Comparison of different pressure waveforms for heat transfer performance of oscillating flow in a circular cylinder

Abstract

In the present investigation, an oscillating motion of unsteady Burgers fluid in a circular cylinder was modeled with different pressure waveforms. Three different waveforms are considered: the case of a trapezoidal, triangular and sinusoidal waveform. Analytical solutions of velocity and temperature distribution are obtained for an oscillating laminar flow, which can be used to analyze the effects of flow type on the heat transfer performance. The limiting cases have been considered to examine the heat transfer performance of four different non-Newtonian fluids. Results show that the heat transfer of the oscillating flow depends on the fluid material parameter, Prandtl number, amplitude oscillating waveform and radial coordinate. The trapezoidal and sinusoidal waveforms of oscillating motion can result in a higher heat transfer performance.

Keywords

Trapezoidal waveform ; Triangular waveform ; Sinusoidal waveform ; Transient flow ; Analytical solution

Nomenclature

A_n- Constant defined by Eq. (23)

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B_n- Constant defined by Eq. (24)

C_n- Constant defined by Eq. (42)

c_p- Specific heat at constant pressure

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D_n- Constant defined by Eq. (43)

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F- Eigenfunction

H- Constant defined by Eq. (25)

H₁- Function defined by Eq. (26)

H₂- Function defined by Eq. (27)

I- Identity tensor

J_n- Bessel function of order n

k- Thermal conductivity, W/mK

k₀ ,…, k₃- Real constants

p- Pressure, N/m²

Pr- Prandtl number

q- Heat flux, W/m²

r- Radius, m

s_m , s_q , - Eigenvalues

S- Extra-stress tensor

Re- Reynolds number

t- Time, s

T- Cauchy stress tensor

u- Velocity, m/s

z- Axial position, m

Greek letters

γ- Amplitude of pressure fluctuation, m

λ₁- Relaxation time

λ₂- Material constant of Burgers' fluid

λ₃- Retardation time

µ- Dynamic viscosity, N-s/m²

ν- Kinematic viscosity, N-s-m/kg

ρ- Density, kg/m³

ω- Angular velocity, rad/s

Θ- Temperature, K

Superscripts

• - Dimensional conditions

Subscripts

s- Steady state

t- Transient state

w- Wall

1. Introduction

Ruckmongathan [1] showed that besides the multisteps, other waveforms like trapezoidal and triangular exist in electronics for reducing power dissipation in liquid crystal displays. Most of the electronic components and systems continue reduction in size while growing in energy density resulting in the need for more efficient thermal management. Oscillating heat cylinder occurs in many electronic devices, and most of the interest in this subject is the ability to establish extra-high effective thermal and manage high energy rich in heat fluxes [2] , [3] , [4] , [5]  and [6] . Yu et al. [7] developed an analytical solution to determine pulsating laminar heat convection in a circular tube with constant heat flux. The results show that both the temperature profile and the Nusselt number fluctuate periodically about the solution for steady laminar convection, with the fluctuation amplitude depending on the dimensionless pulsation frequency, the amplitude of the pressure, and the Prandtl number. Wang and Zhang [8] analyzed convection heat transfer of pulsing turbulent flow in a pipe with constant wall temperature and high velocity oscillating amplitudes. Their results showed that the heat transfer enhancement is principally impacted by Womersley number and velocity oscillation amplitude. Pendyala et al. [9] conducted experimental studies on the oscillation flow and heat transfer in a vertical tube with a constant amplitude. The results indicated that the influence of an externally imposed periodic oscillation on the heat transfer is stronger at lower flow rates. Akdag and Ozguc [10] analyzed experimentally the heat transfer flow with constant heat flux and oscillating flow inside a vertical annular liquid column. The results demonstrated that heat transfer increases with increasing both the amplitude of the oscillation and frequency. Liu et al. [11] solved the energy equation of circular micro-channels, which considers axial heat conduction, velocity slip, temperature jump, viscous dissipation and thermal entrance effect. The design criterion for whether the axial heat conduction and viscous dissipation should be considered in engineering is given by studying their contributions to average Nusselt number.

