Equation (A.1.2) is known as the continuity equation. In this document we are interested in incompressible fluids in which the volume of any fluid element is time invariant when moving with flow i.e.,
d dt
Z
Ω0
dx= 0. (A.1.3)
On the other hand with the help of Lagrangian technique, which will be discussed in detail while deriving the equation of motion, we have the following equation for any kind of fluid, whether it is compressible or incompressible,
d
For an arbitrary Ω0, equations (A.1.3) and (A.1.4) implies the following incompress-ibility condition
∇ ·u= 0. (A.1.5)
If we assume that the mass density ρ is a constant function of x and t over the whole flow region then the continuity equation (A.1.2) reduces to the incompress-ibility condition (A.1.5). However the reciprocal is not true. For interested readers we refer to [3].
A.2 Equation of Motion
Considering a fluid element occupying domain Ω at timet. The fluid element moves with the flow velocityu(x, t) and reaches the domain Ω0 at time t0, while obviously t0 > t. Newton’s second law of motion applied on the fluid element gives us
d
Wheref denotes a density of volume forces per mass unit and S denotes a density of surface forces per surface unit. The surface forces S are expressed with a 2×2 symmetric tensorσ, known as stress tensor in the following way
S =σν. (A.2.2)
Hereνis an outward unit normal vector and the stress tensorσis the combination of normal stressesσnand shear stressesτncorresponding to the normal and tangential forces applied on the some surface element, respectively. In short we can say that the surface forces are the result of the forces ofσ applied in the outward unit normal vectorn. Following [5] and [15] we can write the stress tensor for the viscous fluid as
σ =−˜pI+A (A.2.3)
120 Chapter A. Background from Fluid Dynamics
Here ˜pI is the in determinant part of the stress tensor, where ˜p is the pressure and I is the identity matrix. For Newtonian fluids the matrix A is given as
A= 2˜µ As a remark we would like to mention that the viscosity parameters ˜µ(dynamic viscosity) and ˜µ1 (second viscosity) play an important role in the modeling of equa-tion (A.2.4). The first term of the right hand side represents the viscous effects associated to volume invariant deformation while the second term is responsible for the volume dilations due to viscous effects. In general viscosity coefficients are the functions of temperature, but we are interested only in isothermal flows in which the temperature is uniformly constant in the flow region Ω. With the help of equation of continuity (A.1.5), the equation (A.2.4) is reduced to
A= 2˜µD. (A.2.5)
HereD is the deformation rate tensor defined as 2D=∇u+ (∇u)T
Now making use of equation (A.2.2) and with the help of the divergence theorem, the second term on the right hand side of equation (A.2.1) can be written as
Z
From equations (A.2.3) and (A.2.4), keeping in mind that∇ ·aI =∇a, we obtain
∇ ·σ =−∇p˜+ ˜µ∆u (A.2.7)
Here ∆ is a Laplace operator. Insert (A.2.7) in equation (A.2.6) we finally obtain Z Now we evaluate the left hand side of equation (A.2.1). We denote the time difference by ∆t=t0−t, then by the classical definitions of derivatives, we have
d For a small ∆t, we may have the infinitesimal one-parameter coordinate transfor-mation in the following way
x0=x+ ∆tu(x, t) +O(∆t2).
A.2 Equation of Motion 121
Neglecting terms of orderO(∆t2), we have
x0=x+ ∆t u(x, t). (A.2.10)
Using equation (A.2.10), we change the variables of the first integrand on the right hand side of equation (A.2.9), such as
d
In the above equation ∂x∂x0 denotes the Jacobian matrix of the transformation defined in (A.2.10). With straight forward calculation starting from equation (A.2.10) yields
∂x0
∂x =I+ ∆t∇u (A.2.12)
Now taking the determinant of equation (A.2.12), we obtain
∂xi. Under the assumption that ∆tis small, we again neglect the terms with (∆t2) in equation (A.2.13) and with incompressibility we have
∂x0
∂x = 1 (A.2.14)
Now using the first order Taylor series expansion on (ρu)(x+ ∆tu, t+ ∆t) gives u(x+ ∆tu, t+ ∆t) =u+ ∆t(u· ∇)u+ ∆t∂u
∂t. (A.2.15)
Thus in the view of equations (A.2.14) and (A.2.15), equation (A.2.11) takes the following form Applying the limit on equation (A.2.16) we are left with
d
122 Chapter A. Background from Fluid Dynamics
Making use of equation of continuity (A.1.5), the last equation can also be written
as d
dt Z
Ω
ρudx=ρ Z
Ω
∂u
∂t + (u· ∇)udx. (A.2.17) Now, finally substituting back (A.2.8) and (A.2.17) in equation (A.2.1) and taking advantage of an arbitrary domain Ω we have the following equation of motion for incompressible fluids
ρ∂u
∂t +ρ(u· ∇)u=−∇˜p+ ˜µ∆u+ρf. (A.2.18) We modify the above equation (A.2.18) such that,
∂u
∂t + (u· ∇)u=−∇p+µ∆u+f, (A.2.19) herep= ˜p/ρand µ:= ˜µ/ρis the kinematic viscosity.
