On dual solutions occurring in mixed convection in a porous medium
01 Jun 1986-Journal of Engineering Mathematics (Kluwer Academic Publishers)-Vol. 20, Iss: 2, pp 171-179
TL;DR: In this paper, the dual solutions to an equation, which arose previously in mixed convection in a porous medium, occuring for the parameter α in the range 0 < α < α0 are considered.
Abstract: The dual solutions to an equation, which arose previously in mixed convection in a porous medium, occuring for the parameter α in the range 0 < α < α0 are considered. It is shown that the lower branch of solutions terminates at α=0 with an essential singularity. It is also shown that both branches of solutions bifurcate out of the single solution at α=0 with an amplitude proportional to (α0-α)1/2. Then, by considering a simple time-dependent problem, it is shown that the upper branch of solutions is stable and the lower branch unstable, with the change in temporal stability at α=α0 being equivalent to the bifurcation at that point.
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TL;DR: In this paper, the simultaneous effects of normal transpiration through and tangential movement of a semi-infinite plate on self-similar boundary layer flow beneath a uniform free stream is considered.
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TL;DR: In this paper, the steady boundary-layer flow near the stagnation point on an impermeable vertical surface with slip that is embedded in a fluid-saturated porous medium is investigated.
Abstract: The steady boundary-layer flow near the stagnation point on an impermeable vertical surface with slip that is embedded in a fluid-saturated porous medium is investigated. Using appropriate similarity variables, the governing system of partial differential equations is transformed into a system of ordinary differential equations. This system is then solved numerically. The features of the flow and the heat transfer characteristics for different values of the governing parameters, namely, the Darcy–Brinkman, Γ, mixed convection, λ, and slip, γ, parameters, are analysed and discussed in detail for the cases of assisting and opposing flows. It is found that dual solutions exist for assisting flows, as well as those usually reported in the literature for opposing flows. A stability analysis of the steady flow solutions encountered for different values of the mixed convection parameter λ is performed using a linear temporal stability analysis. This analysis reveals that for γ = 0 (slip absent) and Γ = 1 the lower solution branch is unstable while the upper solution branch is stable.
507 citations
TL;DR: In this paper, the steady two-dimensional stagnation point flow of a micropolar fluid over a shrinking sheet in its own plane was analyzed and the features of the flow characteristics were analyzed and discussed.
Abstract: An analysis is carried out to study the steady two-dimensional stagnation-point flow of a micropolar fluid over a shrinking sheet in its own plane. The shrinking velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation point. The features of the flow characteristics are analyzed and discussed. Different from a stretching sheet, it is found that the solutions for a shrinking sheet are nonunique.
234 citations
Cites background or methods from "On dual solutions occurring in mixe..."
...A full stability analysis, along the lines described by Merkin (1985), Merrill et al. (2006), Paullet and Weidman (2007), and Harris et al. (2009), requires an unsteady flow, whereas our problem is a steady one....
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...Miklavc̆ic̆ and Wang (2006) studied the properties of a viscous flow due to a shrinking sheet with suction. The flow is unlikely to exist unless adequate suction on the boundary is imposed, since vorticity of the shrinking sheet is not confined within a boundary layer (Miklavc̆ic̆ and Wang, 2006). This problem was then extended by Sajid and Hayat (2009) and Fang and Zhang (2009) to magneto hydrodynamic (MHD) flow....
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...Miklavc̆ic̆ and Wang (2006) studied the properties of a viscous flow due to a shrinking sheet with suction....
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TL;DR: In this paper, the steady mixed convection boundary layer flow past a vertical flat plate embedded in a porous medium filled with nanofluids is studied using different types of nanoparticles as Cu (cuprom), Al2O3 (aluminium), and TiO2 (titanium)
234 citations
TL;DR: In this article, the behavior of the nanofluid was investigated for three different nanoparticles in the water-base fluid, namely copper, alumina and titania, and it was shown that a dual solution exists for negative values of the unsteadiness parameter A and, as it increases, the skin friction Cfr grows but the heat transfer rate Nur takes a decreasing trend.