Yin and Ma [12] reported the analytical result on an oscillating capillary tube in Newtonian laminar pulsating flows driven by a sinusoidal waveform. Their results show that the oscillating frequency, amplitude, and Prandtl number, are significant factors affecting the heat transfer performance of an oscillating flow in a capillary tube. Yin and Ma [13] investigated Newtonian flow in a tube driven by a triangular pressure waveform and showed how the oscillating flow could result in a different heat transfer coefficient with the corresponding result of a sinusoidal pressure waveform. Abdulhameed et al. [14] proposed the mathematical modeling of unsteady second grade fluid flowing in a capillary tube with sinusoidal pressure waveform and non-homogenous boundary conditions. Exact analytical solutions for the velocity profiles have been obtained in explicit forms using finite Hankel transform. The solutions are written as the sum of the steady and transient solutions for small and large times. For large value of times, the starting solution reduces to the well-known periodic solution that coincides with the corresponding solution of a Newtonian fluid. Khalid et al. [15] studied the problem of unsteady MHD free convection flow of Casson fluid past over an oscillating vertical plate embedded in a porous medium. The governing equations were solved using the Laplace transform technique and exact solutions for velocity and energy are obtained. Gul et al. [16] analyzed the problem of unsteady thin film flow of a second grade fluid over a vertical oscillating belt. The governing equation for velocity field was solved analytically using Adomian decomposition method (ADM) and Optimal asymptotic method (OHAM).

As mentioned above, the previous literature have focused on the Newtonian and second-grade fluids at a sinusoidal and triangular waveform effect on the heat transfer performance of oscillating flow in a capillary tube. The oscillating flow non-Newtonian Burgers' fluid at a trapezoidal, triangular and sinusoidal waveform has not been investigated before. It has been demonstrated that the oscillating heat tube using the thermally excited oscillating motion can significantly enhance the heat transfer performance (Yin and Ma [13] ). The unanswered questions regarding trapezoidal, triangular and sinusoidal waveforms in Burgers' fluid made it necessary to study the effect of oscillating motion on the heat transfer performance of oscillating flow in a circular cylinder. In the current investigation, a Burgers' fluid of an oscillating flow with trapezoidal, triangular and sinusoidal waveforms is modeled using an infinite Fourier series. Using the constant heat flux boundary condition, analytical solutions of velocity and temperature distribution have been obtained. The effects of waveform frequency, waveform amplitude, and Pr number on the heat transfer performance of oscillating flow are analyzed. To demonstrate the unique feature of trapezoidal waveform effect, the sinusoidal waveform and triangular effect on the temperature of the oscillating flow are presented as well.

2. Governing equation of Burgers' fluid

Burgers' model has the ability in successfully capturing various non-Newtonian strange features, e.g. shear thinning/thickening and display of elastic effects. Therefore, it has been the subject of many investigations covering various facets [17] , [18] , [19] , [20]  and [21] .

The constitutive equations for an incompressible homogeneous Burgers' fluid are given by:

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( 1)

where

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( 2)

where T is the Cauchy stress tensor, − pI is the indeterminate spherical stress, S is the extra-stress tensor, λ₁ is the relaxation time, λ₂ is the material constant of Burgers' fluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{d}{dt}}

 is the material time differentiation, L  is the velocity gradient, T  is the transpose operator, µ  is the dynamic viscosity, A  is the first Rivlin-Ericksen tensor and λ3  is the retardation time.      

The above model includes special cases, the Oldroid-B model Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(\mbox{for}\mbox{ }{\lambda }_2=\right. } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. 0\right)

, the Maxwell model Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(\mbox{for}\mbox{ }{\lambda }_2=\right. }

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, the second-grade model Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(\mbox{for}\mbox{ }{\lambda }_1=\right. }

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 and the linear viscous model Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(\mbox{for}\mbox{ }{\lambda }_1=\right. }

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.      

The flow is incompressible and the surface tension effect is not considered

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( 3)

where the velocity and temperature fields are assumed to be of the form:

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( 4)

Under the above considerations, Eq. (3) gives the following dimensional governing equations

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( 5)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \rho c_p\mbox{ }\left(\frac{\partial \Theta }{\partial t}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. u\frac{\partial \Theta }{\partial z}\right)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): k\left(\frac{{\partial }^2\Theta }{\partial r^2}+\frac{1}{r}\frac{\partial \Theta }{\partial r}\right).

( 6)

The boundary conditions corresponding to Eqs. (5) and (6) are

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( 7)

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( 8)

3. Trapezoidal pressure waveform

The oscillating Burgers' flow shown in Fig. 1 is driven by pressure difference with a trapezoidal waveform as in Fig. 2 of amplitude γ and frequency ω , i.e.

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( 9)

Fig. 1. The physical model configuration.

Fig. 2. Triangular pressure waveform configuration.

Here, we need to solve Eqs. (5) and (6) subject to Eqs. (7) and (8) in the case where pressure gradient is given by Eq. (9) .

