Reverse Flow Solution Research Articles

Unsteady separated stagnation-point flows and heat transfer over a plane surface moving normal to the flow impingement

In this paper, we have investigated the momentum and heat transfer in an unsteady separated stagnation-point flow of a viscous fluid over a plane surface advancing towards or receding from the normal stagnation flow which impinges on the surface with a variable strain rate s(t), causing the unsteadiness in this flow problem. The flow and heat transfer characteristics are therefore governed by the flow strength (stagnation) parameter a, unsteadiness parameter β, plate velocity (normal) parameter α and Prandtl number Pr. The current analysis ensures the existence of the self-similar solutions to this flow problem only when the plate velocity varies directly to the square root of the strain rate s(t). A closed form analytic solution of this flow problem is found for the specific relational values of β=2a. The solutions of this flow problem are non unique only under the negative values of β. When the plate surface is advancing towards the normal stagnation flow (i.e., when α> 0), the self-similar boundary layer solution continues for any given values of a and β, whereas for receding of the plate surface from the incoming flow (i.e., for α< 0) the boundary layer solution does not exist after a certain value of α depending upon the values of a and β. In the case where receding of the plate surface occurs, the features of the boundary layer flows depend highly on the sum values of the parameters a and β. For (a+β)> 0, the boundary layer solution ends with an attached flow solution, while it gets terminated at a reverse flow solution for (a+β)< 0. And for (a+β)= 0, the governing boundary layer equations provide us with the trivial (zero) solutions which are unable to capture the free boundary conditions.

Unsteady stagnation-point boundary layer flows of power-law fluids over a porous flat plate

The problem of unsteady stagnation-point flows of power-law fluids over a porous flat plate with mass transfer is considered with a view to examine the rheological behaviors of the fluids. This study is completely based on the four physical parameters, namely, flow strength parameter a, mass transfer parameter d, unsteadiness parameter $$\beta$$ and non-Newtonian power-law index n. For $$d=0$$ , the numerical results of this analysis reveal the existence of two types of solutions - one is attached flow solution (AFS) and the other is reverse flow solution (RFS) in a definite range of n ( $$0< n \le 2$$ ) when (a, $$\beta )= (1, - 1)$$ . The present analysis confirms that the velocity profile for any dilatant fluid $$(n > 1)$$ matches smoothly with the free stream velocity for a suitable amount of blowing $$d(< 0)$$ depending upon the values of n. We will also discuss the asymptotic behaviors of the boundary layer flows for large values of d, i.e., for $$d \rightarrow \pm \infty$$ . The asymptotic analysis ensures the existence of the above two solutions for large values of suction $$d>0$$ , whereas the boundary layer solution is terminated after a certain value of blowing $$d<0$$ , dependent on the values of $$\beta (< 0)$$ and n. Below this critical value of blowing d, this unsteady flow problem also provides us with a solution which does not appear to have a boundary layer character.

Effects of Transpiration on the Unsteady Separated Stagnation-Point Flow and Heat Transfer Over a Moving Porous Plate

Abstract The combined influence of normal transpiration and tangential movement of a porous surface on the unsteady separated stagnation-point flow and heat transfer of a viscous fluid is studied. This study is based on five physical parameters, namely (i) flow strength parameter a, (ii) suction/injection parameter d, (iii) plate velocity parameter λ, (iv) unsteadiness parameter β, and (v) Prandtl number Pr. This analysis shows an interesting relation β = 2a which allows us to derive some closed-form analytic solutions depending upon the unlikely values of d. For suction d &amp;gt; 0, two attached flow solutions (AFS) without point of inflection are found in the range (−ad2/4−3&amp;lt;λ&amp;lt;−3), whereas for λ &amp;gt; −3 only one solution of the same type is found for any given value of d. Besides them, the numerical computations reveal two types of AFS—one without and the other with a point of inflection in the range (−1.24658 ≤λ≤ −1.07) when d = β = 0. The present analysis confirms the nonexistence of the second attached flow solution after a certain value of suction d depending upon the choice of the values of λ in this range. A reverse flow solution (RFS) along with the above two solutions is found for a negative value of β which continues even for large rate of suction d. The asymptotic solutions of this flow problem have also been derived for large values of d which provide with the exact results after a certain value of d depending upon the values of the other parameters.

