TRANSIENT MHD FREE CONVECTION FLOW IN A VERTICAL MICROTUBE

1.6 Dimensionless Quantities 3

 

A wide range of applications of micro-electromechanical systems and nano-technology have given afillip to research area where a non-continuum behavior is present. In this work, we are interested instudying the surface-fluid interaction where slip flow regime occurs. In this regard, Knudsen number (Kn) is a deciding factor, which is a measure of molecularmean free path to characteristic length. When theKnudsen number (Kn) is very small, no slip is observedbetween the surface and the fluid and is in tune with the essence of continuum mechanics. Furthermore, when Knudsen number (Kn) lies in the range 0.001-0.1, slip occurs at the surface–fluid interaction and is generally studied under the light of model Maxwell–Smoluchowskn first-order slip boundary conditions. Extensive investigations have been conductedrecently in the field of micro geometry flow, but the literature lacks studies that take into account the role of wall surface curvature on transient magnetohydrodynamic (MHD) free convective flow in vertical microtube. However, to cite a few work in this direction, Khadrawi et al.( 2005) investigated analytically the transient thermal behaviour of a stagnant gas confined in a horizontal micro-channel under the effect of the dual-phase-lag heat conduction model.Khadrawi(2011) studied the transient hydrodynamic and thermal behaviors’ of fluid flow in a vertical porousmicro-channel under the effect of hyperbolic heat conduction model. The unsteady hydrodynamics and thermal behaviour of fluid flow in an open-ended vertical parallel-plate micro-channel are investigated semi analytically under the effect of the dual-phase-lag heat conduction model by Khadrawi and Al-Nimr. (2007) Also,Weng and Chen (2009) studied the impact of wall surface curvature on steady fully developed natural convection flow in an open-ended vertical microtube with an asymmetric heating of annulus surface. Recently, further extended the work of  Weng and Chen (2009)by taking into  account suction/injection on vertical annular micro-channel. It is observed that skin friction decreases at the outer surface of the inner porous cylinder with an increase in fluid–wall interaction parameter reverse at the inner surface of the outer porous cylinder. Avci and Aydin (2009) studied the fully developed mixed convective heat transfer of a Newtonian fluid in a vertical microtube. On the other hand, the MHD phenomenon has received considerable attention during the last two decades due to its importance from the energy generation point of view, and one may envisage MHD generators for power generation. MHD pumps are already in use in chemical energy technology for pumping electrically conducting fluids at some of the atomic energy center. Besides these applications, when the fluid is electrically conducting, the free convection flow is appreciably influenced by an imposed magnetic field. Therefore, to refer to few works in this direction, Sheikholeslami et al. (2014) investigated the magnetic field effect on nano-fluid flow and heat transfer in a semi annulus enclosure via control volume-based finite element method. Sheikholeslami and Gorji-Bandpy(2014) presented the numerical solution for free convection of ferrofluid in a cavity heated from below in the presence of external magnetic field, while the MHD natural convection of nano fluid in a vertical microtube.

This work analyses the transient magnetohydrodynamic free convective flow in vertical microtube in the presence of velocity slip and temperature jump at the inner surface of the cylinder. The effect of various flow parameters entering into the problem such as time, Prandtl number, Hartmann number, rarefaction parameter, and fluid–wall interaction parameter are discussed with the aid of line graphs.

The aim of this project is to study transient MHD Fluid free convection in a vertical microtube. With the following objectives:

The scope of this research encompasses a comprehensive investigation into transient MHD free convection flow in a vertical microtube. It will involves mathematical modeling, validating the results obtained with existing literature or experimental data which can be challenging, especially when dealing with complex transient flows in microtubes.

Dimensionless parameters, such as the Reynolds number(Re), Grashof number(Gr), and Hartmann number(M), are used to characterize the relative importance of various physical effects in the governing equations. They help in understanding the dominant forces and behaviors in the system.

is a dimensionless number used in fluid dynamics and heat transfer to characterize the relative importance of momentum diffusivity (viscosity) to thermal diffusivity in a fluid.

