Analytical solutions of temperature distributions and the Nusselt numbers in forced convection are reported for flow through infinitely long parallel plates, where the upper plate moves in the flow direction with constant velocity and the lower plate is kept stationary. The flow is assumed to be laminar, both hydro-dynamically and thermally fully developed, taking into account the effect of viscous dissipation of the flowing fluid. Both the plates being kept at specified and at different constant heat fluxes are considered as thermal boundary conditions. The solutions obtained from energy equation are in terms of Brinkman number, dimensionless velocity and heat flux ratio. These parameters greatly influence and give complete understanding on heat transfer rates that has potentials for designing and analyzing energy equipment and processes.
Flow of Newtonian fluids through various channels is of practical importance and heat transfer is dependent on flow conditions such as flow geometry and physical properties. Investigations in heat transfer behavior through various channels showed that the effect of viscous dissipation cannot be neglected for some applications, such as flow through micro-channels, small conduits and extrusion at high speeds. The thermal development of forced convection through infinitely long fixed parallel plates, both plates having specified constant heat flux had been investigated [1-5]. For the same but filled by a saturated porous medium, heat transfer analysis was done where the walls were kept at uniform wall temperature with the effect of viscous dissipation and axial conduction taken into account [
For the pipe flow, where the walls are kept either at constant heat flux or constant wall temperature, analytical solution is obtained for both hydro-dynamically and thermally fully developed and thermally developing Newtonian fluid flow, considering the effect of viscous dissipation [9,10].
Analytical solution with the effect of viscous dissipation was derived for Couette-Poiseuille flow of nonlinear visco-elastic fluids and with the simplified Phan-ThienTanner fluid between parallel plates, with stationary plate subjected to constant heat flux and the other plate moving with constant velocity but insulated [11-13]. Numerical solution of fully developed laminar heat transfer of power-law non-Newtonian fluids in plane Couette flow, with constant heat flux at one wall with other wall insulated had been investigated [
A numerical investigation had been done to find the heat transfer for the simultaneously developing steady laminar flow, where the fluid was considered to be viscous non-Newtonian described by a power-law model flowing between two parallel plates with several different thermal boundary conditions [
The Bingham fluid was assumed to be flowing in between two porous parallel plates. With the slip effect at the porous walls, the analytical solutions were obtained for the Couette-Poiseuille flow [
From the literature survey, it is observed that heat transfer analysis with effect of viscous dissipation is not found for the Couette-Poiseuille flow with both the plates being kept at specified but different constant heat fluxes. The heat transfer analysis with one plate moving is a different fundamental problem worth pursuing. This study is necessary specifically in the design of special heat exchangers and other devices where the dimensions have to be kept very small. Hence, the case of lower plate being fixed and the upper plate moving with constant velocity, both being imposed to different but constant heat fluxes is considered. The energy equation is solved leading to expressions in temperature profiles and Nusselt number, that could be useful to industrial applications.
Consider two flat infinitely long parallel plates distanced W or 2 apart, where the upper plate is moving with constant velocity U and the lower plate is fixed. The coordinate system chosen is shown in
The momentum equation in the x-direction is described as
where u is the velocity of the fluid, is the dynamic viscosity, P is the pressure.
The velocity boundary conditions are u = 0 when y = 0 and u = U when y = W.
Using the following dimensionless parameters:
the well-known velocity-distribution is [
where the mean velocity (um) is given by
For the above equation, expression for u is obtained by solving the momentum Equation (1).
The energy equation, including the effect of viscous dissipation, is given by
where the second term on the right-hand side is the viscous-dissipative term. In accordance to the assumption of a thermally fully developed flow with uniformly heated boundary walls, the longitudinal conduction term is neglected in the energy equation [
where and are the upper and lower wall temperatures, respectively.
By taking, introducing the non-dimensional quantity
and defining a dimensionless constant,
and modified Brinkman number as
Equation (5) can be written as
The thermal boundary conditions are
The solution of Equation (10) under the above thermal boundary conditions can be obtained as
To evaluate in the above equation, a third boundary condition is required:
By substituting Equation (13) into Equation (12),
can be expressed as
Therefore, the solution of Equation (10) under the above thermal boundary conditions can be written in a simplified form as
where
In fully developed flow, it is usual to utilize the mean fluid-temperature, , rather than the centerline temperature, when defining the Nusselt number. Thus mean or bulk temperature is given by
with the cross-sectional area of the channel and the denominator on the right-hand side of Equation (17) can be written as
Using Equations (3) and (15), the numerator of Equation (17) can be found. Therefore the dimensionless mean temperature is given by
At this point, the convective heat transfer coefficient can be evaluated by the equation
Defining Nusselt number to be
where Dh is the hydraulic diameter defined by Dh = 2W, the expression for Nusselt number can be shown to be
When q2 = 0,
agreeing with reference [
Explicit expressions for Nusselt number for various values of U*, and are given in the following discussions.
For the purpose of discussion on the behavior of the Couette-Poiseuille flow, two types of graphs based on the analytical solutions are made. The temperature profile in the channel is plotted with variations of various parameters to indicate the heated region, and the Nusselt number is plotted to reveal the heat transfer characteristics of the flow.
The effect of viscous dissipation is seen in the value of modified Brinkman number. It is interesting to observe the behavior of the temperature profiles for various heat flux ratios for a fixed modified Brinkman number and hence to note the effect of viscous dissipation. In