Unsteady separated stagnation-point flow over a permeable surface

In this paper, we have investigated numerically the unsteady separated stagnation-point flow of an incompressible viscous fluid over a porous flat plate subjected to continuous suction or blowing. The analysis covers the complete range of the suction/blowing parameter d, including $$d = 0$$ and $$d \rightarrow \pm \infty $$ , in conjunction with the flow strength parameter $$a (> 0)$$ and the unsteadiness parameter $$\beta $$ . Two types of solutions, namely attached flow solution (AFS) and reverse flow solution (RFS), have been found for a negative value of $$\beta $$ . A novel result that emerges from our analysis is the characteristic features of the boundary layer flows which firmly depend on the sum values of $$(a + \beta )$$ in case of massive blowing $$(d \ll 0)$$ given at the wall. In fact, a negative value of $$(a + \beta )$$ opposes the flow like an adverse pressure gradient and the solution domain ends off with a RFS. On the other hand, a positive value of $$(a + \beta )$$ assists the flow like a favourable pressure gradient for which the solution continues with an AFS. An interesting result of this analysis is that after a certain negative value of $$(a + \beta )$$ , dependent on a and $$\beta $$ , the solution of this flow problem does not exist for blowing, and this trend persists even for suction $$(d>0)$$ . On the other hand, for large positive values of $$(a + \beta )$$ , the attached flow solution exists in case of both strong suction and massive blowing, whereas for small positive values of $$(a + \beta )$$ and especially in case of hard blowing, the solution of the governing boundary layer equation does not appear to have the boundary layer character. Furthermore, when $$(a + \beta ) = 0$$ , the velocity gradient at the wall is zero for both AFS and RFS flows, and ultimately, the governing boundary layer equation produces a trivial solution which does not satisfy the outer boundary condition.

An Unsteady Separated Stagnation-Point Flow Towards a Rigid Flat Plate

This paper discusses an unsteady separated stagnation-point flow of a viscous fluid over a flat plate covering the complete range of the unsteadiness parameter β in combination with the flow strength parameter a (&gt;0). Here, β varies from zero, Hiemenz's steady stagnation-point flow, to large β-limit, for which the governing boundary layer equation reduces to an approximate one in which the convective inertial effects are negligible. An important finding of this study is that the governing boundary layer equation conceives an analytic solution for the specific relation β = 2a. It is found that for a given value of β (≥0) the present flow problem always provides a unique attached flow solution (AFS), whereas for a negative value of β the self-similar boundary layer solution may or may not exist that depends completely on the values of a and β (&lt;0). If the solution exists, it may either be unique or dual or multiple in nature. According to the characteristic features of these solutions, they have been categorized into two classes—one which is AFS and the other is reverse flow solution (RFS). Another interesting finding of this analysis is the asymptotic solution which is more practical than the numerical solutions for large values of β (&gt;0) depending upon the values of a. A novel result which arises from the pressure distribution is that for a positive value of β the pressure is nonmonotonic along the stagnation-point streamline as there is a pressure minimum which moves toward the stagnation-point with an increasing value of β &gt; 0.

On the fluid dynamics of unsteady separated stagnation-point flow of a power-law fluid on the surface of a moving flat plate

The transformation group theoretic approach is applied to perform an analysis of the unsteady separated stagnation-point (USSP) flow over a moving flat plate immersed in a non-Newtonian power-law fluid. This unsteady two-dimensional laminar boundary layer flow of a power-law fluid is governed by the classical Ostwald–de Waele power-law model. The system of governing partial differential equations reduces to an autonomous third order nonlinear ordinary differential equation via two-parameter group theoretic method. By using an efficient shooting method, we have established the existence of two different kinds of solutions, namely: (i) attached flow solution (AFS) and (ii) reverse flow solution (RFS) of this USSP flow problem under a definite negative value of the unsteadiness parameter β with a given value of the acceleration parameter a and for all values of the power-law index n considered in the present study. We have also found a unique AFS solution that is irrespective of n when β≥0. Dual solutions of the governing boundary layer equation are found for all kinds of fluids with the fixed values of a=1 and β=−1. The role of the rheological index n on the displacement thickness, skin-friction coefficient and velocity boundary layer are explored numerically. The results of this analysis reveal that the wall shear stress is always positive for AFS flow and negative for RFS flow for the above values of a and β with any given value of n. A novel result that emerges from this analysis is that the magnitude of the wall shear stress decreases and finally it approaches asymptotically to a constant value with the increase in the non-Newtonian behavior (n>1) of the fluids for both the cases of AFS and RFS.

