### Abstract

The problem of steady mixed convection boundary layer flow over a vertical impermeable flat plate in a porous medium saturated with water at 4°C (maximum density) when the temperature of the plate varies as x^{m} and the velocity outside boundary layer varies as x^{2m}, where x measures the distance from the leading edge of the plate and m is a constant is studied. Both cases of the assisting and the opposing flows are considered. The plate is aligned parallel to a free stream velocity U_{∞} oriented in the upward or downward direction, while the ambient temperature is T_{∞} = T_{m} (temperature at maximum density). The mathematical models for this problem are formulated, analyzed and simplified, and further transformed into non-dimensional form using non-dimensional variables. Next, the system of governing partial differential equations is transformed into a system of ordinary differential equations using the similarity variables. The resulting system of ordinary differential equations is solved numerically using a finite-difference method known as the Keller-box scheme. Numerical results for the non-dimensional skin friction or shear stress, wall heat transfer, as well as the temperature profiles are obtained and discussed for different values of the mixed convection parameter λ and the power index m. All the numerical solutions are presented in the form of tables and figures. The results show that solutions are possible for large values of λ and m for the case of assisting flow. Dual solutions occurred for the case of opposing flow with limited admissible values of λ and m. In addition, separation of boundary layers occurred for opposing flow, and separation is delayed for the case of water at 4°C (maximum density) compared to water at normal temperature.

Original language | English |
---|---|

Pages (from-to) | 359-372 |

Number of pages | 14 |

Journal | Transport in Porous Media |

Volume | 69 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 2007 |

### Fingerprint

### Keywords

- Boundary layer
- Dual solutions
- Maximum density
- Mixed convection
- Porous medium
- Variable wall temperature
- Vertical plate

### ASJC Scopus subject areas

- Chemical Engineering(all)
- Catalysis

### Cite this

**Steady mixed convection boundary layer flow over a vertical flat plate in a porous medium filled with water at 4°C : Case of variable wall temperature.** / Ling, S. C.; Mohd. Nazar, Roslinda; Pop, I.

Research output: Contribution to journal › Article

*Transport in Porous Media*, vol. 69, no. 3, pp. 359-372. https://doi.org/10.1007/s11242-006-9077-0

}

TY - JOUR

T1 - Steady mixed convection boundary layer flow over a vertical flat plate in a porous medium filled with water at 4°C

T2 - Case of variable wall temperature

AU - Ling, S. C.

AU - Mohd. Nazar, Roslinda

AU - Pop, I.

PY - 2007/9

Y1 - 2007/9

N2 - The problem of steady mixed convection boundary layer flow over a vertical impermeable flat plate in a porous medium saturated with water at 4°C (maximum density) when the temperature of the plate varies as xm and the velocity outside boundary layer varies as x2m, where x measures the distance from the leading edge of the plate and m is a constant is studied. Both cases of the assisting and the opposing flows are considered. The plate is aligned parallel to a free stream velocity U∞ oriented in the upward or downward direction, while the ambient temperature is T∞ = Tm (temperature at maximum density). The mathematical models for this problem are formulated, analyzed and simplified, and further transformed into non-dimensional form using non-dimensional variables. Next, the system of governing partial differential equations is transformed into a system of ordinary differential equations using the similarity variables. The resulting system of ordinary differential equations is solved numerically using a finite-difference method known as the Keller-box scheme. Numerical results for the non-dimensional skin friction or shear stress, wall heat transfer, as well as the temperature profiles are obtained and discussed for different values of the mixed convection parameter λ and the power index m. All the numerical solutions are presented in the form of tables and figures. The results show that solutions are possible for large values of λ and m for the case of assisting flow. Dual solutions occurred for the case of opposing flow with limited admissible values of λ and m. In addition, separation of boundary layers occurred for opposing flow, and separation is delayed for the case of water at 4°C (maximum density) compared to water at normal temperature.

AB - The problem of steady mixed convection boundary layer flow over a vertical impermeable flat plate in a porous medium saturated with water at 4°C (maximum density) when the temperature of the plate varies as xm and the velocity outside boundary layer varies as x2m, where x measures the distance from the leading edge of the plate and m is a constant is studied. Both cases of the assisting and the opposing flows are considered. The plate is aligned parallel to a free stream velocity U∞ oriented in the upward or downward direction, while the ambient temperature is T∞ = Tm (temperature at maximum density). The mathematical models for this problem are formulated, analyzed and simplified, and further transformed into non-dimensional form using non-dimensional variables. Next, the system of governing partial differential equations is transformed into a system of ordinary differential equations using the similarity variables. The resulting system of ordinary differential equations is solved numerically using a finite-difference method known as the Keller-box scheme. Numerical results for the non-dimensional skin friction or shear stress, wall heat transfer, as well as the temperature profiles are obtained and discussed for different values of the mixed convection parameter λ and the power index m. All the numerical solutions are presented in the form of tables and figures. The results show that solutions are possible for large values of λ and m for the case of assisting flow. Dual solutions occurred for the case of opposing flow with limited admissible values of λ and m. In addition, separation of boundary layers occurred for opposing flow, and separation is delayed for the case of water at 4°C (maximum density) compared to water at normal temperature.

KW - Boundary layer

KW - Dual solutions

KW - Maximum density

KW - Mixed convection

KW - Porous medium

KW - Variable wall temperature

KW - Vertical plate

UR - http://www.scopus.com/inward/record.url?scp=38149051233&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38149051233&partnerID=8YFLogxK

U2 - 10.1007/s11242-006-9077-0

DO - 10.1007/s11242-006-9077-0

M3 - Article

AN - SCOPUS:38149051233

VL - 69

SP - 359

EP - 372

JO - Transport in Porous Media

JF - Transport in Porous Media

SN - 0169-3913

IS - 3

ER -