Consider the following nondimensional quantities

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( 10)

Using Eq. (10) in Eqs. (5) and (6) , the dimensionless momentum and energy equations (after dropping the * notation):

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( 11)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Pr\frac{\partial \Theta \left(z,\mbox{ }r,\mbox{ }t\right)}{\partial t}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\partial }^2\Theta }{\partial r^2}+\frac{1}{r}\frac{\partial \Theta }{\partial r}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 2u\left(r,\mbox{ }t\right)\frac{\partial \Theta \left(z,\mbox{ }r,\mbox{ }t\right)}{\partial z},

( 12)

while the boundary conditions in dimensionless form are

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( 13)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u=0\mbox{ }\mbox{and}\mbox{ }\frac{\partial \Theta }{\partial r}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1\mbox{ }\mbox{at}\mbox{ }r=1.

( 14)

Consider the fluid velocity into two components: a steady u_s , and transient u_t parts i.e.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u\left(r,\mbox{ }t\right)=u_s\mbox{ }\left(r\right)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_t\mbox{ }\left(r,\mbox{ }t\right).

( 15)

Using Eq. (15) into Eq. (11) , we obtained steady and transient components:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\partial }^2u_s}{\partial r^2}+\frac{1}{r}\frac{\partial u_s}{\partial r}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -8,

( 16)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\partial }^2u_t}{\partial r^2}+\frac{1}{r}\frac{\partial u_t}{\partial r}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(1+{\lambda }_1\frac{\partial }{\partial t}+{\lambda }_2\frac{{\partial }^2}{\partial t^2}\right)\frac{\partial u_t}{\partial t}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\lambda }_3\frac{\partial }{\partial t}\left(\frac{{\partial }^2u_t}{\partial r^2}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \frac{1}{r}\frac{\partial u_t}{\partial r}\right) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }-\mbox{ }\frac{256\gamma }{{\pi }^2}\left(1+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\lambda }_1\frac{\partial }{\partial t}+{\lambda }_2\frac{{\partial }^2}{\partial t^2}\right) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }\left(\sum_{n=1}^{\infty }\left(\frac{sin\frac{\pi }{8}\left(2n-1\right)}{{\left(2n-1\right)}^2}sin\left(\left(2n-1\right)\omega t\right)\right)\right),

( 17)

The steady solution of Eq. (16) is followed as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_s=2\left(1-r^2\right).

( 18)

Applying the Bessel transform [22] , [23]  and [24] to Eq. (17) , we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -s_m^2J_0\mbox{ }\left[u_t\mbox{ }\left(r,\mbox{ }t\right)\right]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(1+{\lambda }_1\frac{\partial }{\partial t}+{\lambda }_2\frac{{\partial }^2}{\partial t^2}\right)\frac{\partial J_0\mbox{ }\left[u_t\mbox{ }\left(r,\mbox{ }t\right)\right]}{\partial t}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\lambda }_3s_m^2\frac{\partial }{\partial t}J_0\mbox{ }\left[u_t\mbox{ }\left(r,\mbox{ }t\right)\right] Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }-\mbox{ }\left[\frac{256\gamma }{{\pi }^2}\left(1+\right. \right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \left. {\lambda }_1\frac{\partial }{\partial t}+\right. \right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \left. {\lambda }_2\frac{{\partial }^2}{\partial t^2}\right)\sum_{n=1}^{\infty }\left(\frac{sin\frac{\pi }{8}\left(2n-1\right)}{{\left(2n-1\right)}^2}sin\left(\left(2n-1\right)\omega t\right)\right)\right]\frac{J_1\mbox{ }\left({\lambda }_n\right)}{s_m}.

( 19)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle J_0\mbox{ }\left[u_t\mbox{ }\left(r,\mbox{ }t\right)\right]=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\int }_0^1\mbox{ }ru_t\mbox{ }\left(r,\mbox{ }t\right)J_0\mbox{ }\left(s_mr\right)dr

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle J_0\mbox{ }\left(s_m\right)=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0

, Denote Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F\left({\lambda }_m\mbox{​},\mbox{ }t\right)=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): J_0\mbox{ }\left[u_t\mbox{ }\left(r,\mbox{ }t\right)\right]

, then Eq. (19)  becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F\left(s_m\mbox{​},\mbox{ }t\right)+\left({\lambda }_3+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \frac{1}{s_m^2}\right)\frac{\partial F\left(s_m\mbox{​},\mbox{ }t\right)}{\partial t}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\lambda }_1}{s_m^2}\frac{{\partial }^2F\left(s_m\mbox{​},\mbox{ }t\right)}{\partial t^2}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\lambda }_2}{s_m^2}\frac{{\partial }^3F\left(s_m\mbox{​},\mbox{ }t\right)}{\partial t^3} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }=\left[\frac{256\gamma }{{\pi }^2}\left(1+\right. \right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \left. {\lambda }_1\frac{\partial }{\partial t}+\right. \right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \left. {\lambda }_2\frac{{\partial }^2}{\partial t^2}\right)\sum_{n=1}^{\infty }\left(\frac{sin\left(\frac{\pi }{8}\left(2n-1\right)\right)}{{\left(2n-1\right)}^2}sin\left(\left(2n-1\right)\omega t\right)\right)\right]\frac{J_1\mbox{ }\left(s_m\right)}{s_m^3}.

