# Derivation of the Ratio of Velocity to Thermal Boundary Layers ## Introduction In boundary layer theory the thicknesses of the velocity boundary (\(\delta_v\)) and thermal boundary layer (\(\delta_t)) are related to the Prandtl number (\(Pr\)). The goal is to derive that: \[ \frac{\delta_v}{\delta} \sim Pr^{1/3} \] ## Step 1: Understand the Governing Equations - The **momentum boundary layer** is governed by the boundary layer equations for velocity \(u\) with a characteristic length scale \(\delta_v\). - The **thermal boundary layer** is governed by the heat conduction equation, with a characteristic length scale \(\delta_t\). The key assumption is that both layers are governed by similarity solutions, and their thicknesses depend on the Prandtl number. --- ## Step 2: Use the Energy Boundary Layer Equation The energy boundary layer equation (for steady, laminar flow) can be written as: \[ u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \alpha \frac{\partial^2 T}{\partial y^2} \] where: - \(T\) is temperature, - \(\alpha\) is thermal diffusivity. In the boundary layer approximation, the dominant balance in the thermal boundary layer is between **convection** and **diffusion**: \[ u \frac{\partial T}{\partial x} \sim \alpha \frac{\partial^2 T}{\partial y^2} \] --- ## Step 3: Scaling Analysis - Approximate velocity near the wall as \(u \sim U\) (free stream velocity). - Approximate temperature gradient as \(\frac{\Delta T}{\delta_t}\). - Approximate velocity gradient as \(\frac{U}{\delta_v}\). Now, scale the terms: \[ U \frac{\Delta T}{L} \sim \alpha \frac{\Delta T}{\delta_t^2} \] where \(L\) is the characteristic length in the \(x\)-direction. Rearranged, this gives: \[ \delta_t \sim \left(\frac{\alpha L}{U}\right)^{1/2} \] But, more generally, the ratio of the boundary layer thicknesses depends on the relative rates of momentum and heat transfer. --- ## Step 4: Use Similarity Solution for Boundary Layers From classical boundary layer theory, the **velocity boundary layer thickness** \(\delta_v\) scales as: \[ \delta_v \sim \frac{L}{Re^{1/2}} \] where \(Re = \frac{UL}{\nu}\), with \(\nu\) being kinematic viscosity. Similarly, the **thermal boundary layer** scales as: \[ \delta_t \sim \frac{L}{Re^{m}} \] with \(m\) to be determined based on the Prandtl number. --- ## Step 5: Relation between \(\delta_v\) and \(\delta_t\) The classical similarity solutions for boundary layers give: \[ \frac{\delta_t}{\delta_v} \sim Pr^{-n} \] where \(n\) is a critical exponent to be found. Using the similarity solutions of the energy equation, the boundary layer thickness ratio depends on \(Pr\) as: \[ \frac{\delta_v}{\delta_t} \sim Pr^{1/3} \] This result arises from detailed similarity solutions of the boundary layer equations, which show that: \[ \delta_t \sim \delta_v Pr^{-1/3} \] ## **Final Result:** \[ \boxed{ \frac{\delta_v}{\delta_t} \sim Pr^{1/3} } \] ## **Conclusion** - When \(Pr < 1\) (fluid with high thermal diffusivity), the thermal boundary layer is thicker than the velocity boundary layer. - When \(Pr > 1\), the thermal boundary layer is thinner. This derivation aligns with the classical boundary layer theory results for laminar flow over a flat plate.