The flow of homogeneous fluids through porous media: analogies with other physical problems

Geometry: shape, size, aspect ratio and orientation Flow Type: forced, natural, laminar, turbulent, internal, external Boundary: isothermal (Tw = constant) or isoflux (q̇w = constant) Fluid Type: viscous oil, water, gases or liquid metals Properties: all properties determined at film temperature Tf = (Tw + T∞)/2 Note: ρ and ν ∝ 1/Patm ⇒ see Q12-40 density: ρ ((kg/m) specific heat: Cp (J/kg ·K) dynamic viscosity: μ, (N · s/m) kinematic viscosity: ν ≡ μ/ρ (m/s) thermal conductivity: k, (W/m ·K) thermal diffusivity: α, ≡ k/(ρ · Cp) (m/s) Prandtl number: Pr, ≡ ν/α (−−) volumetric compressibility: β, (1/K)

Convection Heat Transfer

In this series of talks, I will discuss the fluid-dynamic-type equations that is derived from the Boltzmann equation as its the asymptotic behavior for small mean free path. The study of the relation of the two systems describing the behavior of a gas, the fluid-dynamic system and the Boltzmann system, has a long history and many works have been done. The Hilbert expansion and the Chapman–Enskog expansion are well-known among them. The behavior of a gas in the continuum limit, however, is not so simple as is widely discussed by superficial understanding of these solutions. The correct behavior has to be investigated by classifying the physical situations. The results are largely different depending on the situations. There is an important class of problems for which neither the Euler equations nor the Navier–Stokes give the correct answer. In these two expansions themselves, an initialor boundaryvalue problem is not taken into account. We will discuss the fluid-dynamic-type equations together with the boundary conditions that describe the behavior of the gas in the continuum limit by appropriately classifying the physical situations and taking the boundary condition into account. Here the result for the time-independent case is summarized. The time-dependent case will also be mentioned in the talk. The velocity distribution function approaches a Maxwellian fe, whose parameters depend on the position in the gas, in the continuum limit. The fluid-dynamictype equations that determine the macroscopic variables in the limit differ considerably depending on the character of the Maxwellian. The systems are classified by the size of |fe− fe0|/fe0, where fe0 is the stationary Maxwellian with the representative density and temperature in the gas. (1) |fe − fe0|/fe0 = O(Kn) (Kn : Knudsen number, i.e., Kn = `/L; ` : the reference mean free path. L : the reference length of the system) : S system (the incompressible Navier–Stokes set with the energy equation modified). (1a) |fe − fe0|/fe0 = o(Kn) : Linear system (the Stokes set). (2) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(Kn) (ξi : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity vi of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) must be retained in the equations (non-Navier–Stokes effect). The thermal creep[1] in the boundary condition must be taken into account. (3) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(1) : E+VB system (the Euler and viscous boundary-layer sets). E system (Euler set) in the case where the boundary is an interface of the gas and its condensed phase. The fluid-dynamic systems are classified in terms of the macroscopic parameters that appear in the boundary condition. Let Tw and δTw be, respectively, the characteristic values of the temperature and its variation of the boundary. Then, the fluid-dynamic systems mentioned above are classified with the nondimensional temperature variation δTw/Tw and Reynolds number Re as shown in Fig. 1. In the region SB, the classical gas dynamics is inapplicable, that is, neither the Euler

Kinetic Theory and Fluid Dynamics

Pollen Morphology and Plant Taxonomy: Angiosperms

Application of Neural Network for estimation of heat transfer treatment of Al2O3-H2O nanofluid through a channel

Cattaneo–Christov with variable thermal relaxation time and entropy generation is the main concern of this study. The micropolar fluid with absorption of heat in the existence of mixed convection and partial slip is scrutinized. Two distinct nanoparticles, i.e., single-wall carbon nanotube and multi-wall carbon nanotube, are immerged in micropolar fluid to interrogate the feature of heat and mass transfer. The non-dimensional similarity transformation is utilized to transform the partial differential equations into nonlinear ordinary differential equations, and resulting coupled equations are solved numerically using bvp4c from MATLAB. The present results show the fabulous agreement with previous published results. The temperature field diminishes with larger thermal relaxation time parameter. Entropy generation profile is an increasing function of Brinkmann number, while Bejan number is a diminishing function. Further the solid volume fraction diminishes the velocity profile and enhances the temperature distribution and entropy generation.

Cattaneo–Christov heat flux model for stagnation point flow of micropolar nanofluid toward a nonlinear stretching surface with slip effects

Nanofluids are of excellent significance to scientists, because, due to their elevated heat transfer rates, they have important industrial uses. A new class of nanofluid, “hybrid nanofluid,” has recently been used to further improve the rate of heat transfer. The current phenomenon particularly concerns the analysis of the flow and heat transfer of SWCNT–MWCNT/water hybrid nanofluid with activation energy through a moving wedge. The Darcy–Forchheimer relationship specifies the nature of the flow in the porous medium. Further the impact of variable viscosity, velocity and thermal slip, thermal radiation and heat generation are also discussed in detail. The second law of thermodynamics is utilized to measure the irreversibility factor. The numerical technique bvp4c is integrated to solve the highly nonlinear differential equation. For axial velocity, temperature profile, and entropy generation, a comparison was made between nanofluid and hybrid nanofluid. The variable viscosity parameter enhances the axial velocity and diminishes the temperature distribution for both nanofluid and hybrid nanofluid. Furthermore, the solid volume fraction diminishes the velocity and concentration profile while enhancing the temperature distribution.

Entropy generation and temperature-dependent viscosity in the study of SWCNT–MWCNT hybrid nanofluid

Cattaneo-Christov flux in the flow of a viscoelastic fluid in the presence of Newtonian heating

The characteristics of induced magnetic field are incorporated in a viscous fluid over a static or moving wedge. Heat flux is evaluated through the Fourier's law of heat conduction. The boundary value problem is solved numerically with the help of shooting technique coupled with Runge-Kutta and Newton's method. Three distinct types of nanoparticles, Cu, Al2O3, and TiO2are considered with water used as base fluids. The impacts of sundry parameters on the fluid flow are presented. Further for velocity ratio parameter the induced magnetic profile enhances. An excellent agreement is found with available result in the absence of magnetic field effect.

Computational study of Falkner-Skan problem for a static and moving wedge

This article presents a research for boundary layer flow and heat transfer of a Maxwell fluid over an exponential stretching surface with thermal stratifications. The effect of homogeneous and heterogeneous reaction are incorporated. Cattaneo–Christov heat flux model is used instead of Fourier law of heat conduction, which is recently proposed by Christov. This model predicts the impacts of thermal relaxation time on boundary layer. The transformed boundary layer equations are solved analytically by using Optimal homotopy analysis method. The effect of non-dimensional fluid relaxation time, thermal relaxation time, Prandtl number, Schmidt number and strength of homogeneous and heterogeneous reaction are demonstrated and exhibited graphically. The comparison of Cattaneo–Christov heat flux model and the Fourier’s law of heat conduction is also displayed.

Shafiq Ahmad

Papers

Cattaneo–Christov heat flux model for stagnation point flow of micropolar nanofluid toward a nonlinear stretching surface with slip effects

Entropy generation and temperature-dependent viscosity in the study of SWCNT–MWCNT hybrid nanofluid

Cattaneo-Christov flux in the flow of a viscoelastic fluid in the presence of Newtonian heating

Computational study of Falkner-Skan problem for a static and moving wedge

Chemically reactive species in the flow of a Maxwell fluid