( 20)

Assuming the solution of Eq. (20) of the form:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F\left(s_m\mbox{​},\mbox{ }t\right)=\sum_{n=1}^{\infty }\left[A_ncos\left(\left(2n-1\right)\omega t\right)+B_nsin\left(\left(2n-1\right)\omega t\right)\right],\mbox{ }N= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(2n-1\right).

( 21)

then Eq. (20) after rearranging yields:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\left[\begin{array}{c} A_n\mbox{ }\left(N{\pi }^2s_m^3-{\lambda }_1N^3{\pi }^2{\omega }^2s_m\right)+B_n\mbox{ }\left(N^2{\pi }^2\omega s_m-{\lambda }_2N^4{\pi }^2{\omega }^3s_m+{\lambda }_3N^2{\pi }^2\omega s_m^3\right)\\ -256\gamma \omega J_1\mbox{ }\left(s_m\right)sin\left(\frac{\pi }{8}N\right) \end{array}\right]=0}{s_m^3{\pi }^2N}, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\left[\begin{array}{c} B_n\mbox{ }\left(N^2{\pi }^2s_m^3-{\lambda }_1N^4{\pi }^2{\omega }^2s_m\right)-A_n\mbox{ }\left(N^3{\pi }^2\omega s_m-{\lambda }_2N^5{\pi }^2{\omega }^3s_m+{\lambda }_3N^3{\pi }^2\omega s_m^3\right)\\ +256\gamma J_1\mbox{ }\left(s_m\right)\left({\lambda }_2N^2{\omega }^2-1\right)sin\left(\frac{\pi }{8}N\right) \end{array}\right]}{s_m^3{\pi }^2N^2}=0.

( 22)

The expression for A_n and B_n could be obtained as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): A_n=\frac{-256\gamma \omega H_1J_1\mbox{ }\left(s_m\right)sin\left(\frac{\pi }{8}N\right)}{s_mN{\pi }^2H},

( 23)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): B_n=\frac{256\gamma s_mH_2J_1\mbox{ }\left(s_m\right)sin\left(\frac{\pi }{8}N\right)}{N^2{\pi }^2H},

( 24)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): H=s_m^4+N^2{\omega }^2-2{\lambda }_1s_m^2N^2{\omega }^2+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\lambda }_1^2N^4{\omega }^4-2{\lambda }_2N^4{\omega }^4+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\lambda }_2^2N^6{\omega }^6 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }-\mbox{ }2{\lambda }_2{\lambda }_3N^4{\omega }^4s_m^2+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 2{\lambda }_3N^2{\omega }^2s_m^2+{\lambda }_3^2N^2{\omega }^2s_m^4\mbox{​},

( 25)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): H_1=1+{\lambda }_1^2N^2{\omega }^2+{\lambda }_2^2{\omega }^4N^4- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\lambda }_2{\lambda }_3{\omega }^2N^2s_m^2-2{\lambda }_2{\omega }^2N^2- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\lambda }_1s_m^2+{\lambda }_3s_m^2\mbox{​},

( 26)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): H_2=1+{\lambda }_1{\lambda }_3N^2{\omega }^2-{\lambda }_2{\omega }^2N^2\mbox{​}.

( 27)

Hence, the expression for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F\left(s_m\mbox{​},\mbox{ }t\right)}

 can be written as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F\left(s_m\mbox{​},\mbox{ }t\right)=\frac{256\gamma }{{\pi }^2} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }\sum_{n=1}^{\infty }\left(\frac{sin\left(\frac{\pi }{8}N\right)\left(-N\omega H_1cos\left(N\omega t\right)+s_m^2H_2sin\left(N\omega t\right)\right)}{HN}\frac{J_1\mbox{ }\left(s_m\right)}{s_m}\right).

( 28)

Applying the inverse Bessel transform:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_t\mbox{ }\left(r,\mbox{ }t\right)=\frac{512\gamma }{{\pi }^2} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }\sum_{m=1}^{\infty }\left[\sum_{n=1}^{\infty }\left(\frac{sin\left(\frac{\pi }{8}N\right)\left(-N\omega H_1cos\left(N\omega t\right)+s_m^2H_2sin\left(N\omega t\right)\right)}{NH}\right)\frac{J_0\mbox{ }\left(s_mr\right)}{s_mJ_1\mbox{ }\left(s_m\right)}\right].

