Basic Equations, Boundary
Conditions and Dimensionless Parameters
In the foregoing chapter, many basic concepts related to the present investigation and the associated literature survey were provided. In this chapter, the required basic equations, the boundary conditions and the approximations used in investigating the problems in the thesis are introduced. Also, the relevant dimensional parameters that appear in different chapters with their significance are briefly discussed.
2.1 Basic Equations
The problems investigated in the thesis are mainly concerned with convection heat and mass transfer from a vertical stretching sheet embedded in a porous medium in the pres- ence of transverse magnetic field, electric field, thermal radiation, Ohmic and viscous dissipation and chemical reaction. The required equations of motion are the conservation of mass, momentum and energy in porous media.
2.1.1 Conservation of Mass
The law of conservation of mass stems from the principle which states that the amount of fluid flowing into a volume must be equal to the amount of fluid flowing out of that volume, i.e., mass can neither be created nor destroyed within the control volume V . The mathematical equivalence of this physical law is called continuity equation which is same
42
irrespective of the fluid considered. Thus the mass conserved in the control volume V is given by
d dt
∫
V
ρdV = 0. (2.1)
Here ρ is the fluid density field at time t. By Reynold’s transport theorem we have
∫
V
( ∂ρ
∂t + div(ρ− → q )
)
dV = 0. (2.2)
Since the control volume V is being arbitrary for conservation of mass a necessary and sufficient condition is
∂ρ
∂t + div(ρ− → q ) = 0, (2.3)
which for incompressible fluid reduces to
div− → q = 0. (2.4)
It is evident from this form that the density will increase wherever ∇.− → q < 0 (convergence) and decrease wherever ∇.− → q > 0 (divergence); whereas when, ∇.− → q = 0 the density remains constant. There are circumstances, known as incompressible flows, where ∇.− → q = 0 everywhere and therefore ρ never changes.
2.1.2 Conservation of Momentum
The law of conservation of momentum states that rate of change of linear momentum on a system is equal to the net force acting on it. The mathematical equivalence of this physical law is called momentum equation. The momentum equation for Newtonian electrically conducting fluid-saturated high porosity porous medium in the presence of vertical gravitational field is given by
ρ ϵ
∂− → q
∂t + ρ
ϵ 2 (− → q . ∇)− → q = −∇p+ρ− → g +µ 0 ( − → J ×− → H )+µ f ∇ 2 − → q − µ f
k − → q − ρC p
√ k |− → q |− → q +ρ e − → E (2.5) where − → H is the magnetic field, − → J is the current density, − → E is the electric field, µ f is the viscosity of the fluid, − → g is the gravitational acceleration, k is the permeability and ϵ is the porosity of the porous medium, C b is the drag coefficient, p is the pressure, ρ e the density of charge and other quantities have their pre-defined meanings. The following are the body forces:
(a) Buoyancy force due to gravity: ρ− → g
(b) Lorentz force due to magnetic field: µ 0 ( − → J × − → H )
(c) Electric force due to electric field: ρ e − → E
2.1.3 Conservation of Energy
The conservation of energy, in the presence of uniform and non uniform heat source/sink, thermal radiation, Soret effect, viscous and Ohmic dissipations is
∂T
∂t + (− → q . ∇)T = ∂
∂y
(
κ ∂T
∂y
)
− 1 ρC p
∂q r
∂y + Q 0 ρC p
(T − T ∞ ) + 1 ρC p
q ′′′ + D s ∇ 2 C (2.6) where T is the temperature, C p is the specific heat at constant pressure, k is the thermal diffusivity, q r = − 4σ 3k
∗∗∂T
4∂y is the radiative heat flux gradient, σ ∗ is the Stefan-Boltzmann constant, D s = c Dk
Ts
C
pis the diffusion-thermo coefficient, k ∗ is the mean absorption coeffi- cient and other quantities have their usual meanings.
2.1.4 Conservation of Species (or Concentration Equation)
The concentration of species or mass diffusion equation is given by
∂C
∂t + (− → q . ∇)C = D∇ 2 C + D c ∇ 2 T − RC (2.7) where C is the concentration of species, D is the mass diffusion coefficient, D c = Dk T
Tm
is the cross diffusion coefficient.
