Startseite Penetrative convection in a fluid saturated Darcy-Brinkman porous media with LTNE via internal heat source
Artikel Open Access

Penetrative convection in a fluid saturated Darcy-Brinkman porous media with LTNE via internal heat source

  • Amit Mahajan und Reena Nandal EMAIL logo
Veröffentlicht/Copyright: 31. Dezember 2018
Artikel downloaden (PDF)
Veröffentlichen auch Sie bei De Gruyter Brill
Manuskript einreichen Informationen für Autor*innen
Nonlinear Engineering
Aus der Zeitschrift Nonlinear Engineering Band 8 Heft 1

Abstract

The present work involves the study of penetrative convection in an incompressible fluid-saturated porous media with local thermal non-equilibrium. The onset of convection evaluated linearly and nonlinearly for the system influenced by heat extraction and heat generation. Darcy-Brinkman law is employed to model the momentum equation and four type of internal heat generating function are considered which leads to thermo-convective instability within the fluid layer. Linear analysis carried out by using normal mode technique and nonlinear stability analysis has been done by energy method. Due to heat generation within the fluid layer and heat extraction through boundary, the subcritical instability may exist with higher possibility. Effects of various parameters as: inter-phase heat transfer parameter, Darcy-Brinkman number, porosity-modified conductivity ratio, and heat parameter are explored on Darcy-Rayleigh number by Chebyshev pseudospectral method as numerical form and graphical form.

Keywords: heat source; energy method; Brinkman effect; Local thermal non-equilibrium

1 Introduction

Penetrative convection is a phenomenon which occurs due to convective instability arises by unstable equilibrium. When the region of unstable equilibrium is bounded by, the fluid in stable equilibrium, the convective motion penetrates into the neighboring region of stable equilibrium. There are two major reasons for this type of motion, one is velocity and another is non-vanishing tangential stresses. Penetrative convection occurs in many natural occurring astrophysical and geophysical phenomenon. For example, in star convection, causing the granulation pattern observed on the surface, heating, and cooling of lakes, thermocline formation, and radioactive decay and much more as described by Matthews (1988). An extensive literature is available on penetrative convection. Penetrative convection due to the heat sink and nonlinear density in an isotropic porous layer is examined by Carr & Putter (2003). In thermo-convective instability, radiation also plays an important role in a stratified system. Earth’s atmosphere is a good example to understand the phenomena of thermo-convective instability. The solar radiations heat up the surface and air very near to ground become warmer and lighter than upper heavy air and this creates gravitational instability. As convection starts, the warm air penetrates in stable regions. Effect of radiative heating through the lower boundary for a non-Boussinesq fluid layer is determined by Straughan (1991). An experimental study is done by Krishnamurti (1997) to observe the convective instability in a water layer with thymol blue in presence of a chemical concentration-dependent heat source. To understand the stability properties and scaling laws associated with a radiative-convective model, a study is presented by Larson (2000). Hill (2004) performed the stability analysis for a fluid layer which is thermo-convectively unstable due to concentration based internal heat source and absorption of radiations. When a water pool absorbs the solar radiations, it gets heated up and rises to the surface. In this process, it loses its heat to the atmosphere and remains at atmospheric temperature. To improve the ability of solar ponds to store the solar radiation, a stability analysis is done by Hill & Carr (2013a, 2013b).

In porous media convective heat transfer process is found to be having important applications in thermal insulation of building, drying process, geophysical systems, petroleum resources, heat exchangers, chemical reactors, nuclear waste etc. The literature reveals that heat transfer processes in porous medium gained a considerable momentum due to its applicability in various fields. The book by Nield & Bejan (2006) provides the diverse knowledge of this topic. For the study of fluid flow behavior in porous media Darcy’s law is employed but it has limitations in several aspects. It is valid essentially for creeping flow through a long and uniform porous medium of low hydraulic conductivity. But in many technological problems, the fluid flow is very fast and Darcy’s law is not enough to describe the behavior of fluid flow. For such situation, Brinkman (1947) suggested that the classical fractional term must be added to model the momentum equation, and this model known as Brinkman’s model. An enormous work is present in the literature which provides a good understanding of the topic [Postelnicu (2008), Sunil et al. (2010), Shivakumara & Dhananjaya (2015), Umavathi et al. (2017)]. Rees (2002) investigated that how highly porous boundaries (Brinkman) affect convection of Boussinesq fluid in a Darcy-porous medium. Very recently effect of a variety of internal heat generating source is observed by Nandal & Mahajan (2017) for a fluid-saturated Darcy-Brinkman porous media, to examine the effect of heat generation on the stability of the system.

Many investigations conducted on the flow in a porous media hypothesized that the solid matrix and fluid within the pore are in local thermal equilibrium. From an application point of view, the processes involving rapid heating and modeling of nuclear reactor, in fluid and solid matrix there exist temperature gradient so it is very uncertain the existence of local thermal equilibrium for such situations. A huge literature on local thermal non-equilibrium is provided by Straughan (2015). The onset of Darcy-Bénard convection using a local thermal non-equilibrium model is investigated by Banu &Rees (2002). Wang et al. (2014) analyzed the effect of thermal radiations within a porous medium incorporated with local thermal non-equilibrium, for a specific application of solar receivers. The thermo-convective stability in porous media with throughflow and local thermal non-equilibrium with heterogeneity is investigated by Nield & Kuznetsov (2015).

Several studies have considered the onset of penetrative convection in porous media with LTE but due attention has not been given to such studies in the context of LTNE. Such studies find its relevance in many technological and experimental problems. Intend of this work is stability analysis of a fluid layer saturated porous media with LTNE. It is considered that heat generation within the fluid layer and heat extraction through lower boundary induces the thermo-convective instability in the fluid layer. Four heat generating functions are considered for the stability analysis. Nonlinear stability results are compared with the linear instability results to observe whether there exists any subcritical instability or not. Effects of inter-phase heat transfer, Brinkman effect, internal heating and effects of heat extraction through the boundary, diffusivity and thermal conductivity of both the phases are considered to see the variation in critical Darcy-Rayleigh number by linear and nonlinear analysis and the corresponding values are achieved using Chebyshev tau-QZ method and are presented by graphs and numerical results.

