Deewaker, Let me address your first statement about developing hydrodynamic boundary layers in a duct, as I think there are some details that you may not be aware of. The problem of developing hydrodynamic boundary layers in a duct is related to the classic entry flow problem that was studied in detail in the mid 1960's using asymptotic methods (see van Dyke and others), and later the problem was solved using numerical methods. Here is a brief description of the physics.

When a uniform irrotational flow u=U_0 enters a 2-D duct, boundary layers develop on the upper and lower surfaces of the wall, due to the no-slip condition. These boundary layers, which have an interposed irrotational accelerating core u=U(x), grow in thickness until they merge. Note: the inviscid core must accelerate to ensure that mass is conserved. After the boundary layers have merged, the flow then begins its transition to the fully develop planar Poiseuille profile (rectilinear flow). The analysis of this transition process was analyzed by Wilson in 1969. ( see my paper on "Downstrean devlopment of 2-D viscocapillary fllow" for the Wilson reference, and the analysis method). The transition length to fully develop flow after the boundary layers have merged depends on the Reynolds number, as one might expect. So there are really two processes taking place before the flow becomes rectilinear (or fully developed). Normally from an engineering perspective, the termination of the entry flow regime is defined when the boundaty layers have merged, and its length is a function of the Reynolds numbers. A crude approximation for estimating the entry length is to use the formulas from the Blasius solution for boundary layer flow over a flate plate. Of course, this approximation does not account for the accelerating core U(x) that I mentioned above. These efefcts also impact the devlopment of a thermal profile, which I will discuss separately.

If you have uniform heat flux, both the temperature of t he fluid and the wall temperature increase. Thermal development is achieved when the two derivatives of the two temperatures with respect to the z direction are equal. In other words the wall T and and the fluid T increase at the same rate.

Well I suppose you mean at every point at a cross section, the rate of temperature increase is same or in other words linear. Right?

The condition for the thermally fully developed flow is defined as d((Ts-T)/(Ts-Tm))/dx =0. The relative temperature profile does not change with x, but the temperature profile changes along x. With a constant heat flux condition, we can further get dT/dx=dTs/dx=dTm/dx=constant.

Deewaker, Let me address your first statement about developing hydrodynamic boundary layers in a duct, as I think there are some details that you may not be aware of. The problem of developing hydrodynamic boundary layers in a duct is related to the classic entry flow problem that was studied in detail in the mid 1960's using asymptotic methods (see van Dyke and others), and later the problem was solved using numerical methods. Here is a brief description of the physics.

When a uniform irrotational flow u=U_0 enters a 2-D duct, boundary layers develop on the upper and lower surfaces of the wall, due to the no-slip condition. These boundary layers, which have an interposed irrotational accelerating core u=U(x), grow in thickness until they merge. Note: the inviscid core must accelerate to ensure that mass is conserved. After the boundary layers have merged, the flow then begins its transition to the fully develop planar Poiseuille profile (rectilinear flow). The analysis of this transition process was analyzed by Wilson in 1969. ( see my paper on "Downstrean devlopment of 2-D viscocapillary fllow" for the Wilson reference, and the analysis method). The transition length to fully develop flow after the boundary layers have merged depends on the Reynolds number, as one might expect. So there are really two processes taking place before the flow becomes rectilinear (or fully developed). Normally from an engineering perspective, the termination of the entry flow regime is defined when the boundaty layers have merged, and its length is a function of the Reynolds numbers. A crude approximation for estimating the entry length is to use the formulas from the Blasius solution for boundary layer flow over a flate plate. Of course, this approximation does not account for the accelerating core U(x) that I mentioned above. These efefcts also impact the devlopment of a thermal profile, which I will discuss separately.

The simplest problem to consider is a sufficiently long channel (the walls are held at T_0) in which a fluid at uniform T_0 enters and at some distance downstream say x=L, the hydrodynamic boundary layers have merged and the flow is fully developed with a plain Poiseuille profile. Now at x=L+W the thermal BCs imposed at the channel walls are changed from T_0 to q=C (where q=-k\nabla T is the heat flux). Next we will assume that the fluid properties ( viscosity, density) are not strong functions of temperature and can be assumed to be constant. Also we will ignore viscous dissipation. Thus the thermal energy equation becomes decoupled from the flow equations and the temperature is given by a time dependent convective-diffusion equation.

