scholarly journals Analytical Solution of a Non-isothermal Flow in Cylindrical Geometry

2021 ◽  
pp. 55-76
Author(s):  
Chinedu Nwaigwe ◽  
Innocent Uchenna Amadi
Keyword(s):  
Analytical Solution ◽  
Solution Method ◽  
Fluid Velocity ◽  
Slip Boundary Condition ◽  
Cylindrical Geometry ◽  
Cylindrical Domain ◽  
Slip Boundary ◽  
Flow Parameters ◽  
Injection Parameter ◽  
Flow Variables

This study proposes analytical solution to the problem of transport in a Newtonian fluid within a cylindrical domain. The flow is assumed to be dominated along the channel axis, and is taken to be axi-symmetric. No-slip boundary condition is considered for velocity while the temperature and concentration have Dirichlet boundary values. The resulting problem is transformed into a set of non-trivial variable coefficient differential equations in a cylindrical geometry. By adopting the series solution method of Frobenius, the closed-form analytical solutions are derived for the flow variables. We conduct an analysis of the derived model, and showed that, indeed, the flow variables are axi-symmetric. We also state and prove another theorem to show that the derived concentration model is positivity preserving – meaning that it yields positive concentration - provided the boundary value is non-negative. Finally, we present graphical results for the flow variables and discuss the effect of the relevant flow parameters. The results showed that (i) an increase in the cooling parameter, reduces the fluid velocity, (ii) the temperature decreases as the cooling parameter increases   and (iii) an increase in the injection parameter, leads to increase in the concentration.

Vibration and Stability Behaviour of Sandwiched Viscoelastic Pipes Conveying a Non-Newtonian Fluid

2010 ◽  
Author(s):  
Vincent O. S. Olunloyo ◽  
Charles A. Osheku ◽  
Sidikat I. Kuye
Keyword(s):  
Newtonian Fluid ◽  
Fluid Velocity ◽  
Linear Equations ◽  
Steel Pipes ◽  
Flow Parameters ◽  
Internal Fluid ◽  
Reduction Mechanisms ◽  
Internal Fluid Flow ◽  
Flow Induced Vibrations ◽  
Flow Variables

Internal fluid flow parameters in conjunction with elastomechanical properties of conveyance systems have significantly modulated flow induced vibrations in pipeline and riser systems. Recent advances on the mechanics of sandwich elastic systems as effective vibration and noise reduction mechanisms have simulated the possibility of replacing traditional steel pipes with sandwich pipes in deepwater environment. The dynamic behaviour and stability of sandwich elastic pipes conveying a non-Newtonian fluid are investigated in this paper. For this problem, a set of generalised non-linear equations governing the vibration of sandwich pipes held together in pressurised environment and conveying a non-Newtonian fluid is presented. By linearizing the governing partial differential equation matching the problem physics, under slight perturbation of the internal fluid velocity and other flow variables closed form analytical results for the system dual natural frequencies and stability under external excitation are computed for field designs and applications. Results show that for a given length of pipe, beyond the critical velocity, instability increases with the velocity of conveyance.

The torque on a rotating n -bladed vane in a newtonian fluid or linear elastic medium

1992 ◽  
Vol 438 (1902) ◽  
pp. 183-196 ◽  
Keyword(s):  
Boundary Condition ◽  
Newtonian Fluid ◽  
Unit Length ◽  
Fluid Velocity ◽  
Stress Singularity ◽  
Slip Boundary Condition ◽  
Slip Boundary ◽  
Linear Elastic ◽  
Linear Viscoelastic ◽  
Linear Viscoelastic Fluid

The vane is an n -bladed paddle which rotates with angular velocity Ω in a linear viscoelastic fluid. The blades, of zero thickness, are equally spaced around the axis r = 0, and extend from r = 0 to r = a . The problem is assumed two-dimensional. The stress and fluid velocity (or material displacements) are obtained by a Wiener–Hopf technique for the case of a no-slip boundary condition on the surface of the blades, and for the case of zero shear stress on the blades. The torque M (per unit length) required to rotate the vane in an incompressible newtonian fluid of viscosity μ may be approximated as M ≈ 4π μa 2 Ω (1 – n –1 ) to within 1% for the no-slip boundary condition; with the slip boundary condition the same expression is accurate to within 4%. Results are also given for the angular dependence and strength of the stress singularity at the tip of each blade.

Kinematic description of fluids in motion and approximations

2018 ◽  
Author(s):  
Marcel Escudier
Keyword(s):  
Fluid Velocity ◽  
Slip Boundary Condition ◽  
Cross Sectional Area ◽  
Constant Density ◽  
Sectional Area ◽  
Cross Sectional ◽  
Slip Boundary ◽  
Flow Problems ◽  
Mass Flowrate ◽  
Flow Through

In this chapter some of the terminology and simplifications which enable us to begin to describe and analyse practical fluid-flow problems are introduced. The terms ‘fluid particle’ and ‘streamline’ are defined. The principle of conservation of mass applied to steady one-dimensional flow through a streamtube of varying cross-sectional area resulted in the continuity equation. This important equation relates mass flowrate ṁ, volumetric flowrate Q̇, average fluid velocity V̄, fluid density ρ‎, and cross-sectional area A: m = ρ‎ Q̇ = ρ‎AV̅ = constant. For a constant-density fluid this result shows that fluid velocity increases if the cross-sectional area decreases, and vice versa. The no-slip boundary condition, a consequence of which is the boundary layer, is introduced.