2 Mathematical formulation

Here we consider an infinite layer of a thickness d of an incompressible fluid saturated in a homogeneous porous media. The fluid assumes to confined between the layer z = 0, d and respective boundaries conditions are∂Ts∂z=∂Tf∂z=yat z = 0 and Ts = Tf = TU at z = d, where represents the rate of heat extraction through the lower boundary. Boundaries are assumed impermeable and conducting with gravity g acts in negative z-direction and the Boussinesq approximation is employed to drive the density variation. Further, on the momentum equation modified Darcy model is employed and solid-fluid phases are with LTNE. Hence for the proposed model system of flow governing equations (Gasser & Kazimi 1976; Straughan 2015) is given as:

Momentum equation:

(1)ρfε∂q∂t=−∇P−μfKq+μf′∇2q+ρfαTfgk^

Mass balance equation:

(2)∇⋅q=0

Temperature equation (fluid phase):

(3)ε∂Tf∂t+q⋅∇Tf=εκ∇2Tf−hρcfTs−Tf+Qf

Temperature equation (solid phase):

(4)1−ερcs∂Ts∂t=1−εks∇2Ts+hTs−Tf+Qs

where q, ρf , cf , ρs , cs , P, μf , μ'f , t, ε, kf , ks , к and Kare filter velocity, density of the fluid, heat capacity of fluid, density of solid matrix, heat capacity of solid matrix, pressure, the dynamic viscosity of fluid, effective viscosity of fluid, time, porosity, the thermal conductivity of fluid, the thermal conductivity of solid, thermal diffusivity of fluid, and permeability, respectively.

Now, the steady state solution of governing equations is as follows:

(5)qb=0,Pb=Pz,ρb=ρz,Tb=T(z).

here, the basic state followed by subscript ‘b’. The temperature of the solid and fluid phase at the basic state is Tf = Ts. From the equation (3) and (4) it is found that Tf″=QfεκandTs″=Qs1−εks, so the internal heat generation is related by the relationQfεκ=Qs1−εks. This relation allows to define the internal heat generation function in one manner and here we restrict ourselves to Q f.

In a thermodynamical system, internal heat-generating functions are of many types. Roberts (1967) dealt with a constant heat source whereas Straughan (1991) examined the onset of convection in a horizontal layer in the context of three internal heat generating sources, i.e., constant, exponential and non-uniform heat source. In our study the following four type heat-generating functions, i.e. constant, linearly increasing, decreasing and non-uniform (Straughan 2004), are considered:

Case A: Qf = Q

Case B: Qf=Q12+zd

Case C:Qf=Q2+3z22d2−3zd

Case D: Qf=Q1+sin⁡2πzd+sin⁡4πzd

Now, the basic temperature function Tb for the four cases is given as:

(6)Tb=−Q2εκ(z2−d2)+yz+TU−ydcaseA−Qεκz36d+z24−5d212+yz+TU−ydcaseB−Qεκz48d2−z32d+z2−5d28+yz+TU−ydcaseC−Qεκz22−d24π2sin⁡2πzd−d216π2sin⁡4πzd+3d4πz−d22−3d24π+yz+TU−ydcaseD

To find out penetrative convection in the fluid layer, extremum of density is required and this can be followed from Tb. It, in turn, provide the range of heat parameter ξ [defined later in (12)] to observe the effect of penetrative convection. The range of heat parameter obtained from maximization or minimization of the density function is given by (0, 1) for all cases and beyond this range, convection is due to buoyancy forces. Figure 2 displayed the profile of basic temperature functions along with the fluid layer. It can be seen from figure 2 that the small values of ξ creates a high-temperature difference across the fluid layer, which enhances the convective movement from lower to upper parts of the fluid layer, and as the values of ξ increases, the temperature difference becomes small and the fluid layer achieves the stability in the lower parts. To check the system stability we introduced the following perturbed quantities:

Fig. 1 Profile of various heat sources within the fluid layer
Fig. 1

Profile of various heat sources within the fluid layer

Fig. 2 Basic temperature profiles Tb with fluid layer z
Fig. 2

Basic temperature profiles Tb with fluid layer z

(7)q=qb+q′,P=Pb+P′,Tf=Tb+θ,Ts=Tb+ϕ,ρ=ρb+ρ′,

where q' , P' , θ, ϕ, ρ' are the perturbed quantity in velocity, pressure, the temperature of the fluid phase, the temperature of the solid phase, and density respectively. Now, the following non-linear system of perturbation equations (after dropping prime) is obtained:

(8)ρfε∂q∂t=−∇P−μfKq+μf′∇2q+ρfαθgk^
(9)∇⋅q=0
(10)ε∂θ∂t+q⋅∇θ=−dTbdzw+εκ∇2θ+hρcfϕ−θ
(11)1−ερcs∂ϕ∂t=1−εks∇2ϕ−hϕ−θ

In order to analyze the stability of the above system, we first reduce the complexity of the system by using the following scaling parameter,

(12)z=dz∗,q=εkfρcfdq∗,P=μfεkfρcfKP∗,t=d2ρcfkft∗,H=hd2εkf,θ=Qfd2ρcfkfRaθ∗,ϕ=Qfd2ρcfkfRaϕ∗,Va=εμfcfd2kfK,Da=Kd2,A=ρcskfρcfks,D~a=μ′fKμfd2,Ra=ρfαgQfKd3μfεκ2,ψ=εkf1−εks

here, H is inter-phase heat transfer parameter, Va is Vadasz number, Da is Darcy number, A is diffusivity ratio, D̃a is Darcy-Brinkman number, Ra is Darcy-Rayleigh number, and Ψ is porosity-modified conductivity ratio.

Now, the system of non-dimensional perturbed equation (after dropping *) can be written as:

(13)1Va∂q∂t=−∇P−q+D~a∇2q+Raθk^
(14)∇⋅q=0
(15)∂θ∂t+q⋅∇θ=1εmzRaw+∇2θ+Hϕ−θ
(16)A∂ϕ∂t=∇2ϕ−ψHϕ−θ

where

(17)mz=H1z−H2ξcaseAH12z2+z−H2ξcaseBH112z3−32z2+2z−H2ξcaseCH1z−12πcos⁡2πz−14πcos⁡4πz+34π−H2ξcaseD

here, H1 = sign (Qf ) , H2 = sign (y) , andξ=yεκQfdis the heat parameter and the following are the corresponding boundaries for the above system of equation:

(18)q=0onz=0,1
(19)Θ=0,Φ=0onz=1
(20)DΘ=0,DΦ=0onz=0

3 Stability Analysis

For a stability analysis, there are mainly two techniques that may be employed, one is the method of linearized instability and another is the energy method. Infinitesimal small perturbations are assumed to the steady state, in linear analysis and so the impact of nonlinear terms becomes negligible and hence discarded from the set of partial differential equations. The linear theory only provides boundaries for instability; thus it is highly desirable to develop the nonlinear energy stability theory. For obtaining optimal nonlinear thresholds the idea of coupling parameter (Joseph 1965) can be employed for constructing energy functional. It will provide sharpen nonlinear thresholds as compared to the classical approach.

