Hermite-Padé approximation approach to thermal criticality for a reactive third-grade liquid in a channel with isothermal walls

Analysis of transient Generalized Couette flow of a reactive variable viscosity third-grade liquid with asymmetric convective cooling

“…Following [6][7][8][9][10][11][12][13], and neglecting the reacting viscous fluid consumption, the governing equations for the momentum and heat balance can be written as:…”

“…Here u is the fluid axial velocity, T is the temperature, T a is the ambient temperature, T 0 is the fluid initial temperature,P is the modified pressure,t is the time, ρ is the density, k is the thermal conductivity coefficient, c p is the specific heat at constant pressure, h is the heat transfer coefficient, Q is the heat of reaction, A is the rate constant, E is the activation energy, R is the universal gas constant, C 0 is the initial concentration of the reacting species, a is the channel half width, l is Planck's number, K is the Boltzmann's constant, ν is the vibration frequency, α 1 and β 3 are the material coefficients, m is a numerical constant such that m ∈ {−2, 0, 0.5}. The three values chosen for the parameter m represent the numerical exponent for Sensitized, Arrhenius and Bimolecular kinetics respectively (see [7,8,13]). The temperature dependent viscosity (μ) can be expressed as…”

“…Among the many proposed models is the class of third order fluids for which analytical solutions even for the simple flows are not easily obtainable and hence a recourse to numerical methods of solution is unavoidable. Extensive studies have been undertaken involving such fluids [1][2][3][4][5][6][7][8]. Rajagopal [9] studied in detail the general thermodynamics, stability and uniqueness of constitutive models of the differential type with third grade fluid being a special case.…”

Unsteady hydromagnetic flow of a reactive variable viscosity third‐grade fluid in a channel with convective cooling

“…Following 3–5, 12–14 and neglecting the reacting viscous fluid consumption, the governing equations for the momentum and heat balance can be written as subject to the following initial and boundary conditions: where the additional chemical kinetics term in energy balance equation is due to 18. Here, T is the absolute temperature, σ is the fluid electrical conductivity, B 0 ( = µ e H 0 ) is the electromagnetic induction, µ e is the magnetic permeability, H 0 is the intensity of the magnetic field, ρis the density, c p is the specific heat at constant pressure, t is the time, h is the heat transfer coefficient, T 0 is the fluid initial temperature, T a is the ambient temperature, k is the thermal conductivity of the material, Q is the heat of reaction, A is the rate constant, E is the activation energy, R is the universal gas constant, C 0 is the initial concentration of the reactant species, a is the channel half‐width, l is the Planck's number, K is the Boltzmann's constant, ν is the vibration frequency, α 1 and β 3 are the material coefficients, is the modified pressure, m is the numerical exponent such that m ∈{−2, 0, 0.5}, where the three values represent numerical exponents for Sensitized, Arrhenius and Bimolecular kinetics, respectively (see 3, 4, 9). The temperature‐dependent viscosity can be expressed as where b is a viscosity variation parameter and µ 0 is the initial fluid dynamic viscosity at temperature T 0 .…”

“…We use ξ = 1 here so that we are free to choose larger time steps and still converge to the steady solutions. Our algorithm has been tested against the special cases in 9, 14 and has been shown to be capable of accurately reproducing such results.…”