Boundary layer theory

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==Concepts of the Boundary Layer Theory==
==Concepts of the Boundary Layer Theory==
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[[Image:Fig4.2.png|thumb|400 px|alt=Viscous or momentum boundary layer |Viscous or momentum boundary layer.]]
 +
Unless the geometry is very simple, it is difficult to solve for the complete viscous fluid flow around a body.  A full domain numerical solution is time consuming and impractical, because one needs to solve the full Navier-Stokes equations in the full domain, which are nonlinear, elliptic, and complex <ref>Faghri, A., Zhang, Y., and Howell, J. R., 2010, ''Advanced  Heat and Mass Transfer'', Global Digital Press, Columbia, MO.</ref>.
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Unless the geometry is very simple, it is difficult to solve for the complete viscous fluid flow around a bodyA full domain numerical solution is time consuming and impractical, because one needs to solve the full Navier-Stokes equations in the full domain, which are nonlinear, elliptic, and complex.
+
In 1904, Prandtl discovered that for most practical applications, the influence of viscosity is observed in a very thin domain close to the object, as shown in the figure to the rightOutside this region one can assume the flow is inviscid (''μ'' = 0).<ref>Prandtl, L., 1904, Flussigkeitsbewegung bei sehr Reibung, Proceedings of the 3rd International Mathematics Congress Heidelberg, Germany also NACA TM 452, 1928</ref><ref>Schlichting, H. and Gersteu, K., 2000, Boundary layer theory, 8th enlarged and revised ed., Springer-Verlag New York.</ref><ref>White, F.M., 1974, Viscous Fluid Flow, McGraw-Hill, New York, NY.</ref>
-
In 1904, Prandtl discovered that for most practical applications, the influence of viscosity is observed in a very thin domain close to the object, as shown in Fig.1Outside this region one can assume the flow is inviscid (''µ'' = 0) .
+
The thin region where the effect of viscosity is dominant is called the momentum or viscous boundary layerThe solution of boundary layer analysis can be simplified due to the fact that its thickness is much smaller than the characteristic dimension of the object.  The fluid adjacent to the surface of the body has zero relative velocity; ''u<sub>fluid</sub>'' – ''u<sub>surface</sub>'' = 0. This is also called the no slip boundary condition.
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[[Image:Fig4.2.png|thumb|400 px|alt=Viscous or momentum boundary layer |Figure 1: Viscous or momentum boundary layer.]]
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One of the assumptions of [[boundary layer approximations]] is that the fluid velocity at the wall is at rest relative to the surface. This is true except when the fluid pressure is very low, and therefore, the Kundsen number, Kn= ''λ''/''L'', of the fluid molecules is much larger than 1.  For external flow, the flow next to an object can be divided into two parts.  The larger part is related to a free stream of fluid, in which the effect of viscosity is negligible (potential flow theory).  The smaller region is a thin layer next to the surface of the body, in which the effects of molecular transport (such as viscosity, thermal conductivity and mass diffusivity) are very important.
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[[Image:Fig4.3.png|thumb|400 px|alt=Laminar and turbulent boundary layer flow over a flat plate |Laminar and turbulent boundary layer flow over a flat plate.]]
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The thin region where the effect of viscosity is dominant is called the momentum or viscous boundary layer.  The solution of boundary layer analysis can be simplified due to the fact that its thickness is much smaller than the characteristic dimension of the object.  The fluid adjacent to the surface of the body has zero relative velocity; ufluid – usurface = 0. This is also called the no slip boundary condition.
+
-
 
+
-
One of the assumptions of [[boundary layer approximations]] is that the fluid velocity at the wall is at rest relative to the surface. This is true except when the fluid pressure is very low, and therefore, the Kundsen number, Kn= λ/L, of the fluid molecules is much larger than 1.  For external flow, the flow next to an object can be divided into two parts.  The larger part is related to a free stream of fluid, in which the effect of viscosity is negligible (potential flow theory).  The smaller region is a thin layer next to the surface of the body, in which the effects of molecular transport (such as viscosity, thermal conductivity and mass diffusivity) are very important.
+
-
 
