Draft:Buoyancy-Induced Flow

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Buoyancy-Induced Flow Between Parallel Plates

Introduction

In fluid dynamics, the study of buoyancy-driven flow, also referred to as natural convection, is crucial for understanding various physical processes occurring in engineering and environmental systems. This phenomenon arises when fluid motion is induced by density differences caused by temperature variations. Such flows play a pivotal role in applications like cooling of electronic equipment, heat exchangers, and natural ventilation in buildings.

A particularly fundamental problem within this domain is the steady, laminar flow occurring between two vertically aligned, wide, and tall parallel plates where the temperature gradient causes fluid near the heated plate to rise and near the cooled plate to sink. This creates a steady circulation pattern driven solely by buoyancy forces. While this setup is idealized, it provides valuable insights into more complex systems, as it allows for analytical solutions under specific boundary conditions and assumptions.

This analysis draws on established methods from classical works like Bird, Stewart, and Lightfoot (2007), as well as Gebhart et al. (1988), to investigate the velocity and temperature fields in this simple yet informative configuration. By considering assumptions like steady-state flow, neglecting end effects, and applying the Boussinesq approximation, the problem can be effectively modeled and solved to yield key insights into heat transfer rates and flow profiles.  

Considering One-Dimensional Heat Transfer Scenario

The setup consists of two parallel plates: a hot side (labeled with temperature T_H) and a cold side (labeled with temperature T_C). The distance from the center of the setup to each plate is w. The temperature gradient suggests that heat will flow from the hot plate to the cold plate across the gap. The coordinate system indicates that the distance between the two sides is measured along the x-axis, while the y-axis is vertical.

Assumptions

1. Steady-State, Laminar Flow of a Newtonian Fluid The flow is steady, laminar, and governed by a Newtonian fluid with constant viscosity μ, thermal conductivity k, and specific heat capacity Cp​.
2. Wide Geometry in the z-Direction The system is infinitely wide in the z-direction, eliminating edge effects, and ensuring no variation in velocity or temperature in that direction. $v_{z}\equiv 0;\quad {\frac {\partial v}{\partial z}}\equiv 0;\quad {\frac {\partial T}{\partial z}}\equiv 0$
3. Neglect of End Effects End effects are neglected. Flow is confined to the y-direction, with no flow in the x-direction. Temperature variations in the y-direction are assumed negligible.
4. Boussinesq Approximation Density variations are considered only in the buoyancy term, while remaining constant elsewhere, simplifying the governing equations.
5. Neglect of Viscous Dissipation and External Heat Sources Viscous dissipation is ignored, and there are no external heat sources or sinks, so heat transfer is driven purely by temperature gradients.

Continuity Equation

As a result, the velocity component $v_{y}$​ is independent of the y-coordinate. Based on Assumption 1, it is also independent of time, and from Assumption 2, it remains independent of the z-coordinate. Therefore, we can deduce that $v_{y}$​ is only a function of x.

${\cancel {\frac {\partial v_{x}}{\partial x}}}+{\frac {\partial v_{y}}{\partial y}}+{\cancel {\frac {\partial v_{z}}{\partial z}}}=0$

Navier-Stokes Equations

1. x-component Based on Assumption 2 and the fact that the gravitational vector is directed along the negative y-axis, meaning $g_{z}$​= 0, we get the result that the pressure gradient in the z-direction is zero:

${\frac {\delta p}{\delta x}}=0$

2.y-component

$\rho \left({\cancel {\frac {\partial v_{y}}{\partial t}}}+{\cancel {v}}_{x}{\frac {\partial v_{y}}{\partial x}}+v_{y}{\cancel {\frac {\partial v_{y}}{\partial y}}}+{\cancel {v}}_{z}{\frac {\partial v_{y}}{\partial z}}\right)=$$-{\frac {\partial p}{\partial y}}+\rho g_{y}+\mu \left({\frac {\partial ^{2}v_{y}}{\partial x^{2}}}+{\cancel {\frac {\partial ^{2}v_{y}}{\partial y^{2}}}}+{\cancel {\frac {\partial ^{2}v_{y}}{\partial z^{2}}}}\right)$

3.z-component According to Assumption 3, and considering that the gravitational force acts in the negative y direction, where $g_{x}$= 0, the pressure gradient in the x-direction becomes:

${\frac {\delta p}{\delta z}}=0$

Given that $g_{y}$​= −g, we arrive at the equation:

$\mu {\frac {d^{2}v_{y}}{dx^{2}}}={\frac {dp}{dy}}+\rho g$

Here, the partial derivatives have been replaced with ordinary derivatives since v_y depends only on x and p only on y. Due to the presence of flow, the pressure distribution may differ from simple hydrostatic variation. As we shall later observe, the pressure distribution is indeed hydrostatic, corresponding to the average fluid density between the plates.