Analytical Solution of Turbulent Couette Flow by Cosserat Continuum Model and Gradient Theory

2006 ◽  
Author(s):  
H. Ghasvari Jahromi ◽  
G. Atefi ◽  
A. Moosaie ◽  
S. Hormozi ◽  
H. Afshin
Keyword(s):  
Boundary Condition ◽  
Analytical Solution ◽  
Couette Flow ◽  
Continuum Model ◽  
Slip Boundary Condition ◽  
Cosserat Continuum ◽  
Dimensionless Number ◽  
Gradient Theory ◽  
Slip Boundary ◽  
Cosserat Continuum Model

In present paper the theory of the micropolar fluid based on a Cosserat continuum model has been applied for analysis of Couette flow. The obtained results for the velocity field have been compared with known results from experiments done by Reichardt at Max Plank institute for fluids in Gottingen [1,2] and analytical solution of the problem from Gradient theory by Alizadeh [3]. The boundary condition used here was the no slip one and Trostel’s slip boundary condition [4]. A good agreement between experimental results and the results of the problem for Reynolds near 18000 has beeen found. A new dimensionless number introduced that indicates the theoretical relation between cosserat theory and slip theory and their interaction.

Analytical Solution of Turbulent Problems Using Governing Equation of Cosserat Continuum Model

2006 ◽  
Author(s):  
H. Ghasvari-Jahromi ◽  
Gh. Atefi ◽  
A. Moosaie ◽  
S. Hormozi
Keyword(s):  
Boundary Condition ◽  
Analytical Solution ◽  
Continuum Model ◽  
Slip Boundary Condition ◽  
Cosserat Continuum ◽  
Dimensionless Number ◽  
Gradient Theory ◽  
Slip Boundary ◽  
Flow Through ◽  
Cosserat Continuum Model

In present paper the theory of the micropolar fluid based on a Cosserat continuum model has been applied for analysis of Couette flow and turbulent flow through rough pipes. The obtained results for the velocity field have been compared with known results from experiments done by Reichardt at Max Plank institute for fluids in Gottingen [1,2] and analytical solution of the problem from Gradient theory by alizadeh[3] for couette problem and with known results from experiments done by Nikuradse (1932). the boundary condition used here was the no slip one and Trostel's slip boundary condition[4].a good agreement between experimental results and the results of the problem for Reynolds near 18000 has beeen found in the couette case also in this case A new dimensionless number introduced that indicates the theoretical relation between cosserat theory and slip theory and their interaction. The solution has been performed for a Reynolds number of 106 for pipes with different values of roughness and the validity analysis approved by the results of Nikuradse's experiments.

Subsonic gas flow in a straight and uniform microchannel

2002 ◽  
Vol 472 ◽  
pp. 125-151 ◽  
Author(s):  
YITSHAK ZOHAR ◽  
SYLVANUS YUK KWAN LEE ◽  
WING YIN LEE ◽  
LINAN JIANG ◽  
PIN TONG
Keyword(s):  
Gas Flow ◽  
Theoretical Calculations ◽  
Slip Flow ◽  
Velocity Slip ◽  
Slip Boundary Condition ◽  
Perturbation Expansions ◽  
Slip Boundary ◽  
Flow Parameters ◽  
Pressure Distributions ◽  
Flow Effects

A nonlinear equation based on the hydrodynamic equations is solved analytically using perturbation expansions to calculate the flow field of a steady isothermal, compressible and laminar gas flow in either a circular or a planar microchannel. The solution takes into account slip-flow effects explicitly by utilizing the classical velocity-slip boundary condition, assuming the gas properties are known. Consistent expansions provide not only the cross-stream but also the streamwise evolution of the various flow parameters of interest, such as pressure, density and Mach number. The slip-flow effect enters the solution explicitly as a zero-order correction comparable to, though smaller than, the compressible effect. The theoretical calculations are verified in an experimental study of pressure-driven gas flow in a long microchannel of sub-micron height. Standard micromachining techniques were utilized to fabricate the microchannel, with integral pressure microsensors based on the piezoresistivity principle of operation. The integrated microsystem allows accurate measurements of mass flow rates and pressure distributions along the microchannel. Nitrogen, helium and argon were used as the working fluids forced through the microchannel. The experimental results support the theoretical calculations in finding that acceleration and non-parabolic velocity profile effects were found to be negligible. A detailed error analysis is also carried out in an attempt to expose the challenges in conducting accurate measurements in microsystems.

Analysis of the Diameter of Microbubbles Formed in a Cross Flow Microchannel

2009 ◽  
Author(s):  
T. J. John ◽  
B. Mathew ◽  
H. Hegab
Keyword(s):  
Flow Rate ◽  
Gas Flow ◽  
Liquid Flow Rate ◽  
Fluid Velocity ◽  
Cross Flow ◽  
Slip Boundary Condition ◽  
Force Balance ◽  
Gas Channel ◽  
Slip Boundary ◽  
The Moment

Two microfluidic devices for generating microbubbles are considered in the study presented in this paper. The first device consists of a liquid channel and a gas channel that is perpendicular to each other. In this device, the microbubble diameter varies inversely with the liquid flow rate (i.e. with flow velocity) but at the expense of high pressure drop. This device is modified by introducing a solid structure in front of the orifice to become the second device. This modification causes an increase in fluid velocity only in front of the orifice. In this paper a model is developed for determining the diameter of microbubbles generated in both of these devices. The model is developed by balancing the forces acting on the microbubble during growth and at the moment of detachment from the orifice. The detachment of the microbubble is assumed to happen when the sum of all detaching forces equals the net attaching force acting on the microbubble during the growth. Non-slip boundary condition is assumed on the walls of the channels in this model. Based on these assumptions a mathematical model is developed and solved numerically for different values of liquid and gas flow rate to obtain the microbubble diameter at the moment of detachment. A MATLAB® code is developed for solving the force balance equation. The superiority of the second device over the first one is validated by comparing the results obtained from the models in both cases.

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