212 citations
References
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TL;DR: In this paper, the authors considered the flow of a uniform stream past an impermeable vertical surface embedded in a saturated porous medium and which is supplying heat to the porous medium at a constant rate.
Abstract: The flow of a uniform stream past an impermeable vertical surface embedded in a saturated porous medium and which is supplying heat to the porous medium at a constant rate is considered. The cases when the flow and the buoyancy forces are in the same direction and when they are in opposite direction are discussed. In the former case, the flow develops from mainly forced convection near the leading edge to mainly free convection far downstream. Series solutions are derived in both cases and a numerical solution of the equations is used to describe the flow in the intermediate region. In the latter case, the numerical solution indicates that the flow separates downstream of the leading edge and the nature of the solution near this separation point is discussed.
200 citations
"On dual solutions occurring in mixe..." refers background in this paper
...It was shown in [ 1 ] that this equation had just one solution for a ~ a 0 (a 0 = 0.354), while for 0 1 ]....
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...In a previous paper [ 1 ] the author obtained the equation...
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01 Feb 1949
TL;DR: In this article, the velocity profiles for laminar mixing of a high-velocity stream with a region of fluid at rest have been made assuming that the Prandtl number is unity.
Abstract: A theoretical investigation of the velocity profiles for laminar mixing of a high-velocity stream with a region of fluid at rest has been made assuming that the Prandtl number is unity. A method which involves only quadratures is presented for calculating the velocity profile in the mixing layer for an arbitrary value of the free-stream Mach number. Detailed velocity profiles have been calculated for free-stream Mach numbers of 0, 1, 2, 3, and 5. For each Mach number, velocity profiles are presented for both a linear and a 0.76-power variation of viscosity with absolute temperature. The calculations for a linear variation are much simpler than those for a 0.76-power variation. It is shown that by selecting the constant of proportionality in the liner approximation such that it gives the correct value for the viscosity in the high-temperature part of the mixing layer, the resulting velocity profiles are in excellent agreement with those calculated by a 0.76-power variation.
100 citations
24 citations
"On dual solutions occurring in mixe..." refers background or result in this paper
...Merkin and Pop [4] (in fact the equation is the same though the boundary conditions are different) and was solved numerically in the same way as described in [ 4 ]....
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...Merkin and Pop [ 4 ] (in fact the equation is the same though the boundary conditions are different) and was solved numerically in the same way as described in [4]....
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22 citations
"On dual solutions occurring in mixe..." refers result in this paper
...The first case is similar in some respects to the behaviour of the reversed-flow solutions of the Falkner-Skan equation as the Falkner-Skan parameter fl-~ 0 from below, as treated by Brown and Stewartson [ 2 ]....
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TL;DR: In this article, the boundary-layer equations are solved numerically for mainstreams with O((1 − x)−α) near α > 1, where, for the similarity solution at x = 1, the outer boundary condition is approached through exponentially small terms, a straightforward expansion in powers of 1 − x is possible.
Abstract: The boundary-layer equations are solved numerically for mainstreams
\[
U(x) = x(1-x^2)^{-\alpha}\quad {\rm and}\quad U(x) = (1-x)^{-\alpha},
\]
which are both O((1 − x)−α) near x = 1. Series expansions are derived near x = 1. For α > 1, where, for the similarity solution at x = 1, the outer boundary condition is approached through exponentially small terms, a straightforward expansion in powers of 1 − x is possible. For 0 1, and again the outer boundary condition cannot be attained by the higher-order terms in the series. An outer expansion for this case is also derived.
17 citations
"On dual solutions occurring in mixe..." refers methods in this paper
...We require solutions with exponential decay, for we expect the algebraic decay terms to lead to difficulties in the higher-order terms [ 6 ] i.e....
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