3.1 Linear Analysis

Non-linear terms are discarded from non-dimensional system of the perturbed equation (13) – (16) and normal modes (q,θ,ϕ)=qz,Θz,Φzexpσt+iaxx+ayy[Chandrasekhar 1981] introduced to tile the plane (x, y). Here σt represents the temporal growth rate and ax , ay shows the horizontal disturbance in the fluid layer. ConsideringD=ddzand to remove the pressure term operate curlcurl on (13), and the k̂ component is taken. So, we have the following linearized system:

(21)σVaD2−a2W=D~aD2−a22W−D2−a2W−Raa2Θ
(22)σΘ=RaεmzW+D2−a2Θ+HΦ−Θ
(23)σAΦ=D2−a2Φ−ψHΦ−Θ

Boundary conditions corresponding to the above system are given below [Straughan 2015]:

(24)W=0,DW=0,onz=0,1
(25)Θ=0,Φ=0onz=1
(26)DΘ=0,DΦ=0onz=0

For numerical analysis, Chebyshev-pseudospectral method is used. It an efficient numerical technique to find out eigenvalues and eigenfunctions in the system of continuum mechanics. The key point of this method is, one can calculate as many eigenvalues as required, not just the dominant one. Following the method laid out in Kaloni & Lou (2004), the domain of the method is [−1, 1] but the present domain is [0, 1], so a simple transformation is used to match the domain of the method i.e., z is transformed into 2z-1. After transformation the system of linearized equations is:

(27)σVa4D2−a2W=D~a4D2−a22W−4D2−a2W−Raa2Θ
(28)σΘ=RaεnzW+4D2−a2Θ+HΦ−Θ
(29)σAΦ=4D2−a2Φ−ψHΦ−Θ

where

(30)nz=H1z+12−H2ξcaseAH12z+12z+32−H2ξcaseBH1z+1−32z+122+12z+123−H2ξcaseCH1z+12+12πcos⁡πz−14πcos⁡2πz+34π−H2ξcaseD

Next, discretizing equation (27) – (29) following the Chebyshev scheme and hence can be written in form of a system of the matrix as:

(31)σA1X=A2X

where A1 and A2 are complex matrices, given as:

A1=1Va4D2−a2IOOOIOOOAI

and

A2=D~a4D2−a2I2−4D2−a2I−Raa2IORaεnZ4D2−a2I−HIHIOψHI4D2−a2I−ψHI

here and are identity and zero matrices. Results obtained by numerical simulation for the linear instability are presented in section 4.

3.2 Nonlinear stability analysis

Adopting the standard nonlinear energy approach in the stability measure, we first multiply eqs. (13) – (16) with q, θ, ϕ respectively and integrated over V to have:

(32)12Vaddtq2=Rawθ−q2−D~a∇q2
(33)12ddtθ2=Raεmzwθ−∇θ2+Hθϕ−Hθ2
(34)A2ddtϕ2=−∇ϕ2−ψHϕ2+ψHϕθ

To attain steady states (5) nonlinear stability, the energy functional, formed by thermal energy and kinetic energy of perturbations of fluid and solid phase, using coupling parameters λ1 and λ2, is defined as given below:

(35)Et=12θ2+λ1A2ϕ2+λ22Vaq2

Variation in energy functional with respect to variation in time is given by:

(36)ddtEt=12ddtθ2+λ1A2ddtϕ2+λ22Vaddtq2

Using equations (32) – (34) in (36), we get

(37)ddtEt=I−D

where I represents the energy production terms in the system and has the form

(38)I=Raϵ〈m(z)wθ〉+λ2Ra〈wθ〉+H〈ϕθ〉+λ1ψH〈ϕθ〉

and D represents the energy dissipating terms that are as follows:

(39)D=∇θ2+Hθ2+λ1∇ϕ2+ψHλ1ϕ2+λ2q2+D~aλ2∇q2

Now, put

(40)ℜ=maxΩ⁡ID,

where being admissible solution space, such as = {q, θ, ϕ|q, θ, ϕ ϵ L2 [0, 1]. Let, β = 1−ℜ , from the equations (37) it obtains:

(41)ddtEt≤−βD

Next, using Poincaré’s inequality and definition of D, we have the following inequality:

(42)D≥H+π2θ2+ψH+π2λ1ϕ2+1+D~aπ2λ2q2

Combining equations (41) and(42), one shows that for a constant k > 0

(43)ddtEt≤−βkEt

where k=min2H+π2;2H+π2A;2Va1+D~aπ2.

Equations (43) implies that E (t) ≤ E (0) exp (−βkt) which shows that E decays exponentially, yielding the nonlinear stability for θ, ϕ and q.

To obtain nonlinear stability bounds, the variational problem (40) must be solved. Therefore consider ℜ = 1, because it provides the nonlinear stability thresholds so the Euler-Lagrange equations for the maximization problem (40), after using rescalingϕ=1λ1ϕ,q=1λ2qare

(44)2q−2D~a∇2q−Raελ2mzθk^−λ2Raθk^=2∇s
(45)−2∇2θ+2Hθ−Raελ2mzw−λ2Raw−Hλ11+ψλ1ϕ=0
(46)−2∇2ϕ+2ψHϕ−Hλ11+ψλ1θ=0

as q is solenoidal so Lagrange multiplier s is introduced. Next, we operate curlcurl on (44) and consider k̂ component, we have

(47)−2∇2w+2D~a∇4w+Raελ2mz∇12θ+λ2Ra∇12θ=0

Next, the following tiling form is introduced to tile the plane:

(48)w,θ,ϕ=Wz,Θz,Φzexpiaxx+ayy

So, from equations (45), (46) and (47) we have the following system of equations:

(49)2D2−a2W−2D~aD2−a22W+Raελ2mza2Θ+λ2Raa2Θ=0
(50)Raελ2mzW+λ2RaW+2D2−a2Θ−2HΘ+Hλ11+ψλ1Φ=0
(51)Hλ11+ψλ1Θ+2D2−a2Φ−2ψHΦ=0

and boundary conditions correspond to the above system are as follows:

(52)W=0,DW=0,onz=0,1
(53)Θ=0,Φ=0onz=1
(54)DΘ=0,DΦ=0onz=0

Now, following the method explained for linear analysis, the domain is transformed [0, 1] to [−1, 1]. Now, the system of equations (49) – (54) forms an eigenvalue problem of order eight and it is utilized to localize the nonlinear critical Darcy-Rayleigh number RaE, given byRa = max min Ra(λ1, λ2, a). Results derived by Cheby-λ1,λ2a

shev method are presented in section 4.

4 Result and Discussion

The onset of convection in a Darcy-Brinkman porous medium is investigated for a local thermal non-equilibrium model in the presence of internal heat source. The various heat generating functions are taken into consideration which induces instability within the fluid layer when the heat is extracted through the boundary at a constant rate. Stability analysis performed by normal mode technique and energy method and Chebyshev-pseudospectral scheme used to solve generalized eigenvalue problem. A comparison of critical values of Darcy-Rayleigh number and wave number with respect to various parameters is performed for all heat generating functions and corresponding results depicted numerically and

graphically. For the validation of the results, the model is reduced to the LTE limits and in the absence of heat extraction through the lower boundary the results are compared with the existing results and are presented in Table 1. It can be pointed out here that the obtained results appear with the multiple of an additional factor 1/ε, it is because the Rayleigh number is defined in terms of thermal diffusivity к = kf / (ρc)f of the fluid rather than in terms of [εkf + (1 − ") ks(c)f[Kuznetsov & Nield (2010)].