+
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[[Image:Fig4.3.png|thumb|400 px|alt=Laminar and turbulent boundary layer flow over a flat plate |Figure 2: Laminar and turbulent boundary layer flow over a flat plate.]]
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Potential flow theory neglects the effect of viscosity, and therefore, significantly simplifies the Navier-Stokes equations, which provides the solution for the velocity distribution. A disadvantage of the potential theory is that since second order terms are neglected, the effect of viscosity, no slip, and impermeability boundary conditions at the surface cannot be accounted for.  In general, potential flow theory predicts the free stream field accurately, despite its simplicity.  Boundary layer thickness is defined as the distance within the fluid in which most of the velocity change occurs. This thickness is usually defined as the thickness in which the velocity reaches 99% of the free stream velocity, ''u'' = ''0.99U<sub>∞</sub>''.
Potential flow theory neglects the effect of viscosity, and therefore, significantly simplifies the Navier-Stokes equations, which provides the solution for the velocity distribution. A disadvantage of the potential theory is that since second order terms are neglected, the effect of viscosity, no slip, and impermeability boundary conditions at the surface cannot be accounted for.  In general, potential flow theory predicts the free stream field accurately, despite its simplicity.  Boundary layer thickness is defined as the distance within the fluid in which most of the velocity change occurs. This thickness is usually defined as the thickness in which the velocity reaches 99% of the free stream velocity, ''u'' = ''0.99U<sub>∞</sub>''.
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Figure 2 illustrates how the momentum boundary layer thickness changes along the plate. Flow is laminar at relatively small values of x, where it is shown as the laminar boundary layer region.  As x increases, the fluid motion begins to fluctuate.  This is called the transition region.  The boundary layer may be either laminar or turbulent in this region.  Finally, above a certain value of x, the flow will always be turbulent. There is a very thin region next to the wall in the turbulent region where the flow is still laminar, called the laminar sublayer.
+
The figure to the right illustrates how the momentum boundary layer thickness changes along the plate. Flow is laminar at relatively small values of ''x'', where it is shown as the laminar boundary layer region.  As ''x'' increases, the fluid motion begins to fluctuate.  This is called the transition region.  The boundary layer may be either laminar or turbulent in this region.  Finally, above a certain value of ''x'', the flow will always be turbulent. There is a very thin region next to the wall in the turbulent region where the flow is still laminar, called the laminar sublayer.
For flow over a flat plate, experimental data indicates
For flow over a flat plate, experimental data indicates
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{| class="wikitable" border="0"
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{| class="wikitable" border="1"
|-
|-
| Re''<sub>x</sub>'' ≤ 2×10<sup>5</sup>
| Re''<sub>x</sub>'' ≤ 2×10<sup>5</sup>
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|}
|}
-
where
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where <math>\operatorname{Re} _x  = \frac{{\rho U_\infty  x}}{\mu }</math>.
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<big><center><math>\operatorname{Re} _x  = \frac{{\rho U_\infty  x}}
+
-
{\mu }
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</math></center></big>
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==Boundary layer approximation==
==Boundary layer approximation==
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If one assumes that the boundary layer thickness, δ, is very small compared to the characteristic dimension of the object, one can make the assumption that δ is significantly less than L (δ L), where L is the characteristic dimension of the object.  Using scale analysis discussed in Chapter 1, for flow over a flat plate with constant free stream velocity, one can show, in a steady two-dimensional laminar boundary layer representation, that the following conditions are met within the boundary layer region, assuming there are no body forces:
+
If one assumes that the boundary layer thickness, ''δ'', is very small compared to the characteristic dimension of the object, one can make the assumption that δ is significantly less than ''L'' (''δ''<math>\ll </math> ''L''), where ''L'' is the characteristic dimension of the object.  Using [[scale analysis]], for flow over a flat plate with constant free stream velocity, one can show, in a steady two-dimensional laminar boundary layer representation, that the following conditions are met within the boundary layer region, assuming there are no body forces:
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{| class="wikitable" border="0"
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|-
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| width="100%" |<center><math>\frac{{\partial ^2 u}}{{\partial y^2 }} \gg \frac{{\partial ^2 u}}{{\partial x^2 }}</math></center>
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|{{EquationRef|(1)}}
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|}
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<center><math>\frac{{\partial ^2 u}}{{\partial y^2 }} \gg \frac{{\partial ^2 u}}{{\partial x^2 }}</math></center>
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{| class="wikitable" border="0"
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<center><math>u \gg v</math></center>
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|-
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<center><math>\frac{{\partial p}}{{\partial x}} \approx \frac{{dp}}{{dx}}</math></center>
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| width="100%" |<center><math>u \gg v</math></center>
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<center><math>\frac{{\partial p}}{{\partial y}} \approx 0</math></center>
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|{{EquationRef|(2)}}
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|}
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{| class="wikitable" border="0"
 +
|-
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| width="100%" |<center><math>\frac{{\partial p}}{{\partial x}} \approx \frac{{dp}}{{dx}}</math></center>
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|{{EquationRef|(3)}}
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|}
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[[Image:Fig4.4.png|thumb|400 px|alt=Boundary layer concept over a flat plate |Figure 3: Boundary layer concept over a flat plate.]]
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{| class="wikitable" border="0"
 +
|-
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| width="100%" |<center><math>\frac{{\partial p}}{{\partial y}} \approx 0</math></center>
 +
|{{EquationRef|(4)}}
 +
|}
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 +
 