Magnetohydrodynamic unsteady separated stagnation‐point flow of a viscous fluid over a moving plate

An analysis has been made for the unsteady separated stagnation‐point (USSP) flow of an incompressible viscous and electrically conducting fluid over a moving surface in the presence of a transverse magnetic field. The unsteadiness in the flow field is caused by the velocity and the magnetic field, both varying continuously with time t. The effects of Hartmann number M and unsteadiness parameter β on the flow characteristics are explored numerically. Following the method of similarity transformation, we show that there exists a definite range of for a given M, in which the solution to the governing nonlinear ordinary differential equation divulges two different kinds of solutions: one is the attached flow solution (AFS) and the other is the reverse flow solution (RFS). We also show that below a certain negative value of β dependent on M, only the RFS occurs and is continued up to a certain critical value of β. Beyond this critical value no solution exists. Here, emphasis is given on the point as how long would be the existence of RFS flow for a given value of M. An interesting finding emerges from this analysis is that, after a certain value of M dependent on , only the AFS exists and the solution becomes unique. Indeed, the magnetic field itself delays the boundary layer separation and finally stabilizes the flow since the reverse flow can be prevented by applying the suitable amount of magnetic field. Further, for a given positive value of β and for any value of M, the governing differential equation yields only the attached flow solution.

Unsteady separated stagnation-point flow over a moving porous plate in the presence of a variable magnetic field

This paper deals with the numerical study of unsteady separated stagnation point (USSP) flow of an incompressible, viscous, electrically conducting fluid over a porous flat plate which moves in its own plane with a velocity u0(t). A variable magnetic field is applied normal to the plate. The numerical results pertaining to the present study indicate that there exist two solutions for the USSP flow: one is the attached flow solution (AFS) and the other is reverse flow solution (RFS). It results in from the stability analysis that the AFS is stable and physically realizable, while the RFS is not stable and, therefore, not physically realizable. For a stationary plate, the USSP flow is separated for all values of the magnetic parameter (M) and suction/blowing parameter (d) in both AFS and RFS cases. It is found that velocity at a point decreases with the increase in M for RFS but the opposite trend is observed for AFS. The results of the analysis reveal that the number of stagnation points in presence of suction is two while the number of stagnation points in presence of blowing is only one for the RFS. A novel result of the analysis is that for RFS, there is only one stagnation point in absence of magnetic field whereas for a suitable magnetic field, two stagnation points appear.

The boundary layers of an unsteady separated stagnation-point flow of a viscous incompressible fluid over a moving plate

In this paper we have investigated the boundary layer analysis of an unsteady separated stagnation-point (USSP) flow of an incompressible viscous fluid over a flat plate, moving in its own plane with a given speed . The effects of the accelerating parameter a and unsteadiness parameter β on the flow characteristics are explored numerically. Our analysis, based on the similarity solution of the boundary layer equations, indicates that the governing ordinary differential equation, which is non-linear in nature, has either a unique solution, dual solutions or multiple solutions under a negative unsteadiness parameter β with a given value of a. Whatever the number of solutions may be, these solutions are of two types: one is the attached flow solution (AFS) and the other is the reverse flow solution (RFS). A novel result which emerges from our analysis is the relationship between a and β. This relationship essentially gives us the conditions needed for the solutions that exhibit flow separation (where ) and those conditions that exhibit only flow reattachment (where ). Another noteworthy result which arises from the present analysis is the existing number of non-zero stagnation-points inside the flow for the given values of a and β. It is found that this number is exactly two when the velocity gradient at the wall is positive; otherwise this number will only be one. For a stationary plate , this USSP flow is found to be separated for all values of a and β in both cases of AFS and RFS. Finally, we have also established that in the case of AFS flow over a stationary plate, no stagnation-point exists inside the flow, even though the flow becomes separated for all values of a and β.

