$$
\textbf{Velocity}
$$
For two-dimensional laminar flow with negligible gravitational effects, the Navier-Stokes equations can be reduced to:
$$
(1) \quad u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = -\frac{1}{\rho}\frac{\partial p}{\partial x} + v(\frac{\partial^2u}{\partial x^2} \ + \ \frac{\partial^2u}{\partial y^2})
$$
$$
(2) \quad u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} = -\frac{1}{\rho}\frac{\partial p}{\partial y} + v(\frac{\partial^2v}{\partial x^2} \ + \ \frac{\partial^2v}{\partial y^2})
$$
The conservation of mass equation for incompressible flow is:
$$
(3) \quad \frac{\partial u}{\partial x} \ + \ \frac{\partial v}{\partial y} \ = \ 0
$$
With a thin boundary layer, the velocity component normal to the plate is much smaller than the component parallel to the plate. Additionally, the rate of change for a parameter across the boundary layer should much greater than any in the flow's direction, meaning:
$$
v \lt \lt u \quad \text{and} \quad \frac{\partial}{\partial x} \lt \lt \frac{\partial}{\partial y}
$$
These assumptions enable Eqs. 1, 2, and 3, to be simplified to:
$$
(4) \quad \frac{\partial u}{\partial x} \ + \ \frac{\partial v}{\partial y} \ = \ 0
$$
$$
(5) \quad u\frac{\partial u}{\partial x} \ + \ v\frac{\partial u}{\partial y} = v\frac{\partial^2 u}{\partial y^2}
$$
With boundary conditions:
$$
u = v = 0 \ \ \text{at} \ \ y \ = \ 0
$$
$$
u \rightarrow U \ \ \text{as} \ \ y \rightarrow ∞
$$
As the solutions to nonlinear partial differential equations such as Eqs. 4 and 5, are difficult to obtain, the Prandtl/Blasius solution is applied. By using a dimensionless form and assuming the velocity profile should be similar regardless of location along the plate, the dimensionless velocity profile can be represented by:
$$
\frac{u}{U} \ = \ g(\frac{y}{\delta})
$$
where \(g(y/\delta)\) is an unknown function. By applying an order of magnitude analysis on the forces acting on a fluid in the boundary layer:
$$
\delta \propto (\frac{vx}{U})^{1/2}
$$
Now, a dimensionless \(\textit{similarity variable}\) \(\eta = (U/vx)^{1/2}y\) and stream function \(\Psi = (v\ x \ U)^{1/2}f(\eta)\), where \(f = f(\eta)\) is an unknown function, can be applied. For two-dimensional flow, velocity components in terms of the stream function are \(u = \partial \Psi / \partial y\) and \(v = -\partial \Psi/ \partial x\). Thus:
$$
(6) \quad u = Uf'(\eta)
$$
$$
(7) \quad v = (\frac{vU}{4x})^{1/2}(\eta f'-f)
$$
Then Eqs. 6 and 7 can be substituted into Eqs. 4 and 5 and manipulated into the form:
$$
(8) \quad 2f''' + ff'' = 0
$$
With boundary conditions:
$$
(9) \quad f = f' = 0 \ \text{at} \ \eta = 0 \ \ \text{and} \ \ f' \rightarrow 1 \ \text{as} \ \eta \rightarrow ∞
$$
And so the nonlinear partial differential equation and associated boundary conditions have been reduced to a third order ordinary differential equation. From this solution it is found that \(u/U \approx 0.99\) when \(\eta = 5\). Thus the velocity boundary layer is:
$$
(10) \quad \delta = 5\sqrt{\frac{vx}{U}}
$$
With a solution found to the ordinary differential equation and the similarity variable \(\eta\) and the stream function \(\Psi\), as well as the boundary conditions in Eq. 9. The quantity \(\partial u/\partial y|_{y=0}\) can be evaluated according to:
$$
(11) \quad \frac{\partial u}{\partial y}|_{y=0} = U\sqrt{\frac{U}{vx}}\frac{d^2f}{d\eta^2}|_{\eta = 0}
$$
Eq. 11 can then be related to the shear stress at the plate surface through:
$$
(12) \quad \tau_{s,x} = \mu \frac{\partial u}{\partial y}|_{y=0} = \mu U \sqrt{\frac{U}{vx}} \frac{d^2f}{d\eta^2}|_{\eta = 0}
$$
The local coefficient of friction is then:
$$
(13) \quad C_{f,x} = \frac{\tau_{s,x}}{\rho U^2/2}
$$
$$
\textbf{Thermal}
$$
The thermal boundary solution also utilizes the Prandtl/Blasius similarity solution with Eqs. 4 and 5. An additional conservation equation is needed, energy:
$$
(14) \quad u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} = \alpha \frac{\partial^2T}{\partial y^2}
$$
where \(\alpha\) is the thermal diffusivity, related to the Prandtl number through:
$$
(15) \quad Pr = \frac{\nu}{\alpha}
$$
Then dimensionless temperature \(T^* = (T \ - \ T_s)/(T_∞ \ - \ T_s)\) is used with an assumed similarity solution in the form \(T^* \ = \ T^*(\eta)\). Appropriate substitutions and reductions are made with Eq. 14 to yield:
$$
(16) \quad \frac{d^2T^*}{d\eta^2} + \frac{Pr}{2}f\frac{dT^*}{d\eta} = 0
$$
where \(f\) is the solution to the velocity boundary similarity solution. Eq. 16 is subject to the boundary conditions:
$$
(17) \quad T^*(0) = 0 \quad \text{and} \quad T^*(∞) = 1
$$
With these boundary conditions, Eq. 16 can be solved for a ratio of velocity to thermal boundary layer thickness, for Pr \(\geq 0.6\):
$$
(18) \quad \frac{\delta}{\delta_t} \approx Pr^{1/3}
$$
Again for Pr \(\geq 0.6\), the temperature gradient \(dT^*/d\eta|_{\eta=0} \) can be correlated with:
$$
\frac{dT^*}{d\eta}|_{\eta = 0}= 0.332\ Pr^{1/3}
$$
The local convection coefficient can be evaluted through:
$$
(19) \quad h_x = -\frac{T_∞ \ - \ T_s}{T_s \ - \ T_∞}k\frac{\partial T^*}{\partial y}|_{y=0} = k(\frac{U}{vx})^{1/2}\frac{dT^*}{d\eta}|_{\eta=0}
$$
The local Nusselt number can be evaluated through:
$$
(20) \quad Nu_x = \frac{h_xx}{k} = 0.332 \ Re_x^{1/2} Pr^{1/3} \qquad Pr \geq 0.6
$$
References:
-
Bergman, T. L., & Incropera, F. P. Fundamentals of Heat and Mass Transfer ,7th ed., Hoboken, NJ :John Wiley & Sons, Inc., 2011.
-
Munson, Bruce Roy, 1940-. Fundamentals of Fluid Mechanics,7th ed., Hoboken, NJ :John Wiley & Sons, Inc., 2013.
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