( 29)

where J₁ is the Bessel function of first kind of order one and s_m is the eigenvalue of the Bessel function of first kind of order zero.

Consider the temperature Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Theta \left(z,\mbox{ }r,\mbox{ }t\right)}

 field in two components:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Theta \left(z,\mbox{ }r,\mbox{ }t\right)={\Theta }_s\mbox{ }\left(z,\mbox{ }r\right)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\Theta }_t\mbox{ }\left(r,\mbox{ }t\right).

( 30)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\Theta }_s\mbox{ }\left(z,\mbox{ }r\right)}

 is the steady and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\Theta }_t\mbox{ }\left(r,\mbox{ }t\right)}
 is the transient components.      

Substituting Eqs. (15) and (30) into Eq. (12) and separating the result into steady and transient components:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 2u_s\mbox{ }\left(r\right)\frac{\partial {\Theta }_s\mbox{ }\left(z,\mbox{ }r\right)}{\partial z}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\partial }^2{\Theta }_s\mbox{ }\left(z,\mbox{ }r\right)}{\partial r^2}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{1}{r}\frac{\partial {\Theta }_s\mbox{ }\left(z,\mbox{ }r\right)}{\partial r},

( 31)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Pr\frac{\partial {\Theta }_t\mbox{ }\left(r,\mbox{ }t\right)}{\partial t}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\partial }^2{\Theta }_t\mbox{ }\left(r,\mbox{ }t\right)}{\partial r^2}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{1}{r}\frac{\partial {\Theta }_t\mbox{ }\left(r,\mbox{ }t\right)}{\partial r}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 2u_t\mbox{ }\left(r,\mbox{ }t\right)\frac{\partial {\Theta }_s\mbox{ }\left(z,\mbox{ }r\right)}{\partial z}.

( 32)

Assuming Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\Theta }_s\mbox{ }\left(z,\mbox{ }r\right)=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): T_1\mbox{ }\left(z\right)+T_2\mbox{ }\left(r\right)

 and using the expression in Eq. (18) , Eq. (31)  transformed as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{dT_1\mbox{ }\left(z\right)}{dz}=\frac{1}{4\left(1-r^2\right)}\left(\frac{d^2T_2\mbox{ }\left(r\right)}{dr^2}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \frac{1}{r}\frac{dT_2\mbox{ }\left(r\right)}{dr}\right).

( 33)

and this can be separated into two components as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{1}{4\left(1-r^2\right)}\frac{d}{rdr}\left(r\frac{dT_2\mbox{ }\left(r\right)}{dr}\right)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): k_0\mbox{​},

( 34)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{dT_1\mbox{ }\left(z\right)}{dz}=k_0\mbox{​}.

( 35)

where k₀ is the separation constant.

Integrating Eqs. (34) and (35) yields:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): T_2\mbox{ }\left(r\right)=\frac{k_0}{4}r^2\mbox{ }\left(4-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. r^2\right)+k_1lnr+k_2\mbox{​},

( 36)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): T_1\mbox{ }\left(z\right)=z+k_3\mbox{​}.

( 37)

Using the boundary conditions (13) and (14) , it followed that k₀  = 1, k₁  = k₂  = 0 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k_3=-\frac{7}{24}}

. Hence, the solution of the steady temperature is obtained as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\Theta }_s\mbox{ }\left(z,\mbox{ }r\right)=z+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{1}{4}r^2\mbox{ }\left(4-r^2\right)-\frac{7}{27}.

( 38)

Plugging Eqs. (38) and (29) into Eq. (32) and apply the Bessel transform:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F\left(s_q\mbox{​},\mbox{ }t\right)+\frac{Pr}{s_q^2}\frac{\partial F\left(s_q\mbox{​},\mbox{ }t\right)}{\partial t}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\frac{1024\gamma }{s_q^3{\pi }^2} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }\sum_{m=1}^{\infty }\left[\sum_{n=1}^{\infty }\left(\frac{sin\left(\frac{\pi }{8}N\right)\left(-N\omega H_1cos\left(N\omega t\right)+s_m^2H_2sin\left(N\omega t\right)\right)}{NH}\right)\frac{J_0\mbox{ }\left(s_{mr}\right)J_1\mbox{ }\left(s_q\right)}{s_mJ_1\mbox{ }\left(s_m\right)}\right].

