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

This article discusses the important of single wall carbon nanotube (SWCNTs) and multiple walls carbon nanotube (MWCNTs) past a static wedge under the effects of magnetohydrodynamics (MHD). The significance of activation energy, thermal radiation, heat generation and double stratification are also taken into account. Utilizing the BVP-4c function of MATLAB the system of differential equations is solved numerically. To validate our results, a correlation with a previously studied problem is also directed and a brilliant concurrence is found; thus, reliable results are being introduced. The influence of sundry parameter on axial velocity, temperature, and concentration profile are deliberated and studied graphically. The temperature and concentration distribution diminish respectively with thermal and concentration stratified parameter. Further the solid volume fraction enhances the axial velocity and temperature profile for both carbon nanotubes.

Radiative SWCNT and MWCNT nanofluid flow of Falkner-Skan problem with double stratification

Mathematical analysis of bio-convective micropolar nanofluid

A mathematical model is framed to discuss the flow of carbon nanotube-suspended nanofluids with Cattaneo-Christov heat flux and binary chemical reaction. The flow analysis is performed in attendance of heat generation/absorption, energy activation, and buoyancy effects past a nonlinear stretched surface embedded in a non-Darcy permeable medium. A combination of varied nanotubes with base fluids is also taken into account. The Runge-Kutta fifth-order Fehlberg technique is engaged to find the numerical solution of a highly nonlinear problem. The impact of sundry parameters on involved distributions is illustrated graphically with requisite discussion keeping in view their physical aspects. Different tables that comprise numerically calculated values of numerous sundry parameters highlighting their physical significance are also erected. It is witnessed that velocity and temperature profiles are enhanced for mounting values of nanoparticle volume fraction parameters. Further, it is seen that for enhancing th...

Upshot of binary chemical reaction and activation energy on carbon nanotubes with Cattaneo-Christov heat flux and buoyancy effects

This research is made to visualize the boundary layer flow by a curved stretching sheet embedded in porous medium. The geometry is bended (curved), therefore the curvilinear coordinates are used to model the present problem. Fluid is electrically conducting with the presence of uniform magnetic field. The governing non-linear partial differential equation reduces to non-linear ordinary differential equations by using the dimensionless suitable transformations. The numerical solutions are obtained by using the method bvp4c from MATLAB. The effects of curvature parameter, non-dimensional magnetic parameter, and porosity parameter on the velocity field and skin friction coefficient are examined. The skin friction profile enhances with enhancing the values of porosity and magnetic parameter. Comparison of the present results with the existing results in the literature for the flat surface is also given.

Boundary Layer Flow over a Curved Surface Imbedded in Porous Medium

The aim of the present study is to address the impacts of Newtonian heating and homogeneous–heterogeneous (h-h) reactions on the flow of Ag–H2O nanofluid over a cylinder which is stretched in a nonlinear way. The additional effects of magnetohydrodynamics (MHD) and nonlinear thermal radiation are also added features of the problem under consideration. The Shooting technique is betrothed to obtain the numerical solution of the problem which is comprised of highly nonlinear system ordinary differential equations. The sketches of different parameters versus the involved distributions are given with requisite deliberations. The obtained numerical results are matched with an earlier published work and an excellent agreement exists between both. From our obtained results, it is gathered that the temperature profile is enriched with augmented values radiation and curvature parameters. Additionally, the concentration field is a declining function of the strength of h-h reactions.

https://www.mdpi.com/2073-8994/11/2/295/pdf

A Numerical Simulation of Silver–Water Nanofluid Flow with Impacts of Newtonian Heating and Homogeneous–Heterogeneous Reactions Past a Nonlinear Stretched Cylinder

Shafiq Ahmad

Papers

Radiative SWCNT and MWCNT nanofluid flow of Falkner-Skan problem with double stratification

Mathematical analysis of bio-convective micropolar nanofluid

Upshot of binary chemical reaction and activation energy on carbon nanotubes with Cattaneo-Christov heat flux and buoyancy effects

Boundary Layer Flow over a Curved Surface Imbedded in Porous Medium

A Numerical Simulation of Silver–Water Nanofluid Flow with Impacts of Newtonian Heating and Homogeneous–Heterogeneous Reactions Past a Nonlinear Stretched Cylinder