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[[Image:Fig4.4.png|thumb|400 px|alt=Boundary layer concept over a flat plate | Boundary layer concept over a flat plate.]]
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 +
A similar concept exists when there is heat and/or mass transfer between a fluid and the surface of an object. Again, the region in which the effect of temperature or concentration is dominant is, in the most practical case, a region very close to the surface, as shown in the figure to the right. In general, for the case of flow over a flat plate, one can expect three different boundary layer regions with thicknesses ''δ'', ''δ''<sub>T</sub>, and ''δ''<sub>C</sub>, corresponding to momentum, thermal, and concentration boundary layers, respectively.  ''δ'', ''δ''<sub>T</sub>, and ''δ''<sub>C</sub> are not necessarily the same thickness and their values, as will be shown later, depend on the properties of the fluid such as kinematic viscosity (<math>\nu </math>), thermal diffusivity (''α = k/ρc<sub>p</sub>''), specific heat, ''c'', and mass diffusion coefficient, ''D''.<br>
-
A similar concept exists when there is heat and/or mass transfer between a fluid and the surface of an object. Again, the region in which the effect of temperature or concentration is dominant is, in the most practical case, a region very close to the surface, as shown in Fig. 3. In general, for the case of flow over a flat plate, one can expect three different boundary layer regions with thicknesses δ, δT, and δC, corresponding to momentum, thermal, and concentration boundary layers, respectively.  δ, δT, and δC are not necessarily the same thickness and their values, as will be shown later, depend on the properties of the fluid such as kinematic viscosity ( ), thermal diffusivity (α = k/ρcp), specific heat, c, and mass diffusion coefficient, D.
 
Using boundary layer approximation, scale analysis, and order of magnitude, one can show that similar approximations exist for thermal and mass concentration boundary layer analysis.
Using boundary layer approximation, scale analysis, and order of magnitude, one can show that similar approximations exist for thermal and mass concentration boundary layer analysis.
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{| class="wikitable" border="0"
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|-
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| width="100%" |<center><math>\frac{{\partial T}}{{\partial y}} \gg \frac{{\partial T}}{{\partial x}},\quad \frac{{\partial ^2 T}}{{\partial y^2 }} \gg \frac{{\partial ^2 T}}{{\partial x^2 }}</math></center>
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|{{EquationRef|(5)}}
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|}
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{| class="wikitable" border="0"
 +
|-
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| width="100%" |<center><math>\frac{{\partial \omega }}{{\partial y}} \gg \frac{{\partial \omega }}{{\partial x}},\quad \frac{{\partial ^2 \omega }}{{\partial y^2 }} \gg \frac{{\partial ^2 \omega }}{{\partial x^2 }}</math></center>
 +
|{{EquationRef|(6)}}
 +
|}
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<center><math>\frac{{\partial T}}{{\partial y}} \gg \frac{{\partial T}}{{\partial x}},\quad \frac{{\partial ^2 T}}{{\partial y^2 }} \gg \frac{{\partial ^2 T}}{{\partial x^2 }}</math></center>
 
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<center><math>\frac{{\partial \omega }}{{\partial y}} \gg \frac{{\partial \omega }}{{\partial x}},\quad \frac{{\partial ^2 \omega }}{{\partial y^2 }} \gg \frac{{\partial ^2 \omega }}{{\partial x^2 }}</math></center>
 
