In this paper, we study the steady state solutions for viscous reactive flows through channels with a sliding wall. The reaction is assumed to be strongly exothermic under Arrhenius kinetics, neglecting the consumption of the material. Approximate solutions are constructed for the governing nonlinear boundary value problem using regular perturbation techniques together with a special type of Hermite–Padé approximants, and important properties of the temperature field including bifurcations and thermal criticality conditions are discussed.
The study of the thermal effect of a sliding plate on a viscous reacting fluid is extremely important in understanding lubricants hydrodynamics in engineering systems as well as plasma and fluid physics [1], [2], [5], [8] and [12]. Lubricant is a thin viscous film used to prevent solid-to-solid contact during sliding motion. Generally speaking, most lubricants used in both engineering and industrial processes are reactive e.g. hydrocarbon oils, polyglycols, synthetic esters, polyphenylethers, etc., and their efficiency depends largely on the temperature variation from time to time. Hence, it is important to determine the thermal criticality conditions for viscous reactive fluid effectiveness as lubricants. In this particular problem, we have assumed that the pressure gradient is zero and the flow is driven solely by uniform velocity at the upper plate, i.e. the well-known plane Couette flow [2]. The resulting velocity profile is linear with zero value at the lower fixed plate and maximum value at the upper moving plate. Generally, when a fluid is sheared, some of the work done is dissipated as heat. The shear-induced heating gives an inevitable increase in temperature within the fluid. Neglecting the reacting viscous fluid consumption, since the channel is narrow (see Fig. 1) with very small aspect ratio, the equations for the momentum and heat balance in one dimension together with the boundary conditions can be written as [2], [8], [9] and [12]
equation(1)
View the MathML sourced2udȳ2=0,d2Tdȳ2+QC0Ake−ERT+μk(dudȳ)2=0,u(a)=U,T(a)=T0,u(0)=0,T(0)=T0,
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where TT is the absolute temperature, UU the upper wall characteristic velocity, T0T0 the geometry wall temperature, kk the thermal conductivity of the material, QQ the heat of reaction, AA the rate constant, EE the activation energy, RR the universal gas constant, C0C0 the initial concentration of the reactant species, aa the channel width, View the MathML sourceȳ the distance measured in the normal direction and μμ the fluid dynamic viscosity coefficient [1], [2], [3], [5] and [9]. We introduce the following dimensionless variables into Eq. (1):
equation(2)
View the MathML sourceθ=E(T−T0)RT02,y=ȳa,λ=QEAa2C0e−ERT0T02Rk,
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View the MathML sourceW=uU,β=μU2eERT0QAa2C0,ε=RT0E,
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and obtain the dimensionless governing equation together with the corresponding boundary conditions as (neglecting the bar symbol for clarity)
equation(3)
View the MathML sourced2θdy2+λ(e(θ1+εθ)+β)=0,θ(0)=0,θ(1)=0,
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where the fluid velocity profile is given as W(y)=yW(y)=y and λλ, εε, ββ represent the Frank Kamenetskii parameter [5], the activation energy parameter and the viscous heating parameter, respectively. In the following sections, Eq. (3) is solved using both perturbation and multivariate series summation techniques [6], [7], [9], [10] and [11].
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Fig. 1.
Geometry of the problem.
