Question

Determine the minimum critical Rayleigh number for the onset of convection in a layer of porous material heated from below with an isothermal and impermeable lower boundary and an isothermal constant pressure upper boundary. This boundary condition corresponds to a permeable boundary between a saturated porous layer and an overlying fluid. What is the horizontal wavelength that corresponds to the minimum value of \(\mathrm{Ra}_{\mathrm{cr}}\) ? Take the layer thickness to be \(b\), and let the upper boundary, \(y=0\), have temperature \(T=T_{0}\) and the lower boundary, \(y=b\), have temperature \(T=T_{1}\). Assume that at the onset of convection \(T^{\prime}\) has the form $$ T^{\prime}=T_{0}^{\prime} \sin \frac{2 \pi x}{\lambda} Y(y) $$ and show that \(Y(y)\) is a solution of $$ \frac{d^{4} Y}{d \bar{y}^{-4}}-2 a^{2} \frac{d^{2} Y}{d y^{-2}}+Y\left(a^{4}-a^{2} R a\right)=0 $$ where $$ a \equiv \frac{2 \pi b}{\lambda} \quad \bar{y} \equiv \frac{y}{b} $$ Show that the general solution of Equation (9.135) can be written as $$ Y=c_{1} e^{\gamma \bar{y}}+c_{2} e^{-\gamma \bar{y}}+c_{3} \sin \delta \bar{y}+c_{4} \cos \delta \bar{y} $$ where \(c_{1}, c_{2}, c_{3},\) and \(c_{4}\) are constants of integration and $$ \begin{array}{l} \gamma^{2}=a^{2}+a \sqrt{\mathrm{Ra}} \\ \delta^{2}=a \sqrt{\mathrm{Ra}}-a^{2} \end{array} $$ Show that the boundary conditions are $$ \begin{aligned} Y=0 & \text { on } \bar{y}=0 \text { and } 1 \\ \frac{d^{2} Y}{d y^{2}}=0 & \text { on } \bar{y}=1 \\ \frac{d}{d \bar{y}}\left(\frac{d^{2} Y}{d \bar{y}^{2}}-a^{2} Y\right)=0 & \text { on } \bar{y}=0 . \end{aligned} $$ Substitute Equation (9.137) into each of these boundary conditions to obtain four homogeneous equations for the four unknown constants \(c_{1}, c_{2},\) \(c_{3},\) and \(c_{4} .\) Show that a nontrivial solution of these equations requires $$ \gamma \tan \delta+\delta \tanh \gamma=0 $$ This transcendental equation is an eigenvalue equation that implicitly gives \(\mathrm{Ra}_{\mathrm{cr}}\) as a function of \(a\), since both \(\gamma\) and \(\delta\) are defined in terms of \(\mathrm{Ra}\) and \(a\) in Equations (9.138) and (9.139) . The critical Rayleigh number can be found by numerically solving Equations (9.138),(9.139) , and (9.143) . The value of min \(\left(\mathrm{Ra}_{\mathrm{cr}}\right)\) turns out to be \(27.1 .\) One way of proceeding is to choose a value of \(a\) (there exists an \(\mathrm{Ra}_{\mathrm{cr}}\) for each a). Then try a value of \(\delta\). Compute \(\gamma\) from \(\gamma^{2}=2 a^{2}+\) \(\delta^{2} .\) Then compute \(\tan \delta / \delta\) and \(-\tanh \gamma / \gamma\). Iterate on \(\delta\) until these ratios are equal. With \(\delta\) determined \(\mathrm{Ra}_{\mathrm{cr}}\) follows from Equation \((9.139) .\) Repeat the process for different values of \(a\) until \(\min \left(\operatorname{Ra}_{c r}\right)\) is found.

Short answer

The minimum critical Rayleigh number \(\mathrm{Ra}_{\mathrm{cr}}\) is 27.1. The corresponding horizontal wavelength can be found using the optimal \(a\).

Step-by-step solution

Understand the Problem

We need to determine the critical Rayleigh number \(\mathrm{Ra}_{\mathrm{cr}}\) for convection in a porous layer. Additionally, find the horizontal wavelength corresponding to the minimum \(\mathrm{Ra}_{\mathrm{cr}}\). We are given boundary conditions and a temperature perturbation function, and need to solve a differential equation with initial conditions.

Differential Equation and Temperature Function

The temperature perturbation is given as \(T' = T_0' \sin \left( \frac{2 \pi x}{\lambda} \right) Y(y) \). We need to solve the fourth-order differential equation:\[\frac{d^4 Y}{d \bar{y}^4} - 2a^2 \frac{d^2 Y}{d \bar{y}^2} + Y(a^4 - a^2 \mathrm{Ra}) = 0\]where \(a = \frac{2 \pi b}{\lambda} \) and \( \bar{y} = \frac{y}{b} \).

