An Asymptotic Study of the Nusselt-Graetz Problem—Part I: Large x Behavior

1974 ◽  
Vol 96 (3) ◽  
pp. 354-358 ◽  
Author(s):  
H. J. Hickman
Keyword(s):  
Nusselt Number ◽  
Laminar Flow ◽  
Laplace Transform ◽  
The Other ◽  
Circular Pipe ◽  
Parallel Plates ◽  
Newtonian Flow ◽  
Percent Error ◽  
The Laplace Transform ◽  
Analytic Expressions

A method is given for determining the large x behavior of the Nusselt number for a variety of Nusselt-Graetz problems. Exploitation of properties of the Laplace transform of the temperature yields analytic expressions for Nu as explicit functions of the other parameters of the problem. Accurate results (<1 percent error) are deduced for problems involving the laminar flow of a Newtonian flow between parallel plates and in a circular pipe (valid for all values of the wall Nusselt number).

scholarly journals Alenezi Transform–A New Transform to Solve Mathematical Problems

2021 ◽  
pp. 1-8
Author(s):  
Ahmad M. Alenezi
Keyword(s):  
Laplace Transform ◽  
Integral Equations ◽  
Integral Transform ◽  
The Other ◽  
The Laplace Transform ◽  
Mathematical Problems ◽  
Sumudu Transforms

In this paper, we present a new integral transform called Alenezi-transform in the category of Laplace transform. We investigate the characteristic of Alenezi-transform. We discuss this transform with the other transforms like J, Laplace, Elzaki and Sumudu transforms. We can demonstrate that Alenezi transforms are near to the condition of the Laplace transform. We can explain the new Properties of transforms using Alenezi transform. Alenezi transform can be applied to solve differential, Partial and integral equations.

scholarly journals HYDRODYNAMICAL INSTABILITY OF NEWTONIAN FLOW BEFORE AN AXISYMMETRIC SUDDEN CONTRACTION

2021 ◽  
Vol 2021 (2) ◽  
pp. 32-38
Author(s):  
Vadym Orel ◽  
◽  
Bohdan Pitsyshyn ◽  
Tetiana Konyk ◽  
◽  
...  
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Turbulent Flow ◽  
Transition Region ◽  
Step Height ◽  
Circular Pipe ◽  
Sudden Expansion ◽  
Newtonian Flow ◽  
Extreme Dependence ◽  
Sudden Contraction

The sizes of the vortex region before the axisymmetric sudden contraction of the circular pipe at the Newtonian flow have been investigated. Area ratios 0.250 and 0.500 were considered. The sizes of the vortex region have the extreme dependence with a maximum at the transition of the laminar flow into a turbulent flow one. When the Reynolds number at the laminar flow increase, these sizes also increase, and they decrease at the turbulent flow. In both cases, the sizes of the vortex region are proportional to the Reynolds number. A transition region between laminar flow and turbulent flow lies in the range of the Reynolds number from 3000 to 5300 and 750…1300, determined by the diameter of a bigger pipe of sudden expansion and a step height correspondingly

scholarly journals Analytical solutions of heat transfer for laminar flow in rectangular channels

2014 ◽  
Vol 35 (4) ◽  
pp. 29-42 ◽  
Author(s):  
Witold Rybiński ◽  
Jarosław Mikielewicz
Keyword(s):  
Nusselt Number ◽  
Laminar Flow ◽  
Friction Factor ◽  
Cross Section ◽  
Constant Temperature ◽  
Analytical Solutions ◽  
Rectangular Cross Section ◽  
Temperature Profiles ◽  
Mathematical Form ◽  
Parallel Plates

Abstract The paper presents two analytical solutions namely for Fanning friction factor and for Nusselt number of fully developed laminar fluid flow in straight mini channels with rectangular cross-section. This type of channels is common in mini- and microchannel heat exchangers. Analytical formulae, both for velocity and temperature profiles, were obtained in the explicit form of two terms. The first term is an asymptotic solution of laminar flow between parallel plates. The second one is a rapidly convergent series. This series becomes zero as the cross-section aspect ratio goes to infinity. This clear mathematical form is also inherited by the formulae for friction factor and Nusselt number. As the boundary conditions for velocity and temperature profiles no-slip and peripherally constant temperature with axially constant heat flux were assumed (H1 type). The velocity profile is assumed to be independent of the temperature profile. The assumption of constant temperature at the channel’s perimeter is related to the asymptotic case of channel’s wall thermal resistance: infinite in the axial direction and zero in the peripheral one. It represents typical conditions in a minichannel heat exchanger made of metal.

scholarly journals MASS TRANSFER OF FREE CONVECTION FLOW BETWEEN TWO PARALLEL OSCILLATING PLATES.