Table 1

Comparison of the present study with the existing results.

Internal Heat ParameterPresent StudyExisting Results
ξ = 0RaL/ε = 471.385 (for case A)RaL = 471.38466[Borujerdi et al. (2008)]
ξ >> 1RaL/ε = 39.237 (for case A)RaL = 39.478 [Nield and Bejan (2013)]

We start the discussion in the context of heat parameter ξ ranges between (0, 1) for all heat supply functions and provide facts and figures concerning penetrative convection. From the temperature profile presented in Figure 2, it can be seen that lower part of the fluid layer has a high temperature for small values of ξ (heat parameter) but as ξ increases, the upper part of the fluid layer has high temperature thus the stabilizing effect of ξ is expected. In Table 2, 3, 4, 5, obtained values of critical Darcy-Rayleigh numbers by linear approach (RaL) and nonlinear approach (RaE) with variation in heat parameter (ξ),

Table 2

variation of critical Rayleigh number (RaL , RaE) with internal heat parameter (ξ) at fixed values of D̃a, H, A, ψ for case A of the constant heat source

H = 1; A = 0.5; ψ = 0.1H = 10; A = 0.5; ψ = 0.1
ξacLRaLacERaEacLRaLacERaE
˜Da = 0
0.102.7230.222.7229.093.6045.223.5943.71
0.202.8641.062.8338.583.7659.053.7156.19
0.303.2260.933.0655.034.0782.033.9276.40
0.403.9398.103.5085.734.60122.354.28111.67
0.504.79169.004.20147.005.33198.684.87179.10
0.605.94327.295.10283.866.41365.935.78326.32
˜Da = 0.2
0.102.82301.672.82295.333.38457.063.38446.72
0.202.86409.072.87394.093.41609.663.42586.65
0.302.98626.062.97579.653.50902.223.49838.23
0.403.341199.583.21999.193.761601.823.661369.13
0.504.202965.673.682146.544.443596.903.992713.49
0.605.168637.224.015122.765.389885.044.226120.99
˜Da = 0.4
0.102.81568.462.82556.663.36862.583.37843.22
0.202.86770.982.86743.033.401151.163.401107.91
0.302.971181.012.961093.973.481706.063.471584.97
0.403.322274.903.191891.553.733045.343.632596.78
0.504.185711.083.674092.494.416930.603.965178.03
0.605.1416842.823.989780.545.3619282.934.1811700.66
Table 3

variation of critical Rayleigh number (RaL , RaE) with internal heat parameter (ξ) at fixed values of D̃a, H, A, for case B of linearly increasing heat generation

H = 1; A = 0.5; ψ = 0.1H = 10; A = 0.5; ψ = 0.1
ξacLRaLacERaEacLRaLacERaE
˜Da = 00.102.8541.862.8439.303.7760.233.7257.11
0.203.2462.653.0956.294.1383.903.9677.65
0.304.04100.823.5787.434.73124.374.38112.63
0.404.98169.124.35146.125.54196.605.02175.92
0.506.17305.925.42264.676.65339.425.99301.07
0.607.95623.805.70465.498.34666.126.42532.86
˜Da = 0.20.102.86424.262.87408.003.42632.543.42606.97
0.202.99661.342.99608.223.51951.003.51876.73
0.303.411309.323.251067.323.821730.713.691450.40
0.404.343228.633.792293.024.583877.284.082868.81
0.505.328748.984.125324.855.559940.854.336303.40
0.606.8629199.234.039264.567.0431632.864.2411000.73
˜Da = 0.40.102.86799.742.87769.413.401194.603.411146.56
0.202.981248.042.981148.333.491799.173.491658.56
0.303.392486.953.232022.543.793295.603.672753.86
0.404.326232.583.764380.164.557488.974.045484.60
0.505.3117087.634.0810187.415.5319422.314.2912078.41
0.606.8557561.423.9917686.437.0362367.064.1921026.02
Table 4

Variation of critical Rayleigh number (RaL , RaE) with internal heat parameter (ξ) at fixed values of D̃a, H, A, ψ for case C of decreasing heat source

H = 1; A = 0.5; ψ = 0.1H = 10; A = 0.5; ψ = 0.1
ξacLRaLacERaEacLRaLacERaE
˜Da = 00.102.6320.982.6420.593.4832.473.4731.90
0.202.6825.992.6825.273.5339.563.5238.61
0.302.7633.922.7532.473.6250.253.5948.53
0.402.9447.802.8744.523.8067.673.7264.33
0.503.3974.693.1366.934.1498.683.9591.95
0.604.11130.023.61114.124.68160.274.33146.91
˜Da = 0.20.102.78208.982.78207.023.35321.043.35317.78
0.202.79256.272.80252.713.36391.053.36385.30
0.302.82330.702.83323.323.38499.233.38487.77
0.402.88463.792.88444.983.42686.593.42659.32
0.503.03758.313.00692.353.531076.413.51991.80
0.603.541659.523.291333.663.882149.493.701792.55
˜Da = 0.40.102.77393.752.78390.093.33605.633.34599.53
0.202.79482.822.80476.213.34737.753.35727.00
0.302.82623.032.82609.313.37942.013.37920.55
0.402.87873.942.88838.853.411296.263.411244.94
0.503.021430.702.991306.653.512035.893.491875.24
0.603.523156.213.292526.643.854095.243.683401.07
Table 5

Variation of critical Rayleigh number (RaL , RaE) with internal heat parameter (ξ) at fixed values of D̃a, H, A, for case D of nonuniform heat generation

H = 1; A = 0.5; ψ = 0.1H = 10; A = 0.5; ψ = 0.1
ξacLRaLacERaEacLRaLacERaE
˜Da = 00.102.6216.342.6216.093.4525.473.4525.10
0.202.6419.252.6418.853.4829.693.4829.14
0.302.6823.382.6822.693.5335.493.5234.61
0.402.7629.592.7428.333.6243.873.5842.40
0.502.9039.722.8437.173.7656.743.6854.14
0.603.2257.713.0352.284.0078.003.8573.19
˜Da = 0.20.102.77162.622.77161.443.34250.683.34248.71
0.202.78189.962.78188.103.35291.573.35288.51
0.302.80228.242.80225.073.36348.183.36343.09
0.402.82285.492.82279.473.38431.483.38422.15
0.502.86379.892.87366.553.41565.293.41545.71
0.602.96561.152.95523.933.48810.673.47760.81
˜Da = 0.40.102.77306.402.77304.203.33472.873.33469.19
0.202.78357.902.78354.443.33550.023.34544.30
0.302.79429.992.80424.103.34656.853.35647.32
0.402.81537.852.82526.653.36814.123.37796.64
0.502.86715.752.86690.903.391066.993.401030.22
0.602.951057.912.95988.203.461531.803.451437.56

Darcy-Brinkman number (D̃a), at fixed values of porosity-modified conductivity ratio (ψ), inter-phase heat parameter (H), diffusivity ratio (A), are provided. It is found that for a Darcy model (D̃a = 0), as ξ increases, the critical value of Darcy-Rayleigh number increases. Increase in value of heat parameter adds on the stability of the system. The heat extraction through the lower boundary has a counter effect to the internal heat generation, which slows down the heat transfer and results in a delay in convection. The Darcy-Brinkman number has a significant influence on the stability of the system, as the value of D̃a increases, a substantial change occurs in the value of the critical Rayleigh number for both linear and nonlinear analysis. Again, from tables, It is also observed that increase in the value of H, the value of the critical Rayleigh number increases and the onset of convection delays. Cell size gets

contracted for heat supply functions for increasing values of ξ and H.