Energy Equation

To advance the analysis, we now focus on the energy equation governing the temperature field.

$\rho C_{p}\left({\cancel {\frac {\partial T}{\partial t}}}+{\cancel {v}}_{x}{\frac {\partial T}{\partial x}}+v_{y}{\cancel {\frac {\partial T}{\partial y}}}+{\cancel {v}}_{z}{\frac {\partial T}{\partial z}}\right)$$=k\left({\frac {\partial ^{2}T}{\partial x^{2}}}+{\cancel {\frac {\partial ^{2}T}{\partial y^{2}}}}+{\cancel {\frac {\partial ^{2}T}{\partial z^{2}}}}\right)+\mu {\cancel {\Phi }}_{v}+{\cancel {S}}$

The temperature field adheres to the equation:

$k{\frac {d^{2}T}{dx^{2}}}=0$

Here, the partial derivative has been replaced by the ordinary derivative, as the temperature distribution is independent of time (Assumption 1), the x-axis (Assumption 2), and the y-axis (Assumption 3), meaning T is solely a function of x.

Boundary Conditions

The boundary conditions are as follows:

T(-w) = T_H  (Temperature at the left wall)

T(w) = T_C (Temperature at the right wall)

The linear temperature distribution between the two plates can be directly expressed.

$T={\overline {T}}-{\frac {\Delta T}{2}}$${\frac {x}{w}}$

where ${\overline {T}}$ is the arithmetic mean temperature of the given plates given by:

${\overline {T}}={\frac {1}{2}}(T_{C}+T_{H})$

$\Delta T=T_{H}-T_{C}$

Expansion of Density Using Taylor Series

With the temperature field determined, we can return to the y-momentum equation by expressing fluid density as a function of temperature. To do this, we expand the density using a Taylor series around the mean temperature.

$\rho ={\bar {\rho }}+\left({\frac {\partial \rho }{\partial T}}\right)_{p,T={\bar {T}}}\left(T-{\bar {T}}\right)+\dots$

Terms that are quadratic or higher in the temperature difference $T-{\bar {T}}$ are omitted, assuming the temperature differences are small. The partial derivative of density with respect to temperature is evaluated at constant pressure and at the mean temperature.

The coefficient of thermal expansion β is defined as:

$\beta =-{\frac {1}{V}}$$\left({\frac {\partial V}{\partial T}}\right)_{p}$ $=-{\frac {1}{\rho }}$$\left({\frac {\partial \rho }{\partial T}}\right)_{p}$

where V represents the specific volume of the fluid, with the relationship ρV = 1​.

Thus,

$\left({\frac {\partial \rho }{\partial T}}\right)_{p,T={\bar {T}}}=-{\bar {\rho }}{\bar {\beta }}$

The expression for density as a function of temperature is transformed into:

$\rho \approx {\bar {\rho }}$ $-$ ${\bar {\rho }}{\bar {\beta }}$$(T-{\bar {T}})$

This is an approximation, with higher-order terms omitted in the Taylor series.

Velocity Field Calculation

By substituting the temperature field, we return to the y-momentum equation. Next, by substituting $T-{\bar {T}}$ from the temperature field solution and using the expression for density, the y-momentum equation simplifies to the following ordinary differential equation:

$\mu {\frac {d^{2}v_{y}}{dx^{2}}}={\frac {dp}{dy}}+{\bar {\rho }}g\left(1+{\bar {\beta }}{\frac {\Delta T}{2}}{\frac {x}{w}}\right)$

By differentiating this equation with respect to x, the pressure gradient is eliminated, resulting in the third-order differential equation:

${\frac {d^{3}v_{y}}{dx^{3}}}={\frac {{\bar {\rho }}{\bar {\beta }}g\Delta T}{2w\mu }}$

The term on the right-hand side is a constant. Since the third derivative of v_y is constant, the velocity field solution takes the form of a cubic function in x:

$v_{y}(x)=c_{0}+c_{1}x+c_{2}x^{2}+{\frac {{\bar {\rho }}{\bar {\beta }}g\Delta T}{12w\mu }}x^{3}$

Here, c0​, c1, and c2 are integration constants that need to be determined. To determine the constants, we require three boundary conditions. The two no-slip conditions are:

$v_{y}(-w)=0$

$v_{y}(w)=0$

These provide two equations involving c₀​, c₁, and c₂​.