2019 ◽  
Vol 1 (2) ◽  
pp. 118-121
Author(s):  
Fasihah Zulkiflee ◽  
Sharidan Shafie ◽  
Ahmad Qushairi Mohamad
Keyword(s):  
Nusselt Number ◽  
Free Convection ◽  
Skin Friction ◽  
Schmidt Number ◽  
Convection Flow ◽  
Parallel Plates ◽  
Governing Equations ◽  
Free Convection Flow ◽  
Laplace Transform Technique ◽  
The Laplace Transform

This paper investigated unsteady free convection flow between two parallel plates with mass diffusion. One of the plate are considered oscillating. Appropriate non-dimensional variables are used to reduce the dimensional governing equations along with imposed initial and boundary conditions. The exact solution for velocity, temperature and concentration profiles are obtained using the Laplace Transform technique. The graphical results of the solutions are presented to illustrate the behavior of the fluid flow with the influenced of Schmidt number, Prandtl number, oscillating parameter, Grashof and mass Grashof number. The corresponding expressions for skin friction, Nusselt number and Sherwood number are also calculated. It is observed that increasing Prandtl and Schmidt number will increased the Nusselt number but decreased the skin friction.

Heat Transfer in Fully Developed Laminar Flow of Power Law Fluids

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
A. Baptista ◽  
M. A. Alves ◽  
P. M. Coelho
Keyword(s):  
Nusselt Number ◽  
Laminar Flow ◽  
Viscous Dissipation ◽  
Power Law ◽  
Numerical Solutions ◽  
Constant Heat Flux ◽  
Parallel Plates ◽  
Constant Wall Temperature ◽  
Power Law Fluids ◽  
Wide Range

In this work, we present approximate and exact solutions for the temperature profile and Nusselt number under fully developed laminar flow of a power law fluid inside pipes and between parallel plates. Constant wall temperature and negligible axial heat conduction are considered, for both the cases with and without viscous dissipation. For completeness, the corresponding solutions for the related problem of constant heat flux at the wall are also presented. In the absence of viscous dissipation, the solutions obtained are semi-analytic, since they rely upon an iterative procedure. As a benchmark result, to allow comparison with the results obtained with the semi-analytical expressions, we also present highly accurate numerical solutions for the Nusselt number, Nu, based on numerical integration of the energy equation. Also based on these numerical results, simplified correlations for Nu are proposed, valid for a wide range of the power law index.

scholarly journals Differentiation of the Mittag-Leffler Functions with Respect to Parameters in the Laplace Transform Approach

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 657
Author(s):  
Alexander Apelblat
Keyword(s):  
Laplace Transform ◽  
Integral Representations ◽  
The Other ◽  
Gamma Functions ◽  
Inverse Transform ◽  
Convolution Integrals ◽  
The Laplace Transform ◽  
Infinite Power ◽  
Derivatives Of ◽  
Two Parameter

In this work, properties of one- or two-parameter Mittag-Leffler functions are derived using the Laplace transform approach. It is demonstrated that manipulations with the pair direct–inverse transform makes it far more easy than previous methods to derive known and new properties of the Mittag-Leffler functions. Moreover, it is shown that sums of infinite series of the Mittag-Leffler functions can be expressed as convolution integrals, while the derivatives of the Mittag-Leffler functions with respect to their parameters are expressible as double convolution integrals. The derivatives can also be obtained from integral representations of the Mittag-Leffler functions. On the other hand, direct differentiation of the Mittag-Leffler functions with respect to parameters produces an infinite power series, whose coefficients are quotients of the digamma and gamma functions. Closed forms of these series can be derived when the parameters are set to be integers.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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