Where μ is the momentum diffusivity, K is the thermal diffusivity,

This is the most important non-dimensional number in hydromagnetics, first introduced by Hartmann. It is the ratio of electromagnetic force to the viscous force. The hydromagnetic effects are significant if the Hartmann number is significant. It is defined by:

Where B_0 is the magnetic field, L is the characteristic length scale, σ is the electrical conductivity and µ is the viscosity

MHD is the study of the magnetic properties and behavior of electrically conducting fluids, such as plasmas, liquid metals, and saltwater.

Transients in fluid mechanics refer to changes in fluid flow conditions over time.

This refers to the mode of heat or mass transfer in a fluid (liquid or gas) that occurs due to density differences caused by variations in temperature or concentration.

A vertical microtube refers to a small-diameter tube oriented vertically. Microtubes are tiny tubes with diameters in the micrometer range.

Skin friction, also known as wall shear stress, is a force exerted by a fluid in contact with a solid surface, such as a wall or boundary.

where

τ is the skin friction (shear stress) acting on the surface.

μ (mu) is the viscosity of the fluid.

Steady state refers to the condition in a system where various parameters or properties remain constant over time.

Mass flux, also known as mass flow rate, is a measure of the amount of mass (typically in kilograms or other mass units) passing through a defined area or cross-sectional region per unit of time (typically in seconds). Mathematically, mass flux is expressed as

MassFlux=(Mass Flow rate)/Area

Where the mass flow rate is the total mass passing through a specific section of the system per unit time, and the area is the cross-sectional area through which the mass is flowing.

If f(t) represents some expression in t defined for  t≥0, the Laplace Transform of f(t), denoted by L{f(t)}  is defined to be:

Where s is a variable whose value is chosen so as to ensure that the semi-infinite integral converges.

An efficient and accurate method for Laplace inversion of some difficult functions into their corresponding time domain. This method employs the use of Fourier-series to replace the integral by a series which corresponds to the trapezoidal with a specific discretization error. The Fourier integral thus obtained can be approximated by its Riemann-sum. The Riemann-sum approximation for the Laplace inversion contains a single summation in the numerical process. Its degree of accuracy depends on the value of ϵ  as well as the truncation error controlled by N. The value of ϵ  must be chosen so that the Bromwich contour encloses all the branch points. In this method, any function in the s domain can be inverted into time t domain as follows;

where Re refers to the to the ‘real part’ of the summation,  i=√(-1) is the imaginary number, N is the number of terms used in the Riemann-sum approximation and ϵ is the real part of the Bromwich contour that is used in inverting Laplace transforms.

A fluid obeying Newton’s law of viscosity is referred to as  viscous fluid, otherwise is non-viscous (or inviscid).

A fluid whose density is constant irrespective of how much pressure is increased is said to be incompressible, otherwise it is compressible.

Steady flow refers to a situation where the velocity, pressure, and other properties of a fluid remain constant at a given point in space over time. 

Unsteady flow, on the other hand, is when the properties of a fluid at a particular point change with time. 

A Newtonian fluid is a type of fluid that behaves in a simple and predictable manner, following Newton's law of viscosity. In a Newtonian fluid, the relationship between the shear stress (force per unit area) and the rate of deformation (velocity gradient) is linear and constant. Water and most common liquids, as well as air and other gases, are examples of Newtonian fluids.

A non-Newtonian fluid is a type of fluid that does not follow Newton's law of viscosity, and its viscosity or flow behavior changes under different conditions. Non-Newtonian fluids can exhibit various types of behavior, such as becoming more viscous (thickening) or less viscous (thinning) in response to changes in shear stress. Examples of non-Newtonian fluids include ketchup, toothpaste, and blood, as well as some drilling muds and industrial polymers.