These equations are:

$c_{0}+c_{1}w+c_{2}w^{2}+{\frac {{\bar {\rho }}{\bar {\beta }}g\Delta T}{12\mu }}w^{2}=0$

$c_{0}-c_{1}w+c_{2}w^{2}-{\frac {{\bar {\rho }}{\bar {\beta }}g\Delta T}{12\mu }}w^{2}=0$

By adding the two equations, we determine that

$c_{1}=-{\frac {{\bar {\rho }}{\bar {\beta }}g\Delta T}{12\mu }}w$

$c_{0}=-c_{2}w^{2}$

Hence, we can rewrite the solution for the velocity field as follows.

$v_{y}(x)={\frac {{\overline {\rho }}{\overline {\beta }}g\Delta Tw^{2}}{12\mu }}\left({\frac {x^{3}}{w^{3}}}-{\frac {x}{w}}\right)+c_{2}\left(x^{2}-w^{2}\right)$

The third condition is derived from observing that the mass flow rate at any given cross-section, for any arbitrary value of y, must equal zero.

$\int _{-w}^{w}\rho v_{y}(x)\,dx=0$

By applying this condition and using the result from the Taylor series expansion for ρ, combined with the solution for the temperature field, we derive the following.

$\rho ={\overline {\rho }}+{\frac {{\overline {\rho }}{\overline {\beta }}\Delta T}{2}}{\frac {x}{w}}={\overline {\rho }}\left(1+{\frac {{\overline {\beta }}\Delta T}{2}}{\frac {x}{w}}\right)$

Thus,

$\int _{-w}^{w}\rho \,v_{y}(x)\,dx=\int _{-w}^{w}{\bar {\rho }}\left(1+{\frac {{\bar {\beta }}\Delta T}{2}}{\frac {x}{w}}\right)\left[{\frac {{\bar {\rho }}{\bar {\beta }}g\Delta Tw^{2}}{12\mu }}\left({\frac {x^{3}}{w^{3}}}-{\frac {x}{w}}\right)+c_{2}\left(x^{2}-w^{2}\right)\right]dx$

$={\bar {\rho }}\left[-c_{2}\left({\frac {4}{3}}w^{3}\right)+{\frac {{\bar {\beta }}\Delta T}{2}}\left(-{\frac {{\bar {\rho }}{\bar {\beta }}g\Delta Tw^{3}}{45\mu }}\right)\right]$

and by setting the integral on the left-hand side equal to zero, this results in

$c_{2}=-{\frac {{\bar {\rho }}{\bar {\beta ^{2}}}g\Delta T^{2}}{120\mu }}$

Since we retained only terms up to and including O(ΔT) in the Taylor series, the above result for c₂ ~ O(ΔT²) is considered negligible. Therefore, within the order of our approximation, c₂ ​= 0. The final solution for the velocity field is expressed in dimensionless form below to highlight its physical significance.

$\phi (X)={\frac {wv_{y}{\bar {\rho }}}{\mu }}={\frac {\text{Gr}}{12}}\left(X^{3}-X\right)$

Here, the dimensionless coordinate is defined as X = x / w​, where x is the original coordinate, and w is a characteristic length. The dimensionless group Gr is defined as follows.

${\text{Gr}}={\frac {{\bar {\rho }}^{2}{\bar {\beta }}gw^{3}\Delta T}{\mu ^{2}}}$

Pressure Distribution

The pressure gradient in the fluid can be determined by substituting the velocity distribution into the governing equation for v_y(x). This gives:

$\mu {\frac {d^{2}v_{y}}{dx^{2}}}={\frac {{\bar {\rho }}{\bar {\beta }}g\Delta T}{2w}}x$

From this, we obtain the pressure gradient as:

${\frac {dp}{dy}}=-{\bar {\rho }}g$

Thus, the pressure variation is hydrostatic and corresponds to that in a fluid with a uniform density.

References

1. Aung, W.1972 Fully developed laminar free convection between vertical plates heated asymmetrically.Intl J. Heat Mass Transfer15, 1577–1580.

2. R.B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Wiley, 2007.

3. B. Gebhart, Y. Jaluria, R.L. Mahajan, and B. Sammakia, Buoyancy-Induced Flows and Transport, Hemisphere (1988)

4. Eckert, E. R. G. & Carlson, W. O. 1961 Natural convection in an air layer enclosed between two vertical plates with different temperatures. Intl J. Heat Mass Transfer 2, 106–120

5. Gill, A. E. & Davey, A.1969 Instabilities of a buoyancy-driven system.J. Fluid Mech.35, 775–798

Article prepared by

Nistha Jain (Roll no. 21135089), IIT BHU (Varanasi)

Priya Rathore (Roll no. 21134023), IIT BHU (Varanasi)

Sayali Borse (Roll no. 21135118), IIT BHU (Varanasi)

Yashmita Kamboj (Roll no. 21135146), IIT BHU (Varanasi)