Definition A.1. The Navier-Stokes equation for isothermal incompressible Newto-nian fluids are defined by the system of equations consisting of
• The equation of continuity (A.1.5) and the
• The equation of motion (A.2.19)
In order to analyze the behavior of the fluids in detail we need some conditions that strongly depend on the type of the region Ω of the flow . In this monograph we are considering two types of regions. First type of region is bounded while the other type of the region is the compliment of a bounded region. For the first case we only need boundary condition at the bounding walls∂Ω of the region Ω, i.e.,
u(x) =u∗(y), y∈∂Ω. (A.2.20) While for the second case, in addition to condition (A.2.20), we impose the condition that our velocity fieldu(x) at large spatial distances tends to some vector u∞ such that
|x|→0lim u(x) =u∞(y). (A.2.21) The second term on the left hand side of equation (A.2.19) is responsible for the non-linearity of the Navier-Stokes equation. In the last two centuries it is a challeng-ing task for mathematicians to get explicit solutions of these equations. However, there are several numerical methods developed in computational fluid dynamics (CFD) to deal with the Navier-Stokes equation [8].
A.2 Equation of Motion 123
In this monograph we shall investigate the inverse problems of viscous incom-pressible steady fluids with slow motion. The hypotheses of slow motion means that the viscous forces are much stronger than the inertial forces, i.e.,
(u· ∇)u µ∆u →0.
Thus we can neglect the non-linear term from equation (A.2.19) to make it linear.
For mathematical justification we linearize the Navier-Stokes equation by standard tools. Assume that (u0, p0) is the generic solution of the Navier-Stokes equations.
This means that (u0+λu(x), p0+λp(x)) is again the solution of the Navier-Stokes equation. Plugging back in equations (A.1.5) and (A.2.19) we obtain
(u0· ∇)u+λ(u· ∇)u = −∇p+µ∆u+f
∇ ·u = 0.
Now applying the limit λ → 0 we have the linearized form of the Navier-Stokes equation, i.e.,
(u0· ∇)u = −∇p+µ∆u+f, (A.2.22)
∇ ·u = 0. (A.2.23)
These linearized equations are known as the Oseen equation named after Carl Wil-helm Oseen. It is observed the above pair of equations (A.2.22) and (A.2.23) reduces to a well known Stokes equation foru0= 0. Together with conditions (A.2.20) and (A.2.21), Stokes derived a remarkable and explicit solution (u, p) in1851[44]. How-ever this solution fails to demonstrate the wake region behind the obstacle. In1910 Oseen found a paraboloidal wake region behind the obstacle, which is an important breakthrough in the field of fluid dynamics.
124 Chapter A. Background from Fluid Dynamics
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Curriculum Vitae/Lebenslauf
Personal Data
Name: Qazi Muhammad Zaigham Zia
Date of Birth: 21.10.1982
Place of Birth: Rawalpindi (Bhoner Kaswal) Nationality: Pakistan
Marital Status: Married
Parents: Qazi Muhammad Zia-Ur-Rahman Yasmin Younis
Spouse Name: Mubeen Akhtar
Primary and Secondary Education
1985-1994: Mumtaz Memorial Public School, Islam Pura Jabbar, Gujar Khan, District Rawalpindi
1994-1996: Govt. High School, Bewal, Gujar Khan, District Rawalpindi 1996-1998: Govt. Gordon College, Rawalpindi Higher Education
1998-2000: B.Sc. in Mathematics and Statistics from University of the Punjab, Lahore 2001-2002: M.Sc. in Mathematics
from Quaid-i-Azam University Islamabad 2003-2005: M.Phil in Mathematics
from Quaid-i-Azam University Islamabad Occupational Career
2005-2006: Lecturer, Department of Mathematics,
COMSATS Institute of Information Technology, Islamabad