2.1.5 Maxwell’s Equations
In a electrically conducting fluid in the presence of electromagnetic field, Maxwell’s equa- tions are needed in addition to the above mentioned basic equations. The Maxwell’s equations are connected to electric field − → E , the magnetic field − → H , the dielectric field − → D , the magnetic induction − →
B , the current density − →
J , and the distribution of charge density ρ e . These are governed by the following laws:
(i)Gauss’ law
∇.− → E = ρ e
ϵ 0 (2.8)
(ii) Faraday’s law
∇ × − →
E = − ∂− → B
∂t (2.9)
(iii)Ampere’s law
∇ × − → H = − → J + ∂− → D
∂t (2.10)
(iv) Solenoidal property of magnetic field
∇.− →
H = 0 (2.11)
In addition, these equation are supplemented with the equation of continuity of charge
∂ρ e
∂t + (− → q . ∇)ρ e + ∇.− →
J = 0 (2.12)
and the related auxiliary equation are − →
D = ϵ− → E , − →
B = µ 0 − → H , − →
J = σ(− →
E + − → q × − → B ) for poorly conducting fluid, the induced magnetic field is negligible compared to the applied magnetic field. In case of no applied magnetic field, then
∇ × − → E = 0 (2.13)
If we assume that the convection current ρ e − → q is negligible compared to the conduction current. Then, we have
∂ρ e
∂t + ∇.− → J = 0. (2.14)
2.2 Boundary Conditions
The physical configuration considered in the thesis involve stretching sheet embedded in a porous medium, so the following boundary and surface conditions are used to solve the resulting boundary layer equations.
2.2.1 Boundary Conditions on Velocity
(i) For linear stretching sheet
u = u w (x) = ax, v = 0, or v = v 0 (suction/injection) at y = 0 (2.15)
u = 0 as y → ∞. (2.16)
(ii) For non-linear stretching sheet
u = u w (x) = C 1 x m , v = 0, or v = v 0 (suction/injection) (2.17)
or v = v w (x) = C 2 x n at y = 0 (2.18)
u = 0 as y → ∞. (2.19)
2.2.2 Boundary Conditions on Temperature
T = T w (x) = T ∞ + bx, or T = T w (x) = T ∞ + A 0
( x l
) γ
(2.20)
or T = T w (x) = T ∞ + C 3 x γ at y = 0 (PST case) (2.21)
−κ ∂T
∂y = q w = A 1
( x l
) γ
at y = 0 (PHF case) (2.22)
T → T ∞ as y → ∞. (PST or PHF case) (2.23)
2.2.3 Boundary Conditions on Concentration
C = C w (x) = C ∞ + dx or C = C w = C ∞ + C 4 x γ at y = 0 (2.24)
C → C ∞ as y → ∞ (2.25)
2.3 Non-dimensional Parameters
To understand the physical significance of a problem and to apply the results obtained from numerical analysis to practical problems, the non-dimensional parameters play a significant role. The following are the non-dimensional parameters those are used in the problems investigated in the thesis.
2.3.1 Hartmann Number
Hartmann Number is a measure of the Lorentz force to viscous force and is given by M = µ 0 H 0
√ σ µ f
(2.26)
where µ 0 is the magnetic permeability, H 0 is the applied uniform transverse magnetic field, σ is the electrical conductivity and the other quantities having their usual meanings.
2.3.2 Reynolds Number
In fluid mechanics, the Reynolds number is the ratio of inertial forces to viscous forces and
consequently it quantifies the relative importance of these two types of forces for given
flow conditions. Thus, it is used to identify different flow regimes, such as laminar or
turbulent flow. It is one of the most important dimensionless numbers in fluid dynamics
and is used, usually along with other dimensionless numbers, to provide a criterion for
determining dynamic of the flow. It is named after Osborne Reynolds (1842–1912), who proposed it in 1883. Typically it is given as follows:
Re = ρU
µ = U L
ν = Inertial forces
Viscous forces (2.27)
Where υ s is the mean fluid velocity, L is the characteristic length, µ is the dynamic fluid viscosity, ν is the kinematic fluid viscosity ν = µ/ρ, ρ is the fluid density. Laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion, while turbulent flow, on the other hand, occurs at high Reynolds numbers and is dominated by inertial forces, producing random eddies, vortices and other flow fluctuations. The transition between laminar and turbulent flow is often indicated by a critical Reynolds number , which depends on the exact flow configuration and must be determined experimentally. Within a certain range around this point there is a region of gradual transition where the flow is neither fully laminar nor fully turbulent, and predictions of fluid behavior can be difficult. For example, within circular pipes the critical Reynolds number is generally accepted to be 2300, where the Reynolds number is based on the pipe diameter and the mean velocity vs within the pipe, but engineers will avoid any pipe configuration that falls within the range of Reynolds numbers from about 2000 to 3000 to ensure that the flow is either laminar or turbulent. For flow over a flat plate, the characteristic length is the length of the plate and the characteristic velocity is the free stream velocity. In a boundary layer over a flat plate the local regime of the flow is determined by the Reynolds number based on the distance measured from the leading edge of the plate. In this case, the transition to turbulent flow occurs at a Reynolds number of the order of 10 5 to 10 6 .
2.3.3 Prandtl Number
The Prandtl number is a dimensionless number approximating the ratio of momentum diffusivity (viscosity) and thermal diffusivity. It is named after Ludwig Prandtl. It is defined as:
P r = ν
α = Viscous diffusion rate
Thermal diffusion rate (2.28)
Where ν is the kinematic viscosity, α is the thermal diffusivity α = µC κ
p