Because the flow is rectilinear there is only one advection term: Pe(U(y)\partial T / \partial x) in the energy equation multiplied by a Peclet number when appropriate dimensionless variables are used. This is the problem to solve subject to the flux BCs. The thermal BLs grow in time and space. In certain cases you can consider the quasi steady state, so that the BLs grow in space only. I believe a similarity solution can be found, though numerical calculations may be required (Google Leveque solution).

But it is straightforward to see that for a uniform velocity profile there is no fully developed temperature far downstream if heat is being added at both walls. Perhaps you want to look for an asymptotic solution T(x,y)= f(y)Exp[a x] which will satisfy Pe U(y)\partial T/\partial x=\partial^2T/\partial y^2, .........

Many thanks to Prof. Higgins for describing the phenomena in such a simple yet accurate scientific language.

@Prof. Higgins. Thanks for such a detailed explanation. I have a doubt reg. your reply to hydrodynamic region. Correct me if I am wrong in interpreting.

After the BL has merged, there is a certain region due to acceleration of the core that fully developed profile will be attained after some distance downstream and not at the place where BL meet. However, for sake of simplicity we consider when both the BL have met.

If I am correct, then is there any particular name given to the length described from where BL have to met till fully developed profile has attained. The reason for asking this is: if I take gradient along flow direction at each point on a particular cross-section and then define entrance length up-to cross section where gradient along flow direction will be zero at every point, then this length will not be the length where BL have actually merged but where the profile is not fully developed i.e core is also no more accelerating.

Also, say in case of arbitrary shaped duct, say an irregular hexagon whose fully developed profile may not be known analytically, then is it correct to approach using the above method to calculate hydro entrance length.

The accelerating core affects the growth rate of the BL with distance downstream. Once the boundary layers have merged, the flow then begins its adjustment to reach a fully developed profile. This is an eigenvalue problem and the adjustment length depends on the first few eigenvalues. But for practical engineering calculations one can simply assume that full develop flow is reached when the boundary layers have merged.

I don't think there is an approved name for the adjustment length-I would call it the "adjustment length for fully developed flow". One might quantify it by saying that fully developed flow is reached when the centerline velocity is 98% of the expected centerline velocity for fully developed plain Poiseuille flow.

For a duct that has an irregular hexagon as its cross section, the full developed flow satisfies a Poison equation ( \nabla^2 u= G) which will likely have to be solve numerically using finite elements or using a pseudo-spectral method. Estimating the entrance length becomes a real challenge as you will need to solve the 3-d BL equations. But perhaps one could derive some integral form of the 3-d BL equations ( inspired by the von Karman integral method). I would look at classic books on boundary layers starting with "Boundary Layer Theory" by H. Schlichting and K.Gersten, and "Laminar Boundary Layers" by L. Rosenhead.

"Boundary Layer Theory" by H. Schlichting is a really amazing text book. I do recommend it too.

@Prof Higgins.

Thanks for your nice explanation. I would look into the book by Rosenhead and try to look deep into the aspects.

Meanwhile, one more question has come to mind following your answers. If I am correct in stating that when at a cross section velocity gradient in flow direction goes zero at all points then we have fully developed hydro flow,then what will be the definition in case the flow moves in a circular path, like a solenoid kind of thing. of in simple words , a circular tube of a bicycle.

Ha! Now you are moving into a more complicated world. It turns out that there is considerable interest in flows in curved pipes (e.g., portions of a toroid). Blood flow in arteries comes to mind. This problem was first analyzed by Dean in 1927 who showed that there is no unidirectional flow in say the theta direction. Instead, a steady secondary flow in a plane perpendicular to the flow direction is set up, such that at any cross-section of the curved pipe there is a pair of counter rotating eddies/vortices. Hence the particle path lines would be a spiral-like in the flow direction.

Of course, for a flow in a closed torus , one is required to rotate the torus about the z axis to generate flow, and that is a whole new story....

@ Prof Higgins

My curiosity and interest is gaining momentum with your answers and replies. A brief literature recommendation pertaining to above mentioned question will be highly appreciable.