The neutral stability curves for Rayleigh number and wave number for distinct parameters are shown in Figures 3–6. A neutral stability curve, in the expression of critical Darcy-Rayleigh number, provides the stability criterion. In the region above the linear plot, system’s behavior is unstable and in the region below the nonlinear plot, its behavior is stable. These curves show that for a range of wave number the linear and nonlinear thresholds are so close that they are indistinguishable graphically, signifying the suitability of linear theory to predict the nature of the onset of convection. But after a certain range of wave number, nonlinear threshold leave a region of potential subcritical instability. Further, from figure 3 it is observed that as the value of Darcy-Brinkman number increases, quantitative change occurs in the value of the critical Rayleigh number but the qualitative behavior remains the same. Increase in D̃a is related to increasing effective viscosity of the fluid that retards the fluid flow and hence convection delays. Figure 4, represented the neutral curves for changing values of ξ . From these, it is observed that as ξ increases, the critical value of Darcy-Rayleigh number increases. Increase in value of ξ leads to the system stable, the reason for this behavior can be understood from the figure 2 which shows that for the small values of ξ, system is more unstable in the lower parts of the fluid layer due to the large temperature difference across the layer but as the value of ξ increases, the temperature difference across the layer decreases which stabilize the fluid layer. As a result, with increasing values of ξ, convection slows down.

Fig. 3 Neutral stability curves for case A, B, C and D at different values of D̃a and fixed values of H, ξ ,ψ , A
Fig. 3

Neutral stability curves for case A, B, C and D at different values of D̃a and fixed values of H, ξ ,ψ , A

Fig. 4 Neutral stability curves for case A, B, C and D at different values of ξ and fixed values of H, Ψ,D̃a, A
Fig. 4

Neutral stability curves for case A, B, C and D at different values of ξ and fixed values of H, Ψ,D̃a, A

Fig. 5 Neutral stability curves for case A, B, C and D at different values of H and fixed values of Ψ , ξ ,D̃a, A
Fig. 5

Neutral stability curves for case A, B, C and D at different values of H and fixed values of Ψ , ξ ,D̃a, A

Fig. 6 Neutral stability curves for case A, B, C and D at different values of Ψ and fixed values of H, ξ ,D̃a, A
Fig. 6

Neutral stability curves for case A, B, C and D at different values of Ψ and fixed values of H, ξ ,D̃a, A

The increase in the values of the inter-phase heat transfer parameter has a significant impact on the critical Rayleigh number as shown in Figure 5. For small values of H, no significant heat transfer takes place between the solid and fluid phase and hence the properties of the solid phase does not affect the convection and system is in a strong LTNE situation. Next, as H increases, we found a noticeable change in the critical values of the Rayleigh number because the solid properties affect the onset of convcetion. Figure 6 represents that as the value of increases, the value of critical Rayleigh number decreases; this is because of the increase in the value of leads to substantial transfer of heat through the solid and fluid phases. For very small values of H, the critical Rayleigh number is independent of the values of and approaches the LTE results. On the other hand, for comparatively large values of H (for graphical results the value of H is taken equal to 5), a significant change in the values of critical Rayleigh number can be observed with the increasing values of.

5 Conclusion

This work is a detail presentation of stability analysis of a fluid saturating porous medium by linear and nonlinear analyses for four heat supply functions within the fluid layer with heat extraction through the lower boundary. It is observed that the system is more stable at large values of heat parameter as compared to small values, i.e. higher the heat parameter, higher the stability of the system. The two theories (linear and nonlinear) do not show any resemblance in results, it implies the region of subcritical instability exists. For all heat generating functions, it is observed that cell size gets affected with variation in values heat parameter, Darcy-Brinkman number, and interphase heat transfer parameter. Increasing values of Darcy-Brinkman number, heat parameter, and inter-phase heat transfer parameter delays the onset of convection so these parameters found to be having stabilizing effect on the system. The increase in the value of porosity-modified conductivity ratio hastens the onset of convection and has a destabilizing effect on the stability of the system.

Nomenclature

A Diffusivity ratio

ac Critical wave number

acL Critical wave number for linear analysis

acE Critical wave number by nonlinear analysis

Da Darcy number

a Darcy-Brinkman number

d The depth of the fluid layer

g Gravity

H Inter-phase heat transfer parameter

H1 Heat generation within the fluid layer

H2 Heat extraction through lower boundary

K Permeability

P Pressure

Qf Internal heat source for fluid phase

Qs Internal heat source for solid phase

q Filter velocity

Ra Rayleigh number

RaL Rayleigh number by linear analysis

RaE Rayleigh number by nonlinear analysis

Tb Basic temperature

Tf Fluid phase temperature

Ts Solid phase temperature

TU Upper boundary temperature

t Time

Va Vadasz number

Greek Symbols

ξ Porosity

к Thermal diffusivity

ks Thermal conductivity of the solid phase

λ1, λ2 Coupling parameters

μf Dynamic viscosity of the fluid

μ'f Effective viscosity of the fluid

ρf Density of fluid

ρs Density of solid matrix

ρ' Perturbation in density

θ Perturbation in fluid phase temperature

ϕ Perturbation in solid phase temperature

Ψ Porosity-modified conductivity ratio

ξ Heat parameter

y Rate of heat extraction through lower boundary

References

[1] Banu, N., Rees, D.A.S.: Onset of Darcy-Bénard convection using a thermal non-equilibrium model, Int. J. Heat Mass Transfer, 2002, 45, 2221-2228.10.1016/S0017-9310(01)00331-3Suche in Google Scholar

[2] Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res., 1947, 1, 27-34.10.1007/BF02120313Suche in Google Scholar

[3] Borujerdi, A.N., Noghrehabadi, A.R. and Rees, D.A.S., Influence of Darcy number on the onset of convection in a porous layer with a uniform heat source, Int. J. Therm. Sci., 2008, 47, 1020-1025.10.1016/j.ijthermalsci.2007.07.014Suche in Google Scholar

[4] Carr, M., Putter, S.D.: Penetrative convection in a horizontally isotropic porous layer, Continuum Mech. Thermodyn., 2003, 15, 33-43.10.1007/s00161-002-0102-4Suche in Google Scholar

[5] Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability, Dover, New York, 1981.10.1063/1.3058072Suche in Google Scholar

