The half-Space Model Problem for Compressible Fluid Flow

In this paper we consider the solution formula for Stokes equation system without surface tension in half-space. More precisely, we deal with the solution of velocity and density for the model problem. This result is the basic step to estimate the solution operator of the model problem. We investigate the solution operator for the model problem in N-Dimensional Euclidean space ( ℕ ≥ 2 )

between horizontal plates forming a plane capacitor. The droplets have a charge owing to electrification in the spraying process or absorption of ions from the air. By observing under a microscope the rate of fall of a droplet by the effect of its weight alone, we can use Stokes' formula to calculate the radius and hence the mass of the drop (whose density is known). Then, by applying a suitable potential difference across the capacitor, we can bring the droplet to rest, the downward force of gravity being balanced by the upward electrical force on the charged droplet. Knowing the weight of the droplet and the electric field strength, we can calculate the charge on the droplet.
Such measurements show that the charge on the droplets is always an integral multiple of a certain quantity, which is evidently the unit charge.
Several recent studies investigating this model problem have been carried out not only on to find the solution formula but also to estimate the operator solution families of the model problem.
In 2015 Murata [2] has been investigated the Stokes equation with slip boundary condition. She In this paper we consider the solution formula of the Stokes equation in half-space without surface tension using Fourier transform. As we know that the Stokes formula is usually used to determine the viscosity of a liquid or gas from a measurement of the rate of fall of a solid sphere in it . The viscosity may also be assigned by means of Poiseuille's formula, by measuring the rate of outflow of a liquid from a pipe along which it is impelled by a given pressure difference.
To introduce our main result, first of all we introduce the notation. For a scalar-valued function = ( ) and vector-valued function = (x) = 〈v 1 The set of all natural number is denoted by ℕ and ℕ 0 = ℕ ∪ {0}. Let ℱ = ℱ and ℱ −1 = ℱ −1 denote the Fourier transform and the Fourier inverse transform, which defined by respectively. We also write ̂( ) = ℱ [ ]( ) . Let ℒ and ℒ −1 the denote the Laplace transform and the Laplace inverse transform, which defined by with = + ∈ ℂ, respectively.

Literature Review
Let and be the velocity and density field, respectively. We consider the Stokes equation system without surface tension in bounded domain in half-space. We define ℝ + and ℝ 0 (ℕ ≥ 2) be the half-space and its boundary, respectively by The resolvent problem of Stokes equations are being described by the set of equations, where , and h are scalar vector and ( , ρ) is the stress tensor which is defined by and = (0,0, … , −1) stands for the unit outer normal to ℝ + . The doubled deformation ( ) tensor whose ( , ) components are ( ) = ∂ i + ( = / ), the × identity matrix, , and are positive constants and also div = ∑ =1 .
Before we state the main result, first of all we introduce the definition of sobolev space

Research Methodology
In this research methodology, we use literature review of the related articles, especially [7].
In this paper, we put the different approach of the general solution of velocity as in [7]. This
Therefore, we have the new solution formula of , By using equation (16), we can find the new formula of and respectively. Multiplying both side of the fourth equation of (14) by ∑ −1

Conclusion
The conclusion of the article that we can find the solution formula of velocity � and density of the model problem (2). We can see that for the further research, we can estimate the operator families of the solution.