( 39)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F\left(s_q\mbox{​},\mbox{ }t\right)=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): J_0\mbox{ }\left[{\Theta }_t\mbox{ }\left(r,\mbox{ }t\right)\right]

. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F\left(s_q\mbox{​},\mbox{ }t\right)=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\sum }_{n=1}^{\infty }\mbox{ }\left(C_ncos\left(\left(2n-\right. \right. \right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \left. \left. 1\right)\omega t\right)+D_nsin\left(\left(2n-\right. \right. \right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \left. \left. 1\right)\omega t\right)\right)

, N  = 2n  − 1, Eq.  (39)  can be expressed as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(\frac{Pr\omega N}{s_q^2}C_n-\frac{1024\gamma }{s_q^3{\pi }^2}\sum_{m=1}^{\infty }\left(\frac{sin\left(\frac{\pi }{8}N\right)s_m^2H_2}{NH}\frac{J_0\mbox{ }\left(s_{mr}\right)J_1\mbox{ }\left(s_q\right)}{s_mJ_1\mbox{ }\left(s_m\right)}\right)-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. D_n\right)=0,

( 40)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(-\frac{Pr\omega N}{s_q^2}D_n+\frac{1024\gamma }{s_q^3{\pi }^2}\sum_{m=1}^{\infty }\left(\frac{sin\left(\frac{\pi }{8}N\right)\omega H_1}{H}\frac{J_0\mbox{ }\left(s_{mr}\right)J_1\mbox{ }\left(s_q\right)}{s_mJ_1\mbox{ }\left(s_m\right)}\right)-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. C_n\right)=0.

( 41)

Solving Eqs. (40) and (41) for C_n and D_n gives:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): C_n=\frac{1024\gamma }{{\pi }^2}\frac{\left(E_{1n}Pr\omega N+E_{2n}s_q^2\right)}{\left(s_q^4+{Pr}^2{\omega }^2N^2\right)}J_1\mbox{ }\left(s_q\right),

( 42)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): D_n=-\frac{1024\gamma }{{\pi }^2}\frac{\left(E_{1n}s_q^2-E_{2n}Pr\omega N\right)}{\left(s_q^4+{Pr}^2{\omega }^2N^2\right)}J_1\mbox{ }\left(s_q\right),

( 43)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): E_{1n}=\sum_{m=1}^{\infty }\left(\frac{sin\left(\frac{\pi }{8}N\right)s_m^2H_2}{NH}\frac{J_0\mbox{ }\left(s_{mr}\right)}{s_mJ_1\mbox{ }\left(s_m\right)}\right),

( 44)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): E_{2n}=\sum_{m=1}^{\infty }\left(\frac{sin\left(\frac{\pi }{8}N\right)\omega H_1}{H}\frac{J_0\mbox{ }\left(s_{mr}\right)}{s_mJ_1\mbox{ }\left(s_m\right)}\right),

( 45)

The solution of temperature distribution is expressed as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F\left(s_q\mbox{​},\mbox{ }t\right)=\frac{1024\gamma }{{\pi }^2}J_1\mbox{ }\left(s_q\right) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }\sum_{n=1}^{\infty }\left[\frac{\left(E_{1n}Pr\omega N+E_{2n}s_q^2\right)cos\left(N\omega t\right)-\left(E_{1n}s_q^2-E_{2n}Pr\omega N\right)sin\left(N\omega t\right)}{\left(s_q^4+{Pr}^2{\omega }^2N^2\right)}\right].

( 46)

Applying the inverse Bessel transform:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\Theta }_t\mbox{ }\left(r,\mbox{ }t\right)=\frac{2048\gamma }{{\pi }^2} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }\sum_{q=1}^{\infty }\left[\sum_{n=1}^{\infty }\left(\frac{\left(E_{1n}Pr\omega N+E_{2n}s_q^2\right)cos\left(N\omega t\right)-\left(E_{1n}s_q^2-E_{2n}Pr\omega N\right)sin\left(N\omega t\right)}{\left(s_q^4+{Pr}^2{\omega }^2N^2\right)}\right)\frac{J_0\mbox{ }\left(s_qr\right)}{J_1\mbox{ }\left(s_q\right)}\right].

( 47)

4. Triangular pressure waveform

Here, we consider a triangular pressure waveform given by Fig. 3 of amplitude γ and frequency ω is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\frac{\partial p}{\partial z}=\gamma \frac{8}{{\pi }^2}\sum_{n=1}^{\infty }\left[\frac{{\left(-1\right)}^{n-1}}{{\left(2n-1\right)}^2}\right]sin\left(\left(2n-1\right)\omega t\right),

( 48)

Fig. 3. Trapezoidal pressure waveform configuration.

