Fully Developed Flow with Developing Temperature Profile

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Convective heat transfer for steady state, laminar, hydrodynamically developed flow and developing temperature profile in microchannels with both uniform temperature and uniform heat flux boundary conditions were solved by Tunc and Bayazitoglu [1][2] using the integral transform technique that is presented below.

Uniform Temperature

The energy equation assuming fully developed flow, including viscous dissipation and neglecting axial conduction[3], is

u\frac{\partial T}{\partial x}=\frac{\alpha }{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right)+\frac{\nu }{{{c}_{p}}}{{\left( \frac{\partial u}{\partial r} \right)}^{2}}

The boundary and inlet conditions for constant wall temperature are:

\begin{matrix}   {} & {}  \\\end{matrix}T={{T}_{s}}\text{  at  }r={{r}_{o}}


\frac{\partial T}{\partial r}=0\text{  at  }r=0


\begin{matrix}   {} & {}  \\\end{matrix}T={{T}_{0}}\text{  at  }x=0


The fully developed velocity profile with slip boundary condition is used. The slip boundary condition is also used to express the wall temperature jump. The following non-dimensional variables are introduced: for temperature (θ), radial coordinate (r+), axial coordinate (x+), and velocity (u+):

\theta =\frac{T-{{T}_{s}}}{{{T}_{0}}-{{T}_{s}}},\text{ }{{r}^{+}}=\frac{r}{{{r}_{o}}},\text{ }{{x}^{+}}=\frac{x}{L},\text{ }{{u}^{+}}=\frac{u}{{{u}_{m}}}


The non-dimensional energy equation and boundary conditions are obtained through use of the above variables:

\frac{\text{Gz}\left( 1-{{r}^{+}}^{2}+4\text{Kn} \right)}{2\left( 1+8\text{Kn} \right)}\frac{\partial \theta }{\partial {{x}^{+}}}=\frac{1}{{{r}^{+}}}\frac{\partial }{\partial {{r}^{+}}}\left( {{r}^{+}}\frac{\partial \theta }{\partial {{r}^{+}}} \right)+\frac{16\text{Br}}{{{\left( 1+8\text{Kn} \right)}^{2}}}{{r}^{+}}^{2}


\begin{matrix}   {} & {}  \\\end{matrix}\theta =0\text{  at  }{{r}^{+}}=1


\frac{\partial \theta }{\partial {{r}^{+}}}=0\text{  at  }{{r}^{+}}=0


\begin{matrix}   {} & {}  \\\end{matrix}\theta =1\text{  at  }{{x}^{+}}=0


where the Graetz number (Gz) and the Brinkman number (Br) are defined as:

\text{Gz}=\frac{\text{RePr}D}{L}\text{  and  Br}=\frac{\mu u_{m}^{2}}{k\Delta T}


where ΔT is the difference between the temperature of the fluid at the wall, Ts, and at the tube entrance, T0, i.e. ΔT = T0Ts.

Uniform Heat Flux

For the case of constant heat flux at the wall, the following non-dimensional variables are used:

\theta =\frac{T-{{T}_{0}}}{{{{{q}''}}_{w}}{{r}_{0}}/k}\text{  and  Br}=\frac{\mu u_{m}^{2}}{{{{{q}''}}_{w}}D}


Upon use of the above non-dimensional variables, the non-dimensional energy equation for constant wall heat flux can be obtained:

\frac{\text{Gz}\left( 1-{{r}^{+2}}+4\text{Kn} \right)}{2\left( 1+8\text{Kn} \right)}\frac{\partial \theta }{\partial {{x}^{+}}}=\frac{1}{{{r}^{+}}}\frac{\partial }{\partial {{r}^{+}}}\left( {{r}^{+}}\frac{\partial \theta }{\partial {{r}^{+}}} \right)+\frac{16\text{Br}}{{{\left( 1+8\text{Kn} \right)}^{2}}}{{r}^{+2}}


where the centerline symmetric and uniform inlet temperature conditions are the same as eqs. (7) and (8), respectively. However, the boundary condition at the wall is given by \partial \theta /\partial {{r}^{+}}=1.

An integral transform technique based on separation of variables was used by Tunc and Bayazitoglu [1] to solve this problem. An appropriate integral transform pair was developed. Under the transformation, the variable x+ was eliminated from the partial differential governing equation, which transformed the governing equation into an ordinary differential equation.

Configuration for the slip boundary condition in a microchannel.
Configuration for the slip boundary condition in a microchannel.

Jeong and Jeong [4] extended the analysis for application to microchannels of rectangular cross-section, including axial conduction and viscous dissipation.