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where ω is the mass fraction.
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where ''ω'' is the mass fraction.
==Governing Equations for Boundary Layer Approximation==
==Governing Equations for Boundary Layer Approximation==
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As noted before, one can obtain all pertinent information related to momentum, heat, and mass transfer between a surface and a fluid by focusing on the thin region (boundary layer) next to the surface, and solving the governing equations including the boundary conditions. This provides significant simplification irrespective of whether an analytical or numerical approach to solve the physical problem is used.  For most practical applications, the effect of molecular transport due to mass, momentum, energy, and species is dominant in this thin region. The purpose of this section is to develop the transport phenomena equations for boundary layer approximation, starting from the original differential conservation equations developed in Chapter 2.
+
One can obtain all pertinent information related to momentum, heat, and mass transfer between a surface and a fluid by focusing on the thin region (boundary layer) next to the surface, and solving the governing equations including the boundary conditions. This provides significant simplification irrespective of whether an analytical or numerical approach to solve the physical problem is used.  For most practical applications, the effect of molecular transport due to mass, momentum, energy, and species is dominant in this thin region. The purpose of this section is to develop the transport phenomena equations for boundary layer approximation, starting from the original differential conservation equations.<br>
-
Consider flow over a flat plate as shown in Figure 4.4 with constant free stream velocity, temperature, and mass fraction of U∞, T∞, and ω∞, respectively. The surface wall is kept at constant temperature and concentration.  Starting from the differential conservation equations for mass, momentum, energy, and species (as presented in Chapter 2), and assuming two-dimensional, steady, laminar flow, and constant properties in a Cartesian coordinate system, the following results are obtained:
+
Consider flow over a flat plate as shown in the figure to the right with constant free stream velocity, temperature, and mass fraction of ''U<sub>∞</sub>, T<sub>∞</sub>, and ω<sub>∞</sub>'', respectively. The surface wall is kept at constant temperature and concentration.  Starting from the differential conservation equations for mass, momentum, energy, and species, and assuming two-dimensional, steady, laminar flow, and constant properties in a Cartesian coordinate system, the following results are obtained:
Continuity equation
Continuity equation
-
{| class="wikitable" border="1"
+
{| class="wikitable" border="0"
|-
|-
-
| width="90%" |<center><math> \frac{{\partial u}}
+
| width="100%" |<center><math> \frac{{\partial u}}
{{\partial x}} + \frac{{\partial v}}
{{\partial x}} + \frac{{\partial v}}
{{\partial y}} = 0  
{{\partial y}} = 0  
</math></center>
</math></center>
-
| row 1, cell 2
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|{{EquationRef|(7)}}
|}
|}
 +
Momentum equation in the x-direction
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 +
{| class="wikitable" border="0"
 +
|-
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| width="100%" |<center><math>u\frac{{\partial u}}
 +
{{\partial x}} + v\frac{{\partial u}}
 +
{{\partial y}} =  - \frac{1}
 +
{\rho }\frac{{\partial p}}
 +
{{\partial x}} + \nu \left( {\frac{{\partial ^2 u}}
 +
{{\partial x^2 }} + \frac{{\partial ^2 u}}
 +
{{\partial y^2 }}} \right)
 +
</math></center>
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|{{EquationRef|(8)}}
 +
|}
 +
 +
Momentum equation in the y-direction
 +
 +
{| class="wikitable" border="0"
 +
|-
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| width="100%" |<center><math>
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u\frac{{\partial v}}
 +
{{\partial x}} + v\frac{{\partial v}}
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{{\partial y}} =  - \frac{1}
 +
{\rho }\frac{{\partial p}}
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{{\partial y}} + \nu \left( {\frac{{\partial ^2 v}}
 +
{{\partial x^2 }} + \frac{{\partial ^2 v}}
 +
{{\partial y^2 }}} \right)
 +
 +
</math></center>
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|{{EquationRef|(9)}}
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|}
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Energy equation
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{| class="wikitable" border="0"
 +
|-
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| width="100%" |<center><math>
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u\frac{{\partial T}}
 +
{{\partial x}} + v\frac{{\partial T}}
 +
{{\partial y}} = \alpha \left( {\frac{{\partial ^2 T}}
 +
{{\partial x^2 }} + \frac{{\partial ^2 T}}
 +
{{\partial y^2 }}} \right) + \frac{\nu }
 +
{{cp}}\left( {\frac{{\partial u}}
 +
{{\partial y}}} \right)^2
 +
 +
</math></center>
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|{{EquationRef|(10)}}
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|}
 +
 +
Species equation for a binary system
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 +
{| class="wikitable" border="0"
 +
|-
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| width="100%" |<center><math>
 +
u\frac{{\partial \omega _1 }}
 +
{{\partial x}} + v\frac{{\partial \omega _1 }}
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{{\partial y}} = D_{12} \left( {\frac{{\partial ^2 \omega _1 }}
 +
{{\partial x^2 }} + \frac{{\partial ^2 \omega _1 }}
 +
{{\partial y^2 }}} \right)
 +
 +
</math></center>
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|{{EquationRef|(11)}}
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|}
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 +
Assuming there is a simultaneous transfer of momentum, heat and mass, the following boundary conditions are one possibility for conventional applications:
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 +
{| class="wikitable" border="0"
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|-
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| width="100%" |<center><math>
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u\left( {x,0} \right) = 0{\text{  No slip boundary condition}}
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</math></center>
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|{{EquationRef|(12)}}
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|}
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 +
{| class="wikitable" border="0"
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|-
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| width="100%" |<center><math>v(x,0)=\left\{ \begin{align}
 +
  & v_{w}=0\ \text{ Impermeable}\ \text{wall} \\
 +
& v_{w}>0\ \text{ Injection}\ \text{and}\ v_{w}<0\ \text{suction} \\
 +
\end{align} \right.</math>
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</center>
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|{{EquationRef|(13)}}
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|}
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{| class="wikitable" border="0"
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|-
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| width="100%" |<center><math>
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u\left( {x,\infty } \right) = U_\infty  {\text{,    }}U_\infty  {\text{ is the free stream velocity}}
 +
</math></center>
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|{{EquationRef|(14)}}
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|}
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{| class="wikitable" border="0"
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|-
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| width="100%" |<center><math>\text{heat transfer=}\left\{ \begin{align}
 +
  & T_{w}=\text{uniform}\ \text{temperature at the wall} \\
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& -k\left. \frac{\partial T}{\partial y} \right|_{y=0}={q}''_{w}=\text{constant} \\
 +