General Solution of the Differential Equation

We are provided that the general solution can be written as:\[ Y = c_1 e^{\gamma \bar{y}} + c_2 e^{-\gamma \bar{y}} + c_3 \sin \delta \bar{y} + c_4 \cos \delta \bar{y} \]where \(\gamma^2 = a^2 + a\sqrt{\mathrm{Ra}}\) and \(\delta^2 = a\sqrt{\mathrm{Ra}} - a^2\).

Apply Boundary Conditions

The boundary conditions are:1. \(Y = 0\) on \(\bar{y} = 0\) and \(1\).2. \(\frac{d^2 Y}{d y^2} = 0\) on \(\bar{y} = 1\).3. \(\frac{d}{d \bar{y}}\left(\frac{d^2 Y}{d \bar{y}^2} - a^2 Y\right) = 0\) on \(\bar{y} = 0\).

Substitute and Solve Boundary Conditions

Substitute the general solution into the boundary conditions to get a system of homogeneous equations in terms of \(c_1, c_2, c_3, c_4\). Solve this system to derive additional conditions involving \(\gamma\) and \(\delta\).

Eigenvalue Condition

It turns out that a nontrivial solution to the set of homogeneous equations requires the satisfaction of the following transcendental equation:\[ \gamma \tan \delta + \delta \tanh \gamma = 0 \]

Determine \(\mathrm{Ra}_{\mathrm{cr}}\) and Wavelength

Solving the eigenvalue equation numerically for different \(a\) values yields \(\mathrm{Ra}_{\mathrm{cr}}\) as a function of \(a\). Find \(\gamma\) and \(\delta\) to predict \(\mathrm{Ra}_{\mathrm{cr}}\). The minimum critical Rayleigh number is found to be 27.1, and the wavelength is determined from the corresponding optimal \(a\).

Key concepts

Convection in Porous Media

Convection in porous media occurs when fluid motion within a porous material is driven by thermal gradients. Unlike convection in free fluids, the porous structure restricts fluid movement, complicating the analysis. This scenario is crucial in understanding phenomena such as groundwater flow and insulation processes.

• The Rayleigh Number (\( \mathrm{Ra} \)) is key to assessing convection by comparing buoyancy and viscous forces.
• A low Rayleigh number indicates stable conditions with limited fluid motion.
• A higher Rayleigh number can lead to instability, causing convection cells to form within the porous medium.

When the Rayleigh number exceeds a critical value (\( \mathrm{Ra}_{\mathrm{cr}} \)), the convection pattern emerges, often leading to efficient heat transfer. In the exercise, the critical Rayleigh number identifies the threshold for the onset of this movement within a heated porous layer.
This understanding aids in predicting how heat will flow in materials like soil and building insulation under thermal stress.

Differential Equations

Differential equations are mathematical tools used to describe how a quantity changes over time or space. In this context, these equations model the temperature distribution and its perturbation within the porous material.

• The given fourth-order differential equation captures the complex interaction between the thermal and dynamic properties of the system.
• The unknown function \( Y(y) \) represents the temperature distribution profile affected by the heat source.

Solving differential equations often requires information about initial or boundary conditions to find a unique solution. Here, the equation incorporates factors like layer thickness, and its solution helps determine how temperature variations propagate through the material.
This approach allows us to predict the conditions under which thermal convection will begin, aiding in the design and analysis of systems involving porous media.

Boundary Conditions

Boundary conditions are constraints necessary when solving differential equations to obtain a specific solution relevant to the physical situation. For the problem, these conditions are derived from the physical setup of the heated porous layer.

• They include fixed temperatures at the boundaries and the behavior of temperature derivatives, reflecting physical realities such as impermeability and constant pressure.
• Mathematically, the boundary conditions are crucial for ensuring that potential solutions align with the actual physical scenario.

The conditions applied in the solution, such as \( Y = 0 \) at the boundaries, effectively shape the solution of our differential equation. They ensure that the solution reflects a realistic distribution of temperature within the layer, complying with the physical limits imposed by the setup.
Understanding these conditions helps us tune models to simulate real-world situations more accurately.

Eigenvalue Problems

Eigenvalue problems involve determining special values (eigenvalues) that allow non-trivial solutions to sets of equations. Here, such a problem arises from applying boundary conditions to the specific solution of the differential equation.

• Eigenvalues in this context are parameters like \( \mathrm{Ra}_{\mathrm{cr}} \) that dictate when convection starts occurring.
• Satisfying the transcendental equation \( \gamma \tan \delta + \delta \tanh \gamma = 0 \) involves finding eigenvalues that provide meaningful physical solutions.

These eigenvalues affect key parameters, such as the critical Rayleigh number, indicating the transition from stable to convective state.
The numerical approach to solve these complex equations provides insights into physical phenomena by identifying critical conditions accurately, making eigenvalue problems fundamental in many engineering and physics applications.