Using the same methodology in the preceding analysis, we obtain the velocity field and temperature distribution for triangular waveform as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{tri}\mbox{ }\left(r,\mbox{ }t\right)=\frac{64\gamma }{\pi } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }\sum_{m=1}^{\infty }\left[\sum_{n=1}^{\infty }\left(\frac{{\left(-1\right)}^{n-1}\mbox{ }\left(-N\omega H_1cos\left(N\omega t\right)+s_m^2H_2sin\left(N\omega t\right)\right)}{NH}\right)\frac{J_0\mbox{ }\left(s_mr\right)}{J_1\mbox{ }\left(s_m\right)}\right],

( 49)

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\Theta }_{tri}\mbox{ }\left(r,\mbox{ }t\right)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{128\gamma }{\pi } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }\sum_{q=1}^{\infty }\left[\sum_{n=1}^{\infty }\left(\frac{\left(E_{1n}Pr\omega N+E_{2n}s_q^2\right)cos\left(N\omega t\right)-\left(E_{1n}s_q^2-E_{2n}Pr\omega N\right)sin\left(N\omega t\right)}{\left(s_q^4+{Pr}^2{\omega }^2N^2\right)}\right)\frac{J_0\mbox{ }\left(s_qr\right)}{J_1\mbox{ }\left(s_q\right)}\right].

( 50)

Letting λ₁  = λ₂  = λ₃  = 0, into Eqs. (25) , (26)  and (27) , the case of Newtonian fluid, Eqs. (49) and (50) becomes identical with that obtained by Yin and Ma [13] .

5. Sinusoidal pressure waveform

In this section, the oscillating Burgers' flow is driven by pressure difference with a sinusoidal waveform given in Fig. 4 of amplitude γ and frequency ω , i.e.,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\frac{\partial p}{\partial z}=\gamma sin\left(\omega t\right),

( 51)

Fig. 4. Sinusoidal pressure waveform configuration.

Applying the same methodology in the preceding analysis, the velocity and temperature distribution for sinusoidal waveform:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{sin}\mbox{ }\left(r,\mbox{ }t\right)=16\gamma \sum_{m=1}^{\infty }\left[\sum_{n=1}^{\infty }\left(\frac{\left(-\omega H_1cos\left(\omega t\right)+s_m^2H_2sin\left(\omega t\right)\right)}{H}\right)\frac{J_0\mbox{ }\left(s_mr\right)}{J_1\mbox{ }\left(s_m\right)}\right],

( 52)

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\Theta }_{sin}\mbox{ }\left(r,\mbox{ }t\right)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 64\gamma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{ }\sum_{q=1}^{\infty }\left[\sum_{n=1}^{\infty }\left(\frac{\left(E_{1n}Pr\omega +E_{2n}s_q^2\right)cos\left(\omega t\right)-\left(E_{1n}s_q^2-E_{2n}Pr\omega \right)sin\left(\omega t\right)}{\left(s_q^4+{Pr}^2{\omega }^2\right)}\right)\frac{J_0\mbox{ }\left(s_qr\right)}{J_1\mbox{ }\left(s_q\right)}\right].

( 53)

Making λ₁  = λ₂  = λ₃  = 0, into Eqs. (25) , (26)  and (27) , the case of Newtonian fluid, Eqs. (52) and (53) becomes identical with that obtained by Yin and Ma [12] .

6. Results and discussion

In this section, we analyzed the transient temperature distribution for various types of fluids namely, a Newtonian fluid for which λ₁  = λ₂  = λ₃  = 0, a Maxwell fluid for which λ₁  ≠ 0, λ₂  = λ₃  = 0, an Oldroyd-B fluid for which λ₁  ≠ 0, λ₂  = 0, λ₃  ≠ 0, and a Burgers' fluid.

Table 1 shows numerical values for different pressure waveforms at r  = 0.00, λ₁  = 0.25, λ₂  = 0.25, λ₃  = 0.25, and ω  = 1. As can be seen, the performance of the transient temperature distribution is higher for trapezoidal pressure waveform in comparison with triangular and sinusoidal pressure waveforms. It is obvious that the peak of the transient temperature only achieves less compared with trapezoidal and sinusoidal.

Table 1.

