers leading to restricted moisture movement through resulting surface hardened layers and retarded drying rates. In this paper, equations are developed, based on the model of Dworecki et al. [Dworecki, K., Slezak, B., Ornal-Wasik, S., 2005. Evolution of concentration field in a membrane system. Journal of Biochemical and Biophysical Methods 62 (2), 153], for the moisture concentration distributions across the material and the surface hardened layers, and for the respective drying rates. Experimental data are compared with the solutions of the above equations. It is observed that the experimental data fits the solution very satisfactorily. Plots of model equations were compared with experimental data giving correlation coefficients closer to unity for data on slices of different thicknesses of papaya and garlic. It is shown that the drying rates under the case hardened conditions could be characterized by two physical parameters lD and r=KD/Df. The respective values of lD and r evaluated for different sizes of papaya and garlic slices showed consistency among different sizes of the same species indicating a reasonable compatibility with the model.

This paper considers the problem of steady two-dimensional flow of a micropolar fluid impinging obliquely on a flat plate. The flow under consideration is a generalization of the classical modified Hiemenz flow for a micropolar fluid which occurs in the boundary layer near an orthogonal stagnation point. A coordinate decomposition transforms the full governing equations into a primary equation describing the modified Hiemenz flow for a micropolar fluid and an equation for the tangential flow coupled to the primary solution. The solution to the boundary-value problem is governed by two non-dimensional parameters: the material parameter K and the ratio of the microrotation to skin friction parameter n. The obtained ordinary differential equations are solved numerically for some values of the governing parameters. The primary consequence of the free stream obliqueness is the shift of the stagnation point toward the incoming flow.