\end{align} \right.</math>
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</center>
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|{{EquationRef|(15)}}
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|}
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{| class="wikitable" border="0"
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|-
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| width="100%" |<center><math>
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T\left( {x,\infty } \right) = T_\infty  {\text{,  }}T_\infty  {\text{ is the free stream temperature}}
 +
</math></center>
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|{{EquationRef|(16)}}
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|}
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{| class="wikitable" border="0"
 +
|-
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| width="100%" |<center><math>
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\omega _1 \left( {x,0} \right) = \omega _{1,w} {\text{,  }}\omega _{1,w} {\text{ is the constant mass fraction at the wall}}
 +
</math></center>
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|{{EquationRef|(17)}}
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|}
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{| class="wikitable" border="0"
 +
|-
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| width="100%" |<center><math>
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\omega _1 \left( {x,\infty } \right) = \omega _{1,\infty } {\text{,  }}\omega _{1,\infty } {\text{ is the free stream mass fraction}}
 +
</math></center>
 +
|{{EquationRef|(18)}}
 +
|}
 +
 +
At the wall, the temperature or heat flux (or a combination of the temperature and heat flux variation) which may change along the wall, should be known. Equation (15) presents the two limiting cases of constant wall temperature or constant heat flux at the wall.  The normal velocity at the wall is zero for the case of no mass transfer from the wall; however, there are three classes of problems in which ''v<sub>w</sub>'' ≠ 0 at the wall.
 +
<BR><BR>
 +
1. Mass transfer due to phase change, such as condensation or evaporation.  In such cases, temperature, mass fraction, and vw are coupled at the wall through mass and energy balance at the surface of the wall.
 +
<BR>
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2. Injection or blowing of the same fluid through a porous wall (for example to protect the surface from a very high temperature main stream).  In these cases, vw is positive, known, and independent of temperature.
 +
<BR>
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3. Suction of the same fluid through a porous wall (for example to prevent boundary layer separation because of an adverse pressure gradient).  In these cases, vw is also known and independent of temperature, but it is negative.
 +
<BR><BR>
 +
In general, not all mechanisms of transport phenomena (mass, momentum, and energy) occur simultaneously, even though it is possible in many practical applications.  If the transport phenomena is due to momentum and heat and not mass diffusion, then eqs. (7) to (10) are uncoupled for the case of constant properties.  If there is mass transfer by diffusion from the wall however, then <math>v_{w}=f\left( \omega _{1} \right)\ne 0</math> and eqs. (7) through (11) are coupled, even if the properties are constant. The occurrence of mass transfer at the wall may cause vw to be nonzero and can be calculated from Fick’s Law for binary mass diffusion.
 +
The coupling effect between mass and momentum in binary mass diffusion can be easily observed by calculating mass flux at the wall as shown below:
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 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |<center>
 +
<math>{\dot{m}}''_{1}=\rho _{1,w}v_{w}-\left. \rho D_{12}\frac{\partial \omega _{1}}{\partial y} \right|_{y=0}\,=\rho \left( \omega _{1,w}v_{w}-\left. D_{12}\frac{\partial \omega _{1}}{\partial y} \right|_{y=0} \right)</math>
 +
</center>
 +
|{{EquationRef|(19)}}
 +
|}
 +
 +
where ''ω<sub>1,w</sub>'' and ''v<sub>w</sub>'' are mass fraction and the y-component of the velocity at the wall, respectively.  There are five dependent variables (''u, v, T, p, ω<sub>1</sub>'') and two independent variables (''x'' and ''y'').  There are five equations (7)-(11) and five unknowns with appropriate boundary conditions; therefore, it is a well posed problem.
 +
Using [[scale analysis]], order of magnitude analysis, and considering boundary layer assumptions, eqs. (7) through (11) are reduced to the following forms:
 +
 +
Continuity equation:
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 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |<center>
 +
<math>\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0</math>
 +
</center>
 +
|{{EquationRef|(20)}}
 +
|}
 +
 +
Moment equation in the x-direction:
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |<center>
 +
<math>u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{\partial ^{2}u}{\partial y^{2}}</math>
 +
</center>
 +
|{{EquationRef|(21)}}
 +
|}
 +
 +
Energy equation:
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |<center>
 +
<math>u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{\partial ^{2}T}{\partial y^{2}}=\frac{\nu }{\Pr }\frac{\partial ^{2}T}{\partial y^{2}}</math>
 +
</center>
 +
|{{EquationRef|(22)}}
 +
|}
 +
 +
Species equation:
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |<center>
 +
<math>u\frac{\partial \omega _{1}}{\partial x}+v\frac{\partial \omega _{1}}{\partial y}=D_{12}\frac{\partial ^{2}\omega _{1}}{\partial y^{2}}=\frac{\nu }{Sc}\frac{\partial ^{2}\omega _{1}}{\partial y^{2}}</math>
 +
</center>
 +
|{{EquationRef|(23)}}
 +
|}
 +
 +
Using boundary layer approximation for flow over a flat plate with constant free stream velocity, the momentum equation in the y-direction and the pressure gradient in the x-direction were eliminated. Eqs. (20) through (23) are now transformed into a parabolic form rather than the original elliptic form.  There are now four dependent variables (''u, v, T, ω<sub>1</sub>'') and four equations.  This is much easier to solve analytically or numerically because the axial diffusion terms
 +
<math>(\partial ^{2}u/\partial x^{2},\partial ^{2}T/\partial x^{2}\ \text{and }\partial ^{2}\omega _{1}/\partial x^{2})</math> disappear.
 +
 +
==Laminar Boundary Layer Solutions for Momentum, Heat, and Mass Transfer==
 +
 +
There are three basic approaches to solve boundary layer equations for momentum, heat, and mass transfer:
 +
 +
1.  &nbsp;&nbsp;[[Similarity solutions]] <br>
 +
2.  &nbsp;&nbsp;[[Integral solution|Integral methods]] <br>
 +
3.  &nbsp;&nbsp;[[Computational methodologies for forced convection|Full numerical solution]] <br>
 +
 +
First we will consider the [[Similarity solutions|similarity approach]], since it was the original classic method developed to solve boundary layer problems analytically. In this approach, using the fact that in some circumstances velocity is geometrically similar along the flow direction, the conservation partial differential equations are converted to ordinary differential equations.  Similarity methods historically provided significant insight and information about the physical boundary layer phenomenon when computer and numerical methodologies for the solution of partial differential equations were non-existent.
 +
 +
However, there are limitations to the use of similarity methods in terms of applications.  In general, they are only applicable to two-dimensional laminar flow for conventional geometry and boundary conditions with constant properties.
 +
The [[Integral solution|integral methods]] will be presented second. Integral methods are approximate solutions and provide closed form solutions by assuming a profile for velocity, temperature, and concentrations.
 +
Finally, there is the [[Computational methodologies for forced convection|full numerical solution]]. The full umerical solution, along with the availability of proven numerical schemes, practical simulation codes, and high speed computers, are more efficient than other methods for complex problems.
-
== ==
+
==References==
 +
{{Reflist}}
 +
{{Refbegin}}
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 +
{{Refend}}