Numerical values of the peak of the transient temperature distribution with different pressure waveforms.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\Theta }_{\mbox{Sinusoidal}}	t	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\Theta }_{\mbox{Trapezoidal}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\Theta }_{\mbox{Triangular}}
0.00	0.0000	0.0000	0.000
0.20	0.507	0.127	0.199
0.40	0.995	0.255	0.389
0.60	1.000	0.382	0.565
0.80	0.999	0.509	0.717
1.00	0.999	0.637	0.841
1.20	0.999	0.764	0.932
1.40	1.000	0.891	0.985
1.60	1.000	0.982	1.000
1.80	1.000	0.854	0.974
2.00	1.000	0.727	0.909

The effects of physical parameters, such as the Prandtl number, Pr , amplitude of pressure fluctuation, γ , and radial coordinate, r , on the transient temperature distribution for all four types of fluid have been shown in Fig. 5 , Fig. 6 , Fig. 7  and Fig. 8 . In all these figures, waveforms profiles are shown for both trapezoidal, triangular and sinusoidal pressure waveforms. Also, from these figures, it is observed that the variation for waveform effects with the classical Newtonian, Maxwell, Oldroyd-B and Burgers' fluids are distinct. Maxwell fluid has the maximum waveform effect as compared with Burgers' fluid. The waveform effect of an Oldroyd-B fluid is smaller than those for a Burgers' fluid. The Newtonian fluid has the lowest waveform effect.

Fig. 5 (a)–(d) shows the waveform effect on transient temperature distribution of an oscillating motion at Pr  = 0.1, γ  = 0.01, and ω  = 1.0. As shown in Fig. 5 (a)–(d), when the oscillating flow is driven by a trapezoidal waveform pressure difference, the peak of the transient temperature distribution can reach the same level as when is driven by sinusoidal waveform difference. However, when the pressure waveform is triangular with the same frequency and same amplitude, the peak of the transient temperature only achieves less compared with trapezoidal and sinusoidal. This indicates that for an oscillating flow with trapezoidal and sinusoidal waveforms, an optimum frequency and an amplitude exist which can significantly increase the heat transfer performance. When the Prandtl number Pr increases from 0.1 to 1.0, keeping amplitude and frequency fixed, i.e., γ  = 0.01 and ω  = 1.0, the influence of Pr on transient temperature becomes negligibly small as shown in Fig. 6 (a)–(d).

Fig. 5. Waveform effects on temperature distribution at Pr = 0.1 when γ  = 0.01, ω  = 1.0 and r  = 0.0.

Fig. 6. Waveform effects on temperature distribution at Pr = 1.0 when γ  = 0.01, ω  = 1.0 and r  = 0.0.

The effect of waveform amplitude is shown in Fig. 7 (a)–(d). It demonstrates that when the pressure oscillating amplitude increases to γ  = 0.1, keeping the Prandtl number and dimensionless frequency fixed, i.e. Pr  = 1 and ω  = 1.0, the peak transient temperature increases as compared with Fig. 6 (a)–(d) when γ  = 0.01. These results indicate that maximizing the oscillating amplitude can enhance heat transfer for trapezoidal, triangular and sinusoidal pressure waveforms. It is also interesting to observe that the waveform effect in Fig. 7 (a)–(d) is equal to 10 times as that in Fig. 7 (a)–(d). If the radial coordinate is increased, i.e. r  = 0.5, keeping other conditions fixed, i.e. Pr  = 1.0, γ  = 0.01 and ω  = 1.0, as shown in Fig. 8 (a)–(d), it can be found that the peak transient temperature decreases significantly. This indicates that the transient temperature is at the maximum at the surface of the cylinder and it decreases on increasing radial coordinate r to approach constant heat flux.

Fig. 7. Waveform effects on temperature distribution at γ  = 0.1 when Pr = 1.0, ω  = 1.0 and r  = 0.0.

Fig. 8. Waveform effects on temperature distribution at γ  = 0.1 when Pr = 1.0, ω  = 1.0 and r  = 0.5.

7. Conclusions

Analytical solutions of velocity and temperature distributions of a laminar oscillating flow of Burgers' fluid driven by trapezoidal, triangular and sinusoidal pressure waveforms are obtained in explicit form. The models of Newtonian, Maxwell and Oldroyd-B are obtained from the present analysis as the limiting cases. Results show that the peak temperature for trapezoidal and sinusoidal pressure waveforms is very different from triangular pressure waveform. For an oscillating flow that was driven by a trapezoidal waveform, when the radial coordinate decreases or when the amplitude increases, the oscillating motion can enhance heat transfer. However, for small pulsating frequencies, e.g. ω  = 0.1, the influence of Pr on transient temperature becomes negligibly small.

Acknowledgments

We are thankful for the constructive comments and suggestions of the reviewers that help the authors a lot to improve the manuscript quality. The third author acknowledges the Ministry of Higher Education Malaysia (MOHE ) and the Research Management Centre-UTM for the financial support through vote numbers 06H67 and 4F255 .

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