The iterative transformation method for the Sakiadis problem

In a transformation method the numerical solution of a given boundary value problem is obtained by solving one or more related initial value problems. This paper is concerned with the application of the iterative transformation method to the Sakiadis problem. This method is an extension of the Töpfer’s non-iterative algorithm developed as a simple way to solve the celebrated Blasius problem. As shown by this author (Fazio, 1997) the method provides a simple numerical test for the existence and uniqueness of solutions. Here we show how the method can be applied to problems with a homogeneous boundary conditions at infinity and in particular we solve the Sakiadis problem of boundary layer theory. Moreover, we show how to couple our method with Newton’s root-finder. The obtained numerical results compare well with those available in literature. The main aim here is that any method developed for the Blasius, or the Sakiadis, problem might be extended to more challenging or interesting problems. In this context, the iterative transformation method has been recently applied to compute the normal and reverse flow solutions of Stewartson for the Falkner–Skan model (Fazio, 2013).

Unsteady separated stagnation-point flow of an incompressible viscous fluid on the surface of a moving porous plate

Using group-theoretic method, an analysis is presented for a similarity solution of boundary layer equations which represents an unsteady two-dimensional separated stagnation-point (USSP) flow of an incompressible fluid over a porous plate moving in its own plane with speed u0(t). It is observed that the solution to the governing nonlinear ordinary differential equation for the USSP flow admits of two solutions (in contrast with the corresponding steady flow where the solution is unique): one is the attached flow solution (AFS) and the other is the reverse flow solution (RFS). A novel result of the analysis is that in the case of stationary plate (u0(t) = 0), after a certain value of the magnitude of the blowing d (&amp;lt;0) at the plate, only the AFS exists and the solution becomes unique. For a stationary plate (u0(t) = 0), the USSP flow is found to be separated for all values of d in both the cases of AFS and RFS. It is also observed that when u0(t) = 0, in the RFS flow with wall suction d (&amp;gt;0), there are two stagnation-points in the flow but in the presence of blowing d (&amp;lt;0), there is only one stagnation-point in the flow which moves further and further up with increase in |d|. Suction is shown to increase the wall shear stress while blowing has an opposite effect. Streamlines for an USSP flow when u0(t) ≠ 0 are also plotted. It is found that in this case, the USSP flow is not in general separated.

Bifurcations of a weighted-residual integral boundary-layer model for nonlinear dynamics of falling liquid films

The nonlinear dynamics of thin liquid films falling on a vertical plane is investigated numerically using the first-order time-dependent weighted-residual integral boundary layer (WRIBL) equations derived by Ruyer-Quil and Manneville (2000). We find that sufficiently close to the stability threshold of the system with periodic boundary conditions, the emerging waves are of γ1-type. However, beyond a secondary bifurcation threshold, γ2-type waves emerge and can coexist with γ1 waves. The analysis of the WRIBL equations reveals the existence of both periodic traveling wave (TW) and aperiodic non-stationary wave (NSW) flows. The bifurcation structure of WRIBL equations is found to include three distinct regions: (i) linearly stable, (ii) bounded wave flows, (iii) reverse-flow solutions.

Re-entrant corner flows of UCM fluids: the initial formation of lip vortices

We discuss here a third type of boundary-layer structure that arises in steady planar flows at re-entrant corners for the upper convected Maxwell fluid. This structure extends the class of similarity solutions that are associated with the inviscid flow equations and hold in an outer core region local to the corner. In previous work ( Renardy 1995 J. Non-Newtonian Fluid Mech. 58 , 83–39 ; Evans 2005 Proc. R. Soc. A 461 , 117–142 ), single and double layer structures were used to the match outer core similarity solutions to wall boundary layers in which viscometric behaviour is obtained. Here a second double layer structure is discussed, which completes the range of validity for the core similarity solutions. This structure is fundamentally different to the single layer structure in that it only admits reverse flow solutions at the upstream wall, a situation of practical relevance to describe the initial formation of lip vortices in contraction flows.

Reverse flow solutions of the Falkner‐Skan equation for λ &gt; 1

Reverse flow solutions of the Falkner‐Skan equation for λ &gt; 1

Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equation

We present three accurate and efficient numerical schemes for solving the Falkner-Skan equation with positive or negative wall shear. Newton's method is employed, with the aid of the variational equations, in all the schemes and yields quadratic convergence. First, ordinary shooting is used to solve for the case of positive wall shear. Then a nonlinear eigenvalue technique is introduced to solve the inverse problem in which the wall shear is prescribed and the pressure distribution is to be determined. With this approach the reverse flow solutions (i.e., negative wall shear) are obtained. Finally, a parallel shooting method is employed to reduce the sensitivity of the convergence of the iterations to the initial estimates.