The configuration for developed flow with developing temperature profile for rectangular microchannels is similar to the figure on the right. The fluid temperature changes from the value T0 at the entrance, to the value Ts on the walls. The governing energy equation and boundary conditions, including axial conduction and viscous dissipation, for laminar flow are:

\rho {{c}_{p}}u\frac{\partial T}{\partial x}=k\left( \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}} \right)+\mu {{\left( \frac{\partial u}{\partial y} \right)}^{2}}


T = T0 at x = 0


T-{{T}_{w}}=\mp \frac{2-{{F}_{t}}}{{{F}_{t}}}\frac{2\gamma }{\gamma +1}\frac{\lambda }{Pr}\frac{\partial T}{\partial y}\text{  at  }y=\pm H


\frac{\partial T}{\partial y}=0\text{  at  }y=0


where H is half the microchannel height, and the wall length in the x-direction is L. The fully developed velocity profile in the rectangular microchannel is:

u\left( y \right)=-\frac{{{H}^{2}}}{2\mu }\frac{dP}{dx}\left[ 1-{{\left( \frac{y}{H} \right)}^{2}}+8\frac{2-F}{F}\text{Kn} \right]


which satisfies the slip boundary condition:

{{u}_{s}}=\mp \frac{2-F}{F}\lambda \left( \frac{\partial u}{\partial y} \right)\text{  at  }y=\pm H


Defining the following dimensionless variables:

\theta =\frac{T-{{T}_{w}}}{{{T}_{0}}-{{T}_{w}}}\text{, }{{x}^{+}}=\frac{x}{\text{RePr}H}\text{,  }{{y}^{+}}=\frac{y}{H}\text{,  Br}=\frac{\mu u_{m}^{2}}{k\left( {{T}_{0}}-{{T}_{w}} \right)},\text{  Kn}=\frac{\lambda }{{{D}_{h}}}


Equations (16) and (20) are respectively non-dimensionalized as:

\frac{1}{4}{{u}^{+}}\frac{\partial \theta }{\partial {{x}^{+}}}=\frac{1}{P{{e}^{2}}}\frac{{{\partial }^{2}}\theta }{\partial {{x}^{+}}^{2}}+\frac{{{\partial }^{2}}\theta }{\partial {{y}^{+}}^{2}}+\text{Br}{{\left( \frac{\partial {{u}^{+}}}{\partial {{y}^{+}}} \right)}^{2}}



{{u}^{+}}=\frac{u}{{{u}_{m}}}=\frac{3}{2}\frac{\left( 1-{{y}^{+}}^{2}+8\frac{2-F}{F}\text{Kn} \right)}{{{C}_{1}}}





The non-dimensional boundary conditions are:

\begin{matrix}   {} & {}  \\\end{matrix}\theta =1\text{  at  }{{x}^{+}}=0


\theta =-4{{C}_{2}}\frac{\partial \theta }{\partial y}\text{  at  }{{y}^{+}}=1


\frac{\partial \theta }{\partial y}=0\text{  at  }{{y}^{+}}=0



{{C}_{2}}=\frac{2-{{F}_{t}}}{{{F}_{t}}}\frac{2\gamma }{\gamma +1}\frac{\text{Kn}}{\text{Pr}}

The effects of the Knudsen number on the Nusselt number variation along a rectangular microchannel neglecting axial conduction and viscous dissipation are studied by Jeong and Jeong[4]. For Kn = 0, the fully developed Nusselt number is approximately 7.54, which is the result for a pipe of conventional size (classic Graetz problem). The Nusselt number decreases as Kn increases due to the temperature jump at the wall.

When the channel is subjected to a constant wall temperature, as {{x}^{+}}\to \infty , the Nusselt number for \text{Br}\ne 0 is independent of Br and different from that for Br = 0. The thermally fully developed Nusselt number was obtained from the fully developed temperature profile for constant wall temperature and heat flux by [4]. For constant wall temperature the Nusselt number as {{x}^{+}}\to \infty (\text{Br}\ne 0) is

\text{N}{{\text{u}}_{\infty }}=\frac{140{{C}_{1}}}{1+7{{C}_{1}}+140{{C}_{1}}{{C}_{2}}}


And for constant heat flux

\text{N}{{\text{u}}_{\infty }}=\frac{420C_{1}^{4}}{C_{1}^{2}\left( 35C_{1}^{2}+14{{C}_{1}}+2+420C_{1}^{2}{{C}_{2}} \right)+Br\left( 42C_{1}^{2}+33{{C}_{1}}+6 \right)}



{{C}_{1}}=1+12\left( \frac{2-F}{F} \right)\text{Kn}
{{C}_{2}}=\frac{2-{{F}_{t}}}{{{F}_{t}}}\frac{2\gamma }{\gamma +1}\frac{\text{Kn}}{\text{Pr}}.

Unless Br is a large negative number, Nu, is always positive.


  1. 1.0 1.1 Tunc, G., and Bayazitoglu, Y., 2001, “Heat Transfer in Microtubes with Viscous Dissipation,” Int. J. Heat Mass Transfer, Vol. 44, pp. 2395-2403.
  2. Tunc, G., and Bayazitoglu, Y., 2002, “Heat Transfer in Rectangular Microchannels,” Int. J. Heat Mass Transfer, Vol. 45, pp. 765-773.
  3. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
  4. 4.0 4.1 4.2 Jeong, H.E., Jeong, J.T., 2006, “Extended Graetz Problem including Streamwise Conduction and Viscous Dissipation in Microchannel,” Int. J. Heat Mass Transfer, Vol. 49, pp. 2151-2157.