[6] Gasser, R.D., Kazimi, M.S.: Onset of convection in a porous medium with internal heat generation, J. Heat Tranfer., 1976, 98, 49-54.10.1115/1.3450468Suche in Google Scholar

[7] Hill, A.A.: Penetrative convection induced by the absorption of radiation with a nonlinear heat source, Dyn. Atmos. Oceans, 2004, 38, 57-67.10.1016/j.dynatmoce.2004.03.002Suche in Google Scholar

[8] Hill, A.A., Carr, M.: The influence of a fluid-porous interface on the solar pond stability, Advances in water resources, 2013, 52, 1-6.10.1016/j.advwatres.2012.08.009Suche in Google Scholar

[9] Hill, A.A., Carr, M.: Stabilizing solar ponds by utilizing porous material, Advances in water resources, 2013, 60, 1-6.10.1016/j.advwatres.2013.07.005Suche in Google Scholar

[10] Joseph, D.D.: On the stability of Boussinesq equations, Arch. Rational Mech. Anal., 1965, 20, 59-71.10.1007/BF00250190Suche in Google Scholar

[11] Kaloni, P.N., Lou, J.X.: Convective instability of magnetic fluids, Phys. Rev. E, 2004, 70, 1-12.10.1103/PhysRevE.70.026313Suche in Google Scholar PubMed

[12] Krishnamurti, R.: Convection induced by selective absorption of radiation: a laboratory model of conditional instability, Dyn. Atmos. Oceans, 1997, 27, 367-382.10.1016/S0377-0265(97)00020-1Suche in Google Scholar

[13] Kuznetsov, A.V., Nield, D.A.: Effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid, Transp Porous Med., 2010, 83,425-436.10.1007/s11242-009-9452-8Suche in Google Scholar

[14] Larson, V.E.: Stability properties of and scaling laws for a dry radiative-convective atmosphere, Q.J.R. Met. Soc., 2000, 126, 145-171.10.1002/qj.49712656208Suche in Google Scholar

[15] Matthews, P.C.: A model for the onset of penetrative convection, J. Fluid Mech., 1988, 188, 571-583.10.1017/S0022112088000850Suche in Google Scholar

[16] Nandal, R., Mahajan, A.: Linear and nonlinear stability analysis of a Horton-Rogers-Lapwood problem with an internal heat source and Brinkman effects, Transp Porous Med, 2017, 117, 261-280.10.1007/s11242-017-0832-1Suche in Google Scholar

[17] Nield, D.A., Bejan, A.: Convection in Porous Media, Springer, Berlin, 2013.10.1007/978-1-4614-5541-7Suche in Google Scholar

[18] Nield, D.A., Kuznetsov, A.V.: Local thermal non-equilibrium and heterogeneity effects on the onset of convection in a layered porous medium with vertical throughflow, J. Porous Med., 2015, 18, 125-136.10.1615/JPorMedia.v18.i2.40Suche in Google Scholar

[19] Rees, D.A.S.: The onset of Darcy-Brinkman convection in a porous layer: an asymptotic analysis, Int. J. Heat Mass Transfer., 2002, 45, 2213-2220.10.1016/S0017-9310(01)00332-5Suche in Google Scholar

[20] Roberts, P.H.: Convection in horizontal layers with internal heat generation. Theory, J. Fluid Mech., 1967, 30, 33-49.10.1017/S0022112067001284Suche in Google Scholar

[21] Straughan, B.: Continuous dependence on the heat source and non-linear stability in penetrative convection, Int. J. Non-Linear Mechanics, 1991, 26, 221-231.10.1016/0020-7462(91)90053-VSuche in Google Scholar

[22] Straughan, B.: Convection caused by radiation through the layer, IMA J. Appl. Math., 1991, 46, 211-216.10.1093/imamat/46.3.211Suche in Google Scholar

[23] Straughan, B.: The Energy Method, Stability, and Nonlinear Convection. Springer, Berlin, 2004.10.1007/978-0-387-21740-6_6Suche in Google Scholar

[24] Straughan, B.: Convection with Local Thermal Non-Equilibrium and Microfluidic Effects, Springer, 2015.10.1007/978-3-319-13530-4Suche in Google Scholar

[25] Wang, P., Vafai, K., Liu, D.Y.: Analysis of radiative effect under local thermal non-equilibrium conditions in porous media-application to a solar air receiver, Numer. Heat Transfer, Part A, 2014, 65, 931-948.10.1080/10407782.2013.850917Suche in Google Scholar

[26] Postelnicu, A.: The onset of Darcy-Brinkman convection using a thermal nonequilibrium model. Part II, Int. J. Therm. Sci, 2008, 47, 1587-1594.10.1016/j.ijthermalsci.2008.01.002Suche in Google Scholar

[27] Sunil, Sharma, P., Mahajan, A.: Onset of Darcy-Brinkman double-diffusive convection in a magnetized ferrofluid layer using a thermal non-equilibrium model: a nonlinear stability analysis, J. Geophys. Eng., 2010, 7, 417-430.10.1088/1742-2132/7/4/009Suche in Google Scholar

[28] Shivakumara, I.S., Dhananjaya, M.: Penetrative Brinkman convection in an anisotropic porous layer saturated by a nanofluid, Ain Shams Engineering Journal, 2015, 6, 703-713.10.1016/j.asej.2014.12.005Suche in Google Scholar

[29] Umavathi, J.C., Sheremet, M.A.: Onset of double-diffusive convection of a sparsely packed micropolar fluid in a porous medium layer saturated with a nanofluid, Microfluid Nanofluidics, 2017, 21, 128.10.1007/s10404-017-1965-9Suche in Google Scholar

Received: 2018-03-19
Revised: 2018-07-18
Accepted: 2018-09-30
Published Online: 2018-12-31
Published in Print: 2019-01-28

© 2019 A. Mahajan and R. Nandal, published by De Gruyter.

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Artikel in diesem Heft

  1. Chebyshev Operational Matrix Method for Lane-Emden Problem
  2. Concentrating solar power tower technology: present status and outlook
  3. Control of separately excited DC motor with series multi-cells chopper using PI - Petri nets controller
  4. Effect of boundary roughness on nonlinear saturation of Rayleigh-Taylor instability in couple-stress fluid
  5. Effect of Heterogeneity on Imbibition Phenomena in Fluid Flow through Porous Media with Different Porous Materials
  6. Electro-osmotic flow of a third-grade fluid past a channel having stretching walls
  7. Heat transfer effect on MHD flow of a micropolar fluid through porous medium with uniform heat source and radiation
  8. Local convergence for an eighth order method for solving equations and systems of equations
  9. Numerical techniques for behavior of incompressible flow in steady two-dimensional motion due to a linearly stretching of porous sheet based on radial basis functions
  10. Influence of Non-linear Boussinesq Approximation on Natural Convective Flow of a Power-Law Fluid along an Inclined Plate under Convective Thermal Boundary Condition
  11. A reliable analytical approach for a fractional model of advection-dispersion equation
  12. Mass transfer around a slender drop in a nonlinear extensional flow
  13. Hydromagnetic Flow of Heat and Mass Transfer in a Nano Williamson Fluid Past a Vertical Plate With Thermal and Momentum Slip Effects: Numerical Study
  14. A Study on Convective-Radial Fins with Temperature-dependent Thermal Conductivity and Internal Heat Generation
  15. An effective technique for the conformable space-time fractional EW and modified EW equations
  16. Fractional variational iteration method for solving time-fractional Newell-Whitehead-Segel equation
  17. New exact and numerical solutions for the effect of suction or injection on flow of nanofluids past a stretching sheet
  18. Numerical investigation of MHD stagnation-point flow and heat transfer of sodium alginate non-Newtonian nanofluid
  19. A New Finance Chaotic System, its Electronic Circuit Realization, Passivity based Synchronization and an Application to Voice Encryption
  20. Analysis of Heat Transfer and Lifting Force in a Ferro-Nanofluid Based Porous Inclined Slider Bearing with Slip Conditions
  21. Application of QLM-Rational Legendre collocation method towards Eyring-Powell fluid model
  22. Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations
  23. MHD nonaligned stagnation point flow of second grade fluid towards a porous rotating disk
  24. Nonlinear Dynamic Response of an Axially Functionally Graded (AFG) Beam Resting on Nonlinear Elastic Foundation Subjected to Moving Load
  25. Swirling flow of couple stress fluid due to a rotating disk
  26. MHD stagnation point slip flow due to a non-linearly moving surface with effect of non-uniform heat source
  27. Effect of aligned magnetic field on Casson fluid flow over a stretched surface of non-uniform thickness
  28. Nonhomogeneous porosity and thermal diffusivity effects on stability and instability of double-diffusive convection in a porous medium layer: Brinkman Model
  29. Magnetohydrodynamic(MHD) Boundary Layer Flow of Eyring-Powell Nanofluid Past Stretching Cylinder With Cattaneo-Christov Heat Flux Model
  30. On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain
  31. An adaptive mesh method for time dependent singularly perturbed differential-difference equations
  32. On stretched magnetic flow of Carreau nanofluid with slip effects and nonlinear thermal radiation
  33. Rational exponential solutions of conformable space-time fractional equal-width equations
  34. Simultaneous impacts of Joule heating and variable heat source/sink on MHD 3D flow of Carreau-nanoliquids with temperature dependent viscosity
  35. Effect of magnetic field on imbibition phenomenon in fluid flow through fractured porous media with different porous material
  36. Impact of ohmic heating on MHD mixed convection flow of Casson fluid by considering Cross diffusion effect
  37. Mathematical Modelling Comparison of a Reciprocating, a Szorenyi Rotary, and a Wankel Rotary Engine
  38. Surface roughness effect on thermohydrodynamic analysis of journal bearings lubricated with couple stress fluids
  39. Convective conditions and dissipation on Tangent Hyperbolic fluid over a chemically heating exponentially porous sheet
  40. Unsteady Carreau-Casson fluids over a radiated shrinking sheet in a suspension of dust and graphene nanoparticles with non-Fourier heat flux
  41. An efficient numerical algorithm for solving system of Lane–Emden type equations arising in engineering
  42. New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind
  43. Numerical Study of Viscoelastic Micropolar Heat Transfer from a Vertical Cone for Thermal Polymer Coating
  44. Analysis of Bifurcation and Chaos of the Size-dependent Micro–plate Considering Damage
  45. Non-Similar Comutational Solutions for Double-Diffusive MHD Transport Phenomena for Non-Newtnian Nanofluid From a Horizontal Circular Cylinder
  46. Mathematical model on distributed denial of service attack through Internet of things in a network
  47. Postbuckling behavior of functionally graded CNT-reinforced nanocomposite plate with interphase effect
  48. Study of Weakly nonlinear Mass transport in Newtonian Fluid with Applied Magnetic Field under Concentration/Gravity modulation
  49. MHD slip flow of chemically reacting UCM fluid through a dilating channel with heat source/sink
  50. A Study on Non-Newtonian Transport Phenomena in Mhd Fluid Flow From a Vertical Cone With Navier Slip and Convective Heating
  51. Penetrative convection in a fluid saturated Darcy-Brinkman porous media with LTNE via internal heat source
  52. Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity
  53. Semitrailer Steering Control for Improved Articulated Vehicle Manoeuvrability and Stability
  54. Thermomechanical nonlinear stability of pressure-loaded CNT-reinforced composite doubly curved panels resting on elastic foundations
  55. Combination synchronization of fractional order n-chaotic systems using active backstepping design
  56. Vision-Based CubeSat Closed-Loop Formation Control in Close Proximities
  57. Effect of endoscope on the peristaltic transport of a couple stress fluid with heat transfer: Application to biomedicine
  58. Unsteady MHD Non-Newtonian Heat Transfer Nanofluids with Entropy Generation Analysis
  59. Mathematical Modelling of Hydromagnetic Casson non-Newtonian Nanofluid Convection Slip Flow from an Isothermal Sphere
  60. Influence of Joule Heating and Non-Linear Radiation on MHD 3D Dissipating Flow of Casson Nanofluid past a Non-Linear Stretching Sheet
  61. Radiative Flow of Third Grade Non-Newtonian Fluid From A Horizontal Circular Cylinder
  62. Application of Bessel functions and Jacobian free Newton method to solve time-fractional Burger equation
  63. A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform
  64. A multiple-step adaptive pseudospectral method for solving multi-order fractional differential equations
  65. A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses
  66. The expa function method and the conformable time-fractional KdV equations
  67. Comment on the paper: “Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet, M.R. Krishnamurthy, B.J. Gireesha, B.C. Prasannakumara, and Rama Subba Reddy Gorla, Nonlinear Engineering 2016, 5(3), 147-159”
  68. Three-Dimensional Boundary layer Flow and Heat Transfer of a Fluid Particle Suspension over a Stretching Sheet Embedded in a Porous Medium
  69. MHD three dimensional flow of Oldroyd-B nanofluid over a bidirectional stretching sheet: DTM-Padé Solution
  70. MHD Convection Fluid and Heat Transfer in an Inclined Micro-Porous-Channel
https://doi.org/10.1515/nleng-2018-0053
Schlagwörter für diesen Artikel
heat source; energy method; Brinkman effect; Local thermal non-equilibrium
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Chebyshev Operational Matrix Method for Lane-Emden Problem
  2. Concentrating solar power tower technology: present status and outlook
  3. Control of separately excited DC motor with series multi-cells chopper using PI - Petri nets controller
  4. Effect of boundary roughness on nonlinear saturation of Rayleigh-Taylor instability in couple-stress fluid
  5. Effect of Heterogeneity on Imbibition Phenomena in Fluid Flow through Porous Media with Different Porous Materials
  6. Electro-osmotic flow of a third-grade fluid past a channel having stretching walls
  7. Heat transfer effect on MHD flow of a micropolar fluid through porous medium with uniform heat source and radiation
  8. Local convergence for an eighth order method for solving equations and systems of equations
  9. Numerical techniques for behavior of incompressible flow in steady two-dimensional motion due to a linearly stretching of porous sheet based on radial basis functions
  10. Influence of Non-linear Boussinesq Approximation on Natural Convective Flow of a Power-Law Fluid along an Inclined Plate under Convective Thermal Boundary Condition
  11. A reliable analytical approach for a fractional model of advection-dispersion equation
  12. Mass transfer around a slender drop in a nonlinear extensional flow
  13. Hydromagnetic Flow of Heat and Mass Transfer in a Nano Williamson Fluid Past a Vertical Plate With Thermal and Momentum Slip Effects: Numerical Study
  14. A Study on Convective-Radial Fins with Temperature-dependent Thermal Conductivity and Internal Heat Generation
  15. An effective technique for the conformable space-time fractional EW and modified EW equations
  16. Fractional variational iteration method for solving time-fractional Newell-Whitehead-Segel equation
  17. New exact and numerical solutions for the effect of suction or injection on flow of nanofluids past a stretching sheet
  18. Numerical investigation of MHD stagnation-point flow and heat transfer of sodium alginate non-Newtonian nanofluid
  19. A New Finance Chaotic System, its Electronic Circuit Realization, Passivity based Synchronization and an Application to Voice Encryption
  20. Analysis of Heat Transfer and Lifting Force in a Ferro-Nanofluid Based Porous Inclined Slider Bearing with Slip Conditions
  21. Application of QLM-Rational Legendre collocation method towards Eyring-Powell fluid model
  22. Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations
  23. MHD nonaligned stagnation point flow of second grade fluid towards a porous rotating disk
  24. Nonlinear Dynamic Response of an Axially Functionally Graded (AFG) Beam Resting on Nonlinear Elastic Foundation Subjected to Moving Load
  25. Swirling flow of couple stress fluid due to a rotating disk
  26. MHD stagnation point slip flow due to a non-linearly moving surface with effect of non-uniform heat source
  27. Effect of aligned magnetic field on Casson fluid flow over a stretched surface of non-uniform thickness
  28. Nonhomogeneous porosity and thermal diffusivity effects on stability and instability of double-diffusive convection in a porous medium layer: Brinkman Model
  29. Magnetohydrodynamic(MHD) Boundary Layer Flow of Eyring-Powell Nanofluid Past Stretching Cylinder With Cattaneo-Christov Heat Flux Model
  30. On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain
  31. An adaptive mesh method for time dependent singularly perturbed differential-difference equations
  32. On stretched magnetic flow of Carreau nanofluid with slip effects and nonlinear thermal radiation
  33. Rational exponential solutions of conformable space-time fractional equal-width equations
  34. Simultaneous impacts of Joule heating and variable heat source/sink on MHD 3D flow of Carreau-nanoliquids with temperature dependent viscosity
  35. Effect of magnetic field on imbibition phenomenon in fluid flow through fractured porous media with different porous material
  36. Impact of ohmic heating on MHD mixed convection flow of Casson fluid by considering Cross diffusion effect
  37. Mathematical Modelling Comparison of a Reciprocating, a Szorenyi Rotary, and a Wankel Rotary Engine
  38. Surface roughness effect on thermohydrodynamic analysis of journal bearings lubricated with couple stress fluids
  39. Convective conditions and dissipation on Tangent Hyperbolic fluid over a chemically heating exponentially porous sheet
  40. Unsteady Carreau-Casson fluids over a radiated shrinking sheet in a suspension of dust and graphene nanoparticles with non-Fourier heat flux
  41. An efficient numerical algorithm for solving system of Lane–Emden type equations arising in engineering
  42. New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind
  43. Numerical Study of Viscoelastic Micropolar Heat Transfer from a Vertical Cone for Thermal Polymer Coating
  44. Analysis of Bifurcation and Chaos of the Size-dependent Micro–plate Considering Damage
  45. Non-Similar Comutational Solutions for Double-Diffusive MHD Transport Phenomena for Non-Newtnian Nanofluid From a Horizontal Circular Cylinder
  46. Mathematical model on distributed denial of service attack through Internet of things in a network
  47. Postbuckling behavior of functionally graded CNT-reinforced nanocomposite plate with interphase effect
  48. Study of Weakly nonlinear Mass transport in Newtonian Fluid with Applied Magnetic Field under Concentration/Gravity modulation
  49. MHD slip flow of chemically reacting UCM fluid through a dilating channel with heat source/sink
  50. A Study on Non-Newtonian Transport Phenomena in Mhd Fluid Flow From a Vertical Cone With Navier Slip and Convective Heating
  51. Penetrative convection in a fluid saturated Darcy-Brinkman porous media with LTNE via internal heat source
  52. Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity
  53. Semitrailer Steering Control for Improved Articulated Vehicle Manoeuvrability and Stability
  54. Thermomechanical nonlinear stability of pressure-loaded CNT-reinforced composite doubly curved panels resting on elastic foundations
  55. Combination synchronization of fractional order n-chaotic systems using active backstepping design
  56. Vision-Based CubeSat Closed-Loop Formation Control in Close Proximities
  57. Effect of endoscope on the peristaltic transport of a couple stress fluid with heat transfer: Application to biomedicine
  58. Unsteady MHD Non-Newtonian Heat Transfer Nanofluids with Entropy Generation Analysis
  59. Mathematical Modelling of Hydromagnetic Casson non-Newtonian Nanofluid Convection Slip Flow from an Isothermal Sphere
  60. Influence of Joule Heating and Non-Linear Radiation on MHD 3D Dissipating Flow of Casson Nanofluid past a Non-Linear Stretching Sheet
  61. Radiative Flow of Third Grade Non-Newtonian Fluid From A Horizontal Circular Cylinder
  62. Application of Bessel functions and Jacobian free Newton method to solve time-fractional Burger equation
  63. A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform
  64. A multiple-step adaptive pseudospectral method for solving multi-order fractional differential equations
  65. A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses
  66. The expa function method and the conformable time-fractional KdV equations
  67. Comment on the paper: “Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet, M.R. Krishnamurthy, B.J. Gireesha, B.C. Prasannakumara, and Rama Subba Reddy Gorla, Nonlinear Engineering 2016, 5(3), 147-159”
  68. Three-Dimensional Boundary layer Flow and Heat Transfer of a Fluid Particle Suspension over a Stretching Sheet Embedded in a Porous Medium
  69. MHD three dimensional flow of Oldroyd-B nanofluid over a bidirectional stretching sheet: DTM-Padé Solution
  70. MHD Convection Fluid and Heat Transfer in an Inclined Micro-Porous-Channel
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 21.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/nleng-2018-0053/html
Button zum nach oben scrollen