Current revision as of 12:25, 26 July 2010

Contents

Concepts of the Boundary Layer Theory

Viscous or momentum boundary layer
Viscous or momentum boundary layer.

Unless the geometry is very simple, it is difficult to solve for the complete viscous fluid flow around a body. A full domain numerical solution is time consuming and impractical, because one needs to solve the full Navier-Stokes equations in the full domain, which are nonlinear, elliptic, and complex [1].

In 1904, Prandtl discovered that for most practical applications, the influence of viscosity is observed in a very thin domain close to the object, as shown in the figure to the right. Outside this region one can assume the flow is inviscid (μ = 0).[2][3][4]

The thin region where the effect of viscosity is dominant is called the momentum or viscous boundary layer. The solution of boundary layer analysis can be simplified due to the fact that its thickness is much smaller than the characteristic dimension of the object. The fluid adjacent to the surface of the body has zero relative velocity; ufluidusurface = 0. This is also called the no slip boundary condition.

One of the assumptions of boundary layer approximations is that the fluid velocity at the wall is at rest relative to the surface. This is true except when the fluid pressure is very low, and therefore, the Kundsen number, Kn= λ/L, of the fluid molecules is much larger than 1. For external flow, the flow next to an object can be divided into two parts. The larger part is related to a free stream of fluid, in which the effect of viscosity is negligible (potential flow theory). The smaller region is a thin layer next to the surface of the body, in which the effects of molecular transport (such as viscosity, thermal conductivity and mass diffusivity) are very important.

Laminar and turbulent boundary layer flow over a flat plate
Laminar and turbulent boundary layer flow over a flat plate.

Potential flow theory neglects the effect of viscosity, and therefore, significantly simplifies the Navier-Stokes equations, which provides the solution for the velocity distribution. A disadvantage of the potential theory is that since second order terms are neglected, the effect of viscosity, no slip, and impermeability boundary conditions at the surface cannot be accounted for. In general, potential flow theory predicts the free stream field accurately, despite its simplicity. Boundary layer thickness is defined as the distance within the fluid in which most of the velocity change occurs. This thickness is usually defined as the thickness in which the velocity reaches 99% of the free stream velocity, u = 0.99U.

The figure to the right illustrates how the momentum boundary layer thickness changes along the plate. Flow is laminar at relatively small values of x, where it is shown as the laminar boundary layer region. As x increases, the fluid motion begins to fluctuate. This is called the transition region. The boundary layer may be either laminar or turbulent in this region. Finally, above a certain value of x, the flow will always be turbulent. There is a very thin region next to the wall in the turbulent region where the flow is still laminar, called the laminar sublayer.

For flow over a flat plate, experimental data indicates

Rex ≤ 2×105 the flow is laminar
2×105 <Rex <3×106 the flow is in transition
Rex ≥ 3×106 the flow is turbulent

where \operatorname{Re} _x  = \frac{{\rho U_\infty  x}}{\mu }.

Boundary layer approximation

If one assumes that the boundary layer thickness, δ, is very small compared to the characteristic dimension of the object, one can make the assumption that δ is significantly less than L (δ\ll L), where L is the characteristic dimension of the object. Using scale analysis, for flow over a flat plate with constant free stream velocity, one can show, in a steady two-dimensional laminar boundary layer representation, that the following conditions are met within the boundary layer region, assuming there are no body forces:

\frac{{\partial ^2 u}}{{\partial y^2 }} \gg \frac{{\partial ^2 u}}{{\partial x^2 }}
(1)
u \gg v
(2)
\frac{{\partial p}}{{\partial x}} \approx \frac{{dp}}{{dx}}
(3)
\frac{{\partial p}}{{\partial y}} \approx 0
(4)


Boundary layer concept over a flat plate
Boundary layer concept over a flat plate.

A similar concept exists when there is heat and/or mass transfer between a fluid and the surface of an object. Again, the region in which the effect of temperature or concentration is dominant is, in the most practical case, a region very close to the surface, as shown in the figure to the right. In general, for the case of flow over a flat plate, one can expect three different boundary layer regions with thicknesses δ, δT, and δC, corresponding to momentum, thermal, and concentration boundary layers, respectively. δ, δT, and δC are not necessarily the same thickness and their values, as will be shown later, depend on the properties of the fluid such as kinematic viscosity (ν), thermal diffusivity (α = k/ρcp), specific heat, c, and mass diffusion coefficient, D.

Using boundary layer approximation, scale analysis, and order of magnitude, one can show that similar approximations exist for thermal and mass concentration boundary layer analysis.

\frac{{\partial T}}{{\partial y}} \gg \frac{{\partial T}}{{\partial x}},\quad \frac{{\partial ^2 T}}{{\partial y^2 }} \gg \frac{{\partial ^2 T}}{{\partial x^2 }}
(5)
\frac{{\partial \omega }}{{\partial y}} \gg \frac{{\partial \omega }}{{\partial x}},\quad \frac{{\partial ^2 \omega }}{{\partial y^2 }} \gg \frac{{\partial ^2 \omega }}{{\partial x^2 }}
(6)


where ω is the mass fraction.

Governing Equations for Boundary Layer Approximation

One can obtain all pertinent information related to momentum, heat, and mass transfer between a surface and a fluid by focusing on the thin region (boundary layer) next to the surface, and solving the governing equations including the boundary conditions. This provides significant simplification irrespective of whether an analytical or numerical approach to solve the physical problem is used. For most practical applications, the effect of molecular transport due to mass, momentum, energy, and species is dominant in this thin region. The purpose of this section is to develop the transport phenomena equations for boundary layer approximation, starting from the original differential conservation equations.
Consider flow over a flat plate as shown in the figure to the right with constant free stream velocity, temperature, and mass fraction of U, T, and ω, respectively. The surface wall is kept at constant temperature and concentration. Starting from the differential conservation equations for mass, momentum, energy, and species, and assuming two-dimensional, steady, laminar flow, and constant properties in a Cartesian coordinate system, the following results are obtained:

Continuity equation

 \frac{{\partial u}}
{{\partial x}} + \frac{{\partial v}}
{{\partial y}} = 0
(7)

Momentum equation in the x-direction

u\frac{{\partial u}}
{{\partial x}} + v\frac{{\partial u}}
{{\partial y}} =  - \frac{1}
{\rho }\frac{{\partial p}}
{{\partial x}} + \nu \left( {\frac{{\partial ^2 u}}
{{\partial x^2 }} + \frac{{\partial ^2 u}}
{{\partial y^2 }}} \right)
(8)

Momentum equation in the y-direction


u\frac{{\partial v}}
{{\partial x}} + v\frac{{\partial v}}
{{\partial y}} =  - \frac{1}
{\rho }\frac{{\partial p}}
{{\partial y}} + \nu \left( {\frac{{\partial ^2 v}}
{{\partial x^2 }} + \frac{{\partial ^2 v}}
{{\partial y^2 }}} \right)
(9)

Energy equation


u\frac{{\partial T}}
{{\partial x}} + v\frac{{\partial T}}
{{\partial y}} = \alpha \left( {\frac{{\partial ^2 T}}
{{\partial x^2 }} + \frac{{\partial ^2 T}}
{{\partial y^2 }}} \right) + \frac{\nu }
{{cp}}\left( {\frac{{\partial u}}
{{\partial y}}} \right)^2
(10)

Species equation for a binary system


u\frac{{\partial \omega _1 }}
{{\partial x}} + v\frac{{\partial \omega _1 }}
{{\partial y}} = D_{12} \left( {\frac{{\partial ^2 \omega _1 }}
{{\partial x^2 }} + \frac{{\partial ^2 \omega _1 }}
{{\partial y^2 }}} \right)
(11)

Assuming there is a simultaneous transfer of momentum, heat and mass, the following boundary conditions are one possibility for conventional applications:


u\left( {x,0} \right) = 0{\text{  No slip boundary condition}}
(12)
v(x,0)=\left\{ \begin{align}
  & v_{w}=0\ \text{ Impermeable}\ \text{wall} \\ 
 & v_{w}>0\ \text{ Injection}\ \text{and}\ v_{w}<0\ \text{suction} \\ 
\end{align} \right.
(13)

u\left( {x,\infty } \right) = U_\infty  {\text{,    }}U_\infty  {\text{ is the free stream velocity}}
(14)
\text{heat transfer=}\left\{ \begin{align}
  & T_{w}=\text{uniform}\ \text{temperature at the wall} \\ 
 & -k\left. \frac{\partial T}{\partial y} \right|_{y=0}={q}''_{w}=\text{constant} \\ 
\end{align} \right.
(15)

T\left( {x,\infty } \right) = T_\infty  {\text{,   }}T_\infty  {\text{ is the free stream temperature}}
(16)

\omega _1 \left( {x,0} \right) = \omega _{1,w} {\text{,   }}\omega _{1,w} {\text{ is the constant mass fraction at the wall}}
(17)

\omega _1 \left( {x,\infty } \right) = \omega _{1,\infty } {\text{,   }}\omega _{1,\infty } {\text{ is the free stream mass fraction}}
(18)

At the wall, the temperature or heat flux (or a combination of the temperature and heat flux variation) which may change along the wall, should be known. Equation (15) presents the two limiting cases of constant wall temperature or constant heat flux at the wall. The normal velocity at the wall is zero for the case of no mass transfer from the wall; however, there are three classes of problems in which vw ≠ 0 at the wall.

1. Mass transfer due to phase change, such as condensation or evaporation. In such cases, temperature, mass fraction, and vw are coupled at the wall through mass and energy balance at the surface of the wall.
2. Injection or blowing of the same fluid through a porous wall (for example to protect the surface from a very high temperature main stream). In these cases, vw is positive, known, and independent of temperature.
3. Suction of the same fluid through a porous wall (for example to prevent boundary layer separation because of an adverse pressure gradient). In these cases, vw is also known and independent of temperature, but it is negative.

In general, not all mechanisms of transport phenomena (mass, momentum, and energy) occur simultaneously, even though it is possible in many practical applications. If the transport phenomena is due to momentum and heat and not mass diffusion, then eqs. (7) to (10) are uncoupled for the case of constant properties. If there is mass transfer by diffusion from the wall however, then v_{w}=f\left( \omega _{1} \right)\ne 0 and eqs. (7) through (11) are coupled, even if the properties are constant. The occurrence of mass transfer at the wall may cause vw to be nonzero and can be calculated from Fick’s Law for binary mass diffusion. The coupling effect between mass and momentum in binary mass diffusion can be easily observed by calculating mass flux at the wall as shown below:

{\dot{m}}''_{1}=\rho _{1,w}v_{w}-\left. \rho D_{12}\frac{\partial \omega _{1}}{\partial y} \right|_{y=0}\,=\rho \left( \omega _{1,w}v_{w}-\left. D_{12}\frac{\partial \omega _{1}}{\partial y} \right|_{y=0} \right)

(19)

where ω1,w and vw are mass fraction and the y-component of the velocity at the wall, respectively. There are five dependent variables (u, v, T, p, ω1) and two independent variables (x and y). There are five equations (7)-(11) and five unknowns with appropriate boundary conditions; therefore, it is a well posed problem. Using scale analysis, order of magnitude analysis, and considering boundary layer assumptions, eqs. (7) through (11) are reduced to the following forms:

Continuity equation:

\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0

(20)

Moment equation in the x-direction:

u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{\partial ^{2}u}{\partial y^{2}}

(21)

Energy equation:

u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{\partial ^{2}T}{\partial y^{2}}=\frac{\nu }{\Pr }\frac{\partial ^{2}T}{\partial y^{2}}

(22)

Species equation:

u\frac{\partial \omega _{1}}{\partial x}+v\frac{\partial \omega _{1}}{\partial y}=D_{12}\frac{\partial ^{2}\omega _{1}}{\partial y^{2}}=\frac{\nu }{Sc}\frac{\partial ^{2}\omega _{1}}{\partial y^{2}}

(23)

Using boundary layer approximation for flow over a flat plate with constant free stream velocity, the momentum equation in the y-direction and the pressure gradient in the x-direction were eliminated. Eqs. (20) through (23) are now transformed into a parabolic form rather than the original elliptic form. There are now four dependent variables (u, v, T, ω1) and four equations. This is much easier to solve analytically or numerically because the axial diffusion terms (\partial ^{2}u/\partial x^{2},\partial ^{2}T/\partial x^{2}\ \text{and }\partial ^{2}\omega _{1}/\partial x^{2}) disappear.

Laminar Boundary Layer Solutions for Momentum, Heat, and Mass Transfer

There are three basic approaches to solve boundary layer equations for momentum, heat, and mass transfer:

1.   Similarity solutions
2.   Integral methods
3.   Full numerical solution

First we will consider the similarity approach, since it was the original classic method developed to solve boundary layer problems analytically. In this approach, using the fact that in some circumstances velocity is geometrically similar along the flow direction, the conservation partial differential equations are converted to ordinary differential equations. Similarity methods historically provided significant insight and information about the physical boundary layer phenomenon when computer and numerical methodologies for the solution of partial differential equations were non-existent.

However, there are limitations to the use of similarity methods in terms of applications. In general, they are only applicable to two-dimensional laminar flow for conventional geometry and boundary conditions with constant properties.

The integral methods will be presented second. Integral methods are approximate solutions and provide closed form solutions by assuming a profile for velocity, temperature, and concentrations.

Finally, there is the full numerical solution. The full umerical solution, along with the availability of proven numerical schemes, practical simulation codes, and high speed computers, are more efficient than other methods for complex problems.

References

  1. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
  2. Prandtl, L., 1904, Flussigkeitsbewegung bei sehr Reibung, Proceedings of the 3rd International Mathematics Congress Heidelberg, Germany also NACA TM 452, 1928
  3. Schlichting, H. and Gersteu, K., 2000, Boundary layer theory, 8th enlarged and revised ed., Springer-Verlag New York.
  4. White, F.M., 1974, Viscous Fluid